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Continuum Solvation Models in Chemical Physics: From Theory to Applications Edited by BENEDETTA MENNUCCI Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Italy and ROBERTO CAMMI Dipartimento di Chimica Generale ed Inorganica, Chimica Analitica e Chimica Fisica, Università di Parma, Italy This page intentionally left blank Continuum Solvation Models in Chemical Physics This page intentionally left blank Continuum Solvation Models in Chemical Physics: From Theory to Applications Edited by BENEDETTA MENNUCCI Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Italy and ROBERTO CAMMI Dipartimento di Chimica Generale ed Inorganica, Chimica Analitica e Chimica Fisica, Università di Parma, Italy Copyright © 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone +44 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com All Rights Reserved. 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No warranty may be created or extended by any promotional statements for this work. Neither the Publisher nor the Author shall be liable for any damages arising herefrom. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Ltd, 6045 Freemont Blvd, Mississauga, Ontario, L5R 4J3, Canada Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Anniversary Logo Design: Richard J. Pacifico Library of Congress Cataloging in Publication Data Continuum solvation models in chemical physics : from theory to applications / edited by Benedetta Mennucci and Roberto Cammi. p. cm. Includes index. ISBN 978-0-470-02938-1 (cloth) 1. Solvation. 2. Chemistry, Physical and theoretical. I. Mennucci, Benedetta. II. Cammi, Roberto. QD543.C735 2007 541 .34—dc22 2007024029 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-470-02938-1 Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by Antony Rowe, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. Contents List of Contributors vii Preface xi 1 1 1 29 49 2 Modern Theories of Continuum Models 1.1 The Physical Model (Jacopo Tomasi) 1.2 Integral Equation Approaches for Continuum Models (Eric Cancès) 1.3 Cavity Surfaces and their Discretization (Christian Silvio Pomelli) 1.4 A Lagrangian Formulation for Continuum Models (Marco Caricato, Giovanni Scalmani and Michael J. Frisch) 1.5 The Quantum Mechanical Formulation of Continuum Models (Roberto Cammi) 1.6 Nonlocal Solvation Theories (Michail V. Basilevsky and Gennady N. Chuev) 1.7 Continuum Models for Excited States (Benedetta Mennucci) Properties and Spectroscopies 2.1 Computational Modelling of the Solvent–Solute Effect on NMR Molecular Parameters by a Polarizable Continuum Model (Joanna Sadlej and Magdalena Pecul) 2.2 EPR Spectra of Organic Free Radicals in Solution from an Integrated Computational Approach (Vincenzo Barone, Paola Cimino and Michele Pavone) 2.3 Continuum Solvation Approaches to Vibrational Properties (Chiara Cappelli) 2.4 Vibrational Circular Dichroism (Philip J. Stephens and Frank J. Devlin) 2.5 Solvent Effects on Natural Optical Activity (Magdalena Pecul and Kenneth Ruud) 2.6 Raman Optical Activity (Werner Hug) 2.7 Macroscopic Nonlinear Optical Properties from Cavity Models (Roberto Cammi and Benedetta Mennucci) 2.8 Birefringences in Liquids (Antonio Rizzo) 64 82 94 110 125 125 145 167 180 206 220 238 252 vi Contents 2.9 2.10 2.11 3 4 Anisotropic Fluids (Alberta Ferrarini) Homogeneous and Heterogeneous Solvent Models for Nonlinear Optical Properties (Hans Ågren and Kurt V. Mikkelsen) Molecules at Surfaces and Interfaces (Stefano Corni and Luca Frediani) 265 282 300 Chemical Reactivity in the Ground and the Excited State 3.1 First and Second Derivatives of the Free Energy in Solution (Maurizio Cossi and Nadia Rega) 3.2 Solvent Effects in Chemical Equilibria (Ignacio Soteras, Damián Blanco, Oscar Huertas, Axel Bidon-Chanal and F. Javier Luque) 3.3 Transition State Theory and Chemical Reaction Dynamics in Solution (Donald G. Truhlar and Josefredo R. Pliego Jr.) 3.4 Solvation Dynamics (Branka M. Ladanyi) 3.5 The Role of Solvation in Electron Transfer: Theoretical and Computational Aspects (Marshall D. Newton) 3.6 Electron-driven Proton Transfer Processes in the Solvation of Excited States (Wolfgang Domcke and Andrzej L. Sobolewski) 3.7 Nonequilibrium Solvation and Conical Intersections (Damien Laage, Irene Burghardt and James T. Hynes) 3.8 Photochemistry in Condensed Phase (Maurizio Persico and Giovanni Granucci) 3.9 Excitation Energy Transfer and the Role of the Refractive Index (Vanessa M. Huxter and Gregory D. Scholes) 3.10 Modelling Solvent Effects in Photoinduced Energy and Electron Transfers: the Electronic Coupling (Carles Curutchet) 313 Beyond the Continuum Approach 4.1 Conformational Sampling in Solution (Modesto Orozco, Ivan Marchán and Ignacio Soteras) 4.2 The ONIOM Method for Layered Calculations (Thom Vreven and Keiji Morokuma) 4.3 Hybrid Methods for Molecular Properties (Kurt V. Mikkelsen) 4.4 Intermolecular Interactions in Condensed Phases: Experimental Evidence from Vibrational Spectra and Modelling (Alberto Milani, Matteo Tommasini, Mirella Del Zoppo and Chiara Castiglioni) 4.5 An Effective Hamiltonian Method from Simulations: ASEP/MD (Manuel A. Aguilar, Maria L. Sánchez, M. Elena Martín and Ignacio Fdez. Galván) 4.6 A Combination of Electronic Structure and Liquid-state Theory: RISM–SCF/MCSCF Method (Hirofumi Sato) 499 Index 313 323 338 366 389 414 429 450 471 485 499 523 538 558 580 593 607 Contributors Hans Ågren Department of Theoretical Chemistry, Royal Institute of Technology, Stockholm, Sweden. Dpto Química Física, Universidad de Extremadura, Badajoz, Spain Manuel. A. Aguilar Dipartimento di Chimica, Università di Napoli ‘Federico II’, Italy Vincenzo Barone Michail V. Basilevsky Russia Axel Bidon-Chanal de Barcelona, Spain Photochemistry Center, Russian Academy of Sciences, Moscow, Departament de Fisicoquímica, Facultat de Farmacia, Universitat Damián Blanco Departament de Fisicoquímica, Facultat de Farmacia, Universitat de Barcelona, Spain Département de Chimie, Ecole Normale Supérieure, Paris, France Irene Burghardt Roberto Cammi Dipartimento di Chimica Generale ed Inorganica, Chimica Analitica e Chimica Fisica, Università di Parma, Italy Eric Cancès CERMICS, Ecole Nationale des Ponts et Chaussées, Champs-sur-Marne, France Chiara Cappelli Italy Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Marco Caricato Gaussian, Inc., Wallingford, CT, USA Chiara Castiglioni Milano, Italy Gennady N. Chuev Paola Cimino Center for NanoEngineered Materials and Surfaces, Politecnico di Karpov Institute of Physical Chemistry, Moscow, Russia Dipartimento di Chimica, Università di Napoli ‘Federico II’, Italy Stefano Corni INFM-CNR Center on nanoStructures and bioSystems at Surfaces, Modena, Italy Maurizio Cossi Dipartimento di Scienze dell’Ambiente e della Vita, Università del Piemonte Orientale ‘Amedeo Avogadro’, Alessandria, Italy viii List of Contributors Carles Curutchet Dipartimento di Chimica Generale ed Inorganica, Chimica Analitica e Chimica Fisica, Università di Parma, Italy Frank J. Devlin Department of Chemistry, University of Southern California, Los Angeles, CA, USA Wolfgang Domcke Institute of Physical and Theoretical Chemistry, Technical University of Munich, Germany Dipartimento di Scienze Chimiche, Università di Padova, Italy Alberta Ferrarini Department of Chemistry, University of Tromsø, Norway Luca Frediani Gaussian, Inc., Wallingford, CT, USA Michael J. Frisch Ignacio Fdez. Galván Spain Dpto Química Física, Universidad de Extremadura, Badajoz, Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Giovanni Granucci Italy Oscar Huertas Departament de Fisicoquímica, Facultat de Farmacia, Universitat de Barcelona, Spain Werner Hug Department of Chemistry, University of Fribourg, Switzerland Vanessa M. Huxter Lash Miller Chemical Laboratories, Center for Quantum Information and Quantum Control, and Institute for Optical Sciences, University of Toronto, Ontario, Canada James T. Hynes Department of Chemistry and Biochemistry, University of Colorado, Boulder, CO, USA Damien Laage Département de Chimie, Ecole Normale Supérieure, Paris, France Branka M. Ladanyi CO, USA F. Javier Luque Barcelona, Spain Department of Chemistry, Colorado State University, Fort Collins, Departament de Fisicoquímica, Facultat de Farmacia, Universitat de Ivan Marchán Molecular Modelling and Bioinformatics Unit, Institute for Research in Biomedicine and Instituto Nacional de Bioinformàtica-Structural Genomic Node, Barcelona, Spain Dpto Química Física, Universidad de Extremadura, Badajoz, Spain M. Elena Martín Benedetta Mennucci Pisa, Italy Kurt V. Mikkelsen Alberto Milani Milano, Italy Keiji Morokuma Dipartimento di Chimica e Chimica Industriale, Università di Department of Chemistry, University of Copenhagen, Denmark Center for NanoEngineered Materials and Surfaces, Politecnico di Fukui Institute for Fundamental Chemistry, Kyoto University, Japan List of Contributors ix Department of Chemistry, Brookhaven National Laboratory, Marshall D. Newton Upton, NY, USA Modesto Orozco Molecular Modelling and Bioinformatics Unit, Institute for Research in Biomedicine and Instituto Nacional de Bioinformàtica-Structural Genomic Node, Barcelona, Spain Dipartimento di Chimica, Università di Napoli ‘Federico II’, Italy Michele Pavone Magdalena Pecul Department of Chemistry, University of Warsaw, Poland Maurizio Persico Italy Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Josefredo R. Pliego Jr. Gerais, Brazil Departamento de Química, Universidade Federal de Minas Christian Silvio Pomelli Pisa, Italy Dipartimento di Chimica e Chimica Industriale, Università di Nadia Rega Dipartimento di Chimica Università di Napoli ‘Federico II’, Italy Antonio Rizzo Istituto per i Processi Chimico-Fisici del Consiglio Nazionale delle Ricerche, Pisa, Italy Kenneth Ruud Department of Chemistry, University of Tromsø, Norway Joanna Sadlej Department of Chemistry, University of Warsaw, Poland Hirofumi Sato Department of Molecular Engineering, Kyoto University, Japan Maria L. Sánchez Giovanni Scalmani Dpto Química Física. Universidad de Extremadura, Badajoz, Spain Gaussian, Inc., Wallingford, CT, USA Gregory D. Scholes Lash Miller Chemical Laboratories, Center for Quantum Information and Quantum Control, and Institute for Optical Sciences, University of Toronto, Ontario, Canada Andrzej L. Sobolewski Poland Institute of Physics, Polish Academy of Sciences, Warsaw, Ignacio Soteras Departament de Fisicoquímica, Facultat de Farmàcia, Universitat de Barcelona, Spain Philip J. Stephens Angeles, CA, USA Jacopo Tomasi Italy Department of Chemistry, University of Southern California, Los Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Matteo Tommasini Milano, Italy Center for NanoEngineered Materials and Surfaces, Politecnico di x List of Contributors Donald G. Truhlar Department of Chemistry and Supercomputing Institute, University of Minnesota Minneapolis, MN, USA Thom Vreven Gaussian, Inc., Wallingford, CT, USA Mirella Del Zoppo Milano, Italy Center for NanoEngineered Materials and Surfaces, Politecnico di Preface The modeling of liquids and solutions with computational tools is a very complex problem which involves several research groups in different parts of the world. Many alternative theoretical models and computational algorithms have been proposed so far. All these models, however, can be classified in two main classes, namely that using an equivalent description for all the components of the system (the solute and the solvent molecules in a dilute solution, the molecules of the different species forming a mixture, etc.), and the other introducing a focused approach, i.e. a hierarchical approach in which the most interesting part of the system is treated at a much more accurate level than the rest. The first class of models include very different approaches which go from classical Molecular Dynamics (MD) and Monte Carlo (MC) simulations to accurate quantum mechanical (QM) calculations on small-medium clusters to ab-initio MD simulation on larger set of molecules. Also the second class of methods include very different approaches; however, in all of them we can individuate a common aspect, namely the use of a mean-field description for the part of the system encircling the subsystem of real interest. In the application of this class of methods to the study of liquid solutions, the most important mean-field approach is represented by continuum models. In such models, the solute is assumed to be inside a cavity of proper shape and dimension within an infinite continuum dielectric mimicking the solvent. Continuum solvation models are nowadays widespread computational techniques to study solvent effects on energy/geometry/reactivity and properties of very different molecular systems (from small molecules to very large biochemical systems such as proteins and enzymes). Continuum solvation models have a quite long history which goes back to the first versions by Onsager (1936) and Kirkwood (1934), however only recently (starting since the 90s) they have become one of the most used computational techniques in the field of molecular modelling. This has been made possible by two factors which will be presented and discussed in the book, namely the increase in the realism of the model on the one hand, and the coupling with quantum-mechanical approaches on the other. The greater realism has also meant an important evolution in the mathematical formalism and in the computational implementation of the continuum models while the QM reformulation of such models has allowed the study of chemical and physical xii Preface phenomena which were impossible to treat with classical only models. This important evolution of continuum models which has transformed them from empirical or qualitative approaches to accurate and quantitative methods has been realized in the last ten years and only now has real maturity been reached. In addition to this, the literature on successful applications of these models to real chemical systems and problems has become large enough to stately prove the reliability of these models. It thus become very interesting to give to both researchers and students a new book in which the analysis of both theory and applications of continuum models is reviewed. For the first time, solvation continuum models are treated in an up-to-date and coherent way but at the same time using very different points of view coming from experts belonging to very different research fields (mathematicians, theoretical chemists, computational chemists, spectroscopists, etc.). The book is partitioned into four chapters. The first chapter focuses on a specific class of continuum solvation models, namely those using as a descriptor for the solvent polarization an apparent surface charge (ASC) spreading on the molecular cavity which contains the solute. This class of methods is central in the whole book (and especially in this first chapter) as during these last years it has become the preferential approach to account for solvent effects in QM calculations. A particular mention, among ASC methods, is for a specific formulation known as Polarizable Continuum Model (PCM). Nowadays, this acronym no longer represents a single computational method but a family of methods which are now available in various QM computational packages. The physics beyond such a family of PCM models is presented and discussed by Tomasi together with an overview on the main features characterizing these models which will be further analyzed in the following chapters. From a mathematical point of view the PCM models can be unified according to the approach they use to solve the linear partial differential equations determining the electrostatic interactions between solute and solvent. This analysis is presented by Cancès who reviews both the mathematical and the numerical aspects of such an integral equation approach when applied to PCM models. A further analysis of the main numerical aspects related to the computational implementation of such a theory is presented and discussed by Pomelli with particular attention given to the definition of the molecular cavity and the sampling of its surface. The last fundamental aspect characterizing PCM methods, i.e. their quantum mechanical formulation, is presented by Cammi for molecular systems in their ground electronic states and by Mennucci for electronically excited states. In both contributions, particular attention is devoted to the specific aspect characterizing PCM (and similar) approaches, namely the necessity to introduce an effective nonlinear Hamiltonian which describes the solute under the effect of the interactions with its environment and determines how these interactions affect the solute electronic wavefunction and properties. In the other two sections of the chapter two further generalizations of PCM models are presented to spatially and dynamically nonlocal media (Basilevsky & Chuev) and to a Lagrangian formulation which includes the polarization of the medium as a dynamical variable (Caricato, Scalmani & Frisch), respectively. In the first case, the goal is to account for the discreteness of molecular liquids still within a continuum description of Preface xiii the solvent, while in the second case the goal is to describe any kind of time-dependent phenomena exploiting an efficient coupling of continuum models with standard MD simulations, both classical and ab-initio. The second chapter presents extensions and generalizations of continuum solvation models (mostly of PCM type but not exclusively) to the calculation of molecular properties (both dynamic and static) and spectroscopic features of molecular solutes in different environments of increasing complexity. Computational methods to study solvent effects on NMR (Sadlej & Pecul) and EPR (Barone, Cimino & Pavone) parameters are presented and discussed within the PCM as well their generalizations to hybrid continuum/discrete approaches in which the presence of specific interactions (e.g. solute-solvents H-bonds) is explicitly taken into account by including some solvent molecules strongly interacting with the solute. Solvent effects on vibrational spectroscopies are analyzed by Cappelli using classical and quantum mechanical continuum models. In particular, PCM and combined PCM/discrete approaches are used to model reaction and local field effects. Rizzo reviews in a unitary framework computational methods for the study of linear birefringence in condensed phase. In particular, he focuses on the PCM formulation of the Kerr birefringence, due to an external electric field yields, on the Cotton-Mouton effect, due to a magnetic field, and on the Buckingham effect due to an electric-fieldgradient. A parallel analysis is presented for natural optical activity by Pecul & Ruud. They present a brief summary of the theory of optical activity and a review of theoretical studies of solvent effects on these properties, which to a large extent has been done using various polarizable dielectric continuum models. The inclusion of the environment effects for non-linear optical (NLO) properties is presented within the PCM (Cammi & Mennucci) and the multipolar expansion (Ågren & Mikkelsen) solvation models. In the first contribution the attention is focused on the connection between microscopic effective properties and macroscopic NLO susceptibilities, whereas in the latter contribution the analysis is extended to treat heterogeneous dielectric media. The extension of continuum models to complex environments is further analyzed by Ferrarini and Corni & Frediani, respectively. In the first contribution the use of PCM models in anisotropic dielectric media such as liquid crystals is presented in relation to the calculation of response properties and spectroscopies. In the second contribution, PCM formulations to account for gas-liquid or liquid-liquid interfaces, as well for the presence of a meso- or nano-scopic metal body, are presented. In the case of molecular systems close to metal bodies, particular attention is devoted to the description of the surface enhanced effects on their spectroscopic properties. The second chapter ends with two overviews by Stephens & Devlin and by Hug on the theoretical and the physical aspects of two vibrational optical activity spectroscopies (VCD and VROA, respectively). In both overviews the emphasis is more on their basic formalism and the gas-phase quantum chemical calculations than on the analysis of solvent effects. For these spectroscopies, in fact, both the formulation of continuum solvation models and their applications to realistic solvated systems are still in their infancy. The third chapter focuses on the modelization of solvent effects on ground state chemical reactivity and excited state reactive and non-reactive processes. xiv Preface The effects of the surrounding medium on the shape of the potential energy surfaces (PES) is discussed by Cossi & Rega using the PCM formulation of continuum models while Soteras, Blanco, Huertas, Bidon-Chanal, & Luque present an overview of the current status and perspectives of theoretical treatments of solvent effects on chemical equilibria using different versions of continuum solvation model. A different aspect of the modelization of chemical reactivity is given by Truhlar & Pliego. In particular, they describe how continuum models can be used to predict the free energy of activation of chemical reactions and the effective potential for condensed-phase tunneling, and they can therefore be combined with variational transition state theory (VTST) to predict chemical reaction rates. With the other contributions, the focus of the chapter is shifted to electronically excited states and their dynamics and reactivity. The computational and experimental analysis of time dependent solvatochromic shift in fluorescence spectra of solutes is used by Ladanyi to achieve an accurate description of solvation dynamics, i.e., the rate of solvent reorganization in response to a perturbation in solute–solvent interaction. Electron transfer (ET) reactions are analyzed by Newton in terms of continuum solvation models. Their role in the determination of the ET critical parameters (i.e. the solvent reorganization energy and the electronic coupling between the initial and final states) is analyzed using both an equilibrium and nonequilibrium solvation framework. Photoinduced hydrogen-transfer and proton-transfer chemistry in hydrogen-bonded chromophore-solvent clusters are analyzed by Domcke & Sobolevski exploiting an interplay of QM and spectroscopic approaches. Laage, Burghardt & Hynes present and discuss analytic dielectric continuum nonequilibrium solvation treatments of chemical reactions in solution involving conical intersections. Their analysis shows that theories of the rates of mechanisms of the chemical reaction in solution have to incorporate the fact that the solvent can be out of equilibrium with the instantaneous charge distribution of the reacting solutes(s). Persico & Granucci focus on the nonadiabatic dynamics of excited states in condensed phase. Static environmental effects are discussed in terms of the change of the PES with respect to the isolated molecule, while dynamic effects are described in terms of transfer of energy and momentum between the chromophore (or reactive centre) and the surrounding molecules. The third chapter ends with two contributions on the effects of the environment on the excitation energy transfers (EET) between chromophores. In the first contribution, Huxter & Scholes present a review of the recent evolution of theory of EET in condensed phase from their earliest and simple formulation, based on the Forster theory to the most recent advances of theoretical and computational methods based on continuum solvation models. In the second contribution, Curutchet reviews the recent developments of PCM towards accurate theoretical investigations of EET in solution. In particular, the modelization of the various contributions of solvent effects in the chromophore–chromophore electronic coupling is presented using quantummechanical approaches. The fourth chapter presents extensions and generalizations of continuum models to classical molecular dynamics simulations, to layered and to hybrid methods as well as to Preface xv methods which can be considered as alternative to continuum models to account for the environment effects. In more detail, Orozco, Marchán & Soteras review recent implementations of continuum models in the context of MD or MC calculations, to study solvent effects on the conformational space of large, flexible molecules. Vreven & Morokuma outline the formalism of the ONIOM method and how it can be extended to include solvation effects, both implicitly (using a ONIOM-PCM combination) and explicitly (using a ONIOM supra-molecular description). Mikkelsen covers the theoretical background of the multiconfigurational self-consistent field response methods for calculating molecular properties of molecules interacting with a structured environment using a hybrid QM/MM approach. Milani, Tommasini, Del Zoppo & Castiglioni compare Raman and infrared experiments in condensed phase with the results obtained using both a quantum supra-molecular approach and a simplified electrostatic embedding scheme. Aguilar, Sánchez, Martín, & Fdez. Galván review the ASEP/MD method, acronym for Averaged Solvent Electrostatic Potential from Molecular Dynamics, showing how this method combines aspects of quantum mechanics/molecular mechanics (QM/MM) methods with aspects of continuum models. Sato presents an alternative method to both continuum solvation models and hybrid QM/MM or ONIOM approaches. This is represented by the “reference interaction site model” (RISM) formalism when combined to a QM description of the solute to give the RISM-SCF theory. As shown in this brief description of the contents, the book aims to present the main aspects and applications of continuum solvation models in a clear and concise format, which will be useful to the expert researcher but also to Ph.D. students and postdoctoral workers. To this end, the presentation of the various contributions follows a step-by-step scheme in which the physical bases of the models come first followed by an analysis of both mathematical and computational aspects and finally by a review of their applications to different physical–chemical problems. For all the parts of the book two reading levels will thus be possible: one, more introductory, on the given theoretical issue or on the given application, and the other, more detailed (and more technical), on specific physical and numerical aspects involved in each issue and/or application. In such a way, the reader will first be introduced to a given subject through a general description of the problem (with more emphasis on those aspects which are more directly related to the presence of the solvent), and then she/he will discover how continuum models can be extended and generalized to properly describe such a problem. In parallel, possible limitations or incompleteness of these models are pointed out with indications of future developments. Ending this Preface we would like to give our sincere thanks to all the colleagues who are (or have been) part of the PCM group in Pisa in the last years and have also contributed to this book: Chiara Cappelli, Marco Caricato, Stefano Corni, Maurizio Cossi, Luca Frediani, and Christian Pomelli. The final and most important acknowledgement goes however to Professor Jacopo Tomasi who greatly contributed to the formation of our scientific and personal growth. Benedetta Mennucci and Roberto Cammi. This page intentionally left blank 1 Modern Theories of Continuum Models 1.1 The Physical Model Jacopo Tomasi 1.1.1 Introduction As the title indicates, this chapter focuses on methodological problems relating to the description of phenomena of chemical interest occurring in solution, using methods in which a part of the whole material system is described by continuum models. The inclusion in the book of this introductory section has been motivated by the remarkable advances of continuum methods. Their extension to more complex properties and to more complex systems makes it necessary to have a more detailed understanding of the way in which physical concepts have to be further developed to continue this promising line of investigation. The relatively simple procedures in use for three decades to obtain with a limited computational effort the numerical values of some basic properties, such as the solvation energy of a solute in very dilute solution, are no longer sufficient. To appreciate the basic reasons why continuous models are so versatile and promising for more applications, however, we have to consider again the simple systems and the simple properties mentioned above. The best way to gain this initial appreciation is to contrast the procedures given by discrete and continuum methods to obtain the solvation energy in a very dilute solution. Continuum Solvation Models in Chemical Physics: Theory and Applications © 2007 John Wiley & Sons, Ltd Edited by B. Mennucci and R. Cammi 2 Continuum Solvation Models in Chemical Physics 1.1.2 Solvation Energy The Discrete Approach The material model consists of a large assembly of molecules, each well characterized and interacting according to the theory of noncovalent molecular interactions. Within this framework, no dissociation processes, such as those inherently present in water, nor other covalent processes are considered. This material model may be described at different mathematical levels. We start by considering a full quantum mechanical (QM) description in the Born–Oppenheimer approximation and limited to the electronic ground state. The Hamiltonian in the interaction form may be written as: Ĥ tot rM rS = Ĥ M rM + Ĥ S rS + Ĥ SS rS + Ĥ MS rM rS (1.1) In extremely dilute solutions only a single solute molecule M is sufficient and so Ĥ M refers to a single molecule only. The number of solvent molecules S is in principle infinite, but the physics of the system is sufficiently well described by a finite, albeit large, number n of S units. The third term of the Hamiltonian, Ĥ SS , represents the interactions between such molecules, and the last term, Ĥ MS the interactions between M and the n solvent molecules. The coordinates rM rS apply to both electrons and nuclei. Nuclear coordinates have to be explicitly considered, because the mobility of solvent molecules is a very important factor in liquid systems, and changes in their internal geometry, due to the intermolecular interactions, may also play a role. The formulation of the Hamiltonian given in Equation (1.1) has introduced considerable simplifications in the formulation of the problem (the existence of specific molecules and their persistence has been acknowledged) but the computational problem remains formidable. Approximations are unavoidable. The system is described as an assembly of interacting molecules whose motions are governed, in a semiclassical approximation, by a potential energy surface (PES) of extremely large dimensions related to the positions of all the nuclei of the system, internal nuclear motions within single molecule being for the moment still allowed. The approach used for the characterization of small clusters, i.e. searching first for the minimum energy conformation of the PES, cannot be used here. The physics of solvation is remarkably different. Solvation energy and related properties (solvent effects on the solute geometry are an example) are averaged properties and we are compelled to perform a suitable average upon the energies corresponding to all the accessible conformations of the whole molecular system. Statistical thermodynamics gives us the recipes to perform this average. The most appropriate Gibbsian ensemble for our problem is the canonical one (namely the isochoric–isothermal ensemble N,V,T). We remark, in passing, that other ensembles such as the grand canonical one have to be selected for other solvation problems). To determine the partition function necessary to compute the thermodynamic properties of the system, and in particular the solvation energy of M which we are now interested in, of a computer simulation is necessary [1]. We do not enter into the description of Monte Carlo of Molecular Dynamics methods, as these details are not important for our discussion. There are other more general aspects of computer simulations to consider here. Modern Theories of Continuum Models 3 (1) These averaging procedures introduce macroscopic parameters, temperature and density which are not present in the QM formulation of the problem given by the Hamiltonian of Equation (1.1). The use of macroscopic parameters is necessary for the description of molecular systems in a condensed phase, whether one uses a discrete or continuum approach. (2) The use of a thermodynamic description leads to a more precise definition of the energy we are seeking. The correct choice is the Helmholtz free energy A, directly defined in the (N,V,T) ensemble, which in liquids may be replaced by the Gibbs free energy G, which is formally related to the isothermal–isobaric ensemble (N,P,T) corresponding more to the usual conditions of physico-chemical measurements in solution. This remark on the thermodynamic status of the solvation energy is important for several reasons we shall discuss later. We anticipate one of them, namely that the molecular properties we can put in the form of a molecular response must be expressed as partial derivatives of the free energy, a condition often neglected in the calculation of properties based on discrete models. (3) The use of thermodynamically averaged solvent distributions replaces the discrete description with a continuum distribution (expressed as a distribution function). The discrete description of the system, introduced at the start of the procedure, is thus replaced in the final stage by a continuous distribution of statistical nature, from which the solvation energy may be computed. Molecular aspects of the solvation may be recovered at a further stage, especially for the calculation of properties, but a new, less extensive, average should again be applied. The need for computer simulations introduces some constraints in the description of solvent–solvent interactions. A simulation performed with due care requires millions of moves in the Monte Carlo method or an equivalent number of time steps of elementary trajectories in Molecular Dynamics, and each move or step requires a new calculation of the solvent–solvent interactions. Considerations of computer time are necessary, because methodological efforts on the calculation of solvation energies are motivated by the need to have reliable information on this property for a very large number of molecules of different sizes, and the application of methods cannot be limited to a few benchmark examples. There are essentially two different strategies. The first strategy maintains the QM description of the solvent molecules but reduces their number and adopts a different description for other molecules (often adopting a continuum distribution) to take account of bulk effects in the calculation. These QM simulation methods, of which the first and most frequently used is the Car– Parrinello method [2], are in use since several years, and have largely passed the stage of benchmark examples. This strategy is the most satisfactory under the formal aspects we have at present, and will surely be employed more and more with increasing computer power, but will certainly not completely replace, in the foreseeable future, other strategies. The second strategy we mention in this rapid survey replaces the QM description of the solvent–solvent and solute–solvent with a semiclassical description. There is a large variety of semiclassical descriptions for the interactions involving solvent molecules, but we limit ourselves to recall the (1,6,12) site formulation, the most diffuse. The interaction is composed of three terms defined in the formula by the inverse power of the corresponding interaction term (1 stays for coulombic interaction, 6 for dispersion and 12 for repulsion). Interactions are allowed for sites belonging to different molecules and are all of two-body character (in other words all the three- and many-body interactions appearing in the cluster expansion of the Ĥ SS and Ĥ MS terms of the Hamiltonian (1.1) 4 Continuum Solvation Models in Chemical Physics are neglected). The interaction energy is thus expressed as a sum of terms with the general formula AKi BmK K rim where Ai is the ith site of molecule A and Bm the mth site of molecule B. The numerical values of the coefficients are empirically defined, with starting guesses from QM calculations on the dimer and then refined with a variety of methods. This simple form of the interaction potential is appropriate to perform the numerical simulations leading to the numerical expression of the thermally averaged distributions. The continuation of the strategy presents at this point a bifurcation. The solute M may be described with a semiclassical procedure similar to that used for solvent molecules, or with a QM approach. The first method is often called classical (or semiclassical) MM description [3], the second a combined QM/MM approach [4]. The physics of the first method is rather elementary, but notwithstanding this it opened the doors to our present understanding of the solvation of molecules. The second method is markedly more accurate, because the QM description of the solute has the potential of taking into account subtler solvent effects, such as the solvent polarization of the solute electronic polarization and the changes in geometry within M. A different approach to mention here because it has some similarity to QM/MM is called RISM–SCF [5]. It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM–SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, r instead of a full position dependent function r expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged r may lead to erroneous conclusions which have to be corrected in some way [7]. The 3D version we have mentioned partly eliminates these artifacts. The use of radial distribution functions is one of the costs paid by simulations methods to the high computational cost of this approach. The ever increasing availability of computer power has allowed a sizable portion of these shortcomings to be eliminated. In a few years the description of the QM part of QM/MM applications has progressed from a rather crude semiempirical description to ab initio levels now sufficiently accurate to describe with reasonable accuracy solvent effects on molecular properties and reaction mechanisms. A greater availability of computer power has also permitted the introduction of some improvements in the formulation of the site–site potentials we briefly characterized above. Modern Theories of Continuum Models 5 The original force field greatly reduced the number of degrees of freedom to monitor during the simulation and the number of elements in the many-body problem introduced with the Hamiltonian (1.1). Any improvement will inevitably increase the number of degrees of freedom and the number of interaction terms, rapidly leading to unmanageable expressions. For this reason the improvement of the force field have proceeded slowly, following the increment of computer power. In the original force field the internal geometry of the molecule was kept fixed; until now calculations with flexible potential are a rarity. For standard solvents, in which molecule are small and comparatively rigid, this defect is less important than the neglect or incomplete description of polarization effects, In the first QM/MM formulations polarization sites (one for each solvent molecule) were introduced. This effect was expressed in terms of induced dipole moments (one per site) computed as product of a site isotropic polarizability multiplied by an electric field vector generated by all the charge and induced dipoles present in the system. The complexity of the calculation is thus considerably increased because these new terms are not of a two-body character as are the original (1,6,12) terms, and have to be computed iteratively. This commendable effort continues, introducing more than a single polarization site for a molecule, but the final (practical) solution of the problem has not yet been reached. This formulation of the problem in fact neglects several aspects of the physics of the phenomenon, which further analyses have shown to be important, and the error in the description of the polarization response that this methodology gives is of the order of 10–20 %. This error is to a large extent due to the absence of some coupling terms, but the situation is more complex, also including parameters which change from case to case (the nature of the solvent, the presence of a net charge on the solute, the macroscopic parameters T and P, etc). The Continuum Approach We report in this subsection a discussion on some aspects of continuum solvation (CS) methods which seems to us useful to examine how the physics of solvation is described by these models. Other contributions in the book will give more details about their methodology, implementation and use. We consider our recent review [8] to be an appropriate text to complement what is said here. The Hamiltonian for the basic formulation of the problem, to be compared with that given in Equation (1.1), may be written in the following form: Ĥ tot rM = Ĥ M rM + Ĥ MS rM (1.2) The solvent coordinates rS do not appear in Equation (1.2) and this is the basic difference between discrete and continuum models. The Hamiltonian Ĥ tot rM is an effective Hamiltonian, written as two separate terms in Equation (1.2) to facilitate comparison with Equation (1.1) but in actual calculations it is convenient to treat Ĥ tot as a whole, because its structure is very similar to that of Ĥ M in vacuo, in the Hartree–Fock (HF) or Density Functional Theory (DFT) formalism. The passage at higher levels of the QM theory follows the same lines as for isolated molecules. 6 Continuum Solvation Models in Chemical Physics Actually Ĥ MS is a sum of different interaction operators each related to an interaction with a different physical origin. The coupling between interactions is ensured by the iterative solution of the pseudo HF (or DFT) solution of the whole Schrödinger equation. These operators are expressed in terms of solvent response functions based on an averaged continuous solvent distribution. They will be symbolically indicated with the symbols Qx r r where r is a position vector and x stands for one of the interactions. We shall examine later the form of some of the operators, which actually are the kernels of integral equations. In contrast with discrete methods, the thermal average is introduced in the continuum approach at the beginning of the procedure. Computer information on the distribution functions and related properties could be used (and in some cases are actually used), but in the standard formulation the input data only include macroscopic experimental bulk properties, supplemented by geometric molecular information. The physics of the system permits the use of this approximation. In fact the bulk properties of the solvent are slightly perturbed by the inclusion of one solute molecule. The deviations from the bulk properties (which become more important as the mole ratio increases) are small and can be considered at a further stage of the development of the model. In the standard continuum solvation model, exemplified by the Polarizable Continuum Model (PCM) we developed in Pisa [9], the solute–solvent interaction energies are described by four Qx operators, each having a clearly defined physical nature. Each term gives a contribution to the solvation energy which has the nature of a free energy. The free energy of M in solution is thus defined as the sum of these four terms, supplemented by a fifth describing contributions due to thermal motions of the molecular framework; GM = Gcav + Gel + Gdis + Grep + Gtm (1.3) The order of contribution given in Equation (1.3) corresponds to the best order in which a sequence of ‘charging processes’ could be performed. A ‘charging processes’ basically is an integration performed with respect to an appropriate parameter running from zero to the final value which couples a given distribution with a potential function. At the end of the charging process the distribution is modified and used for the following charging process. The best sequence is that in which the residual couplings are minimized. In ab initio PCM three charging processes are unified and described by the solution of the Schrödinger equation, thus avoiding the problem of coupling a sequence of separate charge processes. Only the first, namely that giving the cavity formation energy, is treated separately. The last contribution, describing thermal motions of the solute, is composed of different terms and is treated in a different manner. In spite of the unification of different processes in the calculations, each term will here be separately presented and commented on. Cavity formation energy The first charging process is related to the formation in the pure solvent of a void cavity having the appropriate shape and size to accommodate the solute. The electronic properties of the solute are not used here, only the geometrical nuclear parameters Modern Theories of Continuum Models 7 are employed to define the correct shape and size. The reversible work spent to form the cavity is exerted against the forces giving cohesion to the liquid. Calculations are performed at a given temperature T and a given solvent density. There are different methods, using different solvent parameters, to compute this contribution to the solvation energy. We simply mention two methods: the oldest based on the surface tension of the solute [10], and the newest based on the use of information theory methods [11], without giving details, to focus our attention on the method used in PCM and in other variants of the continuum solvation approach. This method is based on the scaled particle theory (SPT), an integral equation method which is simple and effective. The formulation given by Pierotti [12], extended to cavities of molecular shapes according to a suggestion given by Claverie [13], is adopted in PCM. The parameter characterizing the solvent is the solvent equivalent radius. The expression for Gcav is analytical for a spherical cavity and semi-analytical for cavities defined in terms of atomic solute spheres. We started to use SPT derived cavity formation energies in 1981 [14], with many initial perplexities about the physical correctness of the use of hard spheres also for solvents exhibiting hydrogen bonds or irregular shapes. Fortunately, the cavity formation energy is a term (the only one in the whole expression (1.3)) for which an independent validation of its numerical value is possible. There are at present a sufficiently large number of results, obtained with semiclassical simulations with accurate force field potentials, showing that the SPT approach gives good results for a large variety of solvents and cavity sizes and shapes. The formal operator Qcav is not included in the Hamiltonian (1.2). In the BO approximation we are using, this term is constant as long the geometry of the molecule is unchanged. From this point of view it may be assimilated into the nuclear repulsion Vnn of a single molecule, again in the BO approximation. The cavity formation charging process produces an important change in the solvent distribution. After the charging the portion of space within the cavity has zero density. In the outer space the solvent density can be kept constant assuming the cavity volume is infinitely small with respect to the bulk. Electrostatic energy In ab initio formulations this charging process includes the whole molecular density as well as the electric polarization of the solvent, starting from noninteracting nuclei and electrons that will compose the molecule. This is a variant with respect to the traditional view of first defining with QM calculations the molecular density in vacuo, and then of passing to a different version of the charging process to activate mutual solute–solvent polarization effects. The QM procedure normally adopted follows the first strategy, with a single charging process; the traditional strategy which decouples the charging process is necessary when one has to compute the solvation energy given as the difference between the free energies of the molecule in solution and in vacuo. When the explicit evaluation of the solvation energy is not required, the traditional procedure may be considered to be a waste of computer time, because two geometry optimizations are required. The two strategies lead to the same result, and people wishing to know in advance the structure of the isolated molecule and to look at the changes in geometry and electro-distribution produced by the solvent obviously perform two sets of calculations. 8 Continuum Solvation Models in Chemical Physics In the simplest The medium response function for Gel is the polarization function P. formulation of PCM we are at present considering, the following formulation of the polarization function is used: P = −1 E 4 (1.4) is the electric field generated directly or via the apparent charges spread on were E the cavity surface, and is the scalar permittivity, constant over the whole body of the solvent. The basic electrostatic relation one has to satisfy is given by the Poisson equation. because most of the physics of solution We shall have to reconsider the expression for P, not yet considered in this preliminary presentation is related to the appropriate definition In all systems and for all the properties and phenomena the electrostatic component of P. is the most sensitive to changes in the system and to the quality of the description. The utmost care must be taken to have a reliable description of electrostatic solvent effects. Repulsion energy This term is physically related to the electron exchange contributions appearing when interactions among molecules are described at the QM level. The description of this contribution has been extensively examined for small discrete systems. In CS models there are no discrete representations of solvent molecules, but from the wide experience on dimers and small clusters it is possibly to justify the expression used in PCM where it is introduced a Q̂rep r operator based on the solvent density, the number density of electron pairs in the solvent, the normal component at the cavity surface of the electric field generated by the solute and an overlap function. The resulting operator is one electron in character and it is inserted in the Hamiltonian (1.2) under the form of a discretized surface integral, each belonging to a specific portion (tessera) of the closed surface [15]. The physics of this interaction has perhaps to be reconsidered to accurately describe high pressure effects on solvation. Dispersion energy The dispersion contribution to the interaction energy in small molecular clusters has been extensively studied in the past decades. The expression used in PCM is based on the formulation of the theory expressed in terms of dynamical polarizabilities. The Q̂dis r r operator is reworked as the sum of two operators, mono- and bielectronic, both based on the solvent electronic charge distribution averaged over the whole body of the solvent. For the two-electron term there is the need for two properties of the solvent (its refractive index ns , and the first ionization potential) and for a property of the solute, the average transition energy M . The two operators are inserted in the Hamiltonian (1.2) in the form of a discretized surface integral, with a finite number of elements [15]. The procedure we have outlined for these three terms of Equation (1.3) is of the ab initio type, with the form used for HF (or DFT) calculation for an isolated molecule with the addition of a few new operators, all expressed as one-electron integrals over the expansion basis set (also the two-body dispersion contribution is reduced to the combination of two one-electron integrals). We remark that all the elements of the solute–solvent interaction, cavity formation excluded, are expressed as discretized contributions on the Modern Theories of Continuum Models 9 cavity surface, computed at the same positions. The whole computational framework has a compact form, without detriment of the description of the physical effects. We resume here the nature and number of macroscopic parameters used in this version of PCM: the temperature T , the density S of the solvent, its permittivity (here reduced to a constant T dependent ), and its refractive index nS . Among the constitutive parameters there is the hard sphere radius of the solvent molecule and its first ionization potential IS . When the thermal motion contributions Gtm (on which we do not enter into details) are added we have an ‘absolute’ value of the free energy of M in the given solvent. The reference state is given by the unperturbed solvent and the amount of noninteracting electrons and nuclei necessary to form M. By making the difference with the ‘absolute’ free energy of M in vacuo computed with the same QM procedure (the reference state is given by the necessary amount of noninteracting electrons and nuclei) we have an estimate of the free energy of solvation Gsol M. Comparison with experimental values shows that the results are quite good for larges classes of systems (solutes and solvents). The limited cases in which this agreement is only fair will be considered in the following section. With this statement we conclude our summary of a long and complex journey along formal considerations, models for partial contributions to the energy and developments of computational procedures. No experimental values or well established computational results are available for the separate components (apart from cavity formation energy). However, we have to consider that this empirical evidence of good values of solvation energies for large classes of systems (solutes and solvents) is nothing more that an encouragement to proceed further in the construction of models based on well defined physical bases. The energy is not too sensitive a property and casual compensations among errors of different sign could have improved the results. The approach we have considered presents some features which recommend it for further extensions. Firstly, it is an ab initio method with a low computational cost. A calculation a solution with a good basis set has a computational cost lower that double the analogous calculations for the isolated molecules, and the ratio of computational costs becomes even more favourable in passing to higher levels of the QM theory. Secondly, all the features of modern quantum chemistry can be easily implemented in this model. For example, the standard sequence of molecular calculations often adopted for a better characterization of the molecule (HF, DFT, MP2, CCSD, CCSD(T)) could be adopted (see also the contribution by Cammi in this book). As shown in other chapters of this book, analytical expressions for the derivatives necessary for geometry optimizations and calculations of response properties are now available; the interpretative tools in use for characterizing electronic structures can be employed. The last aspect we stress is the flexibility of the method. Simplified versions are abundant, and they have an important role in computational chemistry, but in this chapter we consider extensions and refinements which introduce in the model other aspects of the physics of solvation. 1.1.3 The Solvent Around the Solute Several possible refinements of the continuum model can be examined using again infinitely dilute solutions. In the basic model we have used a uniform distribution of the 10 Continuum Solvation Models in Chemical Physics solvent, characterized by a constant value of the permittivity. Intuition suggests that local disturbances to this description are more probable near the solute, and there are good reasons to think that such disturbances have a measurable effect on some properties of the solvent. We remark that the agreement with experimental solvation energy data is quite good in general, but there are classes of systems in which a greater deviation has been observed. We could try to examine the extent to which this partial disagreement in the solvation energy is due to a local disturbance of the solvent, but surely other cases of local disturbance are not visible in the solvation energy, a property relatively insensitive to small changes in the interaction potential. To look at these cases, more sensitive indicators are needed, and they are given by other properties, mostly of spectroscopic origin. There is a large variety of phenomena to consider in this section, not all completely understood, related to a large variety of effects, all amenable to the physics of interacting molecular systems, some of general occurrence, others with a character of chemical specificity. A clear cut classification is not possible because often different effects are intermingled, and our exposition will not be systematic but limited to some aspects of greater physical interest. More systematic analyses will be found in other chapters of the book. Nonlinearities in the Dielectric Response Among factors of general occurrence we have omitted in the description of the basic CS model, some are related to refinements of the dielectric theory. The charge distribution of almost all solutes gives rise to strong electric fields. These fields are stronger for charged species, especially those of small size such as atomic ions, but they are also present for neutral molecules exhibiting anisotropies in the charge distributions of chemical groups near the periphery of the molecule. The case of ions has been largely explored, but we shall also consider the case of neutral solutes. The occurrence of strong permanent fields may disturb the linear response regime in the dielectric response we have so far employed. The standard treatment of nonlinear dielectric response is based on the expansion of the dielectric displacement function D generally interrupted at the first correction: in powers of the electric field E, =E + 4 P = + 4 3 E 2 E D (1.5) This expression introduces the third order susceptibility of the medium, a quantity not easy to be accurately determined for the small portions of solvent in which the nonlinearity effect is sizeable. In addition we remark that with the favourable exception of atomic ions which have a spherical symmetry, the solvent layer in question has an irregular shape (not directly amenable to the molecular shape because the chemical groups responsible for nonlinearities are not regularly placed on the molecular surface). For this reason the whole tensorial expression of 3 with a position dependent formulation, should be used. The origin of the effect here represented by 3 can be derived from modelistic considerations. Solvent molecules are mobile entities and their contribution to the dielectric response is a combination of different effects: in particular the orientation of the molecule under the influence of the field, changes in its internal geometry and its vibrational response, and electronic polarization. With static fields of moderate intensity all the cited effects contribute to give a linear response, summarized by the constant value of the permittivity. This molecular description of the dielectric response of a liquid is Modern Theories of Continuum Models 11 locally modified by a strong molecular field: firstly a saturation in the response with a nonlinearity reducing the actual permittivity with respect to that obtained in the linear formulation (still valid at larger distances); secondly a displacement of the first shell molecules toward the solute. Liquids are remarkably incompressible, and a collective displacement with a local increase of density requires an appreciable amount of work against molecular repulsions. However, this effect is possible (measurements in solution are generally performed at fixed pressure), and it is called electrostriction. A third effect is related to possible anisotropies in the molecular polarizability; this contribution is also positive. In conclusion the contribution to the dielectric response given by the third order susceptibility has different sources with opposite signs. Molecular simulations on ions in solution show that both dielectric saturation and electrostriction effects are presumably present and that for ions with a high charge density electric saturation predominates. This suggestion is in agreement with the general consensus that dielectric saturation is the first element to consider in the description of nonlinearities. In spite of the remarkable difficulty in defining a detailed model, the number of computational codes introducing dielectric nonlinearity, especially in the form of dielectric saturation, is quite abundant. We quote here the main approaches; more details can be found in the already quoted review [8]. Layered models The solvent is described as a set of onion-like shells with increasing values of , constant within each shell. The layers approach gained some popularity in the late 1970s, generally applied to semiclassical descriptions of the solute. The electrostatic part has analytical solutions for cavities of regular shape (spheres, ellipsoids) but its use is also possible for irregular cavity shapes and for QM descriptions of the solute. Applications of the approach in this more general formulation have been formulated and used for old versions of PCM, with appreciable results (this is an example of the flexibility of continuum models) [16]. We remark that at each layer separation there are boundary electrostatic conditions equivalent to those present in the single cavity model. Several published papers neglect this coupling, and the error may be sizeable. A correct application leads to an increase of the computational times, and for this reason the approach has been abandoned in PCM because there are more efficient ways to describe the saturation phenomenon. The layered model in PCM has not been abandoned, however, and it has been adopted in more specialized approaches addressing specific phenomena, such as the nonequilibrium solvation, electron transfer reactions, and phenomena related to the behaviour of the liquid in phase separations. A case deserving mention is that of solvation in supercritical liquids in which the standard sequence of values of the dielectric constant in the layers, from lower to higher values, has been reversed to describe electrostriction effects [17]. Position dependent dielectric constant This model has been, and still is, widely used especially for some specific applications. An older use is in the description of dielectric saturation effects around ions. The origin is the Debye model, not completely satisfying and thus subjected over the years to many variants. The spherical symmetry of the problem suggests the use of a distance dependent function r. The functions belonging to this family are often called ‘sigmoidal functions’ because their spatial profile starts from a low value and increases monotonically to reach 12 Continuum Solvation Models in Chemical Physics the bulk value with a sigmoidal shape. The definition of these functions is empirical; the contribution of computer simulations to the validation of these functions has been minimal because the longitudinal component of k (calculations are generally performed in reciprocal space) has, at least in dipolar liquids, a nonmonotonic shape, and the portion of the function at high k values, the most important for the definition of solvent effects on the energy, is rarely computed, and the available data have a low numerical reliability. The r functions are frequently employed for large molecular systems of biological interest, to screen the coulombic interactions between the point charges used in these models. Position dependent models are also in use for interfaces of a planar type, under the form of z functions, where z is the Cartesian coordinate perpendicular to the phase separation surface (see the contribution of Corni and Frediani in this book). Electric saturation effects in the description of neutral solutes in polar media have been strongly advocated by Sandberg et al. [18], who worked out a complete continuum ab initio solvation code containing the r feature and published results of good quality for a large number of solutes. Sandberg et al. remark that PCM calculations do not need corrections for electric saturation, this being due, in their opinion, to the cavity PCM uses. We also quote the proposal, made by Luo and Tucker [19], of a model using a dielectric function with dependence of the dielectric constant on the electric field acting on the given position, used for supercritical liquids, in which the solvent density is particularly sensitive to the local value of external electric fields. Emphasis is given in this model to electrostriction effects. This mention of a family of solvents with particular physical properties prompt us to remark that there are other solvents with special physical quantities requiring some modifications in the methodological formulation of basic PCM. We cite, among others, liquid crystals in which the electric permittivity has an intrinsic tensorial character, and ionic solutions. Both solvents are included in the IEF formulation of the continuum method [20] which is the standard PCM version. Nonlocal dielectric constant The dielectric theory may be expressed in a nonlocal form based on the definition of the susceptibility and permittivity in a form that makes these physical quantities the kernel of appropriate integral equations. The formal definitions of the nonlocal operators ˆ and ˆ can be expressed in the form of their application to a generic Fr function: ˆ Fr = d3 r r r Fr (1.6a) ˆFr = d3 rr r Fr (1.6b) The expression for the polarization is given by = d3 r r r Er Pr (1.7) which shows that the permittivity depends on the field felt at all positions of the medium. Modern Theories of Continuum Models 13 The nonlocal dielectric theory has as a special case the standard local theory. Its fuller formulation permits the introduction in a natural way of statistical concepts, such as the ˆ For correlation length which enters as a basic parameter in the susceptibility kernel . brevity we do not cite many other features making this approach quite useful for the whole field of material systems, not only for solutions. What is of interest here is the description of nonlinear dielectric effects with a linear procedure. Nonlinear dielectrics were introduced in the theory of liquids by Dogonatze and Kornyshev in the 1970s [21]; the reformulation of the theory in more recent years by Basilevsky [22] permits its insertion in the whole machinery of the PCM version of the CS method. The reader is also referred to the contribution of Basilevsky and Chuev dedicated to non-local dielectric solvation models. Specific Solute–Solvent Interactions Interactions between the solute and solvent molecule are always present in solution. Their nature depends on the chemical constitution of the interacting partners, and the rules of interaction are the same of those studied in simpler molecular clusters. However, there is an important difference between the same M–S interaction in the gas phase and in solution. In the gas phase the geometry of M–S tends to correspond to the most favourable conformation, and to disrupt the M–S association it is necessary to expend the energy corresponding to the stabilization energy of the dimer. In solution there is competition between the S molecule interacting with the solute and with other solvent molecules. These interactions may disturb the most favourable conformation of M–S and, more importantly, change the nature of the disruption of S from a dissociation to a an act of replacement. These are naïve considerations, but it is convenient to recall them because in our opinion they are often neglected. An example of the application of this different nature of molecular interactions in solution concerns an aspect we have already mentioned, without comment. Among the energy terms collected into the Gtm term there is the contribution due to the rotation of M. This contribution is certainly not equal to that of the freely rotating molecule in vacuo, but it is even more erroneous to assimilate it into the contributions of a rotor impeded by a barrier equal to that, for example, of a hydrogen bond, the existence of which has been inferred from the chemical composition of the system. During the rotation the hydrogen bond assumed to be present at a given moment will be deformed and replaced by other molecular interactions, quite frequently of a similar nature. A parameterization of the rotational contribution to the free energy has to be based on other parameters. This error has been repeated in several of the early attempts at modelling liquid systems. Solute–solvent local interactions may play a role in several aspects of the solvation effects. The analysis is delicate because finer aspects of the physics of interacting molecules have to be introduced. Let us start with a complement to the naïve considerations exposed few lines above. An important aspect of the local interactions in condensed media subjected to thermal averaging is their persistence. The persistence is clearly related to the strength of the interaction, but it is also related to the collective effects of the nearby molecules. The persistence times span a wide range: from the short times corresponding to librations of 14 Continuum Solvation Models in Chemical Physics the molecule to very long times. We limit our considerations here to short and intermediate persistence times, typical of neutral solutes. When we examine the response properties of the solute, attention has to be paid to comparing the persistence of these local interactions with the time necessary to measure the property. Also measurement times may span a very large interval, depending to the property one is measuring and the technique one is using. Let us consider again the solvation energy, which is a response property. All the standard experimental methods to measure solvation energy require long times. Within such times almost all the local interactions are mediated, losing to a great extent the specificity exhibited for example in a Monte Carlo simulation addressing the definition of the minimal internal energy of the solvation cluster. Only a limited number of interactions of particular strength remain to have an effect on the averaged solvent distribution. This is the case for hydrogen bonding and the effect on the distribution function is the reason for the often repeated remark that continuum methods are unable to describe hydrogen bond effects. Actually this is not true, since for many years it has been well established [23] that the energy of hydrogen bonds is well described by the combination of the electrostatic, repulsion and dispersion terms also used in continuum solvation methods, and this is a fortiori true for the deformed hydrogen bond description given for the averaged solvent. The errors given by calculations that are sometimes performed to support this claim are, to the best of our knowledge, due to a poor implementation of the continuum model [24]. These hydrogen bond interactions do, however, influence other properties. We now examine some examples. Solvent effects are comparatively greater on the vibrational properties of the solute group involved in the hydrogen bond. The continuum method gives a fairly good description of the vibrational solvent shift, but not sufficient to reach spectroscopic accuracy. The same holds for the corresponding intensity. We remark that this small error on these vibrations has no effect of the vibrational component of Gtm , because their contribution to the energy of the relevant distribution function is completely negligible. However, there is a small contribution to the zero point energy. There are a number of other molecular properties that may be affected by these persistent interactions. The more studied properties so far are the electronic excitation energy of a chromophore involved in the permanent interaction, and the magnetic shielding of atoms (notably O and N) directly involved in this interaction, but all the properties exhibiting a local character (for example the nuclear quadrupole resonance) may be subject to similar persistent interactions. Persistent interactions are not limited to hydrogen bonds. We mention for example those appearing in solutions of molecules with a terminal C=O or C≡N group dissolved in liquids such as acetone or dimethylsulfoxide. These solvents prefer at short distances an antiparallel orientation which changes at greater distances to a head-to-tail preferred orientation. The local antiparallel orientation is somewhat reinforced by the interaction with the terminal solute group and it is detected by the PCM calculation of nuclear shielding and vibrational properties. Recent experimental correlation studies [25] have confirmed the orientational behaviour of these solvents found in an indirect way from continuum calculations. The physical effect found in this class of solvent–solute pairs seems to be due to dispersion forces. Modern Theories of Continuum Models 15 Calculations show that the main contribution to the solvent effect on these properties is described by the standard CS method, but there is often a missing part. The entity and percentage weight of this part may change noticeably when the molecular framework of the solute is changed. This is an indirect hint that all the solute charge distribution is in some way involved. Calculations also show that by including in the solute a small number of solvent molecules (i. e. going from M to MSn . with n = 1 2 3 according to the case) the continuum method gives fully satisfactory results. The study of this problem is an example of the usefulness of CS ab initio methods. It is computationally easy to repeat calculations of wavefunction, energy and all the above mentioned properties for MSn solutes with an increasing number n of solvent molecules and to determine at what n value the saturation for this effect is reached. Calculations on MSn systems show other interesting aspects of the problem. The n S molecules must be inserted in the solvent as a supermolecule. In fact MM descriptions or Hartree QM descriptions (without exchange) have no effect on this correction. The quality of the wavefunction seems not to be important for the correction (it is important, however, for the main calculation of the property); calculations with an ONIOM scheme [26] with the solvent molecules kept at a low HF description gives the same accurate description as the full high level QM calculations [24]. These empirical findings show that something is missing in the physics we are using. Analyses of the M wavefunctions seem to indicate that in the cases of a missing contribution to the property there is a flow of electrons from M to S. We have arrived at a point which touches on some basic simplifications taken for granted in all theories regarding weak interactions between molecules. The basis for these continuum models, as well as for the QM/MM methods, is given by the application of the perturbation theory approach to the description of noncovalent interactions. It is worth examining the evolution of these theories. The first steps were taken by Debye around 1920, the theory recast in a QM form in 1927, and developed and refined for some decades, until it was recognized in the middle of the 1970s that a discarded contribution, namely that related to the complete antisymmetry of electrons in the interacting system, was essential. In the following 30 years the perturbation theory was reworked and refined again within this modified theoretical background. It now seems that the extension to more accurate calculations of response properties leads to a critical examination of another of the basic tenets of the standard noncovalent interaction theory, i.e. that the amount of electronic charge within each interaction partner has to be kept fixed in defining the interaction. Chemists are well aware that strong molecular interactions may be accompanied by a flow of electron charge but the evidence they present has been disregarded by physicists. The latter consider this evidence not to represent legitimate noncovalent interactions, with the additional remark that in the case of very small electron transfers the polarization contribution is able to describe such small effects. The problem we have raised seems to be of methodological relevance and to require attention. From the computational point of view the strategy of using MSn clusters we have outlined may be accepted as a reasonable provisional compromise. We recall what we have already said, i.e. that the whole cluster has to be considered as a unique supermolecule, and we add that the problem of extracting from a supermolecule a true molecular observable is not yet fully resolved. In conclusion it may be said that for response 16 Continuum Solvation Models in Chemical Physics properties of solutes exhibiting permanent interactions, active in the measurement, good descriptions are possible, but with a blur in the finest details. 1.1.4 Dynamical Aspects of Solvation We have so far considered static aspects of the solvation phenomena. This is a strong limitation, because dynamical aspects are always present and they often play the dominant role. Our selection of topics to consider in this section will be however severely reduced with respect to the number of phenomena of relevance to the section’s title. The variety is too great. A few considerations will justify our reduction. Firstly, the time scales: phenomena in which the molecular aspect of the solute–solvent interactions is the determinant aspect (a subject central to this book) span about 15 orders of magnitude, and such a sizeable change of time scale implies a change of methodology. Secondly, the variety of scientific fields in which the dynamical behaviour of liquids is of interest: to give an example friction in hydrodynamics and in biological systems has to be treated in different ways. All types of time evolution are present in dynamical solvation effects. It is difficult, and perhaps not convenient, to define a general formulation of the Hamiltonian which can be used to treat all the possible cases. It is better to treat separately more homogeneous families of phenomena. The usual classification into three main types: adiabatic, impulsive and oscillatory, may be of some help. The time dependence of the phenomenon may remain in the solute, and this will be the main case in our survey, but also in the solvent; in both cases the coupling will oblige us to consider the dynamic behaviour of the whole system. We shall limit ourselves here to a selection of phenomena which will be considered in the following contributions for which extensions of the basic equilibrium QM approach are used, mainly phenomena related to spectroscopy. Other phenomena will be considered in the next section. Nonequilibrium Aspect of Spectroscopic Phenomena In going from static to dynamic descriptions we have to introduce an explicit dependence on time in the Hamiltonian. Both terms of the Hamiltonian (1.2) may exhibit time dependence. We limit our attention here to the interaction term. Formally, time dependence may be introduced by replacing the set of response operators collected into Q̂r r with Q̂r r t and maintaining the decomposition of this operator we presented in Section 1.1.2. For simplicity we reduce Q̂r r t to the dielectric component under the t. With this simplification we discard both dielectric nonlocality and nonelecform Pr trostatic terms, which actually play a role in dynamical processes, especially dispersion and nonlocality. The basic aspects of the theory of the behaviour of dielectrics in time dependent electric fields have been known for a long time. We recall some elements useful for our discussion. We start with the time dependent polarization function Pt. This quantity may be expressed in the form of an integral equation: = Pt t − dt Qt − t Et (1.8) Modern Theories of Continuum Models 17 is the Maxwell field. where the kernel Qt − t is the solvent response function and Et In the case of an external sinusoidally varying electric field it is easy to obtain from Pt the frequency dependent permittivity which is a complex function = + i (1.9) Both (called the frequency dependent dielectric constant) and (called the loss factor) play a role in our applications of the theory. In continuum methods we have to use the function of pure liquids. Both components of can be experimentally measured and can also be computed with theoretical methods, but it is convenient to introduce here the physical structure of the spectrum. The intensity of the dielectric absorption is proportional to the imaginary part of . The spectrum consists of separate absorption bands, with moderate overlap and separated by regions of very low intensity (the ‘transparent’ regions). The harmonic decomposition of the spectrum into normal modes shows the dominance of a limited number of classes, each having correlation ranges of approximately the same value. A simplified model consists in using a single collective mode per class. Of course more refined descriptions are possible, and for some phenomena they are necessary. We shall not use these refinements, limiting ourselves to stating that models exist that try to describe better the regions in which there is overlap between classes and models giving a description of the ‘transparent’ regions. The microscopic origin of the collective modes has been identified since a long time. They are reported here with the corresponding typical correlation times (CT): reorientation modes (this is the so-called Debye region, CT > 10−12 s), libration modes (rotations impeded by collisions, CT = 10−13 s), atomic motions (vibrations, CT = 10−14 s), electronic motions CT = 10−16 s. When the frequency of the external field increases, the various components of the polarization we have introduced here become progressively no longer active, because the corresponding motions of the solute lag behind the variation of the electric field. These considerations have to be applied to phenomena in which the ‘external’ field has its origin in the solute (or, better, in the response of the solute to some stimulus). The characteristics of this field (behaviour in time, shape, intensity) strongly depend on the nature of the stimulus and on the properties of the solute. The analysis we have reported of the behaviour of the solvent under the action of a sinusoidal field can here be applied to the Fourier development of the field under examination. It may happen that the Fourier decomposition will reveal a range of frequencies at which experimental determinations are not available: to have a detailed description of the phenomena an extension of the spectrum via simulations should be made. It may also happen that the approximation of a linear response fails; in such cases the theory has to be revisited. It is a problem similar to the one we considered in Section 1.1.2 for the description of static nonlinear solvation of highly charged solutes. Current applications have so far avoided these more detailed formulations of the dielectric relaxation, and the scheme of decomposition into collective modes is simplified to two terms only, which here we denote as ‘fast’ and ‘slow’ P ≈ P fast + P slow (1.10) 18 Continuum Solvation Models in Chemical Physics This partition is known under two names, Pekar and Marcus, and it may actually be expressed in two ways, with different couplings between the various components (see ref. [8]). The two decomposition schemes are equivalent in the linear response regime. This two-mode partition is used for a wide variety of phenomena, characterized by a sudden change in the solute charge distribution (electrons as well as nuclei). We give some examples: a sudden change of state in the solute (electronic, but also vibrational), intermolecular electron and energy transfer, and proton transfer. These examples may be extended to other phenomena, and the examples given may also be partitioned into several classes for which the physics of the problem suggests different ways of using the basic approach. This partition is appropriate to characterize the initial nonequilibrium step of many phenomena, such as those occurring in the spectroscopic domain (but also at intermediate stages, such as the rapid step of proton transfer in chemical reactions). To proceed further in the description of a phenomenon one has to replace the two-mode description with a more appropriate model. An example will clarify this discussion. The electronic transition of a solute is a sudden phenomenon followed by other dynamical stages, with different exit channels. According to QM a sudden perturbation (due to a photon in this case) gives rise to nonzero amplitudes for a manifold of states. This also happens for molecules in solution. The first quantity to be computed is the lowest vertical transition energy. Almost all CS methods (including PCM which probably was the first to do it at ab initio QM level) use a two-mode approximation with the slow component of the polarization vector determined on the ground state electronic distribution P slow GS and the fast one using the electronic distribution of the excited state of interest P fast EX. This fast component is based only on the electronic dielectric relaxation of the solvent and has to be determined with an iterative process which also modifies the effective Hamiltonian in use. As a consequence the two wavefunctions, (GS) and (EX), are computed with two different Hamiltonians. The same happens for the other states in the manifold created by the sudden perturbation. The conclusion is that the amplitudes of such states must be described by an expression more complex than that used in the standard formulation for molecules in vacuo. The QM description of molecules in condensed phases is rich in problems of this type. We stress that the physical basis of the description is correct: the origin of the differences with respect to the standard picture is due to the use of effective Hamiltonians, a feature we cannot abandon. We briefly mention a mathematical problem related to the definition of determinants in CI procedures addressing the improvement of the wavefunctions (ground as well as excited states). This is a question of marginal relevance in our rapid discussion, and the mention of the problem, for which a reasonable solution is possible, is sufficient: more details can be found in the contribution by Mennucci. Let us to continue the discussion of the fate of the electronic excitation. We select the channel that after the initial vertical excitation leads to a fluorescent emission. This spectroscopic signal has been widely studied because it leads to information about the relaxation of the solvent. The other modes of dielectric relaxation become progressively active after the excitation and the effects are measured by the time resolved fluorescent Stokes shift (TDFSS). A detailed analysis of these phenomena is given in the contribution by Ladanyi; here we shall merely make some general comments. Modern Theories of Continuum Models 19 The sequence of the observed frequencies, resolved on the time scale, may be regrouped in a form giving a quantity St which may be related to a time correlation function CE t which represents the ensemble average of solvent fluctuations. St ≡ t − Et − E = 0 − E0 − E CE t ≡ (1.11) < E0 >< Et > < E2 > The relationship between spectroscopic and statistical functions has been exploited for a variety of phenomena related in different ways to the dynamical response of the medium. We cite as examples spectral line broadening, photon echo spectroscopy and phenomena related to TDFSS we are examining here. A variety of methods are used for these studies and we add here methods based on ab initio CS. The basic model is actually the same for all the methods in use: ab initio CS has the feature, not yet implemented in other methods, of using a detailed QM description of the solute properties, allowing a description of effects due to specificities of the solute charge distribution. The expression of the St function contains the combination of three terms, two of which, E0 and E, correspond to the differences of energy with respect to the ground state, computed in the vertical transition approximation using respectively the two-mode nonequilibrium and the equilibrium formulations. The third term, Et, which gives the shape of the correlation function, and which is generally drawn from experimental measurements, may be computed in the continuum framework making use of an auxiliary function expressed as an integral over the whole range of frequencies where the integrand is a function of the imaginary part of [27]. We thus obtain an expression in which the continuum method requires the knowledge of another bulk property of the solvent, the spectrum of . There are experimental determinations of portions of this spectrum for a sizeable number of solvent, and there are empirical analytical formulae which describe well, or passably well, the portions at low and intermediate frequencies, while for the portions at high frequency, shown from calculations to be essential for the determination of the fastest steps of the relaxation process, the best way to proceed is to drawn information from accurate MD simulations. We remark that the spectrum is to a good approximation a property of the solvent alone, and so, once determined, it may be used for many solutes. The formulation of the method we have sketched, thus far applied with some approximations, may in principle also be applied to nonpolar solvents. However, there are practical difficulties to overcome. The mode analysis in nonpolar solvents is less developed and experimental data on the dielectric spectra are scarcer. The solution of using computed values of for the whole spectrum is expensive and computationally delicate. The best way is perhaps to develop for apolar solvents a variant of the reduction of Q̂r r t that we have introduced for polar solvents, which takes into account that in nonpolar solvents the interaction is dominated by nonelectrostatic terms. The reformulation of the theory has not yet been attempted, at least by our group, but in recent versions of the continuum ab initio solvation methods there are the elements to develop and test this new implementation. 20 Continuum Solvation Models in Chemical Physics In our discussion about the TDFSS we have not made mention of the relaxation of the solute after the vertical excitation. This relaxation occurs in all cases, except for atomic solutes. Relaxation times are of the same order of magnitude as those active in the first stages of the relaxation of the solvent, so the two processes are coupled. TDFSS measurements have been used mostly to study the dynamical behaviour of liquids, and for this reason the solutes used in experiments are generally quite rigid. In nature (and in laboratories) there are many examples of relaxation phenomena in which the characterizing part is given by the solute geometry relaxation. We remark that in some cases solvent effects on the relaxation of the excited state geometry are better modelled, to a first approximation, in terms of the solute viscosity [28] also in the presence of permanent dipoles. We are here touching on an aspect of great importance in the description of the dynamical evolution of molecular systems in condensed phases, that of motions in the presence of stochastic fluctuations. We shall consider this aspect in the following section, making use of the Langevin equation approach. 1.1.5 Interactions between Solutes The whole body of chemistry is essentially based on the exploitation of interactions between molecules in a liquid phase. There is an enormous wealth of empirical evidence about the influence of solvents on chemical reactions. Chemists actively exploit this body of evidence in many ways, according to different strategies based on their experience and tuned to their needs. Rarely does a new synthesis start with a preliminary accurate theoretical study. However, there is a progressively increasing trend of using computational tools even in the start-up stage of novel syntheses. Computer derived estimates of the solvent influence on some parameters, essentially relating to chemical equilibria and reaction rates, give hints on the definition of an opportune strategy for the synthesis. A good number of the computational tools of this sort rely on the use of continuum descriptions of the solvent, and for this reason they have to be mentioned here. For pragmatic reasons researchers tend to adopt low cost methods. Reduction of computational cost is achieved by simplifications in the description of the physics of phenomena involved in the reaction process. The confidence gained with more accurate studies on reaction processes helps in this reduction of the physics, which is accompanied by a strong parameterization to increase the reliability of the computed parameters. For the solvation energy, to give an example, there are procedures specialized for given classes of solvents (nonpolar, polar, water), for specific classes of solutes, with different types of molecular descriptor, starting from models with a single descriptor, such as molecular volume or area, progressing then to more complex models combining e.g. molecular volume and noncovalent solute–solvent interactions or volume and dipole-driven electrostatic interactions. This variety of models, of which we have given just a few examples, found their justifications in the results obtained with the methods we have introduced in Section 1.1.1 of this contribution. Because this contribution is dedicated to the physics of solvation and not to computational issues, we do not add other comments on these methods, except to remark that a full understanding of the basic justifications of such methods is necessary to avoid misunderstandings and erroneous conclusions in their use. Modern Theories of Continuum Models 21 Detailed and accurate descriptions of reaction mechanisms, however, have been performed for several years, in some cases with the inclusion of solvent effects. In this section we shall briefly examine some aspects of the solvation physics related to the chemical reaction mechanisms; a more general discussion on chemical reactions in solution is given in the contribution by Truhlar and Pliego. We start by considering the simple extension of the basic material model considered in Section 1.1.1: an infinite isotropic solution, containing as solute just the minimal number of molecules involved in the reaction. For simplicity we consider a bimolecular reaction, giving rise after the chemical interaction two different molecules: A+B → C+D (1.12) This simplification of the model eliminates some preliminary aspects of the process which sometimes have considerable importance, such as the processes bringing into contact separate reactions partners. We shall return later to this point for reactions in solution but let us consider first reactions in gas phase. Noncovalent interactions between the two separate molecules define, in the gas phase analogue of this reactive system, the preferential channels of approach (in the simpler cases there is just one channel leading to the reaction) with shape and strength determined only by these interactions. As a general rule, these channels carry the reactants to a stationary point on the potential energy surface called the initial reaction complex. In solution things are more complex. The reaction partners are no longer free in their translational motion as they are in the gas phase; they have to move in a condensed medium, and their motion is governed by other physical phenomena which for economy of exposition we shall not consider in detail. It is sufficient to recall that the physical models for the most important terms, Brownian motions, diffusion forces, are expressed in their basic form using a continuum description of the medium. Both isolated partners of the reaction (1.12) are solvated, and we may consider, for simplicity, that during an initial stage of mutual approach they both maintain their equilibrium solvation shell, as described in Section 1.1.2. To reach the intimate contact corresponding to the initial reaction complex defined for the in vacuo reaction, the two solvation shells must be distorted and strongly rearranged. In solution there are no simple association processes, but more complex processes in which there is a replacement of molecular associations. The modelling of this process is not immediate. Solute– solvent interaction energies are often of comparable strength, the entropy contributions are considerably greater in solution than in vacuo, and so the description cannot be limited to the comparison of the relative strength of the bimolecular interactions involved in this change of molecular interactions. The consequences may be remarkable. Well known examples are given by bimolecular association processes. These reactions, simpler to study than the standard reactions where bond are broken and formed, presented some ‘surprises’ in the first accurate studies performed some years ago. A typical example is that of the association of two amide molecules. In vacuo a stabilizing interaction supported by hydrogen bonds (one or two, according to the channel and the nature of substituent groups in the amide) leads to a remarkable stability of the dimer. In water this type of interaction is destabilizing, and is replaced by a feeble – interaction leading to a completely different dimer geometry. The reason is that the water–amide H-bond strength 22 Continuum Solvation Models in Chemical Physics is comparable with that of the amide–amide H-bond, and entropy changes strongly hinder the formation of an H-bond association between amide molecules. In addition, owing their chemical nature, reactive groups in reacting molecules often exhibit local solvent interactions stronger than other portions of the same molecule. This fact may shift the initial complex contact to another molecular group with a less strong local solvation, inducing modifications of the reaction mechanism with respect to the gas phase analogue. The computational evidence supporting these general considerations is so far scarce, because to do it the examination of rather complex bimolecular systems is necessary, performed with care and good accuracy. The considerable computational cost suggests waiting for more powerful computers. The problem is well known to people undertaking chemical syntheses; the search for the most appropriate solvents is to a large extent related to such differential interactions. Even greater is the indirect evidence coming from reactions occurring in living organisms; the admirable machinery of biochemical reactions exploits the complex nature of the medium, which cannot be assimilated to bulk isotropic water, to enhance or to hinder reaction mechanisms using a variety of physical effects. Let us return to the examination of reaction mechanisms. For reactions in vacuo the methodology to study the steps following the formation of the initial complex are nowadays sufficiently standardized, to a first approximation. The basic concept in use is that of the potential energy surface (PES). This is not a true physical concept, being related to an approximation in the mathematical machinery of formulation of the quantum mechanical problem, but the Born–Oppenheimer approximation on which the PES is based is remarkably accurate and stable and so we may accept the PES as a physical ingredient of the theory. The definition of the family of PESs for an isolated system is unequivocal. We shall consider here cases in which the attention may be limited to a single PES: that of the electronic ground state. The starting point for the characterization of the mechanism is the search for the stationary point corresponding to the top of the reaction barrier (the transition state, TS). The search for this stationary point is still almost an art, but it is feasible and the validation of the result is based on precise mathematical algorithms. The formal definition of the reaction path (RP), a one-dimensional nonlinear coordinate connecting the initial complex of reagents, TS and the final complex of products, is standardized in a quite acceptable form. The definition leads to the definition of the computational strategy which starts from the geometry of the TS and proceeds with performing calculations along the two directions defined by the coordinate corresponding to the descent from the TS [29]. No additional physical concepts are necessary for this static definition of the mechanism. The strategy is well defined and relatively simple to apply to reactions with a simple PES form, i.e. surfaces with a single TS. Actually the topological structure of the surface may be more complex, with several TSs defining accessory stationary points, some of which correspond to intermediates along the RP, others defining alternative routes. Turning now to the mechanisms in solution, the same strategy apparently seems to be applicable. However, there are important differences making its application more difficult. One complication is related to an approximation adopted in the gas phase model which we have not mentioned in introducing the PES concept. The quantity to use in defining the Modern Theories of Continuum Models 23 geometrical evolution of the system in a reaction is the free energy and not the energy. In the BO approximation both quantities depend parametrically on the nuclear coordinates and can be described as a hypersurface in nuclear coordinate space R. The approximation we have mentioned consists in neglecting entropic contributions in the definition of the geometries corresponding to TS and RP. This is an acceptable simplification for systems in vacuo, but it is not acceptable for systems in solution. To pass from internal energy to free energy there are no conceptual problems but major computational problems for methods based on discrete descriptions of the solvent. Umbrella sampling simulations and constrained molecular dynamics methods, now in use, rely on the previous definition in vacuo of a one-dimensional RP on which point by point a free energy profile is computed. Actually the TS in vacuo may be quite different from the TS in solution. A possible alternative to define the lowest free energy path is the use of a method in which appropriate collective variables are introduced [30]. This RP is then used in a set of umbrella sampling simulations. No analytical derivative methods are in use for discrete solvent models. Things are much simpler in continuum methods. Continuum methods in fact directly give free energies which can be collected in a function GR (which could be also called the FES) continuous over the R space and computationally well defined at every point of this space (as it is for the PES function in vacuo) In continuum models there are computational codes enabling the analytical calculation of derivatives (see also the contribution by Cossi and Rega in this book) necessary for the definition of TS and RP. We shall thus limit ourselves to the examination of GR obtained with continuum methods. As remarked before there are aspects of the early stages of the reaction which it is not convenient to describe with the GR formalism. The approach of the two molecules A and B entering into reaction is modulated and impeded by interactions with the solvent, which at large distances are little affected by A–B interactions. The physical keys for this initial stage of the reaction are given by Brownian motions and diffusion phenomena, two important chapters in the physics of solution, amply studied, originally formulated with continuum descriptions of the solvent, and for which modern continuum methods might give important contributions. For economy in the discussion we shall not treat these themes in this contribution, limiting ourselves to the core of the reaction, the description of which is based on the GR function. Let us suppose we have obtained by an analysis of GR a description of the whole RP in solution making use of the appropriate analytical derivatives. The examination of evolution of the system along the RP starting from the initial complex shows an initial region in which the main effects are to be assigned to conformational changes, accompanied by moderate electronic polarization and changes in the internal geometry of the chemical groups. The decomposition of the forces acting on the nuclei of the QM subsystem (a mathematical procedure that may be performed with tools developed for the semiclassical analysis [31] of isolated molecules and easily inserted into continuum solvation codes [32]) shows that the net solvation force component for some groups of the molecule pushes the group towards the completion of the reaction, while for other groups of the molecule a counteracting effect can occur: the local solvation forces act against the completion of the reaction. On the whole there is a distortion of the mechanism with respect to that found in the absence of solvation forces. 24 Continuum Solvation Models in Chemical Physics Near the TS minor changes in the nuclear geometry are accompanied by marked changes in the electronic distribution; new bonds are formed and others broken in this region. An analogous change in the relative evolution of nuclear and electronic components also happens in vacuo. The differences with respect to reaction in vacuo remain in the solvent, which always plays a role, in some cases quite specific. The specific role of the solvent is evident in reactions in water in which an H atom is transferred from one group to another; in these cases the H transfer is mediated by a bridge of a few water molecules, acting as a catalyst. These water molecules must be inserted in the portion of the system described at the QM level and thus in the definition of the free energy hypersurface GR on an enlarged R space. This is just an example, the best studied example, but the active role of solvent molecules has also been found in other cases. Other solvent molecules, not only water, may play a specific role in the reaction. There is no generally accepted terminology, and we use here a term we coined years ago: that of actively assisting solvent molecules [32]. The enlarging of the R space to include actively assisting solvent molecules is a delicate problem. The cases in which the assisting molecules may be defined in position and number at the level of the initial complex are rare. The empirical solution often adopted is that of obtaining an approximate description of the TS without assisting molecules, and then of adding here, after an accurate analysis, a single solvent molecule in a position in which it may exert an assisting role. This computational task is easy for simple cases, but when the assisting role is exerted by two or more molecules the procedure of insertion has to be repeated on a GR surface becoming progressively more flat. It is worth remarking that this procedure has been initially applied to studies of reaction mechanisms with models in which the solvent was described in terms of a few discrete molecules: the addition of the first active solvent molecule is in this case an easy task, but the addition of more active molecules is more difficult, because the added molecules prefer to interact with other portions of the solute. This optimization artefact rarely occurs in continuum solvation methods, because the solvation of other portions of the molecule is already ensured by the continuum reaction potential. Dynamical Aspects of Chemical Reactions In describing the PES-based approach for molecules in the gas phase we added the remark that the picture of the reaction mechanism we have described was static. The same remark also holds for the description of reactions in solution. In neglecting dynamical aspects we have greatly simplified the tasks of describing and interpreting the reaction mechanism, and at the same time we have lost aspects of the reaction that could be important. Let us consider first the in vacuo cases. Dynamical aspects of the reaction in vacuo may be recovered by resorting to calculations of semiclassical trajectories. A cluster of independent representative points, with accurately selected classical initial conditions, are allowed to perform trajectories according to classical mechanics. The reaction path, which is a static semiclassical concept (the best path for a representative point with infinitely slow motion), is replaced by descriptions of the density of trajectories. A widely employed approach to obtain dynamical information (reaction rate coefficients) is based on modern versions of the Transition State Theory (TST) whose original formulation dates back to 1935. Much work has been done to extend and refine the original TST. Modern Theories of Continuum Models 25 Among the numerous features added to the method we mention the concept of a dividing surface (DS) which separates reactant and product regions in the R space. The DS has to be determined dynamically with one of the proposed procedures. We do not give more details of the complex set of TST procedures thus far developed, each adding new features and new approximations. This methodological activity, pursued by several researchers, has been guided by the activity of Truhlar, initiated in the late 1970s and continuing today. We shall refer to this large body of methodological study with the acronym VTST (variational TST). We do not give here more details on VST which is a quite complex and detailed method. Additional aspects of VTST will be considered later in the context of reactions in solution. The dynamics of reactions in solution must include an appropriate description of the solvent dynamics. To simplify this problem we start with some considerations supported by intuition and by some concepts described in the preceding sections. In the initial stages of the reaction the characteristic time is given by the nuclear motions of the solute, large enough to allow the use of the adiabatic perturbation approximation for the description of motions. In practice this means that the evolution of the system in time may be described with a time independent formalism, with the solvent reaction potential equilibrated at each time step for the appropriate geometry of the solute. Near the TS things change. The rapid evolution of the light components of the system (electrons and H atoms involved in a transfer process) makes the adiabatic approximation questionable. Also the sudden time dependent perturbation we introduced in Section 1.1.3 to describe solvent effects on electronic transitions is not suitable. We are considering here an intermediate case for which the time dependent perturbation theory does not provide simple formulae to support our intuitive considerations. Other descriptions have to be defined. An important physical feature which has to be recovered in these descriptions is related to the influence that dynamical solute–solvent interactions have when the solute passes from the reactant to the product region of GR. The solvent molecules involved are subject to thermal random motions and cannot be categorized as assisting molecules. There are different approaches to the description of these dynamical interactions leading to different computational strategies. We shall briefly examine the two most commonly used approaches. A description of the evolution of the system near the TS is given by the VTST. The most complete description of the method has been given by Truhlar and co-workers [33]; in this book there is a good synopsis by Truhlar and Pliego. The dynamical correlation between solute and solvent molecules is described in VTST in terms of trajectories which are scattered back, contributing in this way to the definition of the dividing surface (DS). The introduction of the DS concept has an important methodological relevance because it changes the dimensionality of the critical quantity of the theory. In fact the TS is defined as a single point on the GR surface, while DS is a surface with 3N − 1 dimensions. This fact, certainly important for reactions in vacuo, assumes a greater importance in solutions, where the free energy landscape at the discrete molecular level exhibits a large number of geometrical configurations quasi-degenerate in energy, all capable of acting as a watershed between reactants and products (this also happens with the reduction of solvent degrees of freedom introduced by the continuum approximation; the explicit assisting solvent molecules are sufficient to 26 Continuum Solvation Models in Chemical Physics introduce a sizeable number of quasi-degenerate configurations). The concept of a single TS point is untenable in almost all chemical reactions in solution. The VTST briefly summarized here has been implemented in a computational code which contains many other features [34]. Among them we cite those related to the description of tunnel effects, to which much attention has been paid in the development of the method (to emphasize this aspect the acronym VTST/OMT has been used, where OMT stays for optimized multidimensional tunnelling). We have not paid attention in the preceding pages to tunnelling effects, which are of extreme importance in molecular biology, but also present and important in many other reactions. Having a code able to describe in an optimized way this physical feature of solutions will in the near future be a necessary requisite for the study of reactions in solution. VTST/OMT also contains many other features. It is a complex code in which a good portion of the complexity is due to the effort of defining suitable approximations with the scope of reducing computational costs without losing a clear identification of the thermodynamic characteristics of all the partial quantities introduced. We are confident that the continued development of the procedure will lead to codes that are simpler to use, but the final goal of having codes containing all the features considered in VTST/OMT, and as easy to use as those now available for the construction of PES in vacuo, seems to us still distant. The other approach we are considering here is based on a description of the dynamical interactions occurring after the passage of the TS (or better of the DS divide) in terms of an additional force of a frictional type related to the time correlation of a random force. This formulation was introduced by Kramers in 1940 [35], in the form of a Langevin equation. The Langevin equation, proposed in 1908 just to treat the above mentioned Brownian motions, has had a tremendous impact on the study of all phenomena in physics exhibiting both fluctuations and irreversibility. In the study of solutions the Kramers formulation was later (1980) extended by Grote and Hynes [36] who introduced a time dependence in the friction coefficient. This was the beginning of the family of Generalized Langevin Equations (GLE) on which much work has been done. We remark that GL and GLE procedures are typically limited to a single coordinate, interpreted as the RP coordinate. The extension to a few more coordinates is possible, but the development of a computational protocol to treat with these procedures the many dimensional problem for polyatomic molecules with many degrees of freedom is a hard task. The great merit of GLE studies is the insight they give on the basic nonequilibrium aspects of simple reactions. Another way of introducing nonequilibrium effects in the dynamical equation is given by the addition to the reaction coordinate a solvent coordinate s which measures deviations from the equilibrium distribution of the solvent, following the approach pioneered in 1956 by Marcus [37]. This coordinate describes with a single parameter the dynamical participation of solvent molecules. The definition of the solvent coordinate s given by Zusman [38] is based on the continuum solvation model, with the two-mode decomposition we have introduced in Equation (1.10). The dynamical coordinate is essentially related to P slow . To complete this short discussion of the dynamics of reactions we remark that continuum models play an important role in the dynamical procedures. The basic underlying static description GR is more easily developed, simple molecular models apart, with a continuum solvation code, and it is more easily extended to include the solvent assisting molecules. Continuum models easily give the vibrations and the elements of Modern Theories of Continuum Models 27 the Hessian matrix (second order partial derivatives with respect to nuclear coordinates) necessary for a topological characterization of the points on the hypersurface. In the dynamical part continuum models may also play a role, and some comments have been given in the preceding pages; here we add that the introduction of noise is possible, even if not yet fully explored. With these remarks we do not claim that the whole computational machinery can be reduced to continuum calculations. A judicious combination of different approaches is probably the best choice. We are at present at a stage in the development of the computational models in which it is still necessary to obtain a further insight on the numerical stability and computational effectiveness of the models in use to describe the various physical effects. Our ultimate goal is, in our opinion, to use this increased knowledge to establish methods and computational protocols that are simpler to use, at the cost of some well selected simplifications in the description of the physical model. References [1] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. [2] R. Car and M. Parrinello, Phys. Rev. Lett., 55 (1985) 2471. [3] See for example: W. Damm, A. Frontera, J. Tirado-Rives and W. L. Jørgensen, J. Comput. Chem., 18 (1997) 1995; W. D. Cornell, P. Cielpak, C. L. Bayly, I. R. Gould, K. M. Merz Jr, D. M. Ferguson, D. C. Soellmeyer, T. Fox, J. W. Caldwell and P. A. Kollman, J. Am. Chem. Soc., 117 (1995) 5179. [4] A. Warshel, M. Levitt, J. Mol. Biol. 103 (1976) 227; J. Gao, Rev. Comput. Chem., 7 (1995) 115. [5] S. Ten-no, F. Hirata and S. Kato, J. Chem. Phys., 100 (1994) 7443. [6] H. Sato, A. Kovalenko and F. Hirata, J. Chem. Phys., 112 (2000) 9463. [7] F. M. Floris, A. Tani and J, Tomasi, Chem.Phys., 169 (1993) 11. [8] J. Tomasi, B. Mennucci and R. Cammi, Chem. Rev., 105 (2005) 2999. [9] S. Miertuš, E. Scrocco and J. Tomasi, Chem. Phys., 55 (1981) 117. [10] H. H. Ulig, J. Phys. Chem., 41 (1937) 1215. [11] G. Hummer, S. Garde, A. E. Garcia, M. E. Paulaitis and L. R. Pratt, Proc. Natl. Acad. Sci., USA, 93 (1996) 8951. [12] R. A. Pierotti, Chem. Rev., 76 (1976) 712. [13] F. Vigne’-Maeder and P. Claverie, J. Am. Chem. Soc., 109 (1987) 24. [14] R. Bonaccorsi, C. Ghio and J. Tomasi, The effect of the solvent on electronic transitions and other properties of molecular solutes, in R. Carbo (ed.), Current Aspects of Quantum Chemistry, Elsevier, Amsterdam, 1982, p. 407. [15] C. Amovilli and B. Mennucci, J. Phys. Chem. B, 101 (1997) 1051. [16] J. Tomasi, G. Alagona, R. Bonaccorsi and C. Ghio, A theoretical model for solvation with some applications to biological sistems, in Z. B. Maksic (ed.), Modelling of Structure and Properties of Molecules, Ellis-Horwood, Chichester, 1987, p. 330. [17] C. Pomelli and J. Tomasi, J. Phys. Chem. A, 101 (1997) 3561. [18] L. Sandberg, R. Casemyr and O. Edholm, J. Phys. Chem. B, 106 (2002) 7889. [19] H. Luo and S. C. Tucker, J. Phys. Chem., 100 (1995) 11165. [20] E. Cancès, B. Mennucci and J. Tomasi, J. Chem. Phys., 107 (1997) 3032. [21] R. R. Dogonadze and A. A. Kornishev, J. Chem. Soc. Faraday Trans., 2, 70 (1974) 1121. [22] V. Basilevsky and D. F. Parsons, J. Chem. Phys., 105 (1996) 3734. 28 Continuum Solvation Models in Chemical Physics [23] G. Alagona, c. Ghio, R. Cammi and J. Tomasi, A Reappraisal of the hydrogen bonding interaction obtained by combining energy decomposition analyses and counterpoise corrections, in J. Maruani (ed.), Moleculaes in Physics, Chemistry, Biology, Vol. II, Kluwer, Dordrecht, 1988, p. 507. [24] J. Tomasi, Theor. Chem. Acc., 112 (2004) 184. [25] S. E. McLain, A. K. Soper and A. Luzar, J. Chem. Phys., 124 (2006) 074502. [26] T. Vreven and K. Morokuma, J. Comput. Chem., 21 (2000) 1419. [27] (a) C. P. Hsu, X. Song and R. A. Marcus, J. Phys. Chem. B, 101 (1997) 2546; (b) M. Caricato, F. Ingrosso, B. Mennucci and J. Tomasi, J. Chem. Phys., 122 (2005) 154501. [28] A. Espagne, D. H. Paik, P. Changenet-Barret, M. M. Martin and A. H. Zewail, Chem. Phys. Chem., 7 (2006) 1717. [29] H. B. Schlegel, Reaction path following, in Encyclopedia of Computational Chemistry, Vol. 4, John Wiley & Sons, Ltd, Chichester, 1998, p. 2432. [30] B. Ensing, A. Laio, M. Parrinello and M. L. Klein, J. Phys. Chem. B, 109 (2005) 6676. [31] G. Alagona, R. Bonaccorsi, C. Ghio, R. Montagnani and J. Tomasi, Pure Appl. Chem., 60 (1988) 231. [32] E. L. Coitiño, J. Tomasi and O. N. Ventura, J. Chem. Soc., Faraday Trans., 90 (1994) 1745. [33] D. G. Truhlar, J. Gao, M. Garcia-Viroca, C. Alhambra, J. Corchado, M. L. Sanchez and T. D. Poulsen, Int J Quantum Chem., 100 (2004) 1136. and references cited therein. [34] Polyrate 9.6, http://comp.chem.umn.edu/polyrate/ [35] H. A. Kramers, Physica 7 (1940) 284. [36] R. F. Grote and J. T. Hynes, J. Chem. Phys., 76 (1980) 2715. [37] R. A. Marcus, J. Chem. Phys., 24 (1956) 966. [38] I. Zusman, Chem. Phys., 49 (1980) 295. 1.2 Integral Equation Approaches for Continuum Models Eric Cancès 1.2.1 Introduction The integral equation approach is a general purpose numerical method for solving mathematical problems involving linear partial differential equations with piecewise constant coefficients. It is commonly used in various fields of science and engineering, such as acoustics, electromagnetism, solid and fluid mechanics, In the context of implicit solvent models, several numerical methods based on integral equations (DPCM, COSMO, IEF, ) have been proposed for calculating reaction potentials and energies. In Section 1.2.3 an integral representation of the reaction potential is derived, under the assumption that the molecular charge distribution is entirely supported inside the cavity C. This representation is then used to reformulate the reaction field energy ER = R3 r VR r dr as an integral on the interface = C: ER = VM (1.13) where VM is the potential generated by the charge distribution in the vacuum, i.e. VM r = R3 r dr r − r (1.14) The surface charge is a solution of an integral equation on , that is of an equation of the form ∀s ∈ kA s s s ds = b s (1.15) where kA is the Green kernel of some integral operator A and where the left-hand side b depends linearly on the charge distribution . The various integral equation methods under examination in this chapter correspond to different choices for A and b . For instance, the original version of COSMO [1] is obtained with kA s s = 1/s − s and b s = −f R3 r /s − r dr , with f = − 1/ + 05. DPCM and IEF are exact (and therefore equivalent) as long as the solute charge lies completely inside the cavity, whereas COSMO is only asymptotically exact in the limit of large dielectric constants. If there is some escaped charge, i.e. if some part of the charge distribution is supported outside the cavity, all these methods are approximations. The error generated by the fact that, in QM calculations, the electronic tail of the solute necessarily spreads outside the cavity, is discussed in Section 1.2.4. 30 Continuum Solvation Models in Chemical Physics The usual discretization methods for integral equations (collocation vs Galerkin, boundary elements) are presented in Section 1.2.5. Section 1.2.6 is concerned with geometry optimization, and more generally with the calculation of observables involving derivatives with respect to the shape of the cavity (shape derivatives). Lastly, the extensions of the standard implicit solvent model to more sophisticated settings (liquid crystals, ionic solvents, metallic surfaces, ) are briefly dealt with in section 1.2.7. 1.2.2 Representation Formula for the Poisson Equation All the integral equation methods discussed in this chapter are based on an integral representation of the reaction potential. Let us state this point precisely. Consider a function W R3 −→ R satisfying ⎧ ⎪ ⎨−W = 0 in C (1.16) −W = 0 outside C ⎪ ⎩ W −→ 0 at infinity Let n s be the outward pointing normal vector at s ∈ . We now assume that the following limits exist for all r ∈ W W r − Wr − ns Wi r = lim+ Wr − ns = lim+ i (1.17) →0 n i →0 Wr + ns − We r W ns = lim (1.18) We r = lim+ Wr + →0 n e →0+ Note that the existence of these limits does not imply that the function W nor its normal derivative are continuous across . On the other hand, they ensure that the jump Ws = Wi s − We s of W at s ∈ is well-defined, and that so is the jump of its normal derivative W W W s − s s = n n i n e We can now state a representation formula for W : for all r ∈ 1 W 1 Wr = s ds − Ws ds r − s n 4r − s 4 ns where we have used the notation 1 1 r − s · n s = = · n s s ns 4r − s 4r − s 4r − s 3 (1.19) Modern Theories of Continuum Models 31 The integral representation (1.19) implies that it is sufficient to know the jumps of W and W/n at the crossing of the interface to know W everywhere in R3 \ . The function W , being a priori discontinuous at the crossing of , does not have a well-defined value on . On the other hand, the following representation formula holds for every s ∈ : Wi s + We s 1 1 W = Ws ds s ds − 2 s − s n 4s − s 4 ns (1.20) Similarly for all s ∈ , 1 2 W 1 W W s = + s ds n i n e n 4s − s ns 2 1 − Ws ds 4s − s ns ns with ns and 2 ns ns 1 4s − s 1 4s − s = s = (1.21) 1 s s − s · n · n s = − 4s − s 4s − s 3 s s − s · n s − s · n n s · n s s +3 3 3 4s − s 4s − s The integral representation formulae (1.20) and (1.21) suggest to introduce the integral operators S D D∗ and N defined for → R and s ∈ by 1 Ss = (1.22) s ds s − s 1 Ds = (1.23) s ds s − s ns 1 D∗ s = (1.24) s ds s − s ns 1 2 Ns = s ds (1.25) s − s ns ns When the interface is regular (C 1 at least), the Green kernels of the operators S D and D∗ exhibit integrable singularities: they behave as 1/s − s when s goes to s (for s −s · ns ∼ s −s · ns ≤ s −s 2 when s is close to s). On the other hand, the Green kernel of the operator N is hypersingular (it behaves as 1/s − s 3 when s is close to s) so that the notations (1.21) and (1.25) are only formal: the integral 2 1 s ds s − s ns ns 32 Continuum Solvation Models in Chemical Physics has to be given the sense of a Cauchy principal value [2]. The operators S D and D∗ play a central role in the usual ASC methods (DPCM, COSMO, IEF). As the operator N does not appear in these methods, we will not further detail its properties. Regarding the operators S D and D∗ , they satisfy the following three properties: • Property 1: on L2 , the operator S is self-adjoint, and D∗ is the adjoint of D. • Property 2: DS = SD∗ . • Property 3: denoting by H s the Sobolev space of index s ∈ R [3], the applications S H s → H s+1 and − D∗ H s → H s for − 2 < < + are bicontinuous isomorphisms for any s ∈ R. We will comment on the practical consequences of these properties in the end of Section ID. At this point, let us only mention that the functional space H 0 coincide with L2 , and that for s ∈ N∗ H s is the set of functions which are in L2 and whose surface derivatives of orders lower than or equal to s all are in L2 . Besides H s+1 ⊂ H s for all s ∈ R, and the larger s, the more regular the functions of H s . The first two properties are algebraic in nature and are used in the formal derivation of the various ASC equations. The third property is concerned with functional analysis. As it is of no use for the formal derivation of ASC methods, it is rarely reported in the chemistry literature. However, it has direct consequences on the comparative numerical performances of the various ASC methods (see Section 1.2.5). In the special case of a spherical cavity, the operators S D D∗ and N have simple expressions. Assume for simplicity that is the unit sphere S 2 . A function u defined on = S 2 can then be expended on the spherical harmonics Ylm (see e.g. [4]): u = + m um l Yl l=0 −l≤m≤l where um l are complex numbers. Recall that the spherical harmonics form a Hilbert basis of L2 S 2 so that, in particular, S2 m Ym l Yl = 0 2 0 m Ym l Yl sin d d = ll mm The operators S D D∗ and N turn out to be diagonal in this basis: for = S 2 (the unit sphere) ⎧ + 4 m m ⎪ ⎨Su = l=0 −l≤m≤l 2l+1 ul Yl Du = D∗ u = − 21 Su ⎪ ⎩ ll+1 m m Nu = −4 + −l≤m≤l 2l+1 ul Yl l=0 Modern Theories of Continuum Models 33 Note that D = D∗ for spherical cavities only. Still in this basis, the Sobolev spaces H s S 2 have a nice, simple, definition + + s 2 m m 2 2s m 2 ul Yl such that uH s = l + 1 ul < + H S = u = l=0 −l≤m≤l l=0 −l≤m≤l The properties of the operators S D and D∗ listed above can then be easily established in the special case when is the unit sphere. For more details on the properties of the operators S D D∗ and N , and in particular on their relation with Calderon projectors, we refer to ref. [2]. We conclude this mathematical section with the useful definitions of single-layer and double-layer potentials. A single-layer potential is a function W which can be written as Wr = s ds r − s ∀r ∈ R3 \ (1.26) with ∈ H −1/2 . A single layer potential fulfils Equations (1.16) and the limits defined by Equations (1.17) and (1.18) exist. By identification with the representation formula (1.19), one finds W W = 0 and = 4 (1.27) n This implies in particular that the potential W is continuous across (and therefore on R3 ), and that Equation (1.26) also holds true for r ∈ . In other words, is solution to the integral equation S = W A double-layer potential is a function W which can be written as 1 py dy ∀x ∈ R3 \ Wx = x − y ny with p ∈ H 1/2 . A single-layer potential fulfils Equations (1.16) and the limits defined by Equations (1.17) and (1.18) exist. By identification with the representation formula (1.19), one finds W =0 W = 4p and n A double-layer potential is continuous on R3 \ but exhibits a discontinuity across the interface . The density p is a solution to the integral equation Np = − W n 34 Continuum Solvation Models in Chemical Physics 1.2.3 Reaction Field Energies of Interior Charges The reaction potential VR is defined as VR = V − VM where V is the unique solution to r = 4M r − · r V (1.28) vanishing at infinity, and where VM r = R3 M r dr r − r denotes the potential generated by M in the vacuum. As r = 1 in C and r = outside C, and as, in this section, M is assumed to be supported inside C, one has ⎧ ⎪ ⎨−V = 4M −V = 0 ⎪ ⎩ V −→ 0 in C outside C at infinity Likewise, the potential VM also satisfies ⎧ ⎪ ⎨−VM = 4M −VM = 0 ⎪ ⎩ VM −→ 0 in C outside C at infinity Hence VR = V − VM is such that ⎧ ⎪ ⎨−VR = 0 −VR = 0 ⎪ ⎩ VR −→ 0 in C outside C at infinity In QM calculations, M is the sum of the nuclear contribution (a linear combination of point charges located inside C) and of a regular function (the electronic density), that, in this section, is assumed to be supported in C. It then follows from standard functional analysis results [3] that for such M , the limits VR i VR e VR /ni , and VR /ne defined by Equations (1.17) and (1.18) exist, and VR is continuous across . We thus infer from the representation formula (1.19) that ∀r ∈ 3 VR r = s ds r − s where 1 VR = 4 n (1.29) Modern Theories of Continuum Models 35 The reaction potential VR is therefore a single-layer potential. In order to calculate the apparent surface charge (ASC) distribution , one makes use on the one hand of the relations VR VR − = 4 n i n e 1 VR VR + = D∗ 2 n i n e and on the other hand of the jump condition (see e.g. ref. [5]) V V 0= − n i n e VR VR V = − + 1 − M n i n e n This leads to the integral equation +1 V 2 − D∗ = M −1 n (1.30) Equation (1.30) is nothing but the DPCM equation [6, 7]. The existence and uniqueness of the solution of Equation (1.30) is ensured by property 3 stated in Section 1.2.2. The reaction field energy ER can then, as announced, be written as an integral over : ER = r VR r dr R3 s = r ds dr r − s R3 r = s dr ds r − s R3 = s VM s ds (1.31) The various IEF equations can be derived from the DPCM Equation (1.30) as follows. Multiplying Equation (1.30) by S on the left-hand side, we get +1 V ∗ S 2 −D = S M −1 n Using the commutation relation SD∗ = DS, we also have +1 V 2 − D S = S M −1 n (1.32) (1.33) 36 Continuum Solvation Models in Chemical Physics Applying the representation formula (1.20) to the function W defined by Wr = 0 if r is in C and Wr = VM r if r is outside C, we find that for all s ∈ , 1 1 1 VM VM s = − d s + (1.34) s VM s ds 2 s − s n 4s − s 4 ns The above relation can be rewritten, using the integral operators S and D, as 2VM = −S VM + DVM n (1.35) Combining Equations (1.32), (1.33) and (1.35) it is possible to construct a whole family of ASC equations, including the original IEF equation [8–10] 1 V 1 (1.36) 2 − DS + S2 + D∗ = −2 − DVM − S M n and the IEFPCM [11], also called SS(V)PE [12, 13], equation −1 +1 ∗ −D = − 2 − DVM S 2 −1 (1.37) Equation (1.37) was obtained independently by Mennucci et al. [11] and by Chipmann [12]. Note that the integral operators involved in the IEF and IEFPCM equations are in fact the same, up to a multiplicative constant, and are symmetric: ∗ ∗ +1 +1 ∗ ∗ −D − D S∗ = 2 S 2 −1 −1 +1 = 2 −D S −1 +1 ∗ = S 2 −D −1 On the other hand, the integral operator of the DPCM Equation (1.30) is not symmetric. Finally, the COSMO model introduced in ref. [1] can be recovered as follows. First, the IEFPCM Equation (1.37) can be rewritten as ⎧ S = −VM ⎨ +1 −1 ⎩ 2 − D∗ = 2 − D∗ −1 (1.38) The COSMO model is an approximation of Equations (1.38) consisting in solving exactly the first of Equations (1.38) and in replacing the second equation by −1 +k Modern Theories of Continuum Models 37 where k is an empirical parameter. In the special case when is the unit sphere, the second of Equations (1.38) can be solved analytically: ⎡ ⎤ 1 − 1 ⎢ 2l + 1 ⎥ ⎢ ⎥ lm = m ⎣ −1 1 ⎦ l + 1 1+ + 1 2l + 1 2 1+ The optimal value for k is k = 1 for l = 0 and k = 2 for l = +. On the other hand, numerical simulations on real molecular systems seem to show that, depending on the charge and shape of the system, the optimal value for k is between k = 0 and k = 1/2. The discrepancy between theoretical arguments and numerical results might originate in the escaped charge problem, that is addressed in the following section. 1.2.4 The Escaped Charge Problem As underlined above, there is no approximation in the integral representation (1.31) of the reaction field energy, provided (i) the charge distribution is entirely supported inside the cavity C and (ii) is computed using the DPCM Equation (1.30), the IEF Equation (1.36) or the IEFPCM Equation (1.37). If condition (i) is not satisfied, the integral equation method presented in the previous section needs to be modified. Proceeding as above, it is easy to show that the total electrostatic potential V solution to Equation (1.28) can be decomposed as s 1 a Vr = VMint r + VMext r + ds r − s where VMint r = C r dr r − r VMext r R3 \C r dr r − r and where a is an apparent surface charge that can be obtained by solving some integral equation involving the operators S D, and/or D∗ , as well as the potentials VMint and VMext and/or their normal derivatives. There is therefore no theoretical obstacle in formulating an exact integral equation method in the presence of escaped charge. In classical molecular dynamics, this program can be easily realized. The main practical difficulty arising in quantum chemistry (in particular with gaussian basis sets) is that there is no convenient way to compute the potentials VMint and VMext . For this reason, quantum chemistry calculations are usually performed using the equations derived under the assumption that the charge distribution is entirely supported inside the cavity. The error due to the escaped charge is either neglected or corrected by some empirical rule. It is important to note that, whereas the DPCM, IEF and IEFPCM are exact (and therefore equivalent) when there is no escaped charge, they are non-equivalent approximations in the presence of escaped charge. Theoretical arguments [12], confirmed by numerical simulations, show that the IEFPCM method behaves very much better than the DPCM method in the presence of escaped charge. 38 Continuum Solvation Models in Chemical Physics The simplest method to evaluate the magnitude of the error due to the escaped charge consists in computing the amount of escaped charge by means of Gauss’s theorem. Denoting by Q = R3 the total charge, the escaped charge is Qs = Q − C = Q+ 1 1 VM VM r dr = Q + s ds 4 C 4 n If Qs /Q exceeds a few percent, it is likely that the calculation will not be very reliable. A more elaborate procedure consists in establishing error estimates. For instance, it is proved in ref. [14] that the exact reaction field energy ER and the IEFPCM estimate of it, denoted by ERIEFPCM , satisfy ER ≤ ERIEFPCM (1.39) and ER ≥ ERIEFPCM − − 1 6 4 1/3 1/3 5/3 −1 ext ext max Qs − S VM VM 5 3 (1.40) where max = supR3 \C . These inequalities are optimal (they reduce to equalities) if the charge distribution is entirely supported in C. Inequality (1.39) means that the IEFPCM method provides an upper bound of the exact reaction field energy. In practice, the lower bound (1.40) can be estimated using calculations performed on the interface [14]. 1.2.5 Discretization Methods The usual numerical methods for solving integral equations can be classified in two groups: the collocation methods and the Galerkin methods. Let us detail each approach for the example of the generic integral equation A = g (1.41) where the unknown belongs to H s , where the right-hand side g is in H s , and where the integral operator A ∈ LH s H s is characterized by the Green kernel kA s s : As = kA s s s ds ∀s ∈ Let us consider a mesh Ti 1≤i≤n on , that, in a first step, will be considered as drawn on the curved surface ; let us denote by si a representative point of the element Ti (e.g. its ‘centre’). The P0 collocation and Galerkin methods for solving Equation (1.41) provide two approximations of in the space Vh of piecewise constant functions whose restriction to each element Ti is constant: • in the collocation method, c is the element of the Vh solution to kA si s c s ds = gsi ∀1 ≤ i ≤ n Modern Theories of Continuum Models 39 • while in the Galerkin method, g is the element of Vh satisfying ∀ ∈ Vh kA s s g s ds s ds = gs s ds (1.42) These two methods lead to the matrix equations Ac · c = gc and Ag · g = gg where Acij = Agij = Tj kA si s ds Ti Tj gci = gsi kA s s ds ds ggi = g Ti ci and gi denoting the values of on Ti under the collocation and Galerkin approximations, respectively. The collocation method is more natural and easier to implement (at least at first sight); for these reasons, it is often used in apparent surface charge calculations; on the other hand, the Galerkin method leads to a symmetric linear system when the operator A is itself symmetric, which may appreciably simplify the numerical resolution of the linear system [15, 16]. Let us remark incidentally that in the Galerkin setting, D∗ gij = Dgji . This symmetry is broken with the collocation method: D∗ cij = Dcji . The approximation methods described above belong to the class of boundary element methods (BEMs). BEMs follow the same lines as finite element methods (FEMs). In both cases, the approximation space is constructed from a mesh. The terminology FEM is usually restricted to the case when the equation to be solved is set on some domain of the ambient space, whereas BEM implicitly means that the equation is set on the boundary of some domain of the ambient space. In most applications, FEMs are used to solve partial differential equations involving local differential operators. On the other hand, BEMs are often used to solve integral equations involving nonlocal operators. In the context of implicit solvent models, two options are open: either solve the (local) partial differential Equation (1.28), complemented with convenient boundary conditions, by FEM on a 3D mesh, or solve one of the (nonlocal) integral equations derived in Section 1.2.3, by BEM on a 2D mesh. In the former case, the resulting linear system is very large, but sparse. In the latter case, it is of much lower size, but full. The particular instances of BEM described above are the simplest ones: on each element Ti of the mesh, the functions of the approximation space are constant. In other words, they are polynomials of order 0, hence the terminology P0 BEM. It is possible to further improve the accuracy of the approximation, while keeping the same mesh, by refining the description of the test functions on each Ti . In Pk BEM, the functions of the approximation space are continuous on and such that their restriction to each Ti is piecewise polynomial of total degree lower than or equal to k in some local map (see ref. [2] for instance). In many applications, a polyhedral approximation ˜ of the surface is used; it is obtained by considering the Ti as planar tesserae (Figure 1.1). 40 Continuum Solvation Models in Chemical Physics Points on the molecular surface Molecular surface Gauss points on Ti Curved triangle Planar trianglei T Figure 1.1 Polyhedral approximation of a molecular surface. This approximation makes easier the computation of the coefficients of the matrices Sgij = Ti Tj 1 ds ds s − s Dgij and = Ti Tj ns 1 s − s ds ds It is indeed to be noticed that the function fS s = T 1 ds s − s has an analytical expression when T is a planar triangle, which allows an inexpensive evaluation of the inner integral Tj . Similarly, the function fD s = T s 1 s − s ds which corresponds to the solid angle formed by the geometric element T and the centre s [2] also admits a simple analytical expression for s ∈ R3 when T is planar. Let us notice that in this case, fD s = 0 for any s ∈ T ; therefore the diagonal elements Dcii and Dgii are all equal to zero under this geometric approximation. In the Galerkin approximation, the outer integration can be performed with an adaptive Gaussian integration method [17], the number of integration points depending on the distance and relative orientation of the elements Ti and Tj . The error induced by the polyhedral approximation can be estimated as follows [18]: • for the resolution of S = g, −1 − ˜ P H −1/2 ≤ C h3/2 H 2 • for the resolution of + D∗ = g −2 < < +, −1 − ˜ P L2 ≤ C h H 1 Modern Theories of Continuum Models 41 where denotes the exact solution of the integral equation on the exact surface ˜ the ˜ h = max diamTi the characteristic size exact solution of the integral equation on , ˜ of the sides of the polyhedron P the orthogonal projection on (which defines a one-to-one application from ˜ to when h is small enough), and C a constant. Let us remark incidentally that the van der Waals, solvent-accessible and solventexcluded molecular surfaces commonly used in apparent surface charge calculations, can be discretized without resorting to a polyhedral approximation. Indeed, these surfaces are made of pieces of spheres and tori and it is therefore possible to mesh and compute integrals on the molecular surfaces since analytical local maps are available [19]. As a matter of illustration, let us write in detail the numerical algorithm for computing ER with the PCM model (1.30) and (1.31) and the Galerkin approximation with P0 planar boundary elements: 1. Mesh an approximation of the cavity surface with planar triangles. 2. Assemble the matrix g Aij g +1 ∗ = 2 −D −1 ij +1 g areaTi areaTj − Dji −1 +1 1 = 2 areaTi areaTj − d s ds −1 s − s Tj Ti ns = 2 by analytical (or numerical) integration on Ti and numerical integration on Tj . 3. Assemble the right-hand side g gi = V M Ti n by numerical integration. 4. Solve the linear system Ag g = gg (1.43) 5. Compute ER by the approximation formula ER ER = app n g i VM i=1 the integrals Ti Ti being calculated numerically. Recall that when the charge densities and are composed of point charges, dipoles, or gaussian–polynomial functions, analytical expressions of the potential VM and the normal derivatives VM /n are available. It can be proved that this numerical method is of order 1 in h = max diamTi . As mentioned above, higher order methods can be obtained by first using curved tesserae instead of planar triangles and then increasing the degree of the polynomial approximation on each tessera (P1 or P2 BEM [2]). 42 Continuum Solvation Models in Chemical Physics The transposition to the above algorithm to the COSMO framework is straightforward. On the other hand, the extention to IEF-type methods require some attention. Indeed, a direct transposition of the above algorithm to the IEFPCM framework leads to the matrix elements Agij = 2 +1 1 Sij − ds ds ds −1 s − s ns Ti Tj 1 s − s For practical calculations, the integral over has to be discretized, which introduces an additional numerical error. An alternative consists in applying the Galerkin approximation to system (1.38), which is equivalent to Equation (1.37). The discretized apparent surface charge is obtained by solving successively the linear systems Sg g = −VM g Bg g = (1.44) −1 g Bg (1.45) with Bg ij = 2 +1 areaTi areaTj − Dgji −1 and B = lim B →+ The computational efficiency of an integral equation method is related to the size, the structure and the conditioning of the linear systems to be solved. Recall that there are basically two strategies to solve an N × N linear system of the form Ax = b. The first option is to store the matrix A and to invert it by a direct method, such as the LU decomposition or the Choleski algorithm [15] (the latter algorithm being restricted to the case when A is symmetric, positive definite). The second option is to solve the linear system Ax = b by an iterative method [16], such as the conjugate gradient algorithm (if A is symmetric, positive definite), or the GMRes or BiCGStab algorithms (in the general case). Iterative methods only require the calculation of matrix–vector products and scalar products. For large systems, the first option is not tractable: the memory occupancy scales as N 2 and the computational time as N 3 . The linear systems associated with the COSMO, DPCM, IEF and IEFPCM methods enjoy a remarkable property that make iterative methods very efficient: as the corresponding matrices A originate from integral operators involving the Poisson kernel 1/r or its derivatives, it is possible to compute matrix–vector products Ay for y ∈ RN , without even assembling the matrix A, in N log N elementary operations, by means of Fast Multipole Methods (FMMs) [20, 21]. The number of conjugate gradient, GMRes or BiCGStab iterations depends on the one hand on the quality of the initial guess, and on the other hand on the conditioning of the linear system. Recall that the conditioning parameter of an invertible matrix A for the · 2 norm defined by A 2 = supx∈RN Ax / x ( · denoting the euclidian norm on RN ) is the real number 2 A = A 2 A−12 . If A is symmetric, definite positive, 2 A = N A/1 A where 0 < 1 A ≤ · · · ≤ N A are the eigenvalues of A. The larger 2 A, the larger the number of iterations. If A is symmetric, it can indeed be Modern Theories of Continuum Models 43 proved that the sequence xk generated by the conjugate gradient algorithm with initial guess x0 converges to the solution x to Ax = b, and that one has the error estimate xk − xA ≤ 2 2 A − 1 2 A + 1 k x0 − xA where y A = Ay y1/2 . Note that 2 A ≥ 1 and that 2 A = 1 if and only if A is the identity matrix, up to a multiplicative constant. Not surprisingly, the conjugate gradient algorithm converges in a single iteration in the latter case. For completeness, let us also mention that the conjugate gradient converges in at most N iterations. It follows from the above arguments that the efficiencies of the various integral equation methods under examination are directly related to the conditioning parameters of the matrices S and − D∗ . It is at that point that the functional analysis properties of the underlying operators S and − D∗ come into play. Indeed, as − D∗ is for all > −2 an isomorphism on L2 and as the P0 BEM test functions are in L2 , the conditioning parameter of the matrix 2 + 1/ − 1 − D∗ g is bounded independently of N. Consequently, the number of iterations needed to solve the PCM Equation (1.30) or Equation (1.45) in the P0 BEM Galerkin approximation does not dramatically vary if the mesh is refined. On the other hand, while the operator S is bounded from L2 to L2 S −1 maps L2 onto H −1 and is therefore an unbounded operator on L2 . This implies that the larger eigenvalue of Sg is bounded independly of the size of the mesh, and that the smallest eigenvalue of Sg goes to zero when the mesh is refined. Hence, the conditioning of Sg goes to infinity when the mesh is refined. This problem is encountered with the COSMO, IEF and IEFPCM methods. In order to prevent the iterative algorithm from breaking down in the limit of large molecular systems and/or fine mesh, preconditioning techniques are needed [16]. In the special case of spherical cavities and regular meshes, analytical estimates of the conditioning parameters of Sg and 2 + 1/ − 1 − D∗ g are available: 2 Sg N 1/2 and 2 2 + 1/ − 1 − D∗ g 2/ + 1. 1.2.6 Derivatives and Geometry Optimization For molecular systems in the vacuum, exact analytical derivatives of the total energy with respect to the nuclear coordinates are available [22] and lead to very efficient local optimization methods [23]. The situation is more involved for solvated systems modelled within the implicit solvent framework. The total energy indeed contains reaction field contributions of the form ER , which are not calculated analytically, but are replaced by numerical approximations ERapp , as described in Section 1.2.5. We assume from now on that both the interface and the charge distributions and depend on n real parameters 1 · · · n . In the geometry optimization problem, the i are the cartesian coordinates of the nuclei. There are several nonequivalent ways to construct approximations of the derivatives of the reaction field energy with respect to the parameters 1 · · · n : 1. One way consists in first calculating analytically the derivatives /i ER of the exact reaction field energy, and then approximating /i ER , yielding the quantities denoted by /i ER app . 44 Continuum Solvation Models in Chemical Physics 2. A second way consists in calculating the derivatives /i ER of the approxiapp mated energy ER . This second approach can be subdivided into three methods: app /i ER can be computed (i) by finite differences, (ii) by deriving analytically the app discrete equations used for the calculation of ER , (iii) by automatic differentiation [24]. Although (ii) and (iii) are theoretically equivalent, they are not in practice: they correspond to two dramatically different implementations of a single mathematical formalism. app The main practical difficulty in optimizing the geometry of solvated molecules arises from the fact that ERapp is not, in general, a continuous function of the parameters i . Discontinuities are indeed introduced by the mesh generator. Efficient, robust geometry optimization procedures for solvated molecules are still to be designed. Let us conclude this section by providing an expression of the analytical derivative E i R at 1 · · · n = ∗1 · · · ∗n valid in the case when and are supported inside the cavity. Let us denote by = ∗1 · · · ∗n , and denote for all s ∈ by Ui · ns = d d s ∗1 · · · ∗i−1 ∗i + t ∗i+1 · · · ∗n dt t=0 the velocity field generated by an infinitesimal variation of ith parameter. In the previous expression, ds · denotes the signed distance between s and ·: dx · = − inf y − x y ∈ · + inf y − x y ∈ · if x ∈ R3 \ ⊆ · if x ∈⊆ · The analytical derivative formula then reads [25] V V ER = M + M + Ui · n i i i 4 (1.46) with = 16 2 + − 1V V −1 (1.47) V s denoting the projection of the vector Vs on the tangent plane to at s. In the limit = +, one has [26] = 16 2 The integral Ui · n = 4 Ui · n 4 Modern Theories of Continuum Models 45 then has a simple physical interpretation: 4 Ui · n is the virtual power of the electrostatic pressure p = 4 exerted on the walls of a perfect conductor [5]. When the permitivity is high (which is typically the case for water) the approximate analytical derivative formula V V 4 ER M + M + Ui · n i i i −1 is reasonably accurate [27]. 1.2.7 Beyond the Standard Dielectric Model The range of application of the integral equation method is not limited to the standard dielectric model. It encompasses the cases of anisotropic dielectrics [8] (liquid crystals), weak ionic solutions [8], metallic surfaces (see ref. [28] and references cited therein), However, it is required that the electrostatic equation outside the cavity is linear, with constant coefficients. For instance, liquid crystals and weak ionic solutions can be modelled by the electrostatic equations −div LC · V = 4 (1.48) −div PB V + PB 2 V = 4 (1.49) and respectively. In Equation (1.48), LC is a 3 × 3 symmetric positive definite matrix whose eigenvectors correspond to the principal axes of the liquid crystal. Equation (1.49) is a linearization of the nonlinear Poisson–Boltzmann equation −div PB V + PB sinh2 V = 4 (1.50) and is valid for weak ionic solutions, in the limit when 2 V is small ( is the Debye length of the ionic solution). It is important to note that the integral equation method is not appropriate for strong ionic solutions, since Equation (1.50) is nonlinear. One can associate with any linear electrostatic equation with constant coefficient, formally denoted by Le V = 4 (Le is a differential operator with constant coefficients), a function Ge r called the Green kernel of the operator Le /4 and defined by Le Ge = 40 where 0 is the Dirac distribution. In particular the Green kernels for Equations (1.48) and (1.49) read Ge r = ⎧ −1 ⎨det LC −1/2 LC r r−1/2 for Equation (1.48) ⎩exp−r r −1 PB for Equation (1.49) 46 Continuum Solvation Models in Chemical Physics In the special cases when LC is the identity matrix, and when PB = 1 and = 0, both Equations (1.48) and (1.49) reduce to the Poisson equation −V = 4, and Ge r = r −1 (r −1 is the Green function of the operator −/4). When the linear isotropic dielectric medium used in the standard model is replaced with a linear homogeneous medium with Green kernel Ge , and when the charge distribution is entirely supported inside the cavity, the reaction potential inside the cavity still has a simple integral representation: ∀r ∈ C V R r = s ds r − s (1.51) The apparent surface charge involved in the above expression satisfies the integral equation 2 − De S + Se 2 + D∗ = −2 − De VM − Se VM n where S and D∗ are given by Equations (1.22) and (1.24) and where Se and De are defined by similar formulae as S and D, replacing s − s −1 with Ge s − s and /ns s −s −1 with ·s Ge r −r ·ns respectively. An important difference between the integral representation formulae (1.29) (standard model) and Equation (1.51) is that Equation (1.29) is valid on the whole space R3 whereas Equation (1.51) only holds true inside the cavity. The reaction field energy of two charge distributions and both supported inside the cavity can nevertheless be obtained remarking that E R = VR = R I3 = r C VR s ds dr r − s C r = s dr ds r − s C r = s dr ds r − s R I 3 = sVM s ds Lastly, let us mention that the integral equation method applies mutatis mutandis to the case of multiple cavities (i.e. to the case when C has several connected components). This situation is encountered when studying chemical reactions in solution. References [1] A. Klamt and G. Schüürman, COSMO: A new approach to dielectric screening in solvents with expressions for the screening energy and its gradient, J. Chem. Soc. Perkin Trans., 2 (1993) 799. Modern Theories of Continuum Models 47 [2] W. Hackbusch, Integral Equations – Theory and Numerical Treatment, Birkhäuser Verlag, (1995). [3] E. H Lieb and M. Loss, Analysis, 2nd edn, American Mathematical Society, New York, (2001). [4] E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational quantum chemistry: a primer, in Ph. Ciarlet and C. Le Bris (eds), Handbook of Numerical Analysis. Volume X: Special Volume: Computational Chemistry, Elsevier, Amsterdam, (2003), pp 3–270. [5] L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, ButterworthHeinemann, (1999). [6] S. Miertuš, E. Scrocco and J. Tomasi, Electrostatic interaction of a solute with a continuum. A direct utilization of ab initio molecular potentials for the prevision of solvent effects, Chem. Phys., 55 (1981) 117. [7] J. Tomasi and M. Persico, Molecular interactions in solution: An overview of methods based on continuous distribution of solvent, Chem. Rev., 94 (1994) 2027. [8] E. Cancès and B. Mennucci, New applications of integral equation methods for solvation continuum models: ionic solutions and liquid crystals, J. Math. Chem., 23 (1998) 309. [9] E. Cancès, B. Mennucci and J. Tomasi, A new integral equation formalism for the polarizable continuum model: theoretical background and applications to isotropic and anisotropic dielectrics, J. Chem. Phys., 107 (1997) 3032. [10] B. Mennucci, E. Cancès and J. Tomasi, Evaluation of solvent effects in isotropic and anisotropic dielectrics, and in ionic solutions with a unified integral equation method: theoretical bases, computational implementation and numerical applications, J. Phys. Chem. B, 101 (1997) 10506. [11] B. Mennucci, R. Cammi and J. Tomasi, Excited states and solvatochromic shifts within a nonequilibrium solvation approach: A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration selfconsistent field level, J. Chem. Phys., 109 (1998) 2798. [12] D. M. Chipmann, Reaction field treatment of charge penetration, J. Chem. Phys., 112 (2000) 5558. [13] E. Cancès and B. Mennucci, Comment on: Reaction field treatment of charge penetration, J. Chem. Phys., 114 (2001) 4744. [14] E. Cancès and B. Mennucci, The escaped charge problem in solvation continuum models, J. Chem. Phys., 115 (2001) 6130. [15] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Ithaca, NY, (1996). [16] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edn, Society for Industrial and Applied Mathematics (2003). [17] P. J. Davis and I. Polonsky, in M. Abramowitz and I. A. Stegun (eds), Handbook of Mathematical Functions, Dover Publications, New York, Chapter 25, (1965) pp 875–924. [18] J. C. Nédélec and J. Planchard, Une méthode variationelle d’éléments finis pour la résolution d’un problème extérieur dans R3 , RAIRO 7 (1973) 105. [19] R. J. Zauhar and R. S. Morgan, Computing the electric potential of biomolecules: applications of a new method of molecular surface triangulation, J. Comput. Chem., 11 (1990) 603. [20] L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numerica 6 (1997) 229. [21] G. Scalmani, V. Barone, K. N. Kudin, C. S. Pomelli, G. E. Scuseria and M. J. Frisch, Achieving linear-scaling computation cost for the polarizable continuum model of solvation, Theoret. Chem. Acc., 111 (2004) 90. [22] J. A. Pople, R. Krishnan, H. B. Schlegel and J. S. Binkley, Derivative studies in Hartree–Fock and Møller–Plesset theories, Int. J. Quantum Chem., 13 (1979) 225. 48 Continuum Solvation Models in Chemical Physics [23] P. Y. Ayala and P. B. Schlegel, A combined method for determining reaction paths, minima and transition state geometries, J. Chem. Phys., 107 (1997) 375. [24] G. Corliss, C. Faure, A. Griewank, L. Hascoet and U. Naumann, (eds), Automatic Differentiation of Algorithms, from Simulation to Optimization, Springer, Heidelberg, (2001). [25] E. Cancès and B. Mennucci, Analytical derivatives for geometry optimization in solvation continuum models I: Theory, J. Chem. Phys., 109 (1998) 249. [26] E. Cancès, PhD Thesis, Ecole Nationale des Ponts et Chaussées (in French), (1998). [27] E. Cancès, B. Mennucci and J. Tomasi, Analytical derivatives for geometry optimization in solvation continuum models II: Numerical applications, J. Chem. Phys., 109 (1998) 260. [28] S. Corni and J. Tomasi, Excitation energies of a molecule close to a metal surface, J. Chem. Phys., 117 (2002) 7266. 1.3 Cavity Surfaces and their Discretization Christian Silvio Pomelli 1.3.1 Introduction In a previous contribution in this book, Cancès has presented the formal background of the integral equation methods for continuum models and has shown how the corresponding equations can be solved using numerical methods. In this chapter the specific aspects of the implementation of such numerical algorithms within the framework of the Polarizable Continuum Model (PCM) [1] family of methods will be considered. As described in the previous contributions by Cancès and by Tomasi, in such a family of methods the solvent effects on the molecular solutes are evaluated by introducing a set of apparent charges representing the polarization of the dielectric medium. These charges are obtained by solving integral equations defined on the domain of the boundary of the cavity which hosts the molecular solute. The solution of such equations can be divided in two main steps. The first step defines a molecule–solvent boundary from the molecular geometry and some solvent-related quantities. This boundary is then discretized in a finite number of small elements called tesserae. This step is independent of the molecular structure theory in use (MM, DFT, MP2, etc.). The second step solves the integral equations using the boundary elements previously introduced. The result of this second step is the evaluation of the various contributions of different physical origin (electrostatic, repulsion, dispersion, cavitation) which determine the solvent reaction field. This second step depends (at least for the electrostatic part) on the level of description of the molecular structure. The main scope of this chapter is to give some numerical and computational details of the machinery that is under the surface of modern continuum solvation models and especially those belonging to the PCM family. Knowledge of the details of the boundary partitioning into elements can help one to avoid numerical troubles especially with large (or complex) molecular systems. A smart choice of the method used to solve discretized integral equations can lead to valuable savings in CPU time and hard disk usage and can permit calculations to be performed on large solvated systems with limited computational resources. This chapter is divided into three main parts: one presents and comments the main aspects related to the definition of the solute cavity and the solvent–solute boundary, the second focuses on the numerical techniques to obtain boundary elements while the third part describes the main numerical procedures to solve the integral equations. 1.3.2 The Cavity and its Surface In continuum solvation methods the molecular cavity is the portion of space within the surrounding medium (solvent) that is occupied by the solute molecule: the boundary of the molecular cavity is called molecular surface. There are several models to define the molecular cavities and their surfaces. Historically, the first models proposed were based on the simplest three-dimensional geometrical 50 Continuum Solvation Models in Chemical Physics shapes: the sphere [2] and the ellipsoid [3]. The radius of the sphere, or the ellipsoid axes, are given as parameters and they are empirically based on the extension in space of the molecule. These simple models, which disregard many of the stereochemical details of the molecule, are still in use as they allow an analytical solution of the electrostatic equations defining the solvent reaction field. A completely different definition is based on the isodensity surface [4], i.e. the surface constituted by the set of points having a specified electronic density value (given as a parameter). The most common way to define molecular cavities, however, is to use a set of interlocking spheres centred on the atoms constituting the molecular solute (Figure 1.2). Based on such a definition of the cavity, we can define different molecular surfaces: Figure 1.2 Definitions of cavities based on interlocking spheres. In black (dashed) the spheres centred on atoms A and B, in red the SAS, in cyan the shared parts of VWS and SES. In green the concave part of SES. In blue the crevice part of VWS. In black (dotted) some positions of tangent solvent probes (see Colour Plate section). (i) The van der Waals surface (VWS) is defined as the surface obtained from a set of interlocking spheres, each centred on an atom or group of atoms and having as radius the corresponding van der Waals radius. Several compilations of van der Waals radii [5, 6] are reported in the literature. The VWS is commonly used to calculate the cavitation contribution to the solvation free energy, namely the energy required to build a void cavity inside the medium (see also the chapter by Tomasi). (ii) The solvent-accessible surface (SAS) [7] is defined as the surface determined by the set of points described by the centre of a spherical solvent probe rolling on the VWS: the radius of the solvent probe is related to the dimensions and the nature of the solvent. From this definition it turns out that the SAS is equivalent to a VWS in which the radius of the solvent probe is added to each atomic radius. The SAS is commonly used to calculate the short-range (dispersive and repulsive) contributions to the solvation free energy. (iii) The solvent-excluded surface (SES) [8] is defined as the surface determined by the set of the tangent (or contact) points described by a spherical solvent probe rolling on the VWS. This surface delimits the portion of space in which the solvent probe cannot enter without intersecting the VWS. The SES appears as the VWS in which the crevices correspondent to sphere–sphere intersection are smoothed; the convex part of the SES is shared with the VWS and is called the contact surface, whereas the part of the surface which is not shared with the VWS is concave and is called the re-entrant surface. The region of the space, which is enclosed in the SES but not in VWS, is called the solvent-excluded volume. Modern Theories of Continuum Models 51 The Solvent-excluded Volume As described above VWS and SAS are easily defined as sets of spheres centred on atoms. This definition, however, does not apply to SES; in this case in fact, the pair of surfaces delimiting the boundary between the excluded volume and the solvent cannot be defined using spheres. There are several algorithms which translate the abstract definition of the SES into a complex solid composed of simple geometrical objects from which the surface can be easily tessellated. The first and most famous algorithm to calculate the SES has been proposed by Connolly [9]: in this algorithm a set of points on the surface of the solvent spherical probe is acquired by rolling the sphere on the VWS and it is further organized in a mesh to build the tessellation. The rolling and sampling procedures has been improved over the years so to give an optimal meshing. The package of Connolly, named MSDOT, is widely used in molecular modeling for visualization of molecules (especially in the field of biochemistry and molecular biology), ESP fitting, and docking but it has been rarely used in combination with continuum solvation methods [10]. In its modern formulation, the Connolly surface presents a full analytical tessellation [11] but the reliability of it and of its differentiability has never been tested with PCM-like calculations. As a matter of fact, in the field of molecular modelling and molecular graphics there are several algorithms to calculate the molecular volume and surface and to visualize them, but the number of tesserae needed to produce a good graphical rendering is larger than that needed for the solution of the PCM equations and none of the rendering/modellingoriented methods yields a differentiable tessellation. Completely different approaches are DefPol and BLMOL. In DefPol [12] a giant polyhedron with triangular faces, built around the whole molecule, is deformed until its vertices lie on the molecular surface. This latter is described by a shape function different from zero only in the space inside the molecular cavity. The shape function is a combination of terms related to single atomic spheres supplemented by terms related to pairs or triples of spheres. The multiple sphere terms take account of the solvent-excluded volume. DefPol can also be used for VWS and SAS, simply by skipping the calculation of twoand three-sphere terms. The method is fast from the numerical point of view, but it is affected by serious numerical problems in computing derivative terms and to be applied to oblong and nonconvex molecular shapes. For these reasons, it is currently not in use. BLMOL [13] is a specialized version of a very general tessellated surfaces package called BLSURF [14]. The BLMOL package partitions the SES in patches and triangulates each of them by using an advancing front algorithm. Each patch represents a connected portion of the surface with homogenous curvature properties (e.g. a fragment of an atomic sphere, a portion of torus generate by the rolling of the solvent probe while tangent to two spheres, etc.). BLMOL requires a dimension of the single triangular tesserae very small with respect to that commonly used in this context; these characteristics and the fact that it is not freely available limit its use. Also, the BLMOL tessellation is in principle differentiable but its derivatives have never been implemented. The last method which will be considered here is the GEPOL, which was first elaborated in Pisa by Tomasi and Pascual-Ahuir [15]. GEPOL will be presented in two steps: in this section we will treat the excluded volume filling, whereas the definition of surface elements will be given in the next section. 52 Continuum Solvation Models in Chemical Physics The GEPOL Approach In GEPOL the excluded volume is approximated by a set of supplementary (or ‘added’) spheres, which are defined through a recursive algorithm. The spheres centred on atoms constitute the first generation of sphere. For each pair of spheres, for which rAB < RA + RB + 2RS where rAB is the distance between the atoms and RA RB RS are the radii of atomic and solvent probe spheres, one or more spheres are added. The centre of the new spheres lies on the segment joining the centres of the two generating spheres and the position and the radius of the spheres are chosen in such a way as to maximize the solventexcluded volume filled by the new sphere. This procedure is repeated recursively with the inclusion of the newly generated spheres in the pair-search procedure: in principle this process should not terminate as it tries to fill a concave space with convex objects. Its termination is determined by two tests, namely: 1. If the radius of the generated sphere is less than a given threshold, such a sphere is not added to the sphere set. 2. If the generated sphere overlaps the existing spheres too much, it is not added to the sphere set. A geometrical parameter is used to decide if this condition is verified, and several versions of this test have been proposed over the years. In Figure 1.3 some examples of ‘added’ sphere patterns are illustrated. It is evident that the number, position and radius of these spheres change with the change of the molecular geometry. The space filling procedure has been upgraded over the years, so to efficiently handle large molecular systems, such as proteins [16], to account for molecular symmetry [17, 18] and to reduce the computational complexity from quadratic to linear [19] by using lists of nearby spheres. (a) (b) (c) Figure 1.3 Generation of GEPOL added spheres. (a) For two close spheres a single sphere intersecting with the two parent ones is generated. (b) For farther spheres, first a sphere that does not intersect with the two parent spheres is generated, then two ‘third generation’ spheres are added between the second generation sphere and each of the two first generation spheres. (c) For any pair of spheres with a large separation, small spheres very overlapped with the primitive ones are generated. This last case occurs only with very loose thresholds for the termination tests. In each case all the added spheres are tangential to the solvent probe spheres tangential to both the atomic spheres. Modern Theories of Continuum Models 53 The definition of excluded volume in GEPOL which is exact only if we consider an infinite generation of supplementary spheres, replaces the complex geometrical structure of torus and curvilinear prisms used in BLSURF, Connolly and DEFPOL by simply extending the set of atomic spheres. This aspect is very important from the computational point of view, because it allows an easy development and implementation of well-defined tessellations. 1.3.3 The Surface Tessellation In order to be suitable in the application of the boundary element method (BEM) procedures required to build the reaction field, a molecular surface must be tessellated. A tessellation is a partition of a surface in subsets named tesserae each with a surface area a, a sampling point s and a unit outward vector n̂ at the sampling point. The tessellation elements a s n̂ are the main quantities used to solve the BEM equations. A differentiable tessellation is defined in such a way that it is possible to analytically calculate derivatives with respect to the molecular geometry. A tessellation is well defined when the tessellation related quantities and their derivatives are stable from the numerical point of view. The kinds of partitions of the surface area that lead to a well defined tessellation are one of the main issues of this contribution and will be discussed in the next section. Tessellation of Spheres The partition of the sphere surface is a well known topic in geometry [20]. Apart from the mathematical speculation, this problem is very important in modern computer graphics for the rendering of spherical objects. An important remark is that for the computation of the reaction field even at high numerical accuracy it is sufficient to partition a surface into a number of elements noticeably smaller than that used in any modern rendering package. In particular, the various versions of GEPOL that have been released through the years use geodesic partition schemes based on polyhedra inscribed into a sphere. The original version exploits a 60 tesserae partition scheme based on a pentakisdodecahedron for all the spheres [16]. A flexible partition scheme has been introduced by using some basic polyhedra, in which the original triangular faces are partitioned through an equilateral division procedure [21] (see Figure 1.4 for details). The equilateral division procedure Figure 1.4 Equilateral division of a triangle. From left to right, divisions of order N = 2 3 6. Each side of the original triangle is divided in N equal parts (in the case of spherical triangles the sides are circumference equatorial arcs). A segment (or an arc) is traced from each division point to the corresponding point on another side, so that the final result is a division of the original triangle in N 2 triangles. 54 Continuum Solvation Models in Chemical Physics replaces each original triangle of the polyhedron with N 2 triangles, being N the order of equilateral division, so that if M is the original number of polyhedron faces, the final one is MN 2 . There are two ways of using this flexible partition scheme, (i) the same partition of the surface is used for each sphere (TsNum), or (ii) a number of tesserae proportional to the sphere surface (TsAre) is used (see Figure 1.5). Figure 1.5 Molecular cavity for H2 CO using the TsNum = 60 option (left) and the TsAre = 02 option (right). Both the cavities respect the C2v symmetry of the molecule. The TsAre option is nowadays the default option in some widely used computational packages. Details on the benefits of the TsAre scheme are reported in the subsection about GEPOL numerical stability. Quantum mechanical computational packages use the molecular symmetry in order to reduce the computational effort. This feature can be used if the point sampling of the cavity surface respects the molecular symmetry. A way of obtaining this requirement consists in partitioning each sphere surface by respecting the molecular symmetry [17]: this can be obtained by using basic polyhedra which subtend the same point group of the molecule, so that the resulting cavity partition is invariant under any geometrical transformation that belongs to the molecular symmetry group. In this way a symmetry-reduced cavity, containing only ‘unique by symmetry’ tesserae is obtained (this procedure is similar to the ‘petite list’ of orbitals used in symmetry-adapted ab initio calculations [22]). Partition of Intersecting Spheres When two or more spheres intersect, some of their tesserae are cut to exclude the portion of their surface that lies inside the other spheres. In GEPOL, this cutting procedure tests whether a tessera intersects a sphere surface (excluding the sphere to which the tessera belongs) and cuts the part of the tessera that lies inside it, so that for any tessera–sphere intersection a part of the tessera is cut away. If the entire tessera lies inside the sphere, it is completely removed from the tesserae list. Such a procedure is repeated for any sphere–tessera pair. The computational cost of this step can be reduced, as for the added sphere generation, if a list of nearby spheres has previously been generated [19]. The first version of the tesserae cutting scheme [23] in GEPOL was based on a simple partition in sub-tesserae. The resulting tessellation was not differentiable. Because a differentiable tessellation is essential to use gradient-based automatic geometry optimization procedures, an analytical calculation of the cut tessera area has been introduced [16]. Modern Theories of Continuum Models 55 The geometrical definitions and equations to be used are those of the generalized spherical polygon [24], which is the portion of spherical surface delimited by one or more planes that pass through the sphere centre. The spherical polygon is generalized if one or more of the planes do not pass through the sphere centre [13]. In contrast to plane polygons, a spherical polygon can have only one or two sides (note that the original uncut tessera is a spherical triangle). Each cutting sphere adds a delimiting plane that does not pass through the centre of the sphere on which the tessera lies. The number of different cases which can arise from the intersection between a spherical triangle and one or more spheres is very large: details on this topic are beyond the scope of this chapter. The two most common cases are illustrated in Figure 1.6. Figure 1.6 Two common cases of intersection. The cutting sphere removes A and B vertices replaced by D and E (left). The result is a smaller (and irregular) triangle. The cutting sphere removes vertex B replaced by D and E (right). The result is an irregular quadrilateral polygon (See Colour Plate section). The final result of the cutting is a generalized spherical polygon, for which the surface area of the tesserae can be analytically calculated [23]. The sampling point is taken as the average of the polygon vertices on the sphere surface. This procedure leads to a differentiable tessellation but suffers from numerical troubles in some cases [25]. Some Difficult Cases The various steps of GEPOL we have described above are not fully reliable from a numerical point of view especially when used in a gradient-based geometry optimization procedure. The contribution of the surface elements to the gradients is calculated considering the variation of shape and area of each tessera with respect to the displacement of the intersecting spheres. The primitive spheres are centred on atoms and thus they follow them in the molecular geometry evolution. The displacement of the added spheres is related to the atoms that appear in their genealogical trees: also, the radii of the added spheres are variable by definition. The evolution of the added spheres during a geometry optimization can lead to their annihilation when their overlap with other spheres and/or their radius falls under the selected thresholds. As a result, this variation of the set of added spheres leads to a discontinuity in the description of the solvent reaction field [21]. Typical cases in which this discontinuity can occur are those in which there is a large variation of the distance between two atoms (dissociations, rearrangements, etc.). This kind of numerical instabilities does not alter the final stationary point reached by the optimization procedure, but it can increase the number of steps needed 56 Continuum Solvation Models in Chemical Physics to reach it. In extreme cases the optimization procedure may enter an infinite loop, in which the molecular geometry walks around the stationary point but never reaches it. This occurs when the distance between the geometrical place of sphere annihilation and the stationary point is very small. This infinite loop is generally characterized by a pseudo-periodic behaviour of the geometry optimization related parameters (energy, displacement and gradient norms, etc.). This problem can be resolved by a manual restart of the optimization procedure, in particular: 1. Choose a step of the optimization procedure located just before the infinite loop. 2. Slightly alter by hand the chosen geometry. 3. Restart the procedure. If this procedure is not successful, a further possibility is to alter the threshold parameters for the sphere annihilation. In more unlucky cases some patient tuning work is required. Another possible source of troubles in GEPOL is the presence of ill-defined tesserae, i.e. a very small tessera and/or a tessera with a complex or oblong shape. Some typical cases are illustrated in Figure 1.7. Ill-defined tesserae can affect the solution of the PCM equations, the convergence of the SCF and the convergence of the geometry optimization procedure. A large part of these problems can be solved by using the TsAre option in the sphere partition procedure and by the usage of group spheres for groups such as CHn n = 1 2 3. A manual inspection and resolution of problems related to ill-defined tesserae is not possible (the zoomed part of the phenol cavity reported in Figure 1.6 is less than the 1 % of the total surface). Fortunately, many GEPOL versions in use have built-in tests and tricks to intercept and remove these numerical troubles [19, 26]. In the few cases in which these automatic procedures do not work, a tuning procedure similar to those proposed for the added spheres problem can be used: in this case the parameter to be altered is the TsAre value. Figure 1.7 A zoomed detail of the phenol cavity near the ring centre. Some typical cut tesserae shapes are shown. A: a tessera with complex cutting but without problematic situations. B: an oblong tessera. C: a very small tessera. D: short edges can cause numerical troubles. B, C, D are cases of ill-defined tesserae (see Colour Plate section). Methods Based on Weighted Sets of Points A completely different approach to solve the possible numerical problems inherent in partition procedures such as those used in GEPOL is to approximate the tesserae areas by weights calculated using a scale function. The word weight is used instead of area Modern Theories of Continuum Models 57 because the quantity introduced here does not have a well-defined geometrical meaning. In this framework, a tessera has a weight w that is initially equal to the uncut tessera surface area if a geodesic sampling of the sphere is adopted. Each nearby sphere scales the weight by a function of the tessera centre to sphere surface distance: wi = w0i ! f si − rj − Rj (1.52) j where fx goes from 1 when the point i is far from the sphere j to 0 when the point i is far from the sphere j. In the intermediate region of space (the switching region) a polynomial function smoothly interpolates between 0 and 1. Two alternative schemes have been proposed in the literature to define the polynomial functions. In the first, due to Karplus [27], the interpolating polynomial is determined by requiring that the values of the polynomial’s first and second derivatives are zero at both ends of the switching region. The lowest limit of the switching region is located inside the sphere j and the upper limit is located outside. Furthermore, the point charges are replaced by spherical gaussian distributions of charge so to avoid singularities for very near points and the exponent of the gaussians is chosen to fit the exact values of the Born equation for spherical ions. In the second approach, the Tessellationless (TsLess) [25], the same conditions at both ends of the switching region apply, supplemented by the requirement that the value of the integral of the polynomial on the switching region is 1, so to avoid any underestimation of the weights of points lying on the switching region. The lowest limit of the switching region is located slightly outside the sphere j and the upper limit at a larger distance from the sphere j. The choice of the switching region in TsLess also solves the problem of very near points without altering the physical nature of point charges. Note that the collocation of a part of the switching region inside the sphere j in the Karplus scheme plays the same role as the polynomial ‘normalization’ in TsLess. The calculation of the switching function is fast and very similar in both approaches. The product in Equation (6) mimics the geometrical properties of the tesserae-cutting scheme: the weight of a point is unaffected by far spheres and goes to zero when it is well buried (Karplus) or very near (TsLess) inside a single sphere. The calculation of weights is simpler than that of analytical areas using the tesserae cutting procedure, and it is also not affected by the numerical troubles described in the previous section. 1.3.4 Solution of the BEM Equations In this section we report the most common formulations of the BEM equations for three different versions of PCM [1], namely IEFPCM (isotropic), CPCM and DPCM. The mathematical and physical significance of these equations are discussed in the contribution by Cances. Here we are interested only in the computational features. The most convenient form of the BEM equations for numerical purposes is [18] Tq = −Gf (1.53) where T and G are matrices depending on the tessellation and on the solvent dielectric constant, q are the PCM charges and the f vector contains the molecular electrostatic 58 Continuum Solvation Models in Chemical Physics potential in the IEFPCM and CPCM formulations and the flux of the electrostatic field through the corresponding tessera in DPCM. The formulae for the elements of the matrices and vectors introduced here are reported in Table 1.1. Table 1.1 Definitions of the matrix elements in the BEM equations formulation F IEFPCM V CPCM V DPCM En BEM equations T −1 A−1 − D S 2 +1 2 Aij = 0 # Sii = 10694 4 ai 1 Sij = si − sj 2A−1 − D −1 I I S −1 A−1 − D∗ +1 Matrix elements Aii = ai G " 1 = Dii = 10694 or Rl ai ∗ ∗ 1 Dii = − a 2 + Dij aj Dii∗ i i=j ! si − sj • n̂i =− si − sj 3 ! si − sj • n̂j Dij = − si − sj 3 Dij∗ Two alternative definitions for the diagonal elements of the D and D∗ matrices have been presented. The first reported in the table is the original one and takes into account the curvature of the tesserae (the inverse of the radius Rl of the sphere to which the tessera belongs). The second formulation is based on electrostatic considerations [28]. The numerical factor 1.0694 has been empirically adjusted in order to reproduce the values given by the exact Born equation for spherical ions [18]. When the attention is focused to the development of the formalism for the calculation of molecular properties and energetic, the most appropriate form of Equation (1.53) is: q = −Kf (1.54) where K = T−1 G. This form easily connects the charges to the molecular electrostatic potential (or field) through a linear operator. When attention is focused on the computational aspects, the form with the T and G matrices is more useful, because T and G have simple analytical formulations. In the cases in which the molecular charge partially lies outside the cavity boundary (practically all the cases in which a QM model is used for the description of the molecule) the polarization weights [18] w= q + q∗ 2 (1.55) have to be calculated instead of the charges. The vector q∗ is the solution of the equation q∗ = −K† f (1.56) Modern Theories of Continuum Models 59 Matrix Inversion As shown above, the straightforward resolution method to obtain the PCM charges is simply to invert the T matrix of Equation (2) and to solve the resulting linear system [29]: q = −T−1 Gf = −Kf (1.57) If the pairs of tesserae sampling points are not too close in space, the T matrix is strictly dominated by the diagonal elements, i.e. Tii > Tij (1.58) i=j because the diagonal elements of T depend on the tesserae area and solvent parameters but the off-diagonal elements depends on the inverse of the distance between the pair of tesserae sampling points. If this condition is fulfilled (this occurs for a well tessellated surface) the charges obtained are fully reliable, as a strictly diagonal dominated matrix is not singular [30]. If there are pairs of very close tesserae (for example tessera i and j), a simple ‘safety’ measure is to annihilate the corresponding diagonal elements, Tij and Tji . Note that the methods based on tesserae weights are implicitly not affected by this problem. Derivatives with respect the molecular geometry can be obtained by differentiating Equation (1.54): q K f =− f −K ! ! ! (1.59) where ! is a molecular coordinate. All the derivatives involved in Equation (1.59) can be calculated analytically. More details on the derivatives of the PCM equations are reported in the chapter by Cossi and Rega. Iterative Computation This is the formulation originally used in continuum models [31] but it has been extensively improved through the years so that it now is the method of choice for calculations in which the computational cost of the ASC calculation is not negligible or serious storage limitations are present. The iterative method uses the Jacobi iterative algorithm [32] to solve the linear set of equations. Jacobi iterations are rapidly convergent if the diagonal term dominates the linear system equation: this is the case of PCM-BEM equations. The matrix T is partitioned in two parts: T0 that contains the diagonal elements and T1 that contains the off diagonal elements. A 0th cycle guess of the charges is given by: q0 = −T−1 0 Gf (1.60) then it is updated by iterating the equation qn = − q0 − T1 qn−1 (1.61) 60 Continuum Solvation Models in Chemical Physics until qn − qn−1 = en < (1.62) where is a threshold value. If this iterative calculation is nested into the SCF cycle then can be safely set to 1–2 degrees of magnitude less than the current SCF error norm. The convergence of the method can be improved by using a slightly different set of charges in Equation (1.61): n−1 n−1 = − q 0 − k qk qn−1 k = 1 (1.63) k=1 k=1 Two proposals have been given to set k . In the DAMP scheme [33] only the n-1 and n-2 coefficients are different from zero: n−1 = 1/en−1 1/en−1 + 1/en−2 (1.64) In the DIIS scheme [33] they are determined by minimizing the error function: 2 n−1 s = k ek k=1 n−1 k = 1 (1.65) k=1 Both schemes are also used as SCF convergence accelerators. The DIIS scheme is particularly efficient when used in conjunction with CPCM and IEFPCM schemes, in which the diagonal dominancy of T is less prominent than in DPCM. DIIS is very efficient from the point of view of CPU times, but it requires the storage of several sets of intermediate charges. DAMP is less efficient but requires the storage of two sets of intermediate charges only. CPU time can be traded versus storage using conjugate gradient schemes [18], which require longer CPU times than DIIS but do not need to store intermediate ASC sets. Another improvement concerns the fast calculations of the A1 qk terms, the only ones that contain two nested cycles on the charges and thus scale quadratically with the number of charges. While the original formulation of the iterative scheme eliminates the need of the storage of T (T1 can be calculated freshly at each iteration), it does not scale linearly with the number of charges. The linear scaling can be achieved by looking at the electrostatic nature of the T1 qk terms: ⎧ qkj = Vsl " qk ⎪ ⎪ s −s ⎪ ⎨j=l j l qkj sj −sl •nl sl "qk T1 qk l = = V 3 nl ⎪ sj −sl ⎪ j = l ⎪ ⎩ Tsl " qk for CPCM for DPCM (1.66) for IEF where Vsl " qk is the electrostatic potential at the tessera l sampling point due to the qk set of charges. Tsl " qk has a more complex expression without a electrostatic meaning Modern Theories of Continuum Models 61 but similar to Vsl " qk . Given these properties, approximated expressions of sl " qk can be obtained using local multipole expansions [34] or the powerful fast multipole method (FMM) [35]. For IEFPCM a custom version of FMM has to be used [36]. An alternative approach to IEFPCM involves a partition of the charges into two contributions, one similar to the CPCM one and the other similar to the DPCM one [34]. Thus, two full iterative procedures have to be performed to calculate the two sets of charges that summed give the final IEFPCM charges. When coupled to linear scaling electrostatic engines like FMM, the storage and CPU time of the iterative method are both linear with respect to the number of tesserae. The iterative method is very sensitive to the cavity quality, especially for CPCM and IEFPCM in which the interaction between two tesserae depends on the inverse of the distance. Some unpublished tests performed by the author on slowly convergent iterative calculations have shown that in the last steps almost all the error norm is due to a few charges that still have very large variations with respect the previous iteration cycle, whereas all the other charge variations are several orders of magnitude smaller. Iterative methods also allow the calculation of derivatives of charges with respect to molecular geometry. By differentiating Equation (1.53), we obtain: T q G f q+T =− f −G ! ! ! ! (1.67) All the quantities can be calculated analytically except q/!, which can easily be computed by applying the iterative scheme to a rearranged Equation (1.67): T q G f T =− f −G − q ! ! ! ! (1.68) The iterative scheme for the derivatives is very similar to that used for the original charges, because the matrix to be partitioned is the same in both cases. A method similar to the iterative, is the partial closure method [37]. It was formulated originally as an approximated extrapolation of the iterative method at infinite number of iterations. A subsequent more general formulation has shown that it is equivalent to use a truncated Taylor expansion with respect to the nondiagonal part of T instead of T−1 in the inversion method. An interpolation of two sets of charges obtained at two consecutive levels of truncations (e.g. to the third and fourth order) accelerates the convergence rate of the power series [38]. This method is no longer in use, because it has shown serious numerical problems with CPCM and IEFPCM. 1.3.5 Conclusions Computational methods have accompanied the development of the Polarizable Continuum Model theory throughout its history. In the building of the molecular cavity and its sampling together with the resolution of the BEM equations we nowadays have a large choice of alternative algorithms, suitable for all kinds of molecular calculations. Linear scaling both in time and space is achieved in both fields. Cavities based on interlocking spheres allow a simple and accurate calculation of tessellation elements, thanks to weight function methods. A question not solved yet is 62 Continuum Solvation Models in Chemical Physics a full smooth description of solvent-excluded volume with the use of spherical objects. Alternatives could be the development of methods based on more complex geometrical shapes and fully differentiable or the use of isodensity methods. The field of the numerical solution of BEM equations does not show nowadays problems of this magnitude. The inversion method is full reliable for small molecular systems and the iterative for large molecular systems. References [1] J. Tomasi, B. Mennucci and R. Cammi, Quantum mechanical continuum solvation models, Chem. Rev., 105 (2005) 2999–3093. [2] L. Onsager, Electric moments of molecules in liquids, J. Am. Chem. Soc., 58 (1936) 1486–1493. [3] J. G. Kirkwood, J. Chem. 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Jensen, Improving the accuracy and efficiency of geometry optimizations with the polarizable continuum model: new energy gradients and molecular surface tessellation, J. Comput. Chem., 25 (2004) 1449–1462. [27] D. M. York and Martin Karplus, A smooth solvation potential based on the conductor-like screening model, J. Phys. Chem., A, 103 (1999) 11060–11079. [28] E. O. Purisima and S. H. Nilar, A simple yet accurate boundary element method for continuum dielectric calculations, J. Comput. Chem., 16 (1995) 681–689. [29] R. Cammi and J. Tomasi, Analytical derivatives for molecular solutes. II. Hartree–Fock energy first and second derivatives with respect to nuclear coordinates, J. Chem. Phys., 101 (1994) 3888–3897. [30] K. Briggs, Diagonally Dominant Matrix, From MathWorld, A Wolfram Web Resource, created by E. W. Weisstein, http://mathworld.wolfram.com/DiagonallyDominantMatrix.html [31] S. Miertus, E. Scrocco and J. Tomasi, Electrostatic interactions of a solute with a continuum. A direct utilization of ab initio molecular potentials for the prevision of solvent effects, Chem. Phys., 55 (1981) 117–129. [32] N. Black, S. Moore and E. W. Weisstein, Jacobi Method from MathWorld, A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiMethod.html. [33] C. S. Pomelli, J. Tomasi and V. Barone, An improved iterative solution to solve the electrostatic problem in the polarizable continuum model, Theor. Chem. Acc., 105 (2001) 446–451. [34] H. Li, C. S. Pomelli and J. H. Jensen, Continuum solvation of large molecules described by QM/MM: a semi-iterative implementation of the PCM/EFP interface, Theor. Chem. Acc., 109 (2003) 71–84. [35] L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT, Cambridge, MA, 1987. [36] N. Rega, M. Cossi and V. Barone, Toward linear scaling in continuum solvent models: a new iterative procedure for energies and geometry optimizations, Chem. Phys. Lett., 293 (1998) 221–229. [37] E. L. Coitiño, J. Tomasi and R. Cammi, On the evaluation of the solvent polarization apparent charges in the polarizable continuum model: A new formulation, J. Comput. Chem., 16 (1995) 20–30. [38] C. S. Pomelli and Tomasi, A new formulation of the PCM solvation method: PCM-QINTn, Theor. Chem. Acc., 96 (1997) 39–43. 1.4 A Lagrangian Formulation for Continuum Models Marco Caricato, Giovanni Scalmani and Michael J. Frisch 1.4.1 Introduction Implicit solvation models have proved themselves very effective in providing a computationally feasible way to simulate the microscopic environment of molecules in solution [1–3]: accurate free energy of solvation can be computed, and the spectroscopic properties of solutes can be corrected to take into account solvent effects. While all implicit solvent models share the same advantage with respect to explicit ones, i.e. the very significant reduction in complexity achieved through the description of the solvent as a uniform continuum, they can be grouped in various ways according to the theoretical framework used to describe the solute, the solvent and the interface between them. In the Generalized Born model [2–5], the solvent is described in a extremely simplified way and there is no mutual polarization between solute and solvent. The Onsager model [6] allows for solute–solvent polarization, but the description of the cavity and of the solvent is still very crude. A more sophisticated description of the solvent is achieved using an Apparent Surface Charge (ASC) [1, 3] placed on the surface of a cavity containing the solute. This cavity, usually of molecular shape, is dug into a polarizable continuum medium and the proper electrostatic problem is solved on the cavity boundary, taking into account the mutual polarization of the solute and solvent. The Polarizable Continuum Model (PCM) [1, 3, 7] belongs to this class of ASC implicit solvent models. Finally, other models [8–10] define the dielectric constant as a function of the point in space around the solute and solve the three-dimensional electrostatic problem, usually by a finite differences method. In recent years many attempts have been made to extend the implicit solvent models to the description of time-dependent phenomena. One of these phenomena is nonequilibrium solvation [3] and it can be described effectively in a very simplified way, despite the fact that it actually depends on the details of the full frequency spectrum of the dielectric constant. Typical examples of nonequilibrium solvation are the absorption of light by the solute which produces an excited state which is no longer in equilibrium with the surrounding polarization of the medium [11–13]. Another example is intermolecular charge transfer within the solute, also leading to a nonequilibrium polarization [14]. In the simplest picture of the nonequilibrium state, only a fraction of the solvent degrees of freedom is able to ‘follow’ the quick change in the electronic structure of the solute, while the ‘slow’ degrees of freedom take a longer time to equilibrate with the new state of the solute. More detailed descriptions of the time evolution of the solvent polarization have been reported [15] and similar results have also been recently achieved in the context of the PCM [13, 14]. Aiming to describe any kind of time-dependent phenomena, it would be highly desirable to couple the standard molecular dynamics (MD) methods, both classical and ab initio, with the implicit solvent model. This can be achieved either by solving the Modern Theories of Continuum Models 65 electrostatic problem at every step of the dynamics or by defining an extended Lagrangian which includes the polarization of the medium as a dynamical variable. In the first scheme, the only significant issue is to ensure that the solvation potential given by the implicit solvent model is a continuous and smooth function of the nuclear coordinates. There are numerous examples of successful application of this strategy in the literature. The Generalized Born method has been effectively coupled with MD using classical force fields and the GB–MD technique is nowadays widely used in classical MD simulations of large molecules and proteins [2, 4, 16]. The Car–Parrinello Lagrangian has been extended by De Angelis and co-workers [17] using an ASC implicit solvent model, namely the conductor-like flavor of the PCM model (CPCM), to include the interaction energy between the solute’s electrostatic potential and the polarization charges. A similar approach has been proposed by Fattebert and Gygi [8–10], also in the context of the Car–Parrinello method. They introduce a dielectric permittivity which is a smooth function of the solute’s density, and solve by finite differences the Poisson equation. The results is the electrostatic potential produced by the polarized medium which interacts with the solute’s electronic density. Finally, Rega recently reported [18] the combination of the Atom-centered Density Matrix Propagation (ADMP) [19] technique with CPCM. All the methods mentioned share two common drawbacks. First, the time dependency of the medium polarization is lost in the sense that it is assumed to evolve much faster than the geometry of the solute. No phenomena involving nonequilibrium solvation can be described in this way. A partial solution to this problem would be the use of mixed implicit–explicit solvent models as proposed be Brancato et al. [20, 21]. The second drawback is the high computational cost involved in solving the electrostatic problem for each nuclear configuration. In particular in the case of solutes described at a classical level, this added cost is exceedingly large with respect to the cost of running the simulation in vacuo and probably also larger than the use of a box of explicit solvent molecules. As previously mentioned, an alternative strategy can be used to couple MD methods and implicit solvent models. The Lagrangian describing the solute can be extended to include the medium polarization as a dynamical variable. Such an approach has the advantage of providing a proper description of the time evolution of the solvent polarization coupled to the evolution of the solute geometry. Also, it is potentially characterized by a lower computational cost since the full electrostatic problem is not solved at each nuclear geometry, but rather the medium polarization is propagated in time and allowed to oscillate around the solution of Poisson’s equation. The main difficulty arising from this scheme is the need for a potential energy functional which is valid, i.e. corresponds to the free energy of the interacting solute–solvent system, for an arbitrary medium polarization, and not only for the polarization that solves the Poisson equation. This functional also needs to be variational with respect to both the geometrical and the polarization degrees of freedom so that, when minimized, the free energy of the system at equilibrium polarization is recovered. Other issues are the potentially strong coupling between the geometrical and polarization variables and the need to assign a fictitious mass to the polarization degrees of freedom. In the following sections we will review the possible choices of free energy functionals for both dielectric and conductor boundary conditions, focusing on their applicability in the context of ASC implicit solvent models. Then in Section 1.4.5 we will present our 66 Continuum Solvation Models in Chemical Physics formulation of a smooth extended Lagrangian for the PCM family of solvation models. Finally, in Sections 1.4.6 and 1.4.7 we report numerical examples and prototypical applications of the PCM extended Lagrangian. Before turning our attention to the free energy functionals, we recall a few fundamental concepts that will be used throughout in the following. We start from the general expression for the electrostatic energy of a charge density 0 in a nonlinear dielectric medium [22]: W= 1 3 D dr E · D 4 0 (1.69) where E is the electrostatic field and D is the electric displacement, defined by: D = E + 4P (1.70) and P is the electric dipole polarization of the medium. In the case of a linear response: D 0 1 E · D = E · D 2 (1.71) so that the electrostatic energy is simply: W= 1 0 d3 r 2 (1.72) where is the total electrostatic potential, E = −, and D = E, where we also assumed the dielectric to be isotropic. When the dielectric is fully polarized, the Poisson equation holds: · = −40 (1.73) 1.4.2 Ad Hoc Functionals In this section we describe some examples of functionals proposed to compute the electrostatic potential , which is used in Equation (1.72) to solve for the electrostatic interaction energy between the charge density 0 and the dielectric medium. This class contains functionals which are not energy functionals, in the sense that their minimization does not lead to the electrostatic free energy, Equation (1.72). However, at the end of the variational process they provide an electrostatic potential (or a polarization) which satisfies Equation (1.73) and thus it can be used to compute the electrostatic energy. Although these functionals can be robust from the numerical point of view, they do not correspond to an energy and this prevents their direct use in MD simulations, as part of an extended Lagrangian, since it would not yield the correct forces. By using the electrostatic potential as the variational parameter York and Karplus [23] proposed two general functionals. The first one can be expressed in the form: W " 0 = 0 d3 r − 1 · · d3 r 8 (1.74) Modern Theories of Continuum Models 67 where 0 and are considered functional parameters. When the first derivative of this functional with respect to is nil, the Poisson differential Equation (1.73) is satisfied. However, for > 0 this functional happens to be concave with respect to the potential, so it is a maximum at the stationary point, since it can be demonstrated that the second derivative is negative. This fact makes the above functional not easy to handle, since normal minimization algorithms cannot be used. In the same paper [23] the authors proposed another functional, namely: e2 " 0 = !2 1 E − −1 E0 d3 r 2 (1.75) in which the unconstrained parameter is still the electrostatic potential. This functional is analogous to the function that is minimized in least-square fitting procedures. The stationary point of this functional is equivalent to that of Equation (1.74) but in this case the functional is convex with respect to , thus the functional in Equation (1.75) must be minimized. The functionals in Equations (1.74) and (1.75) can also be expressed in terms of the variations in the polarization potential pol = − 0 , see ref. [23]. If the solute charge density 0 is completely contained inside a cavity surrounded by the dielectric medium, which mimics the solvent, both the functionals can be variationally optimized constraining the variation of the polarization density to be on the cavity surface. Another variational approach is proposed by Allen et al. [24]. In that work the authors deal with the problem of the ion channels through membranes, in which the roles of the solvent and the solute are interchanged. However, the functional they proposed can be used in general solvation problems. The form of this functional is: W = 1 1 · d3 r − 40 + · d3 r 2 2 (1.76) where = − 1 is the dielectric susceptibility. The authors demonstrated that the minimum of the functional in Equation (1.76) corresponds to the solution of the Poisson equation, Equation (1.73). However the value of the functional in the minimum correspond to minus the electrostatic energy. The functional (1.76) still depends on the total electrostatic potential, but it can be turned into a functional of the polarization charge density, see ref. [24]. When a well defined separation between the dielectric medium and the charge density 0 is assumed, so that the dielectric susceptibility undergoes a step discontinuity on the surface boundary with the dielectric, the induced polarization charge reduces to a surface charge, and the integrals involving this quantity can be reduced to surface integrals [24]. Even if the functionals presented in this section cannot be directly used in the context of ASC implicit solvent models to define an extended Lagrangian for MD simulations, the electrostatic potential obtained at the stationary point can then be used to deduce the electrostatic forces acting on the nuclei. This description of the electrostatic interaction between solute and solvent corresponds to a situation in which the dielectric polarization instantaneously follows the change in the solute charge distribution. This means that at each step of the simulation solute and solvent are in equilibrium. 68 Continuum Solvation Models in Chemical Physics 1.4.3 Free Energy Functionals The theory of electronic polarization in dielectric media [25] provides the framework for the derivation of a free energy functional that meets the requirements set forth in the Introduction. In particular, the additional free energy of the system due to a polarization P(r) can be expressed as [26]: WP = 1 Pr · −1 r · Pr dr 2 1 · Pr · Pr + dr dr 2 r − r − · Pr0 r dr (1.77) where r is the dielectric susceptibility of the medium and 0 r is the potential produced by the charge density 0 r. The above functional is valid for an arbitrary value of the polarization field and has a stationary point at W = 0 Pr = r · Er Pr (1.78) where the electric field E(r) is given by Er = −0 r + · Pr dr r − r (1.79) and D = E + 4P satisfies Poisson’s equation. This stationary point is indeed a minimum as r is a positive definite tensor everywhere. Unfortunately, the functional in Equation (1.77) is not easily applicable in the context of ASC implicit solvation models as the polarization is represented by a vector field. Marchi and co-workers [27,28] have applied Equation (1.79) in the context of classical MD by using a Fourier pseudo-spectral approximation of the polarization vector field. This approach provides a convenient way to evaluate the required integrals over all volume at the price of introducing in the extended Lagrangian a set of polarization field variables all with the same fictitious mass. They also recognized the crucial requirement that both the atomic charge distribution and the position-dependent dielectric constant be continuous functions of the atomic positions and they devised suitable expressions for both. A functional even more general than that in Equation (1.77) was given by Marcus [29] in order to describe a system where only a portion of the polarization is in equilibrium. However, also in this case, the functional is in terms of three-dimensional polarization fields and thus it cannot be readily introduced in an ASC implicit solvation model. Recently, Attard [30] proposed a different approach which provides a variational formulation of the electrostatic potential in dielectric continua. His formulation of the free energy functional starts from Equation (1.77), which he justifies using a maximum entropy argument. He defines a fictitious surface charge, s, located on the cavity boundary. The charge s, which produces an electric field f , contributes together with the solute Modern Theories of Continuum Models 69 density charge to polarize the dielectric, producing an apparent surface charge . When the mutual polarization between the solute, the fictitious charge and the dielectric reaches an equilibrium, the fictitious and the induced surface charges are expected to coincide s = . Defining the electrostatic potential produced by the surface charge , a free energy functional can be written in the form: Ws = 1 1$ drrr + drr r − fr 2 2 (1.80) where the expression for the potentials are: $ r sr + dr r − r r − r $ r r r = dr + dr r − r r − r fr = dr (1.81) (1.82) To the best of our knowledge, this is the only free energy functional that can be readily introduced in an ASC implicit solvent model as it involves only surface integrals in terms of the independent polarization variable which is no longer a three-dimensional field, but instead assumes the form of a surface charge distribution on the dielectric boundary. 1.4.4 Free Energy Functional for the Conductor-like Model In the case of the conductor-like model a free energy functional that meets the requirements set forth in the Introduction is readily available. Indeed, in the limit of a conductor, the potential must vanish inside and at the boundary of the medium. Thus the functional can be written in the form: W = − $ dr dr $ rr 1 $ rr dr dr + 2 r − r r − r (1.83) The minimization of this functional satisfies the condition at the boundary: $ r W r = − dr + dr =0 r − r r − r (1.84) This functional is also physically motivated as it expresses the balance of two terms: a favorable (negative) solute–solvent interaction energy and an unfavorable (positive) solvent–solvent interaction. At equilibrium the second term is equal to half of the first as expected also from basic electrostatic arguments. Despite the simple form of Equation (1.83), the detailed formulation of an extended Lagrangian for CPCM is not a straightforward matter and its implementation remains challenging from the technical point of view. Nevertheless, is has been attempted with some success by Senn and co-workers [31] for the COSMO–ASC model in the framework of the Car–Parrinello ab initio MD method. They were able to ensure the continuity of the cavity discretization with respect to the atomic positions, but they stopped short of providing a truly continuous description of the polarization surface charge as suggested, 70 Continuum Solvation Models in Chemical Physics for example, by York and Karplus [23]. This led to the need for different time steps for atomic degrees of freedom and polarization charges, and to the use of micropropagation steps for the latter. 1.4.5 A Smooth Lagrangian Formulation of the PCM Free Energy Functional The strategy to obtain a Lagrangian formulation of PCM is to consider the PCM apparent charges as a set of dynamic variables, exactly as the solute nuclear coordinates. The algorithm proposed in the present chapter is applied within the MM framework, since it allows a simplified notation and faster calculations. However, we point out that it can be straightforwardly extended to QM calculations. It has to be noted that only the values and not the positions of the PCM charges now become independent on the nuclear coordinates. In fact we still keep the PCM cavity as a series of interlocking spheres centered on the nuclei. Thus when a nucleus moves the sphere centered on it also moves and the surface elements located on that sphere move as well. Here we also point out that, when two intersecting spheres move following the motions of the nuclei which they are centered on, the number of surface elements exposed to the solvent changes, and only the apparent charges exposed to the solvent contribute to the free energy. With the term ‘exposed’ we mean the apparent charges which are not inside the volume of the cavity (i.e. which are in a region of the surface of a sphere which is covered by an adjacent intersecting sphere). We stress that, though the term ‘exposed’ is not rigorous, as the charges are not exposed to the solvent but they are the solvent, we continue to use that term to distinguish the surface elements which contribute to the free energy from those that do not contribute. CPCM Functional As outlined in Section IV, in the conductor-like version of PCM we have a simple expression of the energy functional, Equation (1.15). It can be discretized as: Wr q = −qV + qSq 2 − 1 (1.85) where the matrix S represents the electrostatic potential induced by the apparent charges on the surface cavity [3]. The last term on the right hand side represents the polarization of the dielectric medium. In this form the value of the variables q does not explicitly depend on r, while this dependence is present for the electrostatic potential V and for the PCM matrix S, though we omit it in the equation. When the free energy is minimized (at least with respect to the PCM apparent charges variables) then q satisfy the PCM system of equations and the second term in the above equation becomes exactly one half of the qV term. As said above, minimizing the functional in Equation (1.85) with respect to the charges q is equivalent to solve the CPCM system of equations: Sq = V −1 (1.86) However the present strategy also implies new technical difficulties. The first obstacle is represented by the diagonal elements of the matrix S Sii = fi /ai , as they contain Modern Theories of Continuum Models 71 the area of the surface element i ai , as the denominator. If the surface element i is in the region of the intersection between two spheres, the gradient of the energy functional with respect to the charge qi can become very large when ai becomes small, leading to numerical instabilities of the optimization algorithm. A more important source of instability is that, as the solute geometry changes during a geometry optimization or an MD trajectory, some charges becomes buried while some others become exposed to the solvent, following the motion of the spheres where they are located. This fact leads to a discontinuity in the energy derivatives (with respect to both the nuclear and the charges degrees of freedom), as the number of dynamic variables changes. Furthermore this appearance and disappearance of the PCM charges can represent a more severe source of instability in a MD simulation, because no forces act on the charges inside the cavity (because there are not terms of the gradients which involve these charges). This fact means that, when a charge is buried and after a time interval t it is again exposed to the solvent, its value could be arbitrarily large, leading to a nonconservative behavior of the energy. To overcome both these problems we introduce a new set of variables q̄, which have this relation with the PCM charges: qi = q̄i a1/2 i (1.87) where ai is the area of the ith surface element. Thus the value of the charge qi is nonzero when the area ai is not zero, i.e. when the ith surface element is exposed to the solvent. The opposite relation q̄i = qi a−1/2 is valid only if ai is nonzero. This charges q̄ are a sort i of area-weighted apparent surface charges and their definition is in a way reminiscent of that of the mass-weighted nuclear coordinates. During the optimization the value of the q̄ can be nonzero even if the corresponding surface element is inside the cavity: we call these shadow q̄ q̄sh . Thus we introduce an energy term involving the shadow q̄ in the energy functionals and this term has to vanish when the functional is minimized. Still using the CPCM formalism, the functional is now given by: Wr q̄ = −A1/2 q̄V + q̄A1/2 SA1/2 q̄ + f q̄2 2 − 1 2 − 1 sh (1.88) where the last term is a sort of self-interaction of the shadow charges involving the diagonal term of the matrix S Sii = fi /ai . We note that for these terms the dependence on the area of the surface elements ai disappears when we pass from q to q̄. Moreover as the fi elements are positive the last term is positive and the only way to minimize it is to set to zero all the shadow q̄. The form of this self-interaction term for CPCM seems very plausible if we consider an extended CPCM system of equations, analogous to that in Equation (1.86), collecting both the exposed and the shadow charges q̄. Starting from the CPCM equation (switching from q to q̄): A1/2 SA1/2 q̄ = A1/2 V −1 72 Continuum Solvation Models in Chemical Physics which can be also written in an extended form: ext S̄ q̄ = V̄ext −1 (1.89) where: ⎡ / ⎢ ⎢/ ⎢ ⎢ ext S̄ = ⎢/ ⎢ ⎢ ⎣ / / S̄ / / / ⎤ 0 f 0 f 0 ⎡ ⎤ / ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ext ⎢V̄⎥ ⎥ V̄ = ⎢ / ⎥ ⎢ ⎥ 0⎥ ⎥ ⎣ ⎦ ⎦ 0 f (1.90) The nonzero block in the upper left corner of the matrix S̄ext interacts with the charges exposed to the solvent. When the minimization of the functional (1.88) is complete the vector of the q̄ will look: ⎡ ⎤ / ⎢q̄⎥ ext ⎥ q̄ = ⎢ ⎣/⎦ 0 (1.91) The last technical but essential note is that, as the description of the charges in the regions of the intersection of the spheres represents a critical numerical issue, we found that the use of the Karplus smoothing scheme [23], recently extended to the various PCM versions by Scalmani and Frisch [32], is crucial to allow for a smooth behavior of the electrostatic potential and thus of the free energy functional in those regions. We point out that, though the expression of the matrix S is a bit more complicated in the PCM formulation which uses the Karplus weights [23], the expressions presented in the present contribution are still valid for the purpose of illustrating our approach. DPCM Functional The case of the dielectric version of PCM (DPCM) is more complicated than CPCM, as the system of equations which must be solved to compute the apparent surface charges is [3]: 2 + 1 −1 A − D∗ q = −E⊥ −1 (1.92) and so a different approach must be used. Following the work of Attard [30], we discretize the integrals involved in the functional Ws , Equation (1.80), which assumes the following form: 1 1 Wq r = V† q̃q + q̃q − q† Sq̃q 2 2 (1.93) Modern Theories of Continuum Models 73 where q and q̃ are the charges corresponding to the surface charge s and respectively. We omitted the dependence on the nuclear coordinates r, but we emphasized the dependence of q̃ on q. This explicit dependence is: q̃ = − −1 −1 A E⊥ − D∗ q − q 4 2 (1.94) When the mutual polarization reaches equilibrium then q satisfies the DPCM equations and q̃ = q, as can be demonstrated looking at Equation (1.94). The value of the functional (1.93) is then: 1 Wq r = V† q̃q 2 (1.95) which is the expression of the equilibrium free energy of the system. To verify that the constraints, i.e. the DPCM equations, on the functional are satisfied, the derivative of the functional with respect to the charges must be equal to zero. By using the relation between the solute potential and the normal component of the solute electrostatic field [3]: 2I − DA V = SAE⊥ (1.96) which holds only in the case that all the solute charge density is contained inside the cavity, and assuming that: DAS = SAD∗ (1.97) which holds exactly for the integral operators but is still very accurate when their matrix representation is used, the final expression for the functional partial derivative is: W = q −1 4 2 2I − DA SA + 1 −1 2 A − D∗ q + E ⊥ −1 (1.98) This equation can be equal to zero only if the term in the last parentheses is equal to zero, which is equivalent to satisfying the DPCM equations. We note that the term in the second parentheses cannot be equal to zero since Equation (1.96) holds. At the end of the optimization, when both the derivatives of the functional with respect to the nuclear coordinates and the PCM charges are nil, the electrostatic equations for the dielectric are satisfied and the equilibrium DPCM charges are obtained. Now we reintroduce the q̄, which are necessary to take into account the contribution to the functional from the charges buried inside the cavity. The relation between the barred charges and the normal ones is: q = q̄A1/2 ¯ q̃ = q̃A 1/2 (1.99) (1.100) 74 Continuum Solvation Models in Chemical Physics The relation between the q̃¯ and q̄ then becomes: ! −1 − 1 1/2 q̃¯ = − E⊥ − D∗ q̄A1/2 − A q̄ 4 2 (1.101) When the surface element i moves into the cavity, the corresponding area ai = 0, so Equation (1.101) becomes: −1 2 − gi q̄i q̃¯ i = 4 (1.102) since the diagonal element of the matrix D∗ is defined as Dii∗ = gi /ai . The contribution of the shadow charges to the energy functional is: ! 1 1 ¯ ¯ fi q̃i q̃i − q̄i = 2 i 2 −1 4 2 fi 2 − gi 2 q̄i2 + 1 −1 f 2 − gi q̄i2 2 4 i (1.103) which is a positive term, since 2 − gi is positive, as can be demonstrated looking at ref. [33]. Also, as the fi elements are positive, the only way to minimize this term is to set q̄i = 0 and thus, at the end of the optimization the charges inside the cavity do not contribute to the energy as expected. 1.4.6 Prototypical Application: Simultaneous Optimization of Geometry and Reaction Field A first important application of this new strategy is constituted by the geometry optimizations. In fact, the internal energy of the system in the MD methods coincides with the energy functional which has to be minimized in the geometry optimizations, and the same derivatives of the energy with respect to the nuclear coordinates are involved. We stress that our interest focuses on the technical issues and not on the specific characteristics of the systems we use as test molecules. The calculation were performed with a development version of Gaussian [34]. We choose three test molecules: formaldehyde, proline and 2-phenylphenoxide. The structure of these systems is shown in Figure 1.8. The calculations were performed in vacuo and in water solution, with the C and the D versions of PCM with the standard and the simultaneous approaches. Here we note that we used the same solute-shaped cavity for all the optimizations of each system. The force field we used for all the calculations, both in vacuo and in solution, is the UFF [35] and the nuclear charges at the initial point were estimated with the QEq [36] algorithm. As we are not interested in obtaining results comparable with experimental data or with other calculations, but only in the PCM results with the different optimization schemes, the choice of the force field is not a critical point. The only requirement is that we performed all the calculations with the same force field. In Table 1.2 the energy for the three molecules in vacuo and in solution are reported. The data show that the approach of simultaneously optimizing the geometry and the polarization succeeds in providing the same minimum geometry found with the standard approach. Modern Theories of Continuum Models 75 O C OH O– O C HN (a) (b) (c) Figure 1.8 Structure of (a) formaldehyde, (b) proline and (c) 2-phenylphenoxide. Table 1.2 Energy kcal mol −1 for the three molecules in Figure 1.8 in vacuo and in solution are reported. CPCM and DPCM indicate the calculations performed with the standard version of the models, sCPCM and sDPCM indicate the calculations where the geometry and the polarization are optimized simultaneously H2 CO vacuum CPCM sCPCM DPCM sDPCM 000 −486 −486 −484 −484 Proline 10179 9324 9323 9327 9327 Phenoxide 5131 472 472 570 567 Thus now we discuss the features of the CPCM and DPCM free energy functionals presented in Sections 1.4.5 and 1.4.6 in terms of their computational cost with respect to the standard approaches. We outline that this comparison is qualitative since it is based only on some of the parameters that influence the final computational time and we are also limiting our discussion on the small molecules presented in this section. The bottleneck of a calculation in solution is the evaluation of the polarization which, in the case of PCM, corresponds to the evaluation of the apparent surface charges. In particular, the bottleneck is represented by the evaluation of the products between the integral matrices of the electrostatic potential (matrix S in Equation (1.8.6)) or of the normal component of the electric field (matrix D∗ in Equation (1.92)) and the apparent charges vector q. Thus the criterion we use to compare the standard and the simultaneous approach is based on the number of matrix products (Sq or D∗ q) necessary in the whole optimization process. We also remind the reader that the dimension of the matrices is equal to the square of the number of the surface elements. The advantage of the new strategy is that, for each step of the optimization, a small and constant number of matrix–vector products are necessary (three for CPCM and nine for DPCM). In contrast, for the standard approach the evaluation of the apparent charges 76 Continuum Solvation Models in Chemical Physics requires many more matrix–vector products to be solved for the charges using an iterative approach [37], plus some others for the evaluation of the gradients. We must point out that, if the PCM matrices are small enough to be kept in memory during the iterative solution of the PCM equations, the computational time needed to compute the apparent charges greatly reduces. However, this is not likely to be possible for large molecules. We used the conjugate gradient algorithm for the geometry optimization, since it is cheap, so it is a good choice for MM calculations. However, this choice may not be the best one when the simultaneous optimization is performed, since this algorithm does not take into account the coupling between the two different sets of variables (the nuclear coordinate and the solvent charges), because the Hessian (or at least an estimation of the Hessian) is not computed. With all those assumptions and limitations in mind we can analyze the number of matrix–vector products necessary to perform the geometry optimization for the three model molecules, reported in Table 1.3. Table 1.3 Estimation of the number of matrix–vector products necessary to optimize the systems in Figure 1.8. The number in parentheses represents the steps necessary to reach the minimum geometry. The first energy is computed by solving the PCM equations for all the schemes. CPCM and DPCM indicate the calculations performed with the standard version of the models, sCPCM and sDPCM indicate the calculations where the geometry and the polarization are optimized simultaneously CPCM sCPCM DPCM sDPCM H2 CO Proline Phenoxide ∼ 180 7 ∼ 60 13 ∼ 180 7 ∼ 400 44 ∼ 1280 31 ∼ 910 290 ∼ 10550 319 ∼ 25250 2805 ∼ 3700 92 ∼ 950 303 ∼ 11100 336 ∼ 30000 3334 Let us start the analysis from CPCM. From Table 1.3 it is evident that the energy functional performs better than the standard scheme, even if a very simple optimization algorithm is used and even if the two sets of variables are treated on the same footings. So even if the number of steps necessary to reach the minimum geometry is larger for the simultaneous scheme than for the usual one, as expected since in the first case the variables are many more, the total number of matrix–vector products is lower. Thus one can expect that, with a better choice of the optimization algorithm, the number of steps should greatly decrease for the simultaneous approach, especially in areas of the energy surface close to the minimum, and this approach should become more convenient than the usual one even for smaller molecules. The situation is the opposite when we consider the DPCM results. Indeed in this case, even if the ratio between the number of steps for the simultaneous and the standard scheme is comparable to the ratio in the CPCM case (for the two larger molecules) the sDPCM scheme requires a larger computational effort than the DPCM one. This is due to the more complicated expression for the DPCM free energy functional, Equation (1.93), than for the CPCM one, Equation (1.88). The functional (1.93) appears difficult to deal with from a numerical point of view. Numerical instabilities are probably arising from a strong coupling between the two different sets of variables which must be better investigated. Moreover, the potential energy surface in the DPCM case looks more complicated than in Modern Theories of Continuum Models 77 the CPCM case, as can be seen by comparing the number of steps necessary to reach the convergence, even when the standard scheme is used for both methods. Numerical issues are particularly severe close to the energy minimum, and the optimization algorithm oscillates for many steps around the minimum before reaching it. This behavior prevents the use of the DPCM free energy functional with molecules larger than those proposed in this section. Furthermore the coupling between the two sets of variables, on the other hand, makes the separation of the nuclear normal modes from the charges oscillations difficult; thus in the next section only dynamics simulation performed with the CPCM functional are presented. The data shown in this section demonstrate that the simultaneous optimization of the solute geometry and the solvent polarization is possible and it provides the same results as the normal approach. In the case of CPCM it already performs better than the normal scheme, even with a simple optimization algorithm, and it will probably be the best choice when large molecules are studied (when the PCM matrices cannot be kept in memory). This functional can thus be directly used to perform MD simulations in solution without considering explicit solvent molecules but still taking into account the dynamics of the solvent. On the other hand, the DPCM functional presents numerical difficulties that must be studied and overcome in order to allow its use for dynamic simulations in solution. 1.4.7 Prototypical Application: Time Propagation of Geometry and Reaction Field In this section we compare the behavior of the CPCM extended Lagrangian classical dynamics with a dynamics in which the charges are equilibrated, i.e. the PCM system of equations is solved at each time step. The main point which differentiates the two dynamics is that, when an extended Lagrangian is introduced, the solvent apparent charges, or better the area-weighted apparent charges q̄, have their own time evolution. A kinetic energy term appears, which takes into account the velocity of the changes in the space of the charges values, and a fictitious mass must also be defined. This mass can be tuned to obtain different responses of the solvent to the changes in the solute geometry. In the simulations we present in this section we assigned the same mass to all the charges independently of where they are located on the cavity surface. We chose this mass in such a way that the charges are light enough to rapidly follow the motion of the solute. In this way we managed to run an equilibrium dynamics by using the same time step used for the dynamics in which the PCM equations are solved at each step. The latter can be seen as a dynamics in which the charges are infinitely light, so they instantaneously equilibrate with the solute charge distribution at each time step. The advantage of the new approach is that the number of matrix–vector products is greatly reduced, as also shown in the previous section, so it is possible to run much longer trajectories. We studied two of the test molecules used in the previous section (formaldehyde and phenoxide) in water. As far as the formaldehyde dynamics is concerned we will analyze the energy conservation as well as the oscillations of the potential energy. As for the phenoxide we will examine the solvent shift in the normal mode frequencies. The formaldehyde dynamics ran for 25 ps, with a time step of 0.1 fs. Figure 1.9 reports the results obtained with the charges equilibrated at each step and with the extended 78 Continuum Solvation Models in Chemical Physics (a) (b) Figure 1.9 Total and potential energy (au) of formaldehyde in water, (a) with the PCM charges equilibrated at each time step and (b) with the PCM extended Lagrangian formulation. Lagrangian, respectively. At first we note that the total energy is conserved in both the dynamics, with oscillations orders of magnitude smaller than the oscillations of the potential energy. The latter presents on the other hand a behavior that is quite different in the two cases. For the case in which the charges are equilibrated at each step, the oscillations are quite large, of the order of 35 × 10−3 au, and they last for the whole trajectory. On the other hand, for the extended Lagrangian approach, after an initial period Modern Theories of Continuum Models 79 of equilibration, the potential energy oscillations are smaller. The initial equilibration is due to the fact that we started the dynamics with a nil velocity of the charges. The smaller oscillations of the energy are probably due to the mass of the charges, which drags the motion of the nuclei. However, this mass is small enough to prevent an overlap of the nuclear vibrational frequencies with the solvent charges ones. This example shows that also in the case of MD simulations, the extended Lagrangian approach is promising, in the sense that it provides a more stable expression for the potential energy, allowing a better energy conservation. It is also less computationally demanding, because the charges are propagated with the solute nuclear coordinates, thus no linear system must be solved at each point. We stress that, contrary to the formulation proposed in ref. [31], in our formulation the solvent charges are propagated with the same time step of the nuclei and no micropropagations are necessary. In Figure 1.10 the low frequency region of the spectrum of phenoxide is presented. It is obtained by the Fourier transform of the velocity–velocity autocorrelation function, after a trajectory of 20 ps in vacuo and 4 ps in solution with the two approaches. The time step is 0.1 fs. We consider the first four vibrational frequencies, which present the largest solvent shift. The harmonic values of these frequencies, computed analytically in vacuo and in solution at the equilibrium geometries, are reported in Table 1.4. The first and the fourth frequencies, which are those with the larger shifts, correspond to the torsion of the dihedral angle between the two rings and to the motion out of plane of the oxygen, respectively. Figure 1.10 Vibrational spectra (frequencies in cm−1 ) of the phenoxide molecule in vacuo and in solution obtained by the MD simulation. The intensities were scaled in order to fit on the same scale. The results in Figure 1.3, even if the picks are not completely resolved because the dynamics were probably too short, show that the two approaches in solution match. 80 Continuum Solvation Models in Chemical Physics Table 1.4 Analytical first four harmonic vibrational frequencies cm−1 of the phenoxide molecule in vacuo and in solution vacuum water 1 2 3 862 767 1031 987 1499 1430 4 2450 2349 Moreover the shifts in the frequencies passing from the gas phase to the solution are qualitatively correct (we did not consider any anharmonicities in the analytical calculations). Thus also in the case of a larger test molecule, the extended Lagrangian formulation of CPCM is successful in describing the solvation effect. 1.4.8 Conclusion and Perspectives The aim of this contribution was to review the efforts that have been made so far in the formulation of a Lagrangian for the implicit solvation model. The goal is to provide a simple and computationally efficient way to describe the very complex phenomenon of solvation, which involve a large number of molecules, by using a strongly reduced set of degrees of freedom. Among the approaches presented in this contribution, those that seem more appealing are based on free energy functionals, since they can be directly used in molecular dynamics simulation. We used this approach to define the functional for CPCM and DPCM in Section 1.4.5. As for the former, its simple expression makes it feasible to be used with medium sized molecules for simultaneous optimization of geometry and polarization and also to perform MD simulations. The latter, on the other hand, presents numerical difficulties that must be overcome to make it generally useful. Although much work must yet be done to understand the features and the limitations of these functionals, their range of applicability and their accuracy, we consider the results presented in this contribution as encouraging. Acknowledgments The authors would like to thank Prof. Berny Schlegel for his contribution in the discussion that led to the idea of the area-weighted apparent surface charges. Also we would like to thank Prof. Benedetta Mennucci for her continuing interest and her encouragement. References [1] [2] [3] [4] [5] [6] [7] [8] [9] J. Tomasi and M. Persico, Chem. Rev., 94 (1994) 2027. C. J. Cramer and D. G. Truhlar, Chem. Rev., 99 (1999) 2161. J. Tomasi, B. 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Pomelli, G. E. Scuseria and M. J. Frisch, Theor. Chem. Acc., 90 (2004) 111. 1.5 The Quantum Mechanical Formulation of Continuum Models Roberto Cammi 1.5.1 Introduction The quantum mechanical (QM) (time-independent) problem for the continuum solvation methods refers to the solution of the Schrödinger equation for the effective Hamiltonian of a molecular solute embedded in the solvent reaction field [1–5]. In this section we review the most relevant aspects of such a QM effective problem, comment on the differences with respect to the parallel problem for isolated molecules, and describe the extensions of the QM solvation models to the methods of modern quantum chemistry. Such extensions constitute a field of activity of increasing relevance in many of the quantum chemistry programs [6]. In our discussion the usual Born–Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. The structure of the contribution is as follows. In Section 1.5.2 we discuss the structure of effective nonlinear Hamiltonians for the solute. In Section 1.5.3 we present a two-step formulation of the QM problem, with the corresponding Hartree–Fock (HF) equation. In Section 1.5.4 we introduce the fundamental energetic quantity for the QM solvation models while in Section 1.5.5 extensions beyond the HF approximation are presented and discussed. 1.5.2 The Structure of the Effective Hamiltonian %eff , for the solute has already been introduced in The effective electronic Hamiltonian, H the contribution by Tomasi. It describes the solute under the effect of the interactions with its environment and determines how these interactions affect the solute electronic wavefunction and properties. The corresponding effective Schrödinger equation reads Ĥeff = E (1.104) Modern Theories of Continuum Models 83 Ĥeff is composed by two terms, the Hamiltonian of the solute HM0 (i.e., the molecular part M of the continuum model) and the solute–solvent interaction term V̂int : Ĥeff = ĤM0 + V̂int (1.105) The structure of V̂int depends, in general, on the nature of the solute–solvent interaction considered by the solvation model. As already noted in the contribution by Tomasi, a good solvation model must describe in a balanced way all the four fundamental components of the solute–solvent interaction: electrostatic, dispersion, repulsion, charge transfer. However, we limit our exposition to the electrostatic components, this being components of central relevance, also for historical reason, for the development of QM continuum models. This is not a severe limitation. As a matter of fact, the QM problem associated with the solute–solvent electrostatic component defines a general framework in which all the other solute–solvent interaction components may be easily collocated, without altering the nature of the QM problem [5]. The operatorial form of V̂int depends on the method employed to solve the electrostatic problem which has to be nested into the QM Equation (1.104) to determine the reaction potential produced by the polarized solvent on the solute. Here we shall consider the more general case of V̂int corresponding to the ASC version of the continuum solvation models (see the contribution by Cancès). The operator V̂int can be divided into four terms having a similarity to the two-, one-, and zero-electron terms present in the Hamiltonian of the solute. To show it we consider the solute–solvent interaction energy Uint given as the integral of the reaction potential times the whole charge distribution M , conveniently divided into electronic and nuclear components M r = eM r + nM r. The reaction potential has, as sources, the two components of M and thus it is composed of two terms, one stemming from the electronic distribution of the solute M and one from the corresponding nuclear distribution. As a result, Uint is partitioned into four terms: Uint = U ee +U en +U ne +U nn (1.106) where U xy corresponds to the interaction energy between the component of the interaction x potential having as source xM r, namely Vint , and the charge distribution yM r. Following this formalism, three different QM operators appear, namely V̂ nn V̂ ne (it may be shown that U ne and U en are formally identical), and V̂ ee . These have a correspondence, respectively, to zero-, one-, and two-electron terms of HM0 . We note that the zero-order term gives rise to an energetic contribution U nn which is analogous to the nuclei–nuclei repulsion energy Vnn and thus it is generally added as a constant energy shift term in HM0 . The conclusion of this analysis is that we may define four operators (reduced in practice to two, plus a constant term) which constitute the operator V̂int of Equation (1.105). To make the exposition of V̂int more explicit we present here the Schrödinger equation with the introduction of a new formalism: & ' Ĥeff > = ĤM0 + #̂er V̂rR + #̂er V̂rrR < #̂er > >= E > (1.107) 84 Continuum Solvation Models in Chemical Physics With the superscript R we indicate that the corresponding operator is related to the solvent reaction potential, and with the subscripts r and rr the one- or two-electron nature of the operator. The convention of summation over repeated indices followed by integration has been adopted. ˆ er is the electron density operator and ˆ er V̂rR is the operator which describes the two components of the interaction energy we have previously called U en and U ne . In more advanced formulations of continuum models going beyond the electrostatic description, other components are collected in this term. V̂rR is sometimes called the solvent permanent potential, to emphasize the fact that in performing an iterative calculation of > in the BO approximation this potential remains unchanged. The ˆ er V̂rrR < ˆ er > operator corresponds to the energy contribution that we previously called U ee . This operator changes during the iterative solution of the equation. V̂rrR is said to be the response function of the reaction potential. It is important to note that this term induces a nonlinear character to Equation (1.107). Once again, in passing from the basic electrostatic model to more advanced formulations other contributions are collected in this term. The constant energy terms corresponding to U nn and to nuclear repulsion are not reported in Equation (1.107). Summing up, the structure of the effective Hamiltonian of Equation (1.107) makes explicit the nonlinear nature of the QM problem, due to the solute–solvent interaction operator depending on the wavefunction, via the expectation value of the electronic density operator. The consequences of the nonlinearity of the QM problem may be essentially reduced to two aspects: (i) the necessity of an iterative solution of the Schrödinger Equation (1.107) and (ii) the necessity to introduce a new fundamental energetic quantity, not described by the effective molecular Hamiltonian. The contrast with the corresponding QM problem for an isolated molecule is evident. 1.5.3 A Two-Step Formulation of the QM Problem: Polarization Charges and the Hartree–Fock Equation As said before, the nonlinear nature of the effective Hamiltonian implies that the Effective Schrödinger Equation (1.107) must be solved by an iterative process. The procedure, which represents the essence of any QM continuum solvation method, terminates when a convergence between the interaction reaction field of the solvent and the charge distribution of the solute is reached. The most naive formulation of these processes, which corresponds to the mutual interaction between real and apparent charges, is that used in the first version of the Polarizable Continuum Model developed in Pisa, also denoted as DCPM [2]. We recall it here, as it is helpful in the understanding of the basic aspects of the mutual polarization process. One starts from a given approximation of #eM (let us call it #0M ) that could be a guess, or the correct description of #eM without the solvent, and obtains a provisional description of the apparent surface charge density, or better, of a set of apparent point charges that we denote here qkoo . These charges are not correct, even for a fixed unpolarized description of the solute charge density because their mutual interaction has not been considered in this zero-order description. To get this contribution, called mutual polarization of the apparent charges, an iterative cycle of the PCM equation (including the self-polarization of each qk ) must be performed at fixed #0M (see the contribution by Pomelli for more details). The result is a new set of charges qkof , where f stands for Modern Theories of Continuum Models 85 final. The qkof charges are used to define the first approximation to Vint , and a first QM cycle is performed to solve Equation (1.104). With the new #1M the inner loop of mutual ASC polarization is performed again giving rise to a qk1f set of charges. The procedure is continued until self-consistency. We remark that, in this formulation, we have collected into a single set of one-electron operators all the interaction operators we have defined in the preceding section, and, in parallel, we have put in the qk set both the apparent charges related to the electrons and nuclei of M. This is an apparent simplification as all the operators are indeed present. It is interesting here to note that this nesting of the electrostatic problem in the QM framework is performed in a similar way in all continuum QM solvation codes. Following a canonical order to get molecular wavefunctions, we introduce here the Hartree–Fock (HF) level of the two-step approach described above. In this framework we have to define the Fock operator for our model. We adopt here an expansion of this operator over a finite basis set and thus all the operators are given in terms of their matrices in such a basis. The Fock matrix reads: F = h0 + G0 P + hR + XR P (1.108) The first two terms correspond to ĤM0 , the third to #̂er V̂rR and, the last to #̂er V̂rrR < #̂er >. Assuming the reader’s familiarity with the standard HF procedure and formalism, we recall that all the square matrices of Equation (1.108) have the dimensions of the expansion basis set, and that P is the matrix formulation of the one-electron density function over the same basis set. According to the standard conventions P has been placed as a sort of argument to G0 to recall that each element of G0 depends on P. For analogy, we have made explicit a similar dependence for the elements of XR . We also remark that the standard HF equation is nonlinear in character and that in the development of this method its nonlinearity is properly treated. The new term XR P adds an additional non-linearity of different origin but of similar formal nature, that has to be treated in an appropriate way. This fact was not immediately recognized in the old versions of continuum QM methods, giving rise to debates about the correct use of the solute–solvent interaction energy. This point will be treated in the next section. It should be noted that, as in the previous analysis of the Schrödinger Equation (1.104), in the Fock matrix expression (1.108) we have used a single term to describe the oneelectron solvent term. We remark, however, that in the original formulation two matrices, jR and yR , were used, namely: R j$ = V$ sk q n sk (1.109) e V n sk q$ sk (1.110) k R = y$ k In both expressions the summation runs over all the tesserae (each tessera is a single site where apparent charges are located), V$ sk is the potential of the $∗ elementary charge distribution computed at the tessera’s representative point, V n sk is the potential given by the nuclear charges, computed again at the same point, q n sk is the apparent e charge at position sk deriving from the solute nuclear charge distribution, and q$ sk is 86 Continuum Solvation Models in Chemical Physics the apparent charge, at the same position, deriving from $∗ . The two matrices (1.109) and (1.110) are formally identical, as said before, and thus in Equation (1.108) we have replaced them with the single matrix: hR = ! 1 R j + yR 2 (1.111) We note that in computational practice, the more computationally effective expression (6) is generally used. The elements of the second solvent term in the Fock matrix (1.108) can be put in the following form: R = X$ V$ sk q e sk (1.112) k with q e sk = e P$ q$ sk (1.113) $ In this way, we have rewritten all the solvent interaction elements of the Fock matrix in terms of the unknown q e and q n apparent charges (the last, not being modified in the SCF cycle, can be separately computed at the beginning of the calculation). 1.5.4 The Basic Energetic Quantity: the Free Energy Functional The second, and more far reaching, implication of the nonlinearity of the QM problem in continuum models involves the fundamental energetic quantity for these models. To understand this point better it is convenient to compare the standard variational approach for an HF calculation on an isolated molecules with the HF approach for molecules in solution. For an isolated molecule the Fock operator: F0 = h0 + G0 P (1.114) is used ( to determine the variational approximation to the ground state exact wave function 0 corresponding to the system specified by Ĥ 0 . This is determined by minimizing HF the appropriate energy functional E, namely * ) 0 (1.115) = HF Ĥ 0 HF EHF or, in a matrix form: 1 0 = tr Ph0 + tr PG0 P + Vnn EHF 2 (1.116) where we have used same formalism used in the previous section and we have introduced the trace operator (tr). Obviously, the nuclear repulsion energy, Vnn , in the BO approximation is a constant factor. We note that in Equation (1.116) there is a factor 1/2 in Modern Theories of Continuum Models 87 front of the two-electron contribution. This factor is justified in textbooks by the need to avoid a double counting of the interactions, but this double counting has its origin in the nonlinearity of the HF equation. Let us now pass to continuum models. As for the isolated molecule, also here the ( S new Fock operator defined in Equation (1.108) and determining the new solution HF is obtained by minimizing an appropriate functional. However, now the kernel of this functional is not the Hamiltonian Ĥeff given in Equation (1.105) but rather Ĥeff − V̂int /2 and thus the energy of the system is given by , + 1 GSHF = HF Ĥeff − V̂int HF 2 (1.117) 0 , reads: which, expressed in a matrix form similar to that used for EHF 1 1 1 1 GSHF = tr Ph0 + tr PG0 P + tr Pj + y + tr PXP + Vnn + Unn 2 2 2 2 (1.118) where the solvent matrices, j, y and X are those defined in Equations (1.109), (1.110) and (1.112) (here we have only dropped the ‘R’ superscript). We have also added one half of the solute–solvent interaction term related exclusively to nuclei, which in the BO approximation is constant. Similar expressions and properties of the free energy functional (1.118) hold for all other levels of the QM molecular theory: the factor 21 is present in all cases of linear dielectric responses. More generally, the wavefunctions that make the free energy functional (1.117) stationary are also solutions of the effective Schrödinger Equation (1.107). The change of the basic energy functional arises from the nonlinear nature of the effective Hamiltonian. This Hamiltonian has in fact an explicit dependence on the charge e distribution S ( - S of the solute, expressed in terms of M , which is the one-body contraction of , and thus it is nonlinear. It must be added that this nonlinearity is of the first HF HF order, in the sense that the interaction operator depends only on the first power of eM . Some comments about nonlinearities in the Hamiltonian may be added here. The case we are considering here is called scalar nonlinearity (in the mathematical literature it is also called ‘nonlocal nonlinearity’) [7]: this means that the operators are of the form Pu = Au uBu where A, B are linear operators and < > is the inner product in a Hilbert space. The literature on scalar nonlinearities applied to chemical problems is quite scarce (we cite here a few papers [2, 8]) but the results justified by this approach are of universal use in solvation methods. The symbol G used for the energy functional emphasizes the fact that this energy has the status of a free energy. The explicit identification of the functional (1.117) with the free energy of the solute–solvent system was first done by Yomosa [2], on the basis of electrostatic arguments. In the Tomasi–Persico 1994 review [4a] alternative justifications for the factor 21 in the expression of the energy were given starting from perturbation theory, statistical thermodynamics, and classical electrostatics, all valid for a linear response of the dielectric. We report here only a consideration based on classical electrostatics. Half of the work required to insert a charge distribution (i.e. a molecule) into a cavity within a dielectric 88 Continuum Solvation Models in Chemical Physics corresponds to the polarization of the dielectric itself and it cannot be recovered by taking the molecule away and restoring it to its initial position. This one half of the work expended is irreversible, and it has to be subtracted from the energy of the insertion process to obtain the free energy (or the chemical potential). Let us now return to the HF level to illustrate some properties which follow from the variational formulation in terms of the free energy. 1.5.5 QM Descriptions Beyond the HF Approximation In the past few years, a great effort has been devoted to the extensions of solvation models to QM techniques of increasing accuracy. All these computational extensions have been based on a reformulation of the various QM theories describing electron correlation so as to include in a proper way the effects of the nonlinearity of the solvation model by assuming the free-energy functional as the basic energetic quantity. Most of these extensions have involved electron correlation methods based on variational approaches (DFT, MCSCF, CI,VB). These methods can be easily formulated by optimizing the free energy functional (1.117), expressed as a function of the appropriate variational parameters, as in the case of the HF approximation. In contrast, for nonvariational methods such as the Moller–Plesset theory or Coupled-Cluster, the parallel extension to solvation model is less straightforward. DFT Density Functional Theory does not require specific modifications, in relation to the solvation terms [9], with respect to the Hartree–Fock formalism presented in the previous section. DFT also absorbs all the properties of the HF approach concerning the analytical derivatives of the free energy functional (see also the contribution by Cossi and Rega), and as a matter of fact continuum solvation methods coupled to DFT are becoming the routine approach for studies of solvated systems. MCSCF Applications of continuum solvation approaches to MCSCF wavefunctions have required a more developed formulation with respect to the HF or DFT level. Even for an isolated molecule, the optimization of MCSFCF wavefunctions represents a difficult computational problem, owing to the marked nonlinearity of the MCSCF energy with respect to the orbital and configurational variational parameters. Only with the introduction of second-order optimization methods and of the variational parameters expressed in an exponential form, has the calculation of MCSCF wavefunction became routine. Thus, the requirements of the development of a second-order optimization method has been mandatory for any successful extension of the MCSCF approach to continuum solvation methods. In 1988 Mikkelsen et al. [10] pioneered the second-order MCSCF within a multipole continuum model approach in a spherical cavity. Aguilar et al. [11] proposed the first implementation of the MCSCF method for the DPCM solvation model in 1991, and their PCM–MCSCF method has been the basis of many extensions to more robust second-order MCSCF optimization algorithms [12]. It is worth recalling here that the building blocks of a second-order MCSCF optimization scheme, the electronic gradient and Hessian, are also the key elements in the development of MCSCF response methods (see the contribution by Ågren and Modern Theories of Continuum Models 89 Mikkelsen). Linear and nonlinear response functions have been implemented at the MCSCF level by Mikkelsen et al. [13], for a spherical cavity, and by Cammi et al. [14] and by Frediani et al. [15] for the PCM solvation models. CI The conceptual simplicity of the configuration interaction (CI) approaches has attracted the interest of researchers working in the field of solvation methods [2,16,17] to introduce electron correlation effects. However, despite this apparent simplicity, the application of the CI scheme to solvation models raises some delicate issues, not present for isolated molecules. The nonlinear nature of the Hamiltonian implies a nonlinear character of the CI equations which must be solved through an iteration procedure, usually based on the two-step procedure described above. At each step of the iteration, the solvent-induced component of the effective Hamiltonian is computed by exploiting the first-order density matrix (i.e. the expansion CI coefficients) of the preceding step. In addition, the dependence of the solvent reaction field on the solute wavefunction requires, for a correct application of this scheme, a separate calculation involving an iteration optimized on the specific state (ground or excited) of interest. This procedure has been adopted by several authors [17] (see also the contribution by Mennucci). A further issue arises in the CI solvation models, because CI wavefunction is not completely variational (the orbital variational parameter have a fixed value during the CI coefficient optimization). In contrast with completely variational methods (HF/MFSCF), the CI approach presents two nonequivalent ways of evaluating the value of a first-order observable, such as the electronic density of the nonlinear term of the effective Hamiltonian (Equation 1.107). The first approach (the so called unrelaxed density method) evaluates the electronic density as an expectation value using the CI wavefunction coefficients. In contrast, the second approach, the so-called ‘relaxed density’ method, evaluates the electronic density as a derivative of the free-energy functional [18]. As a consequence, there should be two nonequivalent approaches to the calculation of the solvent reaction field induced by the molecular solute. The ‘unrelaxed’ density approach is by far the simplest to implement and all the CI solvation models described above have been based on this method. The CI ‘relaxed’ density approach [18] should give a more accurate evaluation of the reaction field, but because of its more involved computational character it has been rarely applied in CI solvation models. The only notably exception is the CI methods proposed by Wiberg at al. in 1991 [19] within the framework of the Onsager reaction field model. In their approach, the electric dipole moment of the solute determining the solvent reaction field is not given by an expectation value but instead it is computed as a derivative of the solute energy with respect to a uniform electric field. VB Methods The powerful interpretative framework of the Valence Bond (VB) theory has been exploited in several couplings and extensions with continuum models. We mention here the most relevant in the present context. Amovilli et al. [20] presented a method to carry out VB analysis of complete active space-self consistent field wave functions in aqueous solution by using the DPCM approach [3]. A Generalized Valence Bond perfect pairing (GVB–PP) level 90 Continuum Solvation Models in Chemical Physics combined with a continuum description of the solvent using the DelPhi code [21] to obtain a numerical solution of the electrostatic problem as been developed by Honig et al. [22]. Bianco et al. [23] proposed a direct VB wavefunction method combined with a PCM approach to study chemical reactions in solution. Their approach is based on a CI expansion of the wavefunction in terms of VB resonance structures, treated as diabatic electronic states. Each diabatic component is assumed to be unchanged by the interaction with the solvent: the solvent effects are exclusively reflected by the variation of the coefficients of the VB expansion. The advantage of this choice is related to its easy interpretability. The method has been applied to the study of the several SN1/2 reactions. Another method from the same PCM family of solvation methods, namely the IEF– PCM [24] (see also the contribution by Cances), has recently been used to develop an ab initio VB solvation method [25]. According to this approach, in order to incorporate solvent effect into the VB scheme, the state wavefunction is expressed in the usual terms as a linear combination of VB structures, but now these VB structures are optimized and interact with one another in the presence of a polarizing field of the solvent. The Schrödinger equation for the VB structures is then solved directly by a self-consistent procedure. MPn methods The quest for methods able to account for the effects of dynamical correlation in continuum solvation models has lead to several proposals of Møller–Plesset methods for the descriptions of the solute. The question of the electron correlation in solvation models deserves a few words of comment. The introduction of correlation modifies the total electronic charge distribution, with respect to the HF reference, and as a consequence the solvent reaction potential is also changed. On the other hand, the polarization induced by the solvent through the reaction field modifies the electron correlation effects. The decoupling of these effects may give useful information about the solvent effects on the molecular properties of the solute. In this regard, the correlation methods based on the perturbation theory give both a conceptual and a computational framework. However, their extension to solvation models involves several difficulties and has been somewhat controversial. This is reflected in the numerous variant of the MPn methods for continuum solvation models. Perturbation theory within solvation schemes has been originally considered by Tapia and Goscinski [1b] at the CNDO level. An ab initio version of the Møller–Plesset perturbation theory within the DPCM solvation approach was introduced years ago by Olivares et al. [26] following the above intuitive considerations based on the fact that the electron correlation which modifies both the HF solute charge distribution and the solvent reaction potential depending on it can be back-modified by the solvent. To decouple these combined effects the authors introduced three alternative schemes: (1) MPn–PTE: the noniterative ‘energy-only’ scheme (PTE), where the solvated HF orbitals are used to calculate MPn correlation correction; (2) the density-only scheme (PTD) where the vacuum MPn correlated density matrix is used to evaluate the reaction field; (3) the iterative (PTED) scheme, where the correlated electronic density is used to make the reaction field self-consistent. Modern Theories of Continuum Models 91 PTE and PTD describe, respectively, the effects of the solvation on the electron correlation on the solvent polarization and vice versa; the PTED scheme leads instead to a comprehensive description of these two separate effects, revealing coupling between them. However, the PTDE scheme is not suitable for the calculation of analytical derivatives, even at the lowest order of the MP perturbation theory. All the alternative variants of the MPn may be implemented using a ‘relaxed’ density matrix or a ‘unrelaxed’ density matrix, in analogy with the CI solvation methods. In the first case the correlated electronic density is computed as a first derivatives of the free energy, while in the second case only the MPn perturbative wavefunction amplitudes are necessary. An analysis of the ‘unrelaxed’ MPn methods in continuum solvation models has been performed by Angyan [27]. By rigorous application of the perturbation theory for a nonlinear Hamiltonian, as is the case for continuum models, it has been shown that the nth-order correction to the free energy is based on the (n-1)th-order ‘unrelaxed’ density. This means that the correct MP2 solute–solvent energy has to be calculated with the solvent reaction field due to the Hartree–Fock electron density, as is the case of the PTE scheme. Following this analysis the PTED scheme at the MPn level is not analogous to standard vacuum Mller–Plesset perturbation theory as terms higher than the nth order are included. Other MP2 based solvent methods consist of the Onsager MP2–SCRF [19], within a ‘relaxed’ density scheme analogous to the PTDE scheme, and a multipole MP2SCRF model [28], based on a iterative ‘unrelaxed’ approach. The analytical gradients and Hessian of the free energy at MP2–PTE level, has been developed within the PCM framework [29]. Coupled-cluster Methods Although the correlative methods based on the coupled-cluster (CC) ansatz are among the most accurate approaches for molecules in vacuum, their extension to introduce the interactions between a molecule and a surrounding solvent have not yet reached a satisfactory stage. The main complexity in coupling CC to solvation methods comes from the evaluation of the electronic density, or of the related observables, needed for the calculation of the reaction field. Within the CC scheme the electronic density can only be evaluated by a ‘relaxed’ approach, which implies the evaluation of the first derivative of the free energy functional. As discussed previously for the cases of the CI and MPn approaches, this leads to a more involved formalism. The only example of a CC solvation model appearing so far in the literature is the CC/SCRF method developed by Christiansen and Mikkelsen [30] using the multipole solvation approach; the same scheme has also been extended to the CC response method including both equilibrium and nonequilibrium solvation [31]. The CC/SCRF method, exploiting the general concept of variational Lagrangian commonly used in quantum chemistry, defines a coupled-clusters Lagrangian in terms of the free energy functional (14) which leads to a set self-consistent equations. However, the need to evaluate the electric dipole moments of the solute as a first derivative of the Lagragian requires the introduction of set of auxiliary CC parameters, which have to be determined in addition to the CC amplitude. 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Chuev 1.6.1 Introduction In this chapter we consider the extension of continuum solvent models to nonlocal theories in the framework of the linear response approximation (LRA). Such an approximation is mainly applicable to electrostatic solute–solvent interactions, which usually obey the LRA with reasonable accuracy. The presentation is confined to this case. The medium effects are introduced in terms of (the dielectric permittivity) or (the susceptibility). At space point r conventional electrostatic expressions relate the electric field strength Er , the dielectric displacement Dr and the polarization field Pr as D= E" P= E" = 1 + 4 (1.119) Generally, and are tensorial quantities. They reduce to scalars in the case of isotropic media, and then describe the longitudinal polarization effects. Our presentation is devoted to this simple transparent case. Complications introduced by anisotropic phenomena are not considered; they do not change the main idea of nonlocal theory only making the notation cumbersome. According to the nonlocal theory the vector fields Er Dr and Pr in Equations (1.119) can be treated as time dependent and they obey the Maxwell equations [1]. Within the LRA, most general expressions are valid: Dr t = Pr t = d3 r dt r r t t Er t (1.120) d3 r dt r r t t Er t ˆ r t, we reformulate Equations (1.120) By introducing the integral operators ˆ r t" in the contracted form ∧ D = E" ∧ P= E (1.121) ∧ ∧ complemented by the relation between susceptibility operators: = I + 4 , where I is the identity operator. In the most common uniform case (both temporal and spatial) the integral kernels depend only on differences of their arguments: r r t t = r − r t − t " r r t t = r − r t − t " (1.122) r r t t = r − r t − t + 4 r − r t − t Modern Theories of Continuum Models 95 Within this additional constraint the Fourier transforms are useful: = k = k r + itr t d3 rdt expik r + it r t d3 r dt expik (1.123) % where r = r − r and t = t − t (k and are wavevector and frequency variables). r . Correspondingly Scalar products of vectors k and r are denoted as k = Ek = Dk = Pk r + itEr td3 rdt expik r td3 rdt expik+itD k (1.124) r +itPr r td3 rdt expik domain Equations (1.120)–(1.122) reduce to In the k = k Ek Dk = k Ek Pk (1.125) This looks quite similar to the conventional electrostatic Equations (1.119) with the and k become inevitable complication that the susceptibility functions k complex valued. Consequently, although the applied electric field Ek can be always and Pk are complex. Under treated as a real quantity, the response fields Dk certain constraints on k and , Equations (1.120), (1.121) and (1.125) can be considered as solutions to time-dependent electrodynamic (Maxwell) equations. This is a legitimate approximation provided relativistic (i.e. magnetic) effects are negligible. We follow this approach, which will be called the ‘quasilectrostatic approximation’ in the forthcoming 0 = k and text. It becomes exact in the true electrostatic limit = 0. Then k 0 = k represent pure effects of spatial dispersion. In practical implementations k temporal (or frequency) dispersion and spatial dispersion effects are often treated separately, sometimes being combined within simple models. We follow this strategy in the present contribution. The technique of complex-valued dielectric functions was originally applied to solvation problems by Ovchinnikov and Ovchinnikova [2] in the context of the electron transfer the familiar golden rule rate expression theory. They reformulated in terms of k for electron transfer [3]. This idea, thoroughly elaborated and extended by Dogonadze, Kuznetsov and their associates [4–7], constitutes a background for subsequent nonlocal solvation theories. 96 Continuum Solvation Models in Chemical Physics 1.6.2 Temporal (or Frequency) Dispersion We consider as an example the second relation of Equations (1.120) and (1.125) withdrawing from them both k-dependence and anisotropic effects: Pt = + d tEt − " P= E (1.126) − Note that the identity + + tEd = t − Ed − − is valid and its first version accepted in Equation (1.126) is more convenient. Fields P and E depend on space points whereas susceptibility is r-independent. Equation (1.126) is nonlocal in the time domain which means that the response Pt is determined by the whole evolution of E over the period t − . The causality principle requires that the response cannot precede the input signal. This implies the condition = 0 < 0, i.e. the susceptibility must be a step function. With positive , the response Pt lags behind the driving force Et − [8]. Another condition arises because P and E are real in the time domain. Combined with the causality this establishes the following form of the complex susceptibility function [9] = texpitdt= 1 +i 2 " 1 − = 1 2 −= − 2 0 (1.127) Hence, 1 and 2 are real and, respectively, even and odd functions of frequency. As a consequence of this property, the important Kramers–Kronig relation arises [9, 10]: 2 2 1 = P d 2 − 2 (1.128) 0 here symbol P denotes the principal value of the integral. Note that the real part of the susceptibility, i.e. 1 is responsible for dielectric screening effects whereas the imaginary component 2 accounts for the absorption of the radiation field. Frequency regions where 2 vanishes and = 1 are called transparency regions. No energy is absorbed here. Provided is located in a transparency region, the Krames–Kronig relation holds for as well as for 1 . This is always true for = 0, so the static permittivity can be expressed as 0 = 0 = 2 d 0 We assume here that the integrand behaves properly when → 0. (1.129) Modern Theories of Continuum Models 97 Typically, another transparency region exists: a < < where , called the optical frequency, denotes the lower bound of the electronic (optical) absorption spectrum. Provided this transparency band is wide (say, 0 / < 102 ; typically ≥ 1016 s−1 and a ≈ 1013 –1014 s−1 ), one can define the optical dielectric permittivity, = 1 + 4 . The real quantity is defined for < a 2 2 2 2 d d = 2 − 2 We obtain as a consequence: 2 2 P d + 2 − 2 a 1 (1.130) 0 The following interpretation can be suggested [3–5] for Equation (1.129), which is exact, and Equation (1.130), which is approximate but becomes exact when = 0. The static dielectric screening effects arise due to the accumulation of the radiation absorption over the whole frequency range. Within the LRA, solvent behaves as an ensemble of harmonic oscillatory modes with frequency which is much higher than the frequency of the applied field < a . Thereby is a real constant, is local and the corresponding electronic oscillators > are not involved in the observable medium dynamics, being responsible only for screening effects, measurable in terms of the dielectric constant . This is a formulation of the adiabatic approximation for electronic modes. On the other hand, the oscillators which are sluggish < a behave as dynamically active ones and produce retardation effects as expressed by Equation (1.126). They govern the solvent relaxation on time scales >> −1 . 1.6.3 Time-Dependent Polarizable Continuum Model In the solvation theory a reformulation of electrostatic Equations (1.119) is expedient. The solute charge density #r serves as an input variable, i.e. the driving force. The target of a computation is the scalar solvent response potential &r. In the framework of LRA the basic relation &r = K̂r = d3 rKr r r (1.131) is valid, where K̂ is the integral response operator. Its symmetric kernel Kr r = Kr r is called the electrostatic Green function [11]. The expression for Kr r depends on the explicit formulation of a specific problem. Within this framework the input quantities are the dielectric permittivity , the solute charge r and the excluded volume cavity occupied by a solute. The response field is created by the surface charge density r (the apparent charge) arising, as a result of medium polarization, on the cavity surface S: &r = S d2 r r r − r (1.132) 98 Continuum Solvation Models in Chemical Physics A connection to vector fields (1.119) is established by the notion that is equal to the normal component of the polarization vector Pr located on the external side of S. Polarization vanishes in the bulk of the medium provided the dielectric constant does not change there. The apparent charge r found in terms of numerical algorithms [12] is, in turn, a linear functional of r . Its computation is equivalent to a solution of the Poisson equation with proper matching conditions for &r on the boundary of the cavity, i.e. on surface S. This solution, formally expressed as Equation (1.131), is essentially nonlocal in space, although the problem is originally formulated in terms of local Equations (1.119). The spatial nonlocality arises from boundary conditions on S. Simple solutions are available only for spherically symmetrical cases (Born ion or Onsager point dipole). The equilibrium solvation energy is expressed as 1 1 Usolv = &r ·r = ∫ ∫d3 rd3 r r Kr r r 2 2 (1.133) where scalar product &r ·r denotes the volume integral. Let us now consider time-dependent phenomena which can be described in terms of a quasielectrostatic extension of Equation (1.131) based on Equation (1.126): &r = K̂r (1.134) It is assumed that the time-dependent charge r t and response &r t are connected by the linear integral operator K̂ with the time-dependent kernel Kr r t; the quantities in Equation (1.134) are the relevant Fourier transforms. The solution can be found [13] for the special case r t = r 't (1.135) where 't is an arbitrary function of time. We consider the Poisson-like equation ˜ r = K̂r , with a solution similar to Equation (1.132): & ˜ r = & S d2 r r r − r For given value the apparent charge density r is available in terms of the extended PCM procedure with a complex-valued dielectric function , namely, = 1 + i2 where 1 = 1 + 4 1 and 2 = 4 2 with complex-valued susceptibilities defined in Equation (1.127). The complication that both r and ˜ r become complex is inevitable. However, after applying the inverse Fourier trans& form, they become real in the time domain. This is warranted by the symmetry properties, Modern Theories of Continuum Models 99 ˜ r = & ˜ 1 r + i& ˜ 2 r & ˜ 1 r − = the consequence of the causality principle: & ˜ 2 r − = −& ˜ 2 r . All derivations follow those for in Equa˜ 1 r & & tions (1.127)–(1.129). By combining the inverse Fourier transform with the Kramers– Kronig relation (similar to Equation (1.128)) one obtains the real causal function: + 1 ˜ r t = ˜ r exp−itd" & & 2 − 2 ˜ ˜ ˜ r t < 0 = 0 &r t > 0 = &2 r sintd & (1.136) 0 The transformation for r is quite similar. The final solution for the case (1.135) ˜ r , being is straightforward because the procedure implemented for computing & ˜ linear, can be extended for &r as well: &r = &r (, where ( is the Fourier transform of (t. The inverse Fourier transform gives &r t = + ˜ r (t − d. A common selection for (t is the step function (t > 0 = & − 1 (t < 0 = 0. This implies that the solute charge r t is created instantaneously at t = 0 and then remains constant, a situation typical for spectroscopic applications. By taking Equation (1.136) into account we find the basic result: &r t > 0 = t ˜ r d ) (1.137) 0 This approach, based on a complex-valued realization of the PCM algorithm, reduces to a pair of coupled integral equations for real and imaginary parts of apparent charge density for r ) [13]. An alternative technique avoiding explicit treatment of the complex permittivity has been also derived [14, 15]. The kernel Kr r t of operator K̂ does not appear explicitly. However, its matrix be computed for any * can ) elements pair of basis charge densities 1 r and 2 r 1 K̂2 = 1 r &r td3 r, where &r t, given by Equation (1.137), corresponds to r = 2 r . 1.6.4 Formulation of the Spatial Dispersion Theory Spatial dispersion effects are usually considered separately from time dependences and 0 = k and k 0 = k are basic correspond to static limit = 0. Consequently k susceptibility functions. Within the LRA the relation similar to Equation (1.131) is valid. It formally represents a solution to the nonlocal Poisson equation with a k-dependent susceptibility. In computational practice, such solutions are restricted by the approximation that the solvent is uniform and isotropic. It defines in the real space the susceptibility kernel as r r = r − r . The counterpart in the k-domain obtained via Fourier transform, = k, where k = k. The representation for is similar. Parameterization reads k 100 Continuum Solvation Models in Chemical Physics of such functions is a question of practical importance. It is formulated in the k-domain, usually, as a Lorentzian function k = + 0 − 1 + 2 k 2 (1.138) with static and optic values 0 and (Section 1.6.2). The transformation to the real space yields r − r = r − r + 0 − exp−r − r / 42 r − r (1.139) It is seen that serves as a screening length, reflecting a correlation between the neighbouring solvent particles; the local uncorrelated model corresponds to = 0 and r − r ∝ r − r . This notion explains the usually applicable term ‘the correlation length’ [6]. Equation (1.139) implies that the electronic polarization is local, i.e. no correlation exists inside solvent particles, which is an approximation. Originally, the representation similar to Equation (1.138) was applied to another dielectric function [4–6]: 1 1− k 1 1 1 1 + − = 1− 0 1 + * 2 k 2 (1.140) This quantity proves to be proportional to the correlation function of the medium polarization (see Section 1.6.7) and Equation (1.140) has the advantage that its parameters can be extracted from the direct experimental measurements of this correlation function, or from its simulations. Formally Equations (1.138) and (1.140) are equivalent provided = 0 / *, where 0 is the static dielectric constant (see Section 1.6.7). A more refined parameterization allows for the several Lorentzian terms in equations similar to Equations (1.138) and (1.140) [5, 6, 16]. They contain a number of correlation parameters i or *i i = 1 2 ; the interrelations between parameters i and *i depend on this number. Representation of the static susceptibility as Equation (1.138) or its multi-term counterpart returns us to the frequency dispersion theory (Section 1.6.2). Similar to Equation (1.129), it states that for the static case k accumulates additively the contributions from medium polarization modes over the whole frequency absorption spectrum, which is represented by the imaginary part of the complex susceptibility, i.e. the function 2 , or 2 k in the present case. As in Equation (1.130), the electronic (inertialess) modes are separated and assumed to be local. The nonlocality of inertial modes is introduced by means of correlation lengths i or *i , which correspond to medium oscillators confined within a lower frequency ranges and separated from electronic modes by a transparency region. For instance, an appropriate parameterization of water [6, 16] suggests two Lorentzian terms, associated with infrared (vibrational = 1013 –1014 s−1 ) and Debye (orientational = 1011 –1012 s−1 ) absorption. Correlation lengths *i (but not i ) are, roughly speaking, comparable in magnitude with the size of solvent particles. The importance of nonlocal effects is measured by the ratio */Rsol , where Rsol denotes Modern Theories of Continuum Models 101 the characteristic radius which measures the size of the solute (i.e. of its cavity). The limit when this ratio vanishes corresponds to the local continuum medium model: size of solvent particles << 1 size of a solute particle (1.141) By introducing k-dependent susceptibilities one can, at a phenomenological level, imitate the molecular structure of solvent around the solute with any desired degree of accuracy. Invoking isotropic and uniform approximations such as Equations (1.138) or (1.140) constrains the ability of such an approach to a certain degree. In any case, this is an essential extension of structureless local models of solvent. 1.6.5 Spatial Nonlocal Equations We consider the formulation which accounts for the excluded volume of a solute particle. This nonlocal extension of the PCM deals with the stepwise dielectric functions k and k. Their inverse Fourier transforms change on the boundary of the cavity surface: = 1 = 0 inside the cavity and = r − r = r − r outside. The starting ∧ ∧ point is Equation (1.121) where time variable t is suppressed in operators and , and Equation (1.125) where frequency is suppressed. By replacing vector field E = − + by potential + , the Poisson equation appears and it changes its standard form 2 + = −4, ∧ valid only inside the cavity, to + = 0 outside. The boundary conditions require that remains continuous at the cavity surface, but its normal derivative displays a step. Compared to the PCM matching condition, the ∧ matching expression is more complicated because it includes operator and is nonlocal in space. General solutions to this problem have been suggested [17–21]. The algorithm is complicated, requires cumbersome notation and has been actively performed only for simple spatially symmetric cases. We consider below the spherical case as an illustration. The solution is represented [19] as + r = ,r + -r + &r r r 2 g , r = d2 r - r = d r r − r r − r Vi Vl (1.142) &r = S d2 r r r − r (1.143) The vacuum potential ,r is a solution to the ordinary Poisson equation with = 1 in the whole space. The induced potential consists of two components - and & created by the external gr and surface r charge distributions. The normal derivatives /n of the volume potentials , and - are continuous; moreover -/n = 0 on the surface S. The single layer potential &r , however, obeys a singular matching condition on S &/ni − &/ne = 4 r ∈ S, where subscripts i and e denote internal and external sides of the surface. Its presence allows for a step in &/n. With this condition Equations (1.142) and (1.143) describe the solutions valid for the general case, without symmetry restrictions. The equations to be solved comprise a procedure for 102 Continuum Solvation Models in Chemical Physics simultaneously finding unknown functions gr and r . As a supplement to PCM, the volume charge gr and its field -r appear in the nonlocal theory. In the spherically symmetrical Born case we consider the charge r = Qr located at the centre r = 0 of the sphere with radius a. The problem reduces to a single dimension: ,r = ,R gr = gR -r = -R &r = &R = const, and also, (when R = a) ,/R = −Q/a2 -/R = )/ni = 0 )/Re = −4. Spherical coordinates r = R , and r = R , are used here. With this notation, gR and are determined by equations 4 1 Q gR + .RR gR dR = − + 4 .R a a2 (1.144) a = + 1 dR 0aR R a = − 1 Q 1 − 4a2 (1.145) The integral susceptibility kernel is expressed in the form r − r = r − r + ¯ r − r , where ¯ is nonlocal and one-dimensional kernels in Equation (1.144) appear as a result of its averaging over angular variables: ¯ r − r .RR = R2 d, sin d (1.146) ¯ r − r 0R R cos = R2 d, sin d As a result of the spherical symmetry the right-hand parts of Equations (1.146) prove to be angle independent; therefore their calculations can be performed with = , = 0. An analytical solution is available [18, 20] with simple Lorentzian form for the Fourier transform of susceptibility (see Equation (1.138)) with single correlation length ): 1 0 − ¯ k = −1+ (1.147) 4 1 + 2 k 2 The corresponding potentials are: -R < a = 4*g 0 -R > a = 4*2 g0 1 + a/* − expa − R/*/R &R < a = 4a (1.148) &R > a = 4a2 /R with the notation Q a 1 1 g0 = − + 4a − 2 4* a 0 1 + /0 cotha/ 0 / cotha/ − /a 1 Q 1 = − − 4a2 0 0 / cotha/ − /a + /a /0 + 1 (1.149) Modern Theories of Continuum Models 103 The solvation energy is generally expressed as [20] Usolv = 05 ∫ d3 r r -r + &r . For case (1.147) this reduces to the result 1 + 0 / cotha/ − /a 1 Q2 1 Q2 1 Usolv = − 1− − − 2a 2a 0 0 / cotha/ − /a + /a /0 + 1 (1.150) This example shows the degree of complication inherent in the nonlocal extension of the continuum theory even for the simplest Born-like case. In accord with Equation (1.141), the dimensionless parameter /a measures the importance of nonlocality effects; the local Born limit is recovered when /a → 0. The opposite strongly nonlocal limit a/ → 0 corresponds to the unscreened solvation: Usolv = −Q2 1 − 1/ /2a. For the general form of the dielectric function !k a numerical solution for one-dimensional Equation (1.144) is straightforward [19]. However, there exists a principal difficulty hindering such solution when !k has poles on the real k-axis (see Sections 1.6.7 and 1.6.8). This creates oscillating kernels !r − r in the real space. 1.6.6 Uniform Approximation Let us consider the nonlocal Poisson equation ˆ+ = −4 in the uniform space. The singular boundary condition on the surface of the solute cavity is neglected. Note that this condition furnishes the mechanisms of the excluded volume effect. The solute is charged and spherical, i.e. r = R. The solution R is obtained by using Fourier transform [6, 16]; it is valid outside the cavity R > a, + R = 2 dk sinkR k k kR (1.151) 0 Here k is the Fourier transform of R. This Born ion is considered as a conducting sphere with its charge Q being smeared over the surface of its cavity: R = Q/4a2 R − a k = Q sinka/ka. Outside the cavity the electrostatic field created by this charge is fully equivalent to the field due to the point charge Q considered earlier. By this means for R > a 2Q dk sinkR sinka k kR ka + R = (1.152) 0 The solvation energy is obtained from Usolv = 05·+ − , where ,R = Q/R is the vacuum potential. This produces the final result [4, 22]: Q2 1 sinka 2 dk 1 − Usolv = − k ka 0 (1.153) 104 Continuum Solvation Models in Chemical Physics In the same manner [6,16] the interaction energy between a pair of spherical ions (charges Q1 and Q2 with radii a1 and a2 can be derived: 2Q1 Q2 dk sinka1 sinka2 sinkR Usolv R > a1 + a2 = k ka1 ka2 kR (1.154) 0 Here R denotes the distance between the ion centres. The important condition is that the two spheres do not overlap. Equations (1.152)–(1.154) are approximate because of the implicit assumption that uniform potential (1.151) represents the true potential actually existing in the vicinity of the ion. In fact, this expression is perturbed by matching conditions on the boundary, which are neglected. The validity of the uniform approach is illustrated in Figure 1.11 where two solvation energies are compared: that given by Equation (1.153) and another obtained by the exact treatment of Equation (1.150). The dielectric function is k = +0 − /1+2 k2 and uniform result proves to be the excellent fit for this particular case [20]. Figure 1.11 Solvation energy Usolv versus cavity radius a: solid line corresponds to Equation (1.150) [20]; circles to Equation (1.153) [6]; dashed line to the Born theory (0 = 7839 = 17756, (a) = 483 Å, (b) = 072 Å). The approach described can be extended to a more complicated nonspherical case. Similar to Equation (1.154), we consider a neutral system composed of two Born spheres with Q1 = Q and Q2 = −Q. It is usually called ‘the dumbbell’. For the isolated spheres we denote their charge densities as 1 and 2 , their response fields as &1 = 1 − ,1 and &2 = +2 − ,2 , where +i and ,i i = 1 2 are defined similar to the single sphere case. The solvation energy for such system equals to Usolv = 05&1 ·1 + &1 ·2 + &2 ·1 + &2 ·2 . The scalar products mean volume integrals. The reasonable estimate for separate terms in will be Ui = 05 &i ·i i = 1 2 Uint R = &1 ·2 = &2 ·1 , where U1 and U2 are solvation energies obtained in terms of Equation (1.153) whereas the interaction energy is identified with Equation (1.154). In this result we assume that the Modern Theories of Continuum Models 105 electrostatic energy contributions for each ion can be computed neglecting the presence of the neighbouring ion. This assumption is acceptable when the spheres do not overlap. Bearing in mind how complicated are accurate nonlocal solutions, the uniform model comprises a useful practical tool for estimates of nonlocal solvation effects [6, 16]. 1.6.7 Modelling Dielectric Functions The nonlocal theory was originally based on the approximation of k in the form of Equation (1.140) [6, 16], but much effort has been focused on calculations of dielectric function k. Earlier studies have been based on the integral equations theory (IET) [23] and used the mean spherical approximation (MSA) [24] or the hypernetted chain (HNC) model [25]. Using a few fitting parameters (hard sphere radius in the MSA or LennardJones parameters in the HNC and diffusion coefficients), researchers are able to calculate the frequency and spatial dispersions. Concerning the frequency dependence the models are satisfactory to predict accurately the behaviour at low frequencies, while they provide only qualitative effects at optical frequencies [26]. The static dielectric properties of molecular liquids have been studied more intensively on the basis of the IET [27, 28] or molecular dynamics (MD) simulations [29–34]. Figure 1.12 shows the static dielectric function k of water under normal conditions, which is obtained by the MD and by the IET with the employment of the reference interaction site model [35]. As can be seen the IET reproduces the qualitative behaviour of k, although the description of details is less satisfactory due to application of the rigid model of water molecule. 0 1 ε (k) –10 –20 –30 k [A–1] –40 0 2 4 6 8 10 12 −1 Figure 1.12 Dielectric function k for bulk water calculated with the RISM method (dashed line) and for MD simulations (solid line) [35]. The IET as well as the simulations indicate that the dielectric constant increases from the macroscopic dielectric value to infinity and then becomes negative at some value of k. Such exotic pole-like behaviour is not unique and has been reported for the onecomponent plasma and the degenerate electron gas [36]. This overscreening effect leads to 106 Continuum Solvation Models in Chemical Physics repulsion between two unlike charges and attraction between two similar charges at short distances. The overscreening effect is found to have a multi-scale origin. The first reason is trivial and is caused by the discreteness of molecular liquids, when discrete dipoles oriented around an intruding charge provide an overscreening at a submolecular scale. However, the dielectric overscreening may also be due to intermolecular correlations and coupling between polarization and density fluctuations [37]. The profiles of dielectric functions in Figure 1.12 obviously disagree with their Lorentzian models considered in Section 1.6.4, which suggest they have a peak at k = 0. It is expected that Lorentzian peaks can survive in the range of small kk 1 where the accuracy of molecular simulations is insufficient to reveal quite definitely their existence [31]. The question of justifying phenomenological models of k at a microscopic level remains open. The pole structure of k leads to an oscillatory behaviour of the nonlocal kernel r − r and such an oscillating kernel results in an irregular behaviour of the solvation energy as a function of the solute radius, complicating computations of the solvation energy with the use of non-Lorentzian k. On the other hand, the exotic behaviour of k also leads to several interesting and unexpected consequences with important implications. For example, the overscreening effect is believed to be revealed as charge inversion in chemical and biological systems [38] observed as an aggregation of biomolecules. Another example of the exotic behaviour is the insulator– metal transition in metal–ammonia solutions and the associated phase separation. At low metal concentrations, the solutions are nonmetallic and have an intense blue colour, characteristic of the formation of solvated electrons. At intermediate concentrations and below a critical temperature, the two different phases separate within a miscibility gap. At high enough concentrations of metals, the solutions are metallic with a characteristic bronze coloration. As indicated in ref. [39], these phenomena are strongly related to the frequency-dependent dielectric function of the solution. At a finite concentrations, owing to the large frequency-dependent polarizability the solvated electrons induce a polarization catastrophe leading to a markedly increased dielectric constant and the insulator–metal transition. 1.6.8 Applications Among most familiar applications the time-dependent Stokes shift in absorption–emission spectra is essentially an effect governed by the nolocal time evolution of solvation shells surrounding electronically excited states. This phenomenon is discussed in the contribution of Ladanyi to this volume. We only comment here that Sections 1.6.2 and 1.6.3 of the present contribution provide a methodological background for this theme. In such a context, spatial nonlocality is usually ignored, although microscopic solvent models, even the most simplified ones [40–43], actually account for the nonlocal effects. Explicit functions k have been considered in only few publications [44,45] whereas invoking is a standard way to treat the Stokes shift. To get a satisfactory description of the experiment rather sophisticated functions are required [21, 46–49]; simple Debyelike models of are hardly efficient. Applications of spatially nonlocal electrostatic theory are not so numerous. Limited by simple models reducible to a one-dimensional problem, they only include systems obeying spherical or planar symmetry. A traditional treatment of hydration free energies Modern Theories of Continuum Models 107 of small spherical ions within a uniform approximation as considered in Section 1.6.6 is successful. Fitting the experimental data with the refined multi-term Lorentzian spectral functions is surveyed in refs [6,16]. By tuning ion radii and correlation lengths reasonable accuracy is gained. Three-dimensional computations for small ions are also mentioned in ref. [50]. The interfacial solvation effects accompanying electrochemical processes in the vicinity of a planar surface have been studied [6, 16]. Nonlocality is significant at rather small distances between the ‘solute’ (ion or electrode surface) in comparison with the solvent correlation length. The formation of a dynamically ordered water shell is an important factor determining hydration in biological systems. Non-Lorentzian dielectric functions discussed in Section 1.6.7 cannot be directly promote numerical instabilities in applied to treat solvation energies. The poles of k calculations. They have deep physical roots originating from the interference between polarization and density fluctuations in the vicinity of the solute [37]. Attempts to suppress this complication in terms of unusually sophisticated methods have been reported [51,52]. However, simple traditional solutions look more expedient and efficient. Restricting the resolves the problem and provide a consistreatment by purely Lorentzian functions k tent and satisfactory semi-empirical theory for ordinary practical implementations. Lorentzian dielectric functions have also been used to treat solvent reorganization energies in electron transfer reactions [53, 54] within the framework of the uniform approximation. Nonlocal effects reduce their values compared with conventional estimates in terms of the Marcus theory. The role of overscreening has been discussed [55]. However, it is not so obvious how to reveal deviations of this type in experimental data, since nonlinear effects, short rage forces, etc. provide alternative sources of possible complications masking the real physical consequence of spatial dispersion. Still, at least one consequence is certain. This is the nonzero values of reorganization energies in nonpolar solvents (benzene, dioxane, etc) with vanishing permanent dipoles and = 0 [55–57]. Local electrostatic models predict that the solvent reorganization energy must disappear in such solvent but the values of 0.1–0.3 eV have been observed [55–57] and reproduced in molecular level computations [58, 59]. This effect would arise immediately in terms of the nonlocal theory by invoking the simplest Lorentzian models, although no such studies have so far been published. 1.6.9 Conclusions Discreteness of molecular liquids is a source of microscopic inhomogeneity of a solvent revealed as the formation of a structured shell around the solute. Because of the longrange nature of electrostatic interactions, modelling the electrostatic response by molecular simulations taking into account detailed solvent structure requires cumbersome computations. The nonlocal theory can in principle provide a computationally tractable approach and it is therefore a serious candidate for a realistic description of solvent effects. Unfortunately, at its present technical level, the nonlocal approach is considerably more demanding than local continuum schemes such as PCM. A numerical solution of coupled three-dimensional integro-differential equations becomes a formidable task for really interesting large solutes. The absence of available universal computer packages restricts practical implementations of the method. This is why it has been applied mainly 108 Continuum Solvation Models in Chemical Physics to analyse idealized one-dimensional models and to reveal common trends in experiment with the use of additional approximations leading to analytical results. Nevertheless, the concept of spatial dispersion provides a general background for a qualitative understanding of those solvation effects which are beyond the scope of local continuum models. The nonlocal theory creates a bridge between conventional and well developed local approaches and explicit molecular level treatments such as integral equation theory, MC or MD simulations. 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Broadening of the absorption and fluorescence bands results from fluctuations in the structure of the solvation shell around the solute (this effect, called inhomogeneous broadening, superimposes homogeneous broadening because of the existence of continuous set of vibrational sublevels) [1]. Moreover, shifts in absorption and emission bands can be induced by a change in solvent nature or composition; these shifts,called solvatochromic shifts, are experimental evidence of changes in solvation energy. In other words, when a solute is surrounded by solvent molecules, its ground state and its excited state are differently stabilized by solute–solvent interactions, depending on the chemical nature of both solute and solvent molecules [2, 3]. The accurate modelling of excited state formation and relaxation of molecules in solution is a very important problem. Despite this recognized importance and the numerous applications that such a modelling might have not only in photochemical or spectroscopic studies but also in material science and biology, the progress achieved so far is not as great as that achieved for ground state phenomena. This delay in the development of accurate but still computationally feasible strategies to study excited states in solution is due to the complexity of the problem. The modelling of electronically excited molecules when interacting with an external medium, in fact requires the introduction of the concept of time progress, a concept which can be safely neglected in treating most of the properties and processes of solutes in their ground states. In fact, in these cases, and also when introducing reaction processes, one can always reduce the analysis to a completely equilibrated solute–solvent system. In contrast, when attention is shifted towards dynamic phenomena such as those involved in electronic transitions (absorptions and/or emissions), or towards relaxation phenomena such as those which describe the time evolution of the excited state, one has to introduce new models, in which solute and solvent have proper response times which must not be coherent or at least not before very long times. In the previous contributions of this book, an extensive description of continuum solvation models has been given for equilibrated solute–solvent systems. Here, in contrast, an extension of these models will be given in order to describe solvent effects on electronic excitation/de-excitation processes. Different semiclassical schemes [4] have been proposed to evaluate solvatochromic shifts (i.e. the excitation energy difference between gas phase and solution for a given solute) from the properties of the gas phase molecule. These different schemes usually exploit Onsager’s solvation model [5], enclosing the solute in a spherical cavity built in the continuous dielectric representing the solvent and considering the solute as a polarizable dipole. The solvatochromic shifts are finally given in terms of the ground and Modern Theories of Continuum Models 111 excited state dipoles and polarizabilities of the solute considered in the gas phase, and of the static and optical dielectric constant of the solvent. As shown in the other contributions, continuum models have been significantly modified and improved with respect to the older versions; the same improvements have also been achieved for their extensions to the study of vertical excitation/de-excitation processes. These extensions will be reviewed here but before that, a brief overview will be given on the main physical aspects to be accounted for in any theoretical model aimed at reliably reproducing solvatochromic shifts. 1.7.2 Physical Aspects It was mentioned in the Introduction that shifts in absorption and emission bands can be induced by a change in solvent nature or composition. These shifts, called solvatochromic shifts, are experimental evidence of changes in solvation energy and they have been widely used to construct empirical polarity scales for the different solvents. It is worth mentioning here the use of solvatochromism of betaine dyes proposed by Reichardt [6] as a probe of solvent polarity. The exceptionally strong solvatochromism shown by these compounds can be explained by considering that in their ground state they are zwitterions while, upon excitation, electron transfer occurs exactly in the direction of cancelling this charge separation. As a result, the dipole moment which is about 15 D in the ground state becomes almost zero in the excited state and thus solvent interactions change markedly leading to the observed negative solvatochromism. An alternative approach to quantify polarity effects was proposed by Kamlet et al. [7]. According to this approach the positions of the bands in UV–visible absorption and fluorescence spectra can be determined as = 0 + s ∗ + a! + b0 (1.155) where and 0 are the wavenumbers of the band maxima in the solvent considered and in the reference solvent (generally cyclohexane), respectively, ∗ is a measure of the polarity/polarizability effects of the solvent, ! is an index of solvent hydrogen bond donor acidity and 0 is an index of solvent hydrogen bond acceptor basicity. The coefficients s a and b describe the sensitivity of a process to each of the individual contributions. The ∗ scale of Kamlet and Taft deserves special recognition not only because it has been successfully applied in many studies (not limited to UV or fluorescence spectra, and including many other physical or chemical parameters such as reaction rate, equilibrium constant, etc.) but also because it gives a very clear introduction of the problem. Namely, Equation (1.155) indicates that the two main aspect to consider when modelling solvent effects on transition energies are polarity/polarizability effects and hydrogen bonding. Let us briefly analyse these two aspects separately starting from the latter one. Specific Interactions Several examples have shown that specific interactions such as hydrogen bonding interactions should be considered as one of the intrinsic aspects of solvent effects on absorption or fluorescence spectra. 112 Continuum Solvation Models in Chemical Physics A well-known example is the case of n → ∗ transitions in solutes with carbonyl or amide chromophores in protic solvents. In such transitions, the electronic density on the heteroatom (either oxygen of nitrogen) decreases upon excitation. This results in a decrease in the capability of this heteroatom to form hydrogen bonds. The effect on absorption should then be similar to that resulting from a decrease in dipole moment upon excitation, and a blue shift of the absorption spectrum is expected; the higher the strength of hydrogen bonding, the larger the shift. This criterion is convenient for assigning an n ∗ band while the spectral shift can be used to determine the energy of the hydrogen bond. It is easy to predict that the fluorescence emitted from a singlet state n ∗ will be always less sensitive to the ability of the solvent to form hydrogen bonds than absorption. In fact, if n → ∗ excitation causes hydrogen bond breaking, the fluorescence spectrum will only be slightly affected by the ability of the solvent to form hydrogen bonds because emission arises from an n ∗ state without hydrogen bonds. Another case in which hydrogen bonding can play a role is represented by the → ∗ transitions. In these cases, it is often observed that the heteroatom of a heterocycle (e.g. N) is more basic in the excited state than in the ground state. The resulting excited molecule can thus be hydrogen bonded more strongly than the ground state. As a result, → ∗ fluorescence is generally more sensitive to hydrogen bonding than → ∗ absorption. These simple observations clearly show that a change in the ability of a solvent to form hydrogen bonds can affect the nature n ∗ versus ∗ of the lowest singlet state. Some aromatic carbonyl compounds often have low-lying, closely spaced ∗ and n ∗ states. Inversion of these two states can be observed when the polarity and the hydrogen-bonding power of the solvent increases, because the n ∗ state shifts to higher energy whereas the ∗ state shifts to lower energy. This results in an increase in fluorescence quantum yield because radiative emission from n ∗ states is known to be less efficient than from ∗ states. The other consequence is a red shift of the fluorescence spectrum. From these few examples it is apparent that the shifts occurring in hydrogen-bonding solvents are complex and may occur in either direction, but the take-home message is that specific first solvation-shell effects cannot be ignored. On the basis of this picture, one might guess that a good computational prediction of the excitation energies of hydrogenbonding solute–solvent systems is obtained in terms of clusters of solute plus few solvent molecules, namely those interacting with H-bond accepting and donating sites in the solute. In contrast, from many studies it follows that this picture is not completely right, or at least it is incomplete. These analyses in fact show that the supermolecule approach is surely needed to predict the blue or red character of the solvent-induced shifts. However, a better agreement with experimental observation is found when a continuum model is added on top of the aggregates containing the solute and some explicit solvent molecules [8]. This result can be explained by considering that continuum models represent an effective way to include the electrostatic long range effects missing in the cluster-only description. An alternative approach found in the literature producing similar results considers explicitly solvent molecules belonging to the second and outer solvation shells. It is easy to understand that, because of the disordered nature of the solvent, a large number of calculations on different clusters are needed in this type of model to achieve convergency in the statistical sampling. By contrast, the use of a continuum description Modern Theories of Continuum Models 113 allows the consideration of many different solute–solvent configurations to be avoided, as by definition it accounts for an implicit average. Polarity Effects: the Nonequilibrium Solvation In order to analyse bulk polarity effects it is common to represent the electrostatic response of the solvent in terms of the polarization function P. This vectorial function in fact can be directly connected to any electric field (here that produced by the solute) through a single quantity, the susceptibility , or equivalently the permittivity [9]. To apply this picture to solvatochromism we have to consider that the responses of the microscopic constituents of the solvent (molecules, atoms, electrons) required to reach a certain equilibrium value of the polarization have specific characteristic times (CT). When the solute charge distribution varies appreciably within a period of the same order as these CTs, the responses of these constituents will not be sufficiently rapid to build up a new equilibrium polarization, and the actual value of the polarization will lag behind the changing charge distribution. To understand this point better, it is convenient to introduce a partition of the sources of the dynamical behaviour of the medium into two main components. One is represented by the molecular motions inside the solvent due to changes in the charge distribution, and/or in the geometry, of the solute system. The solute when immersed in the solvent produces an electric field inside the bulk of the medium which can modify its structure, for example inducing phenomena of alignment and/or preferential orientation of the solvent molecules around the cavity embedding the solute. These molecular motions are characterized by specific time scales of the order of the rotational and translational times appropriate to the condensed phases. In a analogous way, we can assume that the single solvent molecules are subjected to internal geometrical variations, i.e. vibrations, due to the changes in the solute field; once again these will be described by specific shorter time scales. The translational, the rotational and/or the vibrational motions all involve nuclear displacements and therefore, in the following, they will be collectively indicated as ‘nuclear motions’. The other important component of the dynamical nature of the medium, complementary to the nuclear one, is that induced by motions of the electrons inside each solvent molecule; these motions are extremely fast and they represent the electronic polarization of the solvent. These nuclear and electronic components, owing to their different dynamic behaviour, will give rise to different effects. In particular, the electronic motions can be considered as instantaneous and thus the part of the solvent response they cause is always equilibrated to any change, even if fast, in the charge distribution of the solute. In contrast, solvent nuclear motions, markedly slower, can be delayed with respect to fast changes, and thus they can give rise to solute–solvent systems not completely equilibrated in the time interval of interest in the phenomenon under study. This condition of nonequilibrium will successively evolve towards a more stable and completely equilibrated state in a time interval which will depend on the specific system under scrutiny. If we limit our description to the initial step of the whole process, i.e. the vertical electronic transition (absorption and emission), we can safely assume a Franck–Condon like response of the solvent, exactly as for the solute molecule; the nuclear motions inside and among the solvent molecules will not be able to follow immediately the fast changes in the solute electronic charge distribution and thus the corresponding part of the 114 Continuum Solvation Models in Chemical Physics response (also indicated as inertial) will remain frozen in the state immediately prior to the transition. Within this framework, the polarization can be split into two components (see also the contribution by Tomasi): P P fast + P slow (1.156) where fast indicates the part of the solvent response that always follows the dynamics of the process and slow refers to the remaining slow term. Such splitting in the medium response gives rise to the so called ‘nonequilibrium’ regime. Obviously, what is fast and what is slow depends on the specific dynamic phenomenon under study. In a very fast process such as the vertical transition leading to a change of the solute electronic state via photon absorption or emission, P fast can be reduced to the term related to the response of the solvent electrons, whereas P slow collects all of the other terms related to the various nuclear degrees of freedom of the solvent. This analysis shows that in order to account properly for solvent polarity effects, a solvation model has to be characterized by a larger flexibility with respect to the same model for ground state phenomena. In particular, it should be possible to shift easily from an equilibrium to a nonequilibrium regime according to the specific phenomenon under scrutiny. In the following section, we will show that such a flexibility can be obtained in continuum models and generalized to QM descriptions of the electronic excitations. 1.7.3 Quantum Mechanical Aspects Within the QM continuum solvation framework, as in the case of isolated molecules, it is practice to compute the excitation energies with two different approaches: the state-specific (SS) method and the linear-response (LR) method. The former has a long tradition [10–24], starting from the pioneering paper by Yomosa in 1974 [10], and it is related to the classical theory of solvatochromic effects; the latter has been introduced few years ago in connection with the development of the LR theory for continuum solvation models [25–31]. The state-specific method solves the nonlinear Schrödinger equation for the state of interest (ground and excited state) usually within a multirefence approach (CI, MCSCF or CASSCF descriptions), and it postulates that the transition energies are differences between the corresponding values of the free energy functional, the basic energetic quantity of the QM continuum models. The nonlinear character of the reaction potential requires the introduction in the SS approaches of an iteration procedure not present in parallel calculations on isolated systems. A different analysis applies to the LR approach (in either Tamm–Dancoff, Random Phase Approximation, or Time-dependent DFT version) where the excitation energies are directly determined as singularities of the frequency-dependent linear response functions of the solvated molecule in the ground state, and thus avoiding explicit calculation of the excited state wave function. In this case, the iterative scheme of the SS approaches is no longer necessary, and the whole spectrum of excitation energies can be obtained in a single run as for isolated systems. Although it has been demonstrated that for an isolated molecule the SS and LR methods are equivalent (in the limit of the exact solution of the corresponding equations), Modern Theories of Continuum Models 115 a formal comparison for molecules described by QM continuum models shows that this equivalence is no longer valid. The origin of the LR–SS difference was imputed to the incapability of the nonlinear effective solute Hamiltonian used in these solvation models to correctly describe energy expectation values of mixed solute states, i.e., states that are not stationary. Since in a perturbation approach such as the LR treatment the perturbed state can be seen as a linear combination of zeroth-order states, the inability of the effective Hamiltonian approach to treat mixed states causes an incorrect redistribution of the solvent terms among the various perturbation orders [32]. A simple but effective strategy (‘corrected’ LR, or cLR) aimed at overcoming this intrinsic limit of the nonlinear effective solute Hamiltonian when applied to LR approaches has been first proposed by Caricato et al. [33]. With such a strategy, the statespecific solvent response is recovered within the linear response approach. As a result, the LR–SS differences in vertical excitation energies are greatly reduced (still keeping the computational feasibility of LR schemes). Operative Equations In the previous contributions to this book, it has been shown that by adopting a polarizable continuum description of the solvent, the solute–solvent electrostatic interactions can be described in terms of a solvent reaction potential, V̂ expressed as the electrostatic interaction between an apparent surface charge (ASC) density on the cavity surface which describes the solvent polarization in the presence of the solute nuclei and electrons. In the computational practice a boundary-element method (BEM) is applied by partitioning the cavity surface into Nts discrete elements and by replacing the apparent surface charge density by a collection of point charges qk , placed at the centre of each element sk . We thus obtain: V̂ r = k 1 qs " GS r − sk k (1.157) where r is the electronic coordinate and we have indicated the explicit dependence of the apparent charges q on the solvent dielectric constant and the solute ground state density GS (including the nuclear contribution). The corresponding energetic functional to be minimized becomes: 1 G = Ĥ 0 + V̂ − V̂ 2 (1.158) and its minimization for the ground state gives the equation: ' & Ĥeff = Ĥ 0 + V̂ = E GS (1.159) This approach allows us to rewrite Equation (1.158) as GGS = E GS − 1 V s q s 2 i GS i GS i (1.160) 116 Continuum Solvation Models in Chemical Physics where VGS si is the electrostatic potential produced by the solute in its electronic ground state on the cavity. The free energy expression given in Equation (1.160) for a ground state can be generalized to both an equilibrium and a nonequilibrium excited state K. By rewriting the solute electronic density (in terms of the one-particle density matrix on a given basis set) corresponding to the excited state K as a sum of the GS and a relaxation term P , and by assuming a complete equilibration between the solute in the excited state K and the solvent, we obtain K − GKeq = EGS 1 1 VGS si qGS si + Vsi " P q si " P 2 i 2 i (1.161) where we have defined: ) * K EGS = Keq Ĥ 0 + V̂ GS Keq ) * = Keq Ĥ 0 Keq + VK si qGS si (1.162) i as the excited state energy in the presence of the fixed reaction field of the ground state V̂ GS In the above equations we have exploited the linear dependence of the solvent charges and the corresponding reaction potential on P, namely: VK si = VGS si + Vsi " P qK si = qGS si + q si " P The nonequilibrium equivalent of Equation (1.161) can be obtained using two alternative but equivalent schemes (often associated to the names of Pekar and Marcus). The two schemes are characterized by a different partition of the low and fast contributions of the apparent charges, namely we have [34]: or Partition I qK = qGS + qKel in Partition II qK = qGS + qKdyn (1.163) In PI, the slow and fast indices are replaced by the subscripts or and el referring to ‘orientational’ and ‘electronic’ response of the solvent, respectively, while in PII the subscripts in and dyn refer now to an ‘inertial’ and a ‘dynamic’ polarization response of the solvent, respectively. The differences between the two schemes are related to the fact that, in partition I, the division into slow and fast contributions is done in terms of physical degrees of freedom (namely, those of the solvent nuclei and those of the solvent electrons), whereas in partition II, the concept of dynamic and inertial response is exploited. This formal difference is reflected in the operative equations determining the two contributions to q as, in II, the slow term qin includes not only the contributions due to the slow Modern Theories of Continuum Models 117 degrees of freedom but also the part of the fast component that is in equilibrium with the slow polarization, whereas, in I, the latter component is contained in the fast term qel . This difference is made evident by the electrostatic equations defining the corresponding apparent surface charge densities V V or Partition I − = 4GS n in n out V V Partition II − =0 n in n out As two different partitions of the solvent charges are introduced, in order to obtain equivalent results, we have to use two different expressions for the nonequilibrium free energy, namely: ⎧ neq ! or or ⎪ Gel + Gor − 21 i VGS si qGS si − qKel si ⎨GK = or or Partition I si − 21 i VGS si qGS si Gor = i VK si qGS ⎪ ⎩ 1 0 el Gel = EK + 2 i VK si qK si ⎧ neq ⎪ Gdyn + Gin ⎨GK = in in Partition II Gin = i VK si qGS si − 21 i VGS si qGS si ⎪ ⎩ dyn 1 0 Gdyn = EK + 2 i VK si qK si In order to obtain a more compact formalism, from now on the partition II will be used. By introducing the following partitioning of the charges: dyn qKdyn = qGS + qdyn qKin = (1.164) in qGS after some algebra, we get Kneq Gneq − K = EGS 1 1 VGS si qGS si + Vsi " Pneq qel si " Pneq 2 i 2 i (1.165) which is parallel to that obtained for the equilibrium case but this time the last term is calculated using the dynamic charges qdyn . The vertical transition (free) energy to the excited state K is finally obtained by subtracting the ground state free energy GGS of Equation (1.160) to Gneq of EquaK tion (1.165): K0neq neq + K = EGS 1 Vsi " Pneq qdyn si " Pneq 2 i This equation shows that vertical excitations in solvated systems are obtained as a sum of two terms, the difference in the excited and ground state energies in the presence of a frozen ground state solvent and a relaxation term determined by the mutual polarization 118 Continuum Solvation Models in Chemical Physics of the solute and the solvent after excitation. The latter term is obtained taking into account the fast and slow partition of the solvent response. In the following section we shall show that it is this relaxation term that leads to differences in the two alternative SS and LR approaches State Specific vs. Linear Response The requirement needed to incorporate the solvent effects into a state-specific (multireference) method is fulfilled by using the effective Hamiltonian defined in Equation (1.159). The only specificity to take into account is that in order to calculate V̂ we have to know the density matrix of the electronic state of interest (see the contribution by Cammi for more details). Such nonlinear character of V̂ is generally solved through an iterative procedure [35]: at each iteration the solvent-induced component of the effective Hamiltonian is computed by exploiting Equation (1.157) with the apparent charges determined from the standard ASC equation with the first order density matrix of the preceding step. At each iteration n the free energy of each state K is obtained as GnK = Kn H 0 Kn + 1 n K V̂ Kn−1 Kn 2 i (1.166) where the solvent term V̂ Kn−1 has been obtained using the solute electronic density calculated with the wavefunction of the previous iteration.At convergence n and n−1 must be the same and Equation (1.166) gives the correct free energy of the state K. We note that this procedure is valid for states fully equilibrated with the solvent; the inclusion of the nonequilibrium effects needs in fact some further refinements. In particular, the inclusion of nonequilibrium effects requires a two-step calculation: (i) an equilibrium calculation for the initial electronic state (either ground or excited) from which the slow apparent charges, qs , are obtained and stored for the successive calculation on the final state; (ii) a nonequilibrium calculation performed with the interaction potential V̂ composed by two components: V̂ = Vfixed + Vchang Vfixed is constant as a result of the fixed slow charges qs of the previous calculation, while Vchang changes during the iteration procedure. It is defined in terms of the fast charges qf as obtained from the charge distribution of the solute final state. In order to derive the alternative LR equations, the effective Hamiltonian defined in Equation (1.159) has to be generalized as Heff t = H 0 + V t + Wt (1.167) where Wt is a general time-dependent perturbation term that drives the system and induces a time dependence in the solute–solvent interaction term V . This time dependence originates from dynamic processes involving inertial degrees of freedom of the solvent. The time scale of these processes is orders of magnitude higher than the time scale of the electron dynamics of the solute, and an adiabatic approximation can be used to follow the electronic state of the solute, which can be obtained as an eigenstate of the time-dependent effective Hamiltonian (Equation (1.167)). Modern Theories of Continuum Models 119 As for isolated systems, also for solvated ones, we can express the TD variational wave function t in terms of the time-independent unperturbed variational wave function t = 0 + 0 d + · · · and limit the time-dependent parameter d to its linear term [36]. Instead of working in terms of time, we then consider an oscillatory perturbation and express Wt by its Fourier component. In this framework, the linear term in the parameter assumes the form d = X exp−it + Y expit/2 where the (X, Y) vector is determined by solving the following system: W X + 1 − =0 W Y (1.168) where A B 1 0 − 1 − = B∗ A∗ 0 −1 (1.169) is the inverse of the linear response matrix for the molecular solute. In Equation (1.169) A and B collect the Hessian components of the free energy functional G with respect to the wave function variational parameters. The response matrix depends only on intrinsic characteristics of the solute–solvent system, and it permits one to obtain linear response properties of a solute with respect to any applied perturbation in a unifying and general way. The poles ±n of the response function give an approximation of the transition energies of the molecules in solution; these are obtained as eigenvalues of the system 1 − n Xn =0 Yn (1.170) where Xn Yn are the corresponding transition eigenvectors. This general theory can be made more specific by introducing the explicit form of the wavefunction; in such a way, by using an HF description, we obtain the random phase approximation (RPA) (or TDHF). Within this formalism, the free energy Hessian terms yield Bminj = mn ij + Bminj (1.171) Aminj = mn ij m − i + mj in + Bminj (1.172) where mn ij indicates two-electron repulsion integrals and r orbital energies. Here we have used the standard convention in the labelling of molecular orbitals, that is, i j for occupied and m n for virtual orbitals, respectively. In the definitions (1.171) and (1.172) the effect of the solvent acts in two ways, indirectly by modifying the molecular orbitals and the corresponding orbital energies (they are in fact solutions of the Fock equations including solvent reaction terms) and explicitly through the perturbation term Bminj [26]. This term can be described as the electrostatic interaction between the charge distribution 2m∗ 2i and the dynamic contribution to the solvent reaction potential induced by the charge distribution 2n∗ 2j and it can be written in 120 Continuum Solvation Models in Chemical Physics terms of the vector product between the electrostatic potential and the induced apparent fast charges, determined by the corresponding transition density charge, namely: , + 1 2i q dyn sk " 2 ∗ 2j Bminj = 2m (1.173) n − s r k k where the charges q dyn are calculated according to the partition II (Equation (1.163)) described in the section Operative Equations. A parallel theory can be presented for a DFT description; in this case the term TDDFT is generally used. Within this formalism an analogue of Equation (1.170) is obtained but now the orbitals to be considered are the occupied and virtual Kohn–Sham orbitals and the two-electron repulsion integrals have been replaced by the coupling matrix Kminj containing the Coulomb integrals and the appropriate exchange repulsion integrals determined by the functional used. We note, however, that the explicit solvent term has exactly the same meaning (and the same form) as the Bminj defined in the HF method (see Equation (1.173)). A Linear Response Approach to a State-specific Solvent Response In Equation (1.161) (or equivalently in Equation (1.165) for the nonequilibrium case) we have shown that excited state free energies can be obtained by calculating the frozenK and the relaxation term of the density matrix, P (or Pneq ) where the PCM energy EGS calculation of the relaxed density matrices requires the solution of a nonlinear problem in which the solvent reaction field is dependent on such densities. If we introduce a perturbative scheme and we limit ourselves to the first order, an approximate but effective way to obtain such quantities is represented by the LR scheme as shown in the following equations. K0 K Using an LR scheme, in fact, we can obtain an estimate of EGS = EGS − E GS which represents the difference in the excited and ground state energies in the presence of a frozen ground state solvent as the eigenvalue of the following non-Hermitian eigensystem (1.170) where the orbitals and the corresponding orbital energies used to build A and B matrices have been obtained by solving the SCF problem for the effective Fock (or KS operator), i.e. in the presence of a ground state solvent. The resulting eigenvalue 0K K0 is a good approximation of EGS in the sense that it correctly represents an excitation energy obtained in the presence of a PCM reaction field kept frozen in its GS situation. By using this approximation, the equilibrium and nonequilibrium free energies for the excited state K become: 0 Geq K = GGS + K + 0 Gneq K = GGS + K + 1 Vsi " P q si " P 2 i (1.174) 1 Vsi " Pneq qdyn si " Pneq 2 i (1.175) The only unknown term of Equations (1.174) and (1.175) remains the relaxation part of the density matrix, P (or Pneq ) (and the corresponding apparent charges q or qdyn ). These quantities can be obtained through the extension of LR approaches to analytical energy gradients; here in particular it is worth mentioning the recent formulation Modern Theories of Continuum Models 121 of TDDFT-PCM gradients [37]. In these extensions the so called Z-vector [38] (or relaxed-density) approach is used. The solution of the Z-vector equation as well as the knowledge of eigenvectors XK YK of the linear response system allow one to calculate P for each state K as: P = TK + ZK (1.176) where TK is the unrelaxed density matrix with elements given in terms of the vectors XK YK whereas the Z-vector contribution ZK accounts for orbital relaxation effects. Once P is known we can straightforwardly calculate the corresponding apparent charges qx = qx Px where ⎧ ⎪ ⎨x = Px = P ⎪ ⎩ x q = q ⎧ ⎪ ⎨ x = Px = Pneq ⎪ ⎩ x q = qdyn if an equilibrium regime is assumed if a nonequilibrium regime is assumed By introducing the relaxed density P and the corresponding charges into Equations (1.161) (or (1.165)) we obtain the first-order approximation to the ‘exact’ free energy of the excited state by using a linear response scheme. This is exactly what we have called the ‘corrected’ Linear Response approach (cLR) [33]. The same scheme has been successively generalized to include higher order effects [39]. 1.7.4 Conclusions In this contribution we have presented some specific aspects of the quantum mechanical modelling of electronic transitions in solvated systems. In particular, attention has been focused on the ASC continuum models as in the last years they have become the most popular approach to include solvent effects in QM studies of absorption and emission phenomena. The main issues concerning these kinds of calculations, namely nonequilibrium effects and state-specific versus linear response formulations, have been presented and discussed within the most recent developments of modern continuum models. In these concluding paragraphs it is useful to add that, besides vertical processes, polarizable continuum models can be (and have been) generalized to treat also more complex aspects of the relaxation of the excited state following the vertical excitation, or inversely that of the ground state after emission. These are more general dynamic processes in which solute and solvent dynamic behaviours mutually interact. In other contributions to the book some of these processes (such as excitation energy transfers and excitation-induced electron and proton transfers) are analysed in terms of the available models. Here, however, it is important to stress that in order to account accurately for the time dependence of the solvent response in many dynamic processes new ideas and new computational strategies are still required. A possible direction has recently been proposed in terms of solvent apparent charges continuously depending on time [33, 40]. 122 Continuum Solvation Models in Chemical Physics These are obtained by introducing an explicit time dependence of the permittivity. This dependence, which is specific to each solvent is of a complex nature, cannot in general be represented through an analytic function. What we can do is to derive semiempirical formulae either by applying theoretical models based on measurements of relaxation times (such as that formulated by Debye) or by determining through experiments the behaviour of the permittivity with respect to the frequency of an external applied field. It is evident that these ideas represent only a preliminary indication of a possible direction to follow which is certainly not the only one or maybe not even the best one, but the good news is that something is moving. We are thus quite confident that now it is time for continuum models to take a new important step further and to extend their application to real time-dependent phenomena. However, this extension should not be done independently of the experience achieved in past years on more standard applications of the models to study energy/geometries and properties of solvated systems. From these studies in fact it appears evident that continuum only approaches are often too simplistic and their combinations or couplings with discrete approaches are not only beneficial but in some cases essential. It seems thus necessary to accept from the very beginning that hybrid or combined approaches, mixing not only different levels of calculation (as for example in QM/MM or other similar methods nowadays largely diffused) but also different ‘philosophies’ (as for example continuum and discrete descriptions but also electronic calculations and statistical analyses), represent very promising strategies. 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Mennucci, Theor. Chem. Acc.: Theor., Comput., Model., 116 (2006) 31. This page intentionally left blank 2 Properties and Spectroscopies 2.1 Computational Modelling of the Solvent–Solute Effect on NMR Molecular Parameters by a Polarizable Continuum Model Joanna Sadlej and Magdalena Pecul 2.1.1 Introduction The purpose of this chapter is to present an overview of the computational methods that are utilized to study solvation phenomena in NMR spectroscopy. We limit the review to first-principle (ab initio) calculations, and concentrate on the most widespread solvation model: the polarizable continuum model (PCM), which has been largely described in the previous chapter of this book. NMR spectroscopy is one of the most important techniques available for investigating molecular structure, molecular interactions and the solvation problems. Most NMR measurements are performed on liquid samples (or in the solid state, but this branch of NMR spectroscopy does not concern us here). Such liquid state experiment yield isotropic chemical shifts (related to the nuclear magnetic shielding constants) and scalar spin–spin coupling constants. NMR parameters (in particular NMR chemical shifts) are sensitive to the molecular environment, and only exceptionally the NMR parameters of a molecule in the liquid phase or in solution may be close to those of the gas phase molecule. More often, as for example in aqueous solutions, there are strong interactions between the solute and the solvent, and the difference between gas phase and liquid phase NMR parameters is substantial. Therefore, theoretical methods capable of modelling liquid phase NMR parameters are in great demand. Such simulations help scientists to understand better the relationship between NMR parameters and the structure of liquids, and they are indispensable for a realistic modelling of NMR parameters Continuum Solvation Models in Chemical Physics: Theory and Applications © 2007 John Wiley & Sons, Ltd Edited by B. Mennucci and R. Cammi 126 Continuum Solvation Models in Chemical Physics as a function of conformation (since the solvent and conformational effects are interrelated). This aspect is of particular importance since NMR spectroscopy is nowadays one of the most widespread methods of conformational analysis. In the last 15–20 years first-principle calculations of NMR parameters in solutions have become possible. In this contribution we will outline the current approaches used in these investigations, and briefly review the results obtained using them. The solvation models have been previously reviewed in refs. [1–4] while for ab initio calculations of NMR parameters the reader is referred for example to Ref. [5]. Ref. [6] discusses environmental effects on the NMR parameters. Theoretical bases of continuum models including their mathematical formulation and numerical implementation have already been discussed in the previous chapter of this book. We have therefore restricted our review to the environment effects on the NMR observables, without going into the theory of continuum models. This contribution is divided into five sections. After the Introduction, the definitions of the NMR parameters are recalled in the second section. The third section is focused on methodological aspects of the calculation of the NMR parameters in continuum models. The fourth section reviews calculations of the solvent effects on the nuclear magnetic shielding constants and spin–spin coupling constants by means of continuum models, and the final section presents a survey on the perspectives of this field. 2.1.2 Theory of the Magnetic NMR Parameters for Isolated Molecules Before dealing with solvent effects on the NMR parameters we will briefly present the basic nonrelativistic quantum theory of the NMR parameters, as first derived by Ramsey [7, 8]. Effective NMR Spin Hamiltonian NMR spectra arise from the absorption of electromagnetic radiation by nuclei with nonzero magnetic moment MK , i.e. from radiative transitions between nuclear spin energy levels, which are split in the presence of an external magnetic field B [5, 7, 8]. The empirical interpretation of NMR spectra consists in finding two types of static parameters: shielding constants K (or chemical shifts K ) and spin–spin coupling constants JKL to fit the observed spectrum, using the effective spin Hamiltonian. The general NMR effective spin Hamiltonian has the following form: Ĥ = − K BT 1 − K MK + 1 T M D + KKL ML 2 k=ll K KL (2.1) MK denotes the nuclear magnetic dipole moment operator, obtained by multiplication of the nuclear spin operator IK by the magnetogiric factor K . MK = K IK (2.2) B denotes the external magnetic field. The tensor values appearing in Equation (2.1) have the following meanings: • The NMR shielding constant K describes a modification of the external magnetic field at the nucleus by the presence of electrons. Properties and Spectroscopies 127 • The direct (dipolar) nuclear spin–spin coupling constant DKL represents the classical throughspace interaction of the magnetic moments of nuclei K and L. DKL = 2 0 3RKL RKL − 1RKL 5 4 RKL (2.3) • The reduced indirect (scalar) nuclear spin–spin coupling constant KKL describes the interaction of the magnetic nuclei, transmitted through the surrounding electrons. KKL is four orders of magnitude smaller than DKL . DKL KKL and K are second rank tensors. DKL , in contrast to KKL and K , is a traceless tensor, and vanishes upon spacial averaging. Hence, the parameters relevant for a rapidly tumbling molecule (as in a liquid or in a gas phase) are the isotropic spin–spin coupling constant and the shielding constant. 1 1 Kiso = Tr K = K xx + K yy + K zz 3 3 1 1 iso KKL = TrKKL = KKL xx + KKL yy + KKL zz 3 3 (2.4) (2.5) In experimental practice the isotropic NMR shielding constant is replaced by the NMR chemical shift and the spin–spin coupling constant JKL is used instead of the reduced spin-spin coupling constant KKL . = ref − sam JKL = h K L KKL 22 (2.6) (2.7) ref in Equation (2.6) denotes the isotropic shielding constant of the nucleus in the reference molecule and sam denotes the isotropic shielding constant of the nucleus in the molecule under investigation. Absolute shielding scales, derived from accurate nuclear spin–rotation tensors measured in high resolution microwave techniques, are available for numerous nuclei. NMR Parameters as Energy Derivatives The NMR parameters can be expressed as energy derivatives. The nuclear shielding constant is then equal to K = d2 E B M +1 dB dMK B=0M=0 (2.8) and the reduced nuclear spin–spin coupling constant is equal to d2 E B M KKL = − DKL dMK dML B=0M=0 (2.9) 128 Continuum Solvation Models in Chemical Physics Sum-over-states Expression The time-independent second-order properties, such as the shielding constant or the nuclear spin–spin coupling constant (see Equations (2.8) and (2.9)) can be expressed by means of the perturbation theory as dH dH 2 2 dH 0 dxi n n dxj 0 d E x 0 − = 0 (2.10) dx dx dx dx E −E i j i n=0 j n 0 where the derivatives are taken at zero perturbation x (here magnetic moments of the nuclei MK or induction of external field B). In the case of magnetic properties the first contribution (the expectation value) is called the diamagnetic part and the second sumover-states contribution is known as the paramagnetic part. The phenomenon of NMR properties was analysed theoretically in terms of perturbation theory by Ramsey [7, 8], and the resulting expressions are given below. In the equations below, i and j indices are used for electrons, K and L indices are used for nuclei, is a fine-structure constant ≈ 1/137, and mi is the permanent magnetic moment of the electron. The first derivatives of nonrelativistic molecular electronic Hamiltonian of a system with magnetic nuclei in a static magnetic field with respect to the induction of the external magnetic field B (for zero field) and with respect to the magnetic moments of the nuclei M (also for their zero values), entering the paramagnetic part of the shielding constant and the spin–spin coupling constant are dH = hBorb + hBspn dB dH = hKpso + hKsd + hKfc dMK (2.11) (2.12) For closed-shell systems hBspn vanishes. The orbital operator hBorb = 1 l 2 i iO (2.13) is a singlet operator contributing to the shielding constant. From the three operators obtained by differentiating the Hamiltonian with respect to the nuclear magnetic moments (Equation (2.12)) only the singlet paramagnetic spin–orbital (PSO) operator hKpso = 2 liK 3 i riK (2.14) contributes to the NMR shielding constant. The singlet paramagnetic spin–orbital (PSO) operator (Equation (2.14)), the triplet Fermi contact (FC) operator hKfc = − 82 riK mi 3 i (2.15) Properties and Spectroscopies 129 and the triplet spin–dipole (SD) operator hKsd = 2 2 riK mi − 3 mi · riK riK i 5 riK (2.16) contribute to the spin–spin coupling constant. As a rule, the dominant contribution to the isotropic coupling originates from the FC term. The Diamagnetic Contributions The diamagnetic contributions to the NMR parameters arise from the operators obtained by double differentiation of the Hamiltonian (see Equation (2.10)) d2 H dia = −1 + hBK dBdMK (2.17) d2 H dso = DKL + hKL dMK dML (2.18) dia dso and hKL have the form The diamagnetic operators hBK dia = hBK T 2 riO · riK 1 − riK riO 3 2 i riK (2.19) dso = hKL T 4 riK · riL 1 − riK riL 3 3 2 i riK riL (2.20) and The diamagnetic spin–orbital (DSO) contribution to the spin–spin coupling constants is usually small, but nonnegligible, especially for the proton–proton coupling constants. It should be noted here that for the approximate wavefunctions the partitioning of the NMR shielding constant into the dia- and paramagnetic parts depends on the choice of the gauge origin (see the next paragraph), even when the total shielding constant does not. Therefore this partitioning does not correspond to that introduced by Ramsey [7] and causes some problems with the physical interpretation of dia- and paramagnetic parts [5]. Field-dependent Orbitals The vector potential A of the magnetic field, in contrast to the physical field B, is not defined unambiguously and depends on the choice of the gauge origin. For the approximate wavefunctions this dependence is transferred to the NMR shielding constants. This difficulty can be overcome by using orbitals containing the phase factor, ensuring the independence of the calculated integrals from the choice of the gauge origin for A. The phase factor can be attached to the atomic orbitals, which results in the Londontype atomic orbitals LAOs [9, 10] (also known as GIAO gauge independent atomic orbitals). Also the molecular orbitals can be transformed in this way, which is employed 130 Continuum Solvation Models in Chemical Physics in IGLO [11, 12] (individual gauge for localized orbitals) and LORG [13] (local origin) methods. One can also use CSGT (continuous set of gauge transformations) [14], which defines a gauge as dependent on the position where the induced current is to be calculated (which makes the diamagnetic part vanish analytically). London-type atomic orbitals r B have the form 1 r B = exp − iB × RN − RO · r r 2 (2.21) r is a standard atomic orbital used in the calculations (e.g. Gaussian orbital with spherical harmonics), centred on the nucleus N at a position RN RO is a gauge origin for the vector potential. 2.1.3 Theory of the Magnetic NMR Parameters in Solution General Features of PCM Models The theories behind continuum solvation models have been presented extensively in various reviews [1–4] and in other contributions to this book, so we do not repeat them here, focusing instead on their application to calculations of the NMR parameters. There are three main groups of methods for evaluating the effects of surrounding solvent effects on NMR parameters [3,5]: (I) supermolecular calculations, where both the solute molecule and some neighbouring molecules of the solvent are explicitly included in the quantum mechanics (QM) calculations; (II) continuum models, in which the solvent is modelled as a macroscopic continuum dielectric medium (assumed homogeneous and isotropic) characterized by a scalar dielectric constant. The solute, placed in a cavity in a dielectric medium, is described at the QM level, while the solute–solvent interaction is described as a mutual polarization of solute and solvent; (III) combined molecular dynamics MD/QM approach, where cluster of molecules representing the molecule of interest surrounded by solvent are generated using Monte Carlo simulations or from single configuration (snapshots) of a classical simulation trajectory using MD simulations. Each cluster is treated as a supermolecule in a quantum chemical calculation and the average is obtained to yield the NMR parameters in the liquid phase; thus the solvent maintains its microscopic nature. We have restricted our review mainly to the methods of group (II), i.e. continuum models (including combined methods (I) and (II)), treating the other methods only as a reference frame for them. Methods based on the solvent reaction field philosophy differ mainly in: (i) the cavity shape, and (ii) the way the charge interaction with the medium is calculated. The cavity is differently defined in the various versions of models; it may be a sphere, an ellipsoid or a more complicated shape following the surface of the molecule. The cavity should not contain the solvent molecules, but it contains within its boundaries the solute charge distribution. The solvent reaction potential can be partitioned into several contributions of different physical origin, related to electrostatic, repulsive, induction and dispersion interactions between solute and solvent. In the original polarizable continuum approach only the electrostatic and induction terms are explicitly considered as an interaction potential Properties and Spectroscopies 131 Vr , to be added to the Hamiltonian of the solute molecule in the vacuum in order to obtain the effective Hamiltonian. To compute the electrostatic component of the solvation free energy this model requires the solution of a classical electrostatic Poisson problem. Nowadays, the most popular method of solution of this problem is a polarized continuum model developed primarily by the Pisa group of Tomasi and co-workers [1, 3, 4]. In this approach the cavity surface is divided into a number of small surface elements, where the reaction field is modelled by distributing the charges onto the surface elements, i.e. by creation of apparent surface charges [15–18]. The electrostatic part of the solvent–solute interaction represented by the charge density spread on the cavity surface (apparent surface charges, ASC) gives rise to a specific operator to be added to the Hamiltonian of the isolated system to obtain the final effective Hamiltonian and the related Schrödinger equation: H0 + VR >= E > (2.22) where H0 is the Hamiltonian in the absence of the solvent, and, VR , the solvent operator acting on is defined in terms of the surface apparent charge, and depends on the solute charge distribution. Apart from the ASC–PCM method developed by the Pisa group, there are several other methods based on the polarizable continuum model: the MPE (multipole expansion method) by the Nancy group [19, 20] and by Mikkelsen and co-workers [21, 22], the GBA (generalized Born approximation) by the Minneapolis group – Cramer and Truhlar [23–26] and others. There are currently three different approaches for carrying out ASC–PCM calculations [1, 3]. In the original method, called dielectric D–PCM [18], the magnitude of the point charges is determined on the basis of the dielectric constant of the solvent. The second approach is C–PCM by Cossi and Barone [24], in which the surrounding medium is modelled as a conductor instead of a dielectric. The third, IEF–PCM method (Integral Equation Formalism) by Cances et al., the most recently developed [16], uses a molecular-shaped cavity to define the boundary between solute and dielectric solvent. We have to mention also the COSMO method (COnductorlike Screening MOdel), a modification of the C–PCM method by Klamt and coworkers [26–28]. In the latter part of the review we will restrict our discussion to the methods that actually are used to model solute–solvent interactions in NMR spectroscopy. To characterize the intermolecular interactions it is necessary to take into account the nonelectrostatic terms. There are different approaches to the modelling of repulsion and dispersion interactions. Recently, Amovilli and Mennucci have described an approach where repulsion and dispersion terms are computed self-consistently as part of the reaction field operator [29]. Solvent Effects on the NMR Parameters Solvent effects on nuclear magnetic properties are well known, and have been studied for a long time. Both the NMR shielding constant and the nuclear spin–spin coupling constant depend on the electronic structure of the whole system. This means that both are sensitive to the weak intermolecular interactions between solute and solvent molecules. 132 Continuum Solvation Models in Chemical Physics Shielding constants A good starting point for investigation of the concept of the environment-induced change in the shielding constants is the phenomenological solvent model by Buckingham, where the solvent effect is assumed to be the sum of additive terms [30] = 0 + s = 0 + b + a + w + E (2.23) where 0 is the ro-vibrationally averaged shielding constant for the isolated molecule, while s denotes the contribution to the nuclear shielding constant due to the presence of the solvent. The four terms in the solvent part are defined as follows: b is the contribution from the bulk magnetic susceptibility of the medium, a is the contribution which arises from the anisotropy in the magnetic susceptibility of the solvent molecule, w means the contribution from van der Waals interactions between solute and solvent, and finally E is due to the electric field coming from the charge distribution of the solvent molecules, i.e. it arises from the electrostatic and induction interactions. Our aim is to discuss the environment-induced changes of the NMR parameters arising from intermolecular interaction between the solute and the solvent molecules. We omit the change in the chemical shift due to a difference in the bulk magnetic susceptibility of the solute and the solvent, which depends on the shape of the sample and which can be corrected. From the point of view of the Buckingham formula (Equation (2.23)) only the effect of long-range electrostatic and induction interactions E of the solvent molecule with the reaction field is included in the traditional methods of the (II) group (continuum models). Contrary to that, the supermolecular approach (I) or combined MD/QM methods (III) includes the short-range term a and the long-range w and some of the E term. There have been other phenomenological approaches to rationalize (or even predict) the experimentally observed solvent effect on the chemical shift. Many chemists use the Kamlet–Abbout–Taft (KAT) set of solvatochromic parameters ∗ and [31]. KAT parameters can be used together with the multiple linear analysis to describe the variation in the chemical shift of the solute as the solvent is varied. An extensive study of this type was conducted by Witanowski et al. to interpret the solvent effects on the shielding of 14 N in a large set of compounds (see ref. [32] and references cited therein). For a nitroso aliphatic and aromatic series, solvent-induced shielding was indeed found to depend on the polarity of the solvent. However, other experience with this model suggests the need for caution. Spin–spin coupling constants The solvent effects on the spin–spin coupling constants are less frequently investigated than those on the shielding constats, since they tend to be much smaller. An equation analogous to Equation (2.23) was proposed for the spin–spin constants by Raynes [33]: J s = J m + Jc + Jw + JE (2.24) where Jm , analogous to b is proportional to the bulk magnetizability of the solvent, Jc denotes the influence of specific interactions (e.g. charge transfer or hydrogen bonding), Jw means the dispersion effects and JE denotes the effect of electrostatic contributions. Properties and Spectroscopies 133 A weakness of this model is that the separation of the electrostatic and the so-called specific solute–solvent interactions is not defined. In practice, two main approaches are used to account for solvent effects on the spin–spin coupling constants: the continuum and the supermolecular methods. The combined MD/QM approach is rarely used for the purpose, since calculations of the spin–spin coupling constants are much more time consuming than those of the shielding constants and the MD/QM approach is too expensive for the former. NMR Parameters as Defined in the PCM Model For a molecule in solution described by the PCM model, the nuclear shielding constant and the indirect spin–spin coupling constants are determined as second derivatives of the free energy functional G of the solute–solvent system [34]: K = d2 G dBdMK (2.25) and JKL = h K L d2 G KKL = h K L 2 2 2 2 dMK dML (2.26) G is the fundamental energetic quantity which determines the behaviour of the system in the presence of internal and external perturbations. It includes the changes in internal energy of the solvent arising from the solvent–solute interaction. Thus, the free energy is related to the Schrödinger energy E by G=E− 1 < VR > 2 (2.27) where is the solute wavefunction and VR has been defined in Equation (2.22). The functional to be minimized is constructed as below in the new implementation of the PCM model [29], including the repulsion and dispersion terms Grep and Gdis while in the former PCM scheme the functional included the polarization term Gpol only. G = Gsolute + Gsolvent + Gpol + Grep + Gdis (2.28) The form of the free energy functional G appearing in the Polarizable Continuum Model is discussed in refs [35–37]. Recently, Mennucci and Cammi have extended their integral equation formalism model for medium effects on shielding to the NMR shielding tensor for solutions in liquid crystals [38, 39]. The implementation of various methods for computing solvent effects on the NMR parameters in Gaussian [40] and DALTON [41] has made these methods more popular. From a computational point of view, the effects of the surrounding medium on the NMR parameters can be divided into direct and indirect solvent effects [5]. The direct effects arise from the interaction of the electronic distribution of the solute with the surrounding medium, assuming a fixed molecular geometry, while indirect (secondary) effects are caused by the changes in the solute molecular geometry by the solvent. Experimentally the total effect is observable, while in the computational models they can be separated. 134 Continuum Solvation Models in Chemical Physics 2.1.4 Review of the Numerical Results for Shielding Constants and Spin–Spin Coupling Constants The Shielding Constants The applications of continuum models to the study of solvent induced changes of the shielding constant are numerous. Solvent reaction field calculations differ mainly in the level of theory of the quantum mechanical treatment, the method used for the gauge invariance problem in the calculations of the shielding constants and the approaches used for the calculations of the charge interaction with the medium. Most of the quantum chemical calculations of the nuclear shielding constants have involved two classes of solvation models, which belong to the second group of models (II), namely, the continuum group: (i) the apparent surface charge technique (ASC) in formulation C–PCM and IEF–PCM, and (ii) models based on a multipolar expansion of the reaction filed (MPE). The PCM formalism with its representation of the solvent field through an ASC approach is more flexible as far as the cavity shape is concerned, which permits solvent effects to be taken into account in a more accurate manner. The solvent reaction field calculations involve several different aspects. We would like concentrate on the points required to make these models successful as well as on the facts that limit their accuracy. One of them is the shape of the molecular cavity, which can be modelled spherically or according to the real shape of the solute molecule. First, we discuss the papers in which spherical cavity models were applied. The studies utilizing the solute-shaped cavity models are collected the second group. Finally, the approaches employing explicit treatment of the first-solvation shell molecules combined with the continuum models are discussed. Spherical cavity models Most of the studies employing a spherical cavity have been carried out using the MPE approach of Mikkelsen and co-workers [21, 22]. Mikkelsen and co-workers have studied the dependence of nuclear shieldings and magnetizabilities on the cavity size, the dielectric constants and the order of the multipole expansion for small molecules H2 O CH4 using the GIAO–MCSCF/6-311++G(2d,2p) method (multiconfiguartion self-consistent field, MCSCF) [36]. The cavity radius has been chosen as the distance of the centre of mass from the most distant atom plus the van der Waals radius of that atom. The multipole expansion is converged only after inclusion of six terms. Both direct and indirect (due to the relaxation of the geometry) solvent effects give contributions to the solvent shift. Moreover, the results are quite sensitive to the cavity radius. The linear response GIAO–MCSCF/MPE method has been used also to study the solvent effects on the proton and selenium chemical shifts of H2 Se using the ANO basis sets [35]. A gas-to-liquid downshift of ca. 127 ppm has been observed experimentally for selenium shielding. The calculations reveal the importance of the geometry effects: the bond length is slightly reduced in the dielectric medium, while the bond angle is increased. Large positive solvent shifts have been calculated for the shielding constants of sulfur and nitrogen nuclei in H2 S and HCN, while the shielding constants for carbon in HCN has been found to decrease as the polarity of the medium is increased [42]. Another work of this group is the investigation of the influence of the intermolecular interaction on the shielding constants of acetylene [43]. The reaction field calculations Properties and Spectroscopies 135 have been carried out for several solvents (cyclohexane, benzene, chloroform, acetone, acetonitrile, water). However, in this case the bulk solvent effects on the acetylene 13 C shielding constant estimated by the reaction field method are in disagreement with experiment (they are underestimated by one order of magnitude). This can be attributed to the limitations of the reaction field method alone, since the comparison of the SCF and CASSCF reaction field results indicate that the underestimation of the correlation effects is not the major source of errors. The poor performance of the GIAO/MPE model in this case is probably due to neglect of the influence of the anisotropic magnetizability of close-lying solvent molecules and short-range repulsive terms omitted in classical continuum models. The spherical shape of the cavity may also contribute. There is a lot of experimental data of the 14 N solvent shifts of N -methyl-substituted azoles (pyrroles, pyrazole, triazole, and tetrazoles) compounds and the 14 N shielding is particularly sensitive to solvent influence, so the continuum model calculations of 14 N shielding in these compounds bring interesting results [44]. The authors used the GIAO–MCSCF/MPE response method with a Huzinaga II basis set for the electronic calculations of the 14 N shielding constant [21, 22, 45] for a number of different solvents. As usual, the radius of the spherical cavity has been determined by the sum of the largest distance from the centre of mass to the outermost atom and van der Waals radius of that atom. It has been found that the calculated 14 N shielding constant decreases with the increase of a static dielectric constant. The magnitude of the experimentally found and the computed shifts is generally in agreement, except for systems where specific solute– solvent interactions such as hydrogen bonding affect the nitrogen atoms for which the NMR shielding is considered. The importance of the optimization of the geometry for each dielectric constants in the MPE method at RASSCF/ANO level has been studied by Åstrand et al. [46] for the case of nuclear shielding constants of the fluoromethanes in the gas phase and solution. The anisotropy part of the fluorine shielding of the CH3 F changes sign in comparison to the change observed for fixed geometry calculations. This strongly suggests that it is crucial to optimize the geometry for each dielectric constant. Solute-shaped cavity models The main advantage of the ACS–PCM methods is their great flexibility in the definition of the molecular cavity, which can be modelled according to the real shape of the solute molecule. GIAO–SCF (and CSGT)/6-31G∗ and SCF/6-311+(2d,p) calculations in the framework of the ASC model were performed for the chemical shifts of acetonitrile and nitromethane by Cammi [34]. The solute cavity was defined by interlocking spheres centred on the solute nuclei with the radii equal to 1.2 times the corresponding van der Waals radius. The author found that the 14 N, 13 C and 1 H shielding constants decrease with the increase of the dielectric constant, while the 17 O shielding of nitromethane increases. The solvent indirect effects on the shielding constants of nitromethane are more pronounced than those in acetonitrile. In the case of N both direct and indirect effects have the same sign, while for C and O the two contributions have the opposite sign. The conclusion from this paper is that the ASC model alone is not sufficient to recover the whole solvent effect observed experimentally. The source of its difficulty in reproducing the experimental solvent effect is the lack of the solvent susceptibility anisotropy term 136 Continuum Solvation Models in Chemical Physics denoted in Equation (2.23) as a . This effect can be included by introducing an explicit solvent shell around the solute molecule. A similar system to that discussed in ref. [44] (tetrazine, tetrazole and pyrrole) has been studied by Manalo et al. [47] by means of the CSGT/ASC method at the B3LYP/6311++G(2d,2p) level. The cavity was defined by using the Pauling radius for each solute atom. In this paper the effects of geometric relaxation (indirect effects) are found to be small, and the direct influence of the intensity of the solvent reaction field on the shielding constants dominates. However, the indirect effect has been found to be important for N N -dimethylacetamidine in IEF-PCM calculations [48]. In refs. [49, 50] the need for a good parameterization of the cavity to calculate NMR properties was discussed. One of the largest solvent-induced changes on nitrogen shielding (the cyclohexane-to-water change) of 41 ppm is found for 1,2-diazine [50]. To improve the average agreement between calculated and experimental gas-to-solution shifts, it is found necessary to enlarge the molecular cavity. This has worked well for nonprotic solvents such as DMSO or cyclohexane, but not for water, since for this solvent’s hydrogen bond effects are important and specific terms are required. These calculation have been performed using the GIAO B3LYP/6-311+G(d,p) approach and IEF-PCM formalism. Good results in the interpretation of the solvent effects in the amino acids glycine and alanine have been obtained by means of GIAO-IEF at the B3LYP/6-31G(d) level [51]. The cavity has been formed by interlocking spheres centred on selected nuclei with radii defined according to the topological state of each nucleus (united atom topological model, UATM [52]. The same formalism has been used for cystosine tautomers [53] and 2-amino-3-mercaptopropionamide [54]. The oxygen chemical shifts in N -methylformamide and acetone have been investigated by Barone et al. [55]. The PCM model with standard atom radii has not been able to reproduce the experimental chemical shift in this case. The authors noted the need for a careful parameterization of the cavity in the solvation model. A cavity defined as an isodensity surface has been used for study of the solvent effects on oxygen chemical shifts of the polyoxides CH3 On H and CH3 On CH3 n = 2 3 4 by GIAO-MP2 and GIAOCCSD(T) methods using a reaction field with the self-consistent isodensity polarized continuum approach SCI PCM [56]. Cavity size has also been the subject of investigations by Zhan and Chipman in ref.[57]. GIAO-HF/6-311G(2d,p) calculations have been carried out for nuclear shielding of nitrogen in CH3 CN CH3 NO2 CH3 NCS, with the solvent simulated by means of the PCM model [58]. A solute electronic isodensity contour has been used to define the cavity surface. The main conclusion from this important paper is that, because of the sensitivity of the final results to cavity size, a treatment that also includes volume polarization effects arising from penetration of the solute charge density outside the cavity is very desirable. Mixed continuum–discrete solvation models Let us now review the group of papers discussing the relative weights of the different components in Buckingham equation (Equation (2.23)). Reaction field methods describe only long-range electrostatic interaction, the E term (or, as in IEF-PCM, some of the w term [29]). In order to go beyond the continuum model some solvent molecules Properties and Spectroscopies 137 interacting with the solute have to be treated quantum mechanically or a classical/quantum molecular dynamics simulation of the system should be run to extract a number of configurations of the solute molecules interacting with some solvent molecules from a trajectory. With such supermolecular calculations one can describe the short-range interactions. This approach is now widely used to study gas-to-liquid chemical shifts, which allows us to study the limitations of the PCM model. The combination of the supermolecular approach with the continuum theory is believed to give an effective method of investigation of the solvent effects. The nitrogen shielding constants of pyridine and acetonitrile in chloroform have been studied in an important paper by Mennucci et al. [59] within the B3LYP/6311+G(d,p) model using the GIAO/IEF-PCM framework. The solute–solvent clusters have been obtained through MD shots taken at different simulation times. It has been found that for pyridine the long-range dielectric interactions are the dominant solvent effects (thus solvent shifts in this molecule are successfully reproduced by PCM), while PCM cannot reproduce the experimental results in the case of acetonitrile, where the short-range interactions are important. Taking into account the cluster obtained by the MD simulation gives a good agreement with experimental results. The good performance of the MD/supermolecular approach has later been confirmed by the combined MD/DFT calculations for 14 N and 13 C chemical shifts in nitrobenzaldehyde guanylhydrazones in DMSO by the Pereira group [60] and of nitroamidazoles in water [61]. The combined strategy of calculating the 19 F chemical shifts has been studied for fluorobenzenes [62] in several solvents. Here w has been found to be the dominant contribution to the total solvent-induced change of chemical shift; the authors have neglected the solvent magnetic anisotropy contribution a which is related to the shortrange interactions. To obtain the agreement with the experimental data, the term E has been scaled by a factor of 4.4. Recently Mennucci et al. have studied the competitive effects due to short-range and long-range forces taken into account through a discrete, a continuum or a combined description of the solvent for gallic acid [63] and N -methylacetamide as a model of peptide linkage [64, 65] (using B3LYP/GIAO and the IEF–PCM model). The conclusion from this series of papers is the need for an appropriate consideration of specific effects of those solvent molecules that interact directly with the solute moieties. The inclusion of explicit solvent molecules is crucial, although the long-range effects, described by means of continuum models, are also important. The most fascinating story of the calculations of the solvent-induced changes of the 17 O shielding constant is the simulation of the gas-to-liquid chemical shifts for water. Liquid water continues to be a challenge for prediction of intermolecular effects on shielding. The experimental gas-to-liquid chemical shift in water is −36 ppm for 17 O at room temperature [66] and −4 3 ppm for 1 H [33]. Of the two, the proton gas-to-liquid chemical shift is much easier to calculate. Mikkelsen et al. [37] and Klamt et al. [26] have predicted correctly the proton gas-to-solution chemical shift using quantum chemical calculations for optimized clusters of water molecules with inclusion of the solvent by continuum MPE and COSMO methods, respectively. However, the reaction field models are inadequate for the 17 O chemical shift water problem, even yielding the incorrect sign for the liquid shift of the 17 O shielding constant [67]. 138 Continuum Solvation Models in Chemical Physics An appropriate treatment of molecular properties of liquids requires the molecular motion to be explicitly taken into account using molecular dynamics. Small representative clusters of water molecules have been extracted from such simulations and they have been used to calculate the 17 O shielding constants by Malkin et al. [68]. The results obtained using the MD/DFT approach seem to be promising: the calculated oxygen liquid shift is in qualitative agreement with experiment, although the results depend strongly on the chosen interatomic potential and the cluster size. Pfrommer et al. [69] have used the Car–Parrinello method to model liquid water and hexagonal ice. The authors have found a gas-to-liquid shift of −5 8 ppm for protons and −36 6 ppm for oxygen nuclei, in good agreement with experiment (see above). Recently, Pennanen et al. [70] have presented calculations of the 17 O and 1 H shielding constants for the configurations of water molecules obtained using the first principles molecular dynamics simulation by means of the Car–Parrinello method. Clusters representing the low-density gas and the local liquid structures have been used as input data for B3LYP calculations. The authors obtained gas-to-liquid shifts of −41 2 ppm for 17 O and −5 27 ppm for 1 H. The first supermolecular calculations of the 17 O chemical shifts for small rigid water clusters have been performed by Chesnut [71] at the MP2(all–electron)/6-311+G(d,p) level. Recently, Klein et al. [67] analysed the 1 H and 17 O shieldings in water clusters explicitly by the supermolecular method using hybrid density functional MPW1PW91 in conjunction with the 6-311+G(2d,p) basis set. The authors have found that the 17 O shift is sensitive to the ligand environment [67]. For the oxygen atoms in four–coordinated water molecules in clusters containing the multiple interlocking five–membered rings the 17 O chemical shift approaches the asymptotic value of 272 ppm. This means that the calculated reduction of the 17 O chemical shift from monomer to highly ordered clusters is ca. 55 ppm. However, among these structures there is no model with water molecule surrounded by two hydration shells such as that expected to be formed in liquid water. Such a supermolecular calculation (B3LYP/aug-cc-pCVDZ) has been done in ref. [72]. The 17 O shielding constant decreases as the cluster size increases and these changes are dependent on the ligand environment. The highly dynamic nature of liquid water requires averaging over a distribution of hydrogen-bond geometries. The Spin–Spin Coupling Constants The spin–spin coupling constants, usually less changeable with the environment than the shielding constants [6], have consequently attracted less attention from theoretical chemists. Moreover, they are, as a rule, more difficult to calculate accurately, on account of large triplet instability effects affecting the FC and SD terms and a larger dimension of the perturbation. As a result, only in the recent years ab initio calculations of the spin–spin coupling constants by means of continuum models have been reported in the literature. As in the case of the shielding constants, the continuum model approaches to the calculation of spin–spin coupling constants are divided into those employing a spherical cavity (such as the MPE model of Mikkelsen and co-workers [22, 45, 73]) and those employing a molecule-shaped cavity (IEF-PCM [16], COSMO [26–28]). Spherical cavity models The computational model capable of yielding accurate spin–spin coupling constants is the multiconfigurational self-consistent field (MCSCF) model, and before the advent of density functional theory, spin–spin coupling constants in small systems were often Properties and Spectroscopies 139 calculated by means of it. Linear response MCSCF theory has been combined with the continuum model by Åstrand et al. in the reaction field MPE model of Mikkelsen [22,45, 73], and it has been used to model solvent effects on the spin–spin coupling constants [35] in hydrogen selenide. Åstrand et al. [35] have investigated the relative magnitudes of solvent effects due to polarization of the electronic charge distribution upon solvation and due to geometry changes, and have found that, while for 1 JSeH the former effect prevails, for 2 JHH both are equally important. The MPE model combined with linear response MCSCF theory has been used also to study solvent effects on the spin–spin coupling constants of H2 S [42], HCN [42], acetylene [43], methanol and methylamine [74], and water [75]. The effect of the dielectric medium on the spin–spin couplings in H2 S [42] has been found to be relatively substantial, namely 10 % for 1 JSH and 8 % for 2 JHH [42]. The greatest effect (in absolute terms) has been found for 1 JCH in the case of HCN [42]. The sensitivity of 1 JCH to the molecular environment has been confirmed also in several prior and later studies [43, 74, 76–79]. Once again it has been shown in ref. [42] that for some spin–spin coupling constants (e.g. 2 JHH in H2 S) the geometric relaxation effect may dominate the total change caused by the presence of a dielectric medium. The MPE study of the dielectric environment effects on the spin–spin coupling constants of acetylene [43] allowed for a comparison with experimentally measured gasto-solution shifts for a series of solvents of varying polarity. It has been found in the experimental study that 1 JCC changes considerably with the solvent, and that the changes correlate approximately with the solvent polarity. This tendency has been qualitatively reproduced by the MPE MCSCF linear response calculation, although the calculated changes constitute only approximately 30 % of the experimental shifts. The MPE/MCSCF approach has been employed to study the interplay of solvent and conformation effects on the spin–spin coupling constants in methanol and methylamine [72]. The simulated solvent effects are noticeable for the one-bond coupling constants and for some of the geminal coupling constants but negligible for 3 JHH . The dielectric continuum effects have been found to depend considerably on the molecular conformation in the case of 1 JCH and 2 JHCH . It is worth noting here that the MCSCF results have confirmed the conclusions drawn in ref. [80] from semi-empirical continuum model calculations. The dielectric continuum effects on spin–spin coupling constants have been calculated by means of MPE at the SCF and MCSCF levels for water monomer and dimer [75]. The bulk solvent effect as estimated by this method increases the absolute value of the 1 JOH coupling in water monomer by approximately 4.5 Hz, while the corresponding effects on 1 JOH in water dimer are 2.8 Hz on the coupling constant between the nuclei engaged in hydrogen bond, and approximately 2 Hz on the remaining 1 JOH coupling constants. The overall gas-to-liquid shift, as estimated from the dimer formation effect and bulk solvent effect, is 12 Hz for 1 JOH (as compared to experimental 10 Hz) and 0.4 Hz for 2 JHH (no experimental data available). A similar gas-to-liquid shift of 1 JOH has been obtained by means of supermolecular calculation on rigid water clusters [81]. Another study employing the MPE model (at the SCF computational level) is the calculation of spin–spin coupling constants in methyllithium and lithium dimethylamide [82]. In this case, modelling of the solvent as supermolecular aggregates leads to far better agreement with experimentally measured liquid-state spin–spin coupling constants than 140 Continuum Solvation Models in Chemical Physics does the continuum model, although the latter also improves somewhat on the results obtained for isolated molecules. Solute-shaped cavity models The obvious drawback of Mikkelsen’s MPE model is the spherical shape of the cavity, making the calculations for extended systems such as peptide models or for oblong molecules such as acetylene rather awkward. This is improved in the IEF–PCM model, which is currently most often used to calculate solvent effects on the spin–spin coupling constants. The IEF–PCM model has been adapted for a triplet linear response (necessary for calculation of FC and SD terms of a spin–spin coupling constant) by Ruud et al. [78], and a DFT calculation of solvent effects on the spin–spin coupling constants of benzene has been reported [78]. The numerical results are in good agreement with experiment for the one-bond couplings, provided the geometric relaxation (i.e. indirect effect) is taken into account. The solvent effects on the other coupling constants are very small, and are in general not reproduced by PCM. The solvent effects on the spin–spin coupling constants in acetylene have been recalculated by means of IEF–PCM theory by Pecul and Ruud [77] using the MCSCF and DFT models. Application of IEF-PCM theory with molecule-shaped cavity improves on the the gas-to-solution shifts of 1 JCC and 1 JCH obtained in ref. [43] by means of the MPE model, especially in the case of highly polar solvents (the remarkably large shift of 1 JCC in aqueous solution, −9 8 Hz, is reproduced by IEF–PCM theory but is underestimated in the MPE model). IEF–PCM values of the gas-to-solution shifts also compare favourably with the gas-to-solution shifts obtained by means of the supermolecular approach with rigid supermolecular clusters [77], at least for 1 JCC . There is no qualitative difference between the gas-to-solution shifts calculated at DFT and MCSCF computational levels, which is rewarding considering the widespread use of DFT in PCM calculations of solvent effects on the spin–spin coupling constants. The COSMO model at the DFT level has been used to calculate hydration effects on systems of biological significance: the DNA hairpin molecule [79] and the l-alanyl-lalanine zwitterion [76]. In the first case the PCM results have been compared with the results obtained using explicit solvation (with rigid water molecules), in the second case solvation has also been taken into account by molecular dynamics simulations. Inclusion of solvent effects by means of the COSMO model in both the DNA hairpin molecule and l-alanyl-l-alanine has improved considerably the agreement with experiment, and the accuracy of PCM calculation has been found to be similar to that of models with explicit water molecules. It is also worth mentioning that the sensitivity of 1 JCH couplings to molecular environment has been confirmed once more: the solvent shift of 6.1 Hz has been found for one 1 JCH coupling constant in the guanine unit [77]. Other examples of the application of calculations of spin–spin coupling constants by means of the PCM/DFT model for chemical problems using the IEF-PCM approach are studies of the spin–spin coupling constants in the keto and enol forms of monosubstituted 2-OH-pyridines [83], of the anomeric effect on the 2 JHH and 3 JHH coupling constants in 2-methylthiirane and 2-methyloxirane [83], and of the conformation of pyridine aldehyde derivaties [84]. In these studies, PCM has been used to obtain a more realistic Properties and Spectroscopies 141 simulation of experimental conditions, and has improved considerably the agreement of the theoretical results with experiment. There is a group of spin–spin coupling constants for which inclusion of solvent effects in calculation is crucial; they are spin–spin couplings involving transition metals, as demonstrated in refs. [85–87]. The solvation changes the coupling constant in these systems by more than 100 % [86]. The continuum model in the form of COSMO [26–28] has been employed in conjunction with explicit solvation by water molecules by Autschbach and Le Guennic to model solvent effects on JPt − Tl JPt − C and JTl − C couplings in the complexes NC5 Pt − TlCNn n− n = 0–3 and NC5 Pt − Tl − PtCN5 3− [85]. The two-component relativistic density functional approach, based on the zeroth-order regular approximation (ZORA) Hamiltonian has been employed for calculation of the spin–spin coupling constants. It has been found that the bulk solvent effects included by continuum model are critical in this case: without them, even qualitative agreement with experiment is not achieved, and the trends are reproduced correctly only when both first solvation sphere water molecules are included explicitly and bulk solvation effects are accounted for by continuum model. 2.1.5 Perspective Continuum models are widely used nowadays to simulate solvent effects on the NMR parameters, with varying degree of success. There are several factors which may be responsible for the lack of success of the PCM models, especially for nonpolar solvents. Lack of magnetic effects and imperfect description of dispersion and valence repulsion are probably the most important of these. In most cases continuum models are more reliable for calculation of solvent shifts for the spin–spin coupling constants than for the shielding constants for which combined supermolecular–continuum and MD/QM approaches appear to be more successful. The reason for this is not obvious. It may be connected with the fact that spin–spin coupling constants depend on the electronic density on the nucleus, which experiences the average influence of the solvent, and may be less sensitive to specific interactions. 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Soc., 123 (2001) 3341. 2.2 EPR Spectra of Organic Free Radicals in Solution from an Integrated Computational Approach Vincenzo Barone, Paola Cimino and Michele Pavone 2.2.1 Introduction Organic free radicals take part in a remarkable number of processes of technological and/or biological significance such as polymerizations of increasing technological interest [1, 2] or key reactions involving enzymes or nucleic acids [3, 4]. Since the direct characterization of these generally short-lived species is quite difficult, electron paramagnetic resonance (EPR) spectroscopy has emerged as the most effective technique to detect and characterize organic free radicals in different conditions and environments [5, 6]. Until recent years EPR has been essentially a continuous wave (CW) method, i.e. the samples sitting in a static magnetic field were irradiated by a continuous microwave (MW) electromagnetic field to drive electron spin transitions. Despite the breakthroughs in nanometre and sub-nanometre microwave technologies in the last decade, the prognosis is that a peaceful coexistence between CW and pulse EPR will continue and will be determined entirely by the sample properties and relaxation times. The difference between EPR and NMR spectroscopy, where pulse techniques have completely replaced CW ones, is related to the shorter relaxation times (microseconds in place of milliseconds), which lead to severe technical problems connected to the generation of pulses and the handling of transient signals on the nanosecond time scale. At the same time, for low symmetry species, particularly in frozen solution samples, standard EPR suffers from low spectral resolution due to strong inhomogeneous broadening. Such problems arise, for instance, because several radical species or different magnetic sites of rather similar characteristics are present, or quite small anisotropies of the magnetic tensors do not allow observation of canonical orientations in the powder EPR spectrum. Some of these situations can be dealt with effectively by electron–nuclear double (ENDOR) or even triple (TRIPLE) resonance techniques, which can be seen as variants of NMR on paramagnetic systems, the unpaired electron serving as highly sensitive detector for the NMR transitions. In other circumstances high-field EPR can be of help since unresolved hyperfine interactions do not depend on the magnetic field, whereas the Zeeman interactions are field dependent. Thus, measurements at various field/frequency settings allow different interactions in complex biological systems to be separated. The above experimental developments represent powerful tools for the exploration of molecular structure and dynamics complementary to other techniques. However, as is often the case for spectroscopic techniques, only interactions with effective and reliable computational models allow interpretation in structural and dynamical terms. The tools needed by EPR spectroscopists are from the world of quantum mechanics (QM), as far as the parameters of the spin Hamiltonian are concerned, and from the world of molecular dynamics (MD) and statistical thermodynamics for the simulation of spectral line shapes. The introduction of methods rooted into the Density Functional Theory (DFT) represents a turning point for the calculations of spin-dependent properties [7]. 146 Continuum Solvation Models in Chemical Physics Before DFT, QM calculations of magnetic tensors were either prohibitively expensive even for medium size radicals [8] or not sufficiently reliable for predictive and interpretative purposes. Today, last generation functionals coupled to purposely tailored basis sets allow researchers to compute magnetic tensors in remarkable agreement with their experimental counterparts [7, 9, 10]: computed data can take into proper account both average environmental effects and short-time dynamical contributions such as vibrational averaging from intramolecular vibrations and/or solvent librations [11–13], therefore providing a set of tailored parameters that can be confidently used for further calculations. The other challenging experimental–theoretical match, EPR spectral shape versus probe dynamics, also has a long history. The two limits of essentially fixed molecular orientation as in a crystal, and of rapidly rotating probes in solutions of low viscosity (Redfield limit) [14], have been overcome by methods based on the stochastic Liouville equation (SLE), allowing the simulation of spectra in any regime of motion and in any type of orienting potential [15]. The ongoing integration of the above two aspects, namely improved QM methods for the calculation of magnetic tensors, and effective implementations of SLE approaches for increasing numbers of degrees of freedom, paves the way towards quantitative evaluations of EPR spectra in different phases and large temperature intervals starting from the chemical formula of the radical and the physical parameters of the solvent. In the following sections, we will try to sketch the building blocks of an integrated computational approach to the EPR spectra of organic free radicals in solution and to illustrate the key issues of its application with special reference to one of the important classes of organic free radicals, namely nitroxide derivatives. Besides presenting the main framework of the proposed general model, we will pay special attention to the computation of magnetic parameters, whereas the problem of line shapes will be only briefly illustrated in the last part of this contribution. The selected examples will show that last generation models rooted in the DFT provide an accurate description of the nitroxides’ molecular structure and values of the magnetic parameters in quantitative agreement with experiments. Next, we will see that a suitable theoretical treatment of solvent effects on the magnetic parameters is able to give full account of bulk and specific interactions. In particular, the so-called polarizable continuum model (PCM) performs a remarkable job in reproducing nonspecific solvent effects, whereas in the presence of specific interactions (e.g. solute–solvents H-bonds), it has to be integrated by explicit inclusion of some solvent molecules strongly and specifically interacting with the solute. The resulting discrete/continuum description represents a very versatile tool, that can be adapted to different structural and spectroscopic situations. Which and how many solvent molecules need to be described explicitly is in principle a question that has to be defined on a case by case basis. However, chemical intuition is usually sufficient to provide suitable models, especially because PCM is able effectively to smear out the effect of not too strongly bound solvent molecules. It is noteworthy that recent developments of classical and ab initio dynamics approaches enforcing appropriate boundary conditions are allowing the same general approach to be extended from static to dynamic situations, thus allowing researchers to take into proper account averaging effects issuing from solute vibrations and solvent fluctuations. As mentioned above, longer time-scale dynamical effects determining line shapes require a different approach, whose integration in a consistent general framework is under active development. Properties and Spectroscopies 147 2.2.2 The General Model The calculation of ESR observables can be in principle based on a ‘complete’ Hamiltonian Ĥ ri Rk q , including electronic ri and nuclear Rk coordinates of the paramagnetic probe together with solvent coordinates q : Ĥ ri Rk q = Ĥprobe ri Rk + Ĥprobe−solvent ri Rk q + Ĥsolvent q (2.29) Any spectroscopic observable can then be linked to the density matrix ˆ ri Rk q t governed by the Liouville equation ˆ ri Rk q t = −i Ĥ ri Rk q ˆ ri Rk q t t (2.30) = −L̂ ri Rk q ˆ ri Rk q t Solving Equation (2.30) as a function of time would allow, in principle, a direct evaluation of ˆ ri Rk q t and hence calculation of any molecular property. However, the diverse time scales characterizing different sets of coordinates allow us to introduce a number of generalized adiabatic approximations. In particular, the nuclear coordinates R ≡ Rk can be separated into fast vibrational coordinates Rfast and slow probe coordinates (e.g. overall probe rotations and, if required, large amplitude intramolecular degrees of freedom) Rslow , relaxing at least on a picosecond time scale. Then the probe Hamiltonian is averaged on (i) femtosecond and sub-picosecond dynamics, pertaining to probe electronic coordinates and (ii) picosecond dynamics, pertaining to fast intraprobe degrees of freedom. The averaging on the electron coordinates is the usual implicit procedure for obtaining a spin Hamiltonian from the complete electronic Hamiltonian of the probe. In the frame of the Born–Oppenheimer approximation, the averaging on the picosecond dynamics of nuclear coordinates allows us to introduce in the calculation of magnetic parameters the effect of the vibrational motions, which can be very relevant in some cases [11] The effective probe Hamiltonian obtained in this way is characterized by magnetic tensors. By taking into account only the electron Zeeman and hyperfine interactions, for a probe with a single unpaired electron and N nuclei we can define an averaged magnetic Hamiltonian Ĥ Rslow q : Ĥ Rslow q = e B0 · g Rslow q · Ŝ + e În · An Rslow q · Ŝ n (2.31) + Ĥprobe−−solvent Rslow q + Ĥsolvent q The first term is the Zeeman interaction depending upon the g Rslow q tensor, external magnetic field B0 and electron spin momentum operator Ŝ; the second term is the hyperfine interaction of the nth nucleus and the unpaired electron, defined in terms hyperfine tensor An Rslow q and nuclear spin momentum operator În . The following terms do not affect directly the magnetic properties and account for probe–solvent Ĥprobe−−solvent Rslow q and solvent–solvent Ĥsolvent q interactions. An explicit 148 Continuum Solvation Models in Chemical Physics dependence is left in the magnetic tensor definition from slow probe coordinates (e.g. geometrical dependence upon rotation), and solvent coordinates. The averaged density matrix becomes ˆ Rslow q t = ˆ ri Rk q tri Rfast and the corresponding Liouville equation, in the hypothesis of no residual dynamic effect of averaging with respect to subpicosecond processes, can be simply written as in Equation (2.30) with Ĥ Rslow q instead of Ĥ ri Rk q . Finally, the dependence upon solvent or bath coordinates can be treated at a classical mechanical level, either by solving explicitly the Newtonian dynamics of the explicit set q or by adopting standard statistical thermodynamic arguments leading to an effective averaging of the density matrix with respect to solvent variables ˆ Rslow t = ˆ Rslow q tq . One of the most effective way of dealing with the modified time evolution equation for ˆ Rslow t is represented by the SLE, i.e. by the direct inclusion of motional dynamics in the form of stochastic (Fokker–Planck/diffusive) operators in the Liouvillean governing the time evolution of the system [15] ˆ Rslow t = −i Ĥ Rslow ˆ Rslow t − ˆ ˆ Rslow t = −L̂ Rslow ˆ Rslow t t (2.32) The effective Hamiltonian, averaged with respect to the solvent coordinates, is Ĥ Rslow = e B0 · g Rslow · Ŝ + e În · An Rslow · Ŝ n (2.33) and g Rslow An Rslow are now averaged tensors with respect to solvent coordinates, while ˆ is the stochastic operator modelling the dependence of the reduced density matrix on relaxation processes described by stochastic coordinates Rslow . This is a general scheme, which can allow for additional considerations and further approximations. First, the average with respect to picosecond dynamic processes is carried out, in practice, together with the average with respect to solvent coordinates to allow the QM evaluation of magnetic tensors corrected for solvent effects and for fast vibrational and solvent librational motions. The effective treatment of these aspects represents the heart of this contribution. Dynamics on longer time scales determines spectral line shapes and requires more ‘coarse-grained’ models rooted in a stochastic approach. For semirigid systems the relevant set of stochastic coordinates can be restricted to the set of orientational coordinates Rslow ≡ , which can be described, in turn, in terms of a simple formulation for a diffusive rotator, characterized by a diffusion tensor D [16], i.e. ˆ = Ĵ · D · Ĵ (2.34) where Ĵ is the angular momentum operator for body rotation [18]. Once the effective Liouvillean is defined, the direct calculation of the CW ESR signal is possible without resorting to a complete solution of the SLE by evaluating the spectral density from the expression [15, 17]. I − 0 = 1 Re vi − 0 + iL̂−1 vPeq (2.35) Properties and Spectroscopies 149 where the Liouvillean L̂ acts on a starting vector which is defined as proportional to the x component of the electron spin operator Ŝx . In the following we will discuss the different steps for the application of the above general model with specific reference to nitroxide radicals, which offer a rich and variegated playground in view of their wide field of application and of the richness of experimental data available. 2.2.3 Magnetic Tensors for Isolated Molecules Nitroxide radicals are widely used as spin labels in biology, biochemistry and biophysics to gain information about the structure and the dynamics of biomolecules, membranes, and different nanostructures. Their widespread use is related to an unusual stability, which allows researchers to label specific sites and to detect the most informative EPR parameters (g and hyperfine tensors) that are very sensitive to interactions with the chemical surroundings. Figure 2.1 collects all the radicals used in the following to illustrate the different aspects mentioned in the preceding section. Let us first consider electron–field interactions, governed by the so-called g tensor. The shift with respect to the free-electron value ge = 2 002319 is g = g − ge 13 where 13 is the 3 × 3 unit matrix. Upon complete averaging by rotational motions, only the isotropic part of the g tensor survives, which is given by giso = 13 Trg. Of, course the corresponding shift from the free electron value is giso = giso − ge O O O S N N CH3 CH3 N CH3 DMNO (1) (CH3)3C C(CH3)3 DTBN (2) PROXYL (3) O O O N N N O O N O TEMPO (4) PDT (5) O N H O (CH2)14 CH3 TP (6) Figure 2.1 Structures of the radicals studied. MTPNN (7) 150 Continuum Solvation Models in Chemical Physics g includes three main contributions [19, 20] g = gRMC + gGC + gOZ/SOC (2.36) gRMC and gGC are first-order contributions, which take into account relativistic mass (RMC) and gauge (GC) corrections, respectively. The first term can be expressed as: gRMC = − 2 – T̂ P S (2.37) where is the fine structure constant, S the total spin of the ground state, P – is the spin density matrix, the basis set and T̂ is the kinetic energy operator. The second term is given by: gGC = 1 – P rn rn r0 − rnr r0s T̂ 2S n (2.38) where rn is the position vector of the electron relative to the nucleus n r0 the position vector relative to the gauge origin and rn , depending on the effective charge of the nuclei, will be defined below. These two terms are usually small and have opposite signs so that their contributions tend to cancel out. The last term in Equation (2.36), gOZ/SOC , is a second order contribution arising from the coupling of the orbital Zeeman (OZ) and the spin–orbit coupling (SOC) operators. The OZ contribution in the system Hamiltonian is: ĤOZ = B · l̂ i (2.39) i The gauge origin dependence of this term, arising from the angular momentum l̂ i of the ith electron, can be effectively treated by the so-called Gauge Including Atomic Orbital (GIAO) approach [21, 22]. Finally, the SOC term is a true two-electron operator, which can, however, be approximated by a one-electron operator involving adjusted effective nuclear charges. Several studies have shown that this model operator works fairly well in the case of light atoms, providing results close to those obtained using more refined expressions for the SOC operator [23]. The one-electron approximate SOC operator reads: ĤSOC = rin l̂n i · Ŝ i (2.40) ni where l̂n i is the angular momentum operator the ith electron relative to the nucleus of n and ŝ i its spin operator. The function rin is defined as [24]: n 2 Zeff rin = 2 ri − Rn 3 (2.41) Properties and Spectroscopies 151 n where Zeff is the effective nuclear charge of atom n at position Rn . Starting from zero order Kohn–Sham (KS) spin orbitals, the OZ/SOC contribution is evaluated using the GIAO extension of the coupled perturbed (CP) formalism [10, 21, 22]. The isotropic magnetic properties computed for TEMPO (4 in Figure 2.1) by different methods and basis sets are compared in Table 2.1 with the corresponding experimental values. As mentioned above, a consistent and robust computational protocol must give proper account of the relationships between structural parameters and the molecular properties of interest. In the case of nitroxide derivatives, there are two critical geometrical parameters of the molecular backbone, namely the improper dihedral angle corresponding to the out-of-plane motion of the NO moiety and the nitroxide bond length. In order to gain further insight into the dependence of different molecular properties on these parameters, we have performed a molecular dynamics run for PROXYL (3 in Figure 2.1) in the gas phase and computed the magnetic parameters for a significant number of snapshots. Table 2.1 Isotropic parts of the magnetic tensors of TEMPO obtained by different QM methods are compared with the available experimental data Method/basis set An giso PBE/EPR-II PBE/EPR-III BLYP/EPR-II BLYP/EPR-III PBE0/EPR-II PBE0/EPR-III B3LYP/EPR-II B3LYP/EPR-III QCISD/EPR-II 1052 1050 1098 1122 1267 1274 1243 1268 1494 200594 200611 200603 200624 200619 200632 200626 200644 Experimental 1528a 200594b a In cyclohexane; b in toluene. As shown in Figure 2.2, the isotropic g tensor shift (giso is almost linearly dependent on the NO bond length, whereas it does not display any regular trend with respect to out-of-plane motion. The hyperfine coupling tensor (A) describes the interaction between the electronic spin density and the nuclear magnetic momentum, and can be split into two terms. The first term, usually referred to as Fermi contact interaction, is an isotropic contribution also known as hyperfine coupling constant (HCC), and is related to the spin density at the corresponding nucleus n by [25] An0 = − 8 ge gn n P rkn 3 g0 (2.42) 152 Continuum Solvation Models in Chemical Physics 20 AN / (Gauss) 18 16 14 (a) 12 10 8 Δg iso / (ppm) 4500 4000 (b) 3500 3000 1.25 1.30 1.35 N-O bond distance (Angstrom) 140 160 180 –160 –140 C-N-O-C dihedral angle (degree) Figure 2.2 Computed HCC and isotropic g tensor shift along a Car–Parrinello molecular dynamic trajectory of PROXYL in the gas phase. The second contribution is anisotropic and can be derived from the classical expression for interacting dipoles [26] Anij = − −5 2 ge gn n P rkn rkn ij − 3rkni rknj g0 (2.43) Tensor components of A are usually given in gauss 1 G = 0 1 mT. Since both contributions are governed by one-electron operators, their evaluation is, in principle, quite straightforward. However, hyperfine coupling constants have been among the most challenging quantities for conventional QM approaches for two main reasons. On the one hand, conventional Gaussian basis sets are poorly adapted to describe nuclear cusps and, on the other hand, the overall result derives from the difference between large quantities of opposite sign. However, in recent years, approaches based on the unrestricted Kohn– Sham (UKS) approach to DFT have become the methods of choice for medium to large size systems since they couple a remarkable reliability to reasonable computer requests. In general, coupling some hybrid functionals (B3LYP, PBE0) to purposely tailored basis sets (e.g. EPR-II, EPR-III) performs a remarkable job for both isotropic and dipolar terms [27, 28]. Unfortunately, this is not the case for the nitrogen isotropic hyperfine coupling in nitroxides (hereafter aN ), probably because of a particularly delicate balance between large (and opposite) inner-shell and valence spin polarization (see Tables 2.1 and 2.2). Although hybrid models (PBE0 and B3LYP) represent considerable improvements over conventional functionals, full quantitative agreement has not yet been reached. Properties and Spectroscopies 153 Table 2.2 Atomic spin densities (in a.u.) and nitrogen isotropic hyperfine coupling constants (in gauss) computed for DMNO (1 in Figure 2.1) by different QM methodsa and the EPR-II basis set Method aN N spin density BLYP PBE BP86 B3LYP PBE0 MP2 QCISD 646 607 500 878 927 1436 1213 0461 0461 0459 0464 0465 0580 0453 a O spin density 0516 0515 0515 0528 0531 0414 0451 Planar geometry, NO = 1.28 Å, CN = 1.47 Å, CNC = 120 . Pending ongoing developments of improved functionals, an effective multi-scale scheme (sketched in Figure 2.3) can be profitably used, where the NO moiety is treated at the Quadratic Configuration Interaction Single and Double (QCISD) level of theory and the remaining parts of the system are treated by means of hybrid density functionals: HCC = HCCDFT total system+ HCCQCISD − HCCDFT model system+ < environment > Such an approach provides results that are consistent with experiments with a reasonable computational effort [7]. Figure 2.3 Scheme of QCISD/DFT hyperfine coupling constant (HCC) calculation for the DTBN molecule in vacuo and in aqueous solution: tube represent the model system, balls and sticks represent the rest of the system (see Colour Plate section). It is well known that two main contributions determine the overall isotropic hyperfine coupling of a given atom together with small spin–orbit terms, which are, however, negligible for organic free radicals: (1) The direct (delocalization) contribution, which is always positive and derives from the spin density at the nucleus due to the orbital nominally containing the unpaired electron. (2) The spin polarization, which takes into account the fact that the unpaired electron interacts differently with the two electrons of a spin-paired bond or inner shell, since the exchange interaction is operative only for electrons with parallel spins. The absolute value of this 154 Continuum Solvation Models in Chemical Physics contribution is smaller than that of the direct term, but it becomes dominant when the nucleus leads in, or very close to, a nodal plane of the SOMO, since in this case the direct term is obviously vanishing. Remembering that the direct contribution to the nitrogen hyperfine splitting vanishes for a planar NO moiety and increases strongly with out-of-plane deviations, it is not surprising that, as shown in Figure 2.2, aN values present a clear quadratic dependence on the nitroxide improper dihedral. At the same time, reasonable modifications of the NO bond length have a negligible influence on this parameter. Since, as discussed above, the g tensor shows the opposite behaviour (see Figure 2.2), any successful computational strategy must include accurate determinations of all the geometric parameters. Luckily, except for systematic corrections to nitrogen isotropic hyperfine coupling constants, some hybrid density functionals coupled to purposely tailored basis sets perform a remarkable job in this connection. On these grounds, we can investigate the role of environmental and dynamical effects in determining EPR spectral parameters. 2.2.4 Solvent Effects The most promising general approach to the problem of environmental (e.g. solvent) effects can be based, in our opinion, on a system–bath decomposition. The system includes the part of the solute where the essential of the process to be investigated is localized together with, possibly, the few solvent molecules strongly (and specifically) interacting with it. This part is treated at the electronic level of resolution, and is immersed in a polarizable continuum, mimicking the macroscopic properties of the solvent. The solution process can then be dissected into the creation of a cavity in the solute (spending energy Ecav ), and the successive switching on of dispersion–repulsion (with energy Edis–rep ) and electrostatic (with energy Eel ) interactions with surrounding solvent molecules. The so-called polarizable continuum model (PCM) [29] offers a unified and sound framework for the evaluation of all these contributions for both isotropic and anisotropic solutions. Within the PCM scheme, the solute molecule (possibly supplemented by some strongly bound solvent molecules, to include short-range effects such as, hydrogen bonds) is embedded in a cavity formed by the envelope of spheres centred on the solute atoms. The procedures to assign the atomic radii [30] and to form the cavity [29] have been described in detail together with effective classical approaches for evaluating Ecav and Edis–rep [29, 31]. Here we recall only that the cavity surface is finely subdivided into small tiles (tesserae), and that the solvent reaction field determining the electrostatic contribution is described in terms of apparent point charges appearing in tesserae and self-consistently adjusted with the solute electron density [29, 32]. The solvation charges (q) depend, in turn, on the electrostatic potential (V) on tesserae through a geometrical matrix Qq = QV, related to the position and size of the surface tesserae, so that the free energy in solution G can be written: 1 G = E + VNN + V† QV 2 (2.44) where E is the free-solute energy, but with the electron density polarized by the solvent, and VNN is the repulsion between solute nuclei. Properties and Spectroscopies 155 The core of the model is then the definition of the Q matrix, which in the most recent implementations of PCM depends only on the electrostatic potentials, takes into the proper account the part of the solute electron density outside the molecular cavity, and allows the treatment of conventional, isotropic solutions, and anisotropic media such as liquid crystals. Furthermore, analytical first and second derivatives with respect to geometrical, electric, and magnetic parameters have been coded, thus giving access to proper evaluation of structural, thermodynamic, kinetic, and spectroscopic solvent shifts. Solvent can affect the electronic structure of the solute and, hence, its magnetic properties either directly (e.g. favouring more polar resonance forms) or indirectly through geometry changes. Furthermore, it can influence the dynamical behaviour of the molecule: for example, viscous and/or oriented solvents (such as liquid crystals) can strongly damp the rotational and vibrational motions of the radical. Static aspects will be treated in the following, whereas the last aspect will be tackled in the section devoted to all the dynamical effects. Let us start by illustrating the role of solvent effects on the EPR parameters of 2,2,6,6-tetramethylpiperidine-N -oxyl, TEMPO (4) [33]. The nitrogen isotropic hyperfine coupling constant aN is tuned by the polarity of the medium in which the nitroxide is embedded, as well as by formation of specific hydrogen bonds to the oxygen radical centre. Both factors contribute to a selective stabilization of the charge-separated resonance form of the NO functional group (Figure 2.4) with a consequent increase of aN . Indeed, form II entails a higher spin density on nitrogen, which has a smaller spin–orbit coupling constant than oxygen. N N O I O II Figure 2.4 Main resonance structures of nitroxide radicals. As shown in Figure 2.5, continuum solvent models (PCM) reproduce satisfactorily solvent effects on the aN parameter only for aprotic solvents (bulk effects), whereas there is a noticeable underestimation of solvent shifts for protic solvents (methanol and water). In these media also specific solute–solvent interactions have to be taken into account. In other words, since for solvents with H-bonding ability (methanol and water) the aN of the nitroxide radical is shifted to higher values because of the influence of one or more hydrogen bonds between the solute and the solvent, it becomes necessary to build a model in which nonspecific effects are described in terms of continuum polarizable medium with a dielectric constant typical of the protic solvent under study, whereas specific effects are taken into account through an explicit hydrogen-bonded complex between the radical and some solvent molecules. Figure 2.6 reports the aN values for the complexes formed by TEMPO with phenol, methanol, and water measured experimentally at room temperature, and computed in the gas phase and in solution. The values computed in solution fit the experimental data quite well. 156 Continuum Solvation Models in Chemical Physics 17 A (Gauss) expt calc calc, HB 16 15 14 1.0 2.2 4.9 7.6 8.9 20.7 24.6 32.6 34.9 36.7 46.7 78.4 Dielectric Constant Figure 2.5 Experimental and calculated aN values of TEMPO–choline [4-(N Ndimethyl-N(2-hydroxyethyl))ammonium-2,2,6,6-tetramethylpiperidine-1-oxyl chloride] as a function of the solvent dielectric constant. 17 16.91 16.58 16.51 A (Gauss) 16.34 16.15 expt PCM 1 H-bond PCM + 1 H-bond 2 H-bond PCM + 2H bonds 16.14 16 15 Phenol Methanol Water Figure 2.6 Computed and corresponding experimental aN values (in gauss) for the TEMPO– alcohol complexes in gas and in condensed phases. See text for details (see Colour Plate section). Properties and Spectroscopies 157 From a more general perspective, the example at hand highlights a situation where PCM alone is unable to account fully for solvent effects on spectroscopic properties (e.g. the aN values in solution computed with PCM are 15.70, 15.75 and 15.80 G, versus experimental values of 16.58, 16.15 and 16.91 G for phenol, methanol and water respectively): this is typically related to the presence of strong, specific H-bond interactions. As shown in Figure 2.6, inclusion of specific hydrogen bond effects results in a further increase of the computed aN values, with final results close to their experimental counterparts (16.35, 16.15 and 16.51 G). The accuracy of the cluster/PCM approach is so high that, as shown in Figure 2.6, the computed EPR properties provide valuable indirect information on the nature of the H-bond network around the NO group. In the case of water, computed results in good agreement with experiment are obtained only when two explicit solvent molecules H-bonded to the nitroxyl moiety are introduced; by contrast, a single explicit solvent molecule is required for alcohols. The same approach is able to reproduce the lowering of the isotropic g value observed experimentally when going from nonprotic to protic solvents in terms of the reduced spin density on the oxygen atom: as a matter of fact, formation of intermolecular hydrogen bonds leads to a transfer of spin density from the oxygen to the nitrogen atom. On the one hand, the gxx component (directed along the NO bond) is sensitive to variations in the geometrical parameters of the NO group, including especially the NO bond length and the deviation of the NO bond from the CNC plane; on the other hand, it shows large variations depending on the specific features of inter-molecular H-bonds. One particularly clear effect is the dependence of the g tensor on the dihedral angle CNO H (Figure 2.7). Rotation of the alcohol molecule around the nitroxide group induces a variation of the g value. In turn, this can be traced back to changes in the spin density distribution between nitrogen and oxygen: when the spin density on nitrogen increases, that on oxygen decreases, and the main components of the g tensor (both giso and gxx ) (b) (a) Nitrogen 0.60 Spin Density A N (Gauss) 16.50 16.00 15.50 15.00 14.50 Oxygen 0.55 0.50 0.45 0 0 30 60 90 120 150 180 210 240 270 300 330 Dihedral Angle CNO ... H (c) Dihedral Angle CNO ... H (d) 2.0095 2.00585 2.00580 gxx g iso–tensor 30 60 90 120 150 180 210 240 270 300 330 2.00575 2.0090 2.00570 2.0085 2.00565 0 30 60 90 120 150 180 210 240 270 300 330 Dihedral Angle CNO ... H 0 30 60 90 120 150 180 210 240 270 300 330 Dihedral Angle CNO ... H Figure 2.7 Correlation between diehedral angle CNO H (degrees) of the TEMPO– phenol complex and (A) aN , (B) spin density, (C) giso and (D) gxx values. 158 Continuum Solvation Models in Chemical Physics increase. It is clear that a higher level of accuracy in the description of H-bonding effects on the g tensor would require the computation of the relative energies of all relevant solvent arrangements, followed by proper averaging. The required sampling can result from a systematic exploration, as in the case illustrated above, but, of course, can also be provided by suitable dynamic simulations. The additional effort required to introduce this dynamic level is considerable, but, as will be shown in more detail later on, is often desirable for the accurate computation of spectroscopic parameters. A different example concerns the calculation of the EPR parameters for perdeuterated TEMPONE (5 in Figure 2.1) and TEMPO–palmitate (6 in Figure 1) dissolved in anisotropic media, i.e. n-pentyl (5CB) and n-hexyl (6CB) cyanobiphenyl liquid crystals [34]. The nematic solvents are described through their dielectric tensors, which are given by (value along director) and ⊥ (value perpendicular to director). PCM calculations are carried out with the solute molecule either perfectly aligned or perpendicular to the nematic axis. As shown in Table 2.3 the calculated data are in agreement with the experimental results. Table 2.3 Calculated g and A (gauss) tensors for PDT and TP radicals. Property calculations are performed at the PBE0/6-311 + G∗∗ level using geometries optimized in vacuo at the PBE0/6-31 + G∗ level gxx Experimental 5CB 2.00995a (2.00975)b 6CB 2.00980 (2.00978) Calculated In vacuo 2.01035 (2.00870) PCM-5CB along z 2.01000 (2.00836) 2.01003 along y (2.00837) 2.01003 along x (2.00837) PCM-6CB along z 2.01009 (2.00839) 2.01004 along y (2.00838) 2.01004 along x (2.00838) a gyy gzz Axx Ayy Azz 2.00670 (2.00725) 2.00720 (2.00763) 2.00268 (2.00265) 2.00297 (2.00273) −974 −962 −974 −960 −936 −952 −939 −950 19.12 (19.13) 19.12 (19.10) 2.00644 (2.00642) 2.00232 (2.00380) −854 −871 −825 −843 16.79 (17.43) 2.00637 (2.00635) 2.00637 (2.00635) 2.00637 (2.00635) 2.00226 (2.00375) 2.00226 (2.00375) 2.00226 (2.00375) −900 −919 −900 −919 −901 −919 −877 −898 −877 −898 −877 −898 17.77 (18.18) 17.78 (18.19) 17.78 (18.19) 2.00637 (2.00635) 2.00637 (2.00635) 2.00637 (2.00635) 2.00226 (2.00375) 2.00226 (2.00375) 2.00226 (2.00375) −897 −915 −899 −915 −890 −915 −873 −896 −876 −898 −876 −898 17.71 (18.14) 17.76 (18.14) 17.76 (18.14) For PDT; b for PT. The most important result however, is related to the effective interpretation of the factors influencing the magnetic parameters. Thus, the solvent anisotropy has a very Properties and Spectroscopies 159 limited influence, i.e. the magnetic tensors are fairly independent on the solute orientation with respect to the nematic axis (Table 2.3). On the other hand, solvent effects have a stronger influence on the xx component of the g tensor (stronger polarization effect of the solvent on the NO moiety) and on the isotropic HCC (aN values resulting from PCM computations are in better agreement with experiment than values obtained in vacuo). Moreover, this kind of calculations can also be performed for large molecules by a QM/QM scheme with an appropriate partitioning of the system. This approach provides a good description of the environment surrounding the probe and therefore allows the analysis of experimental anisotropies for solutes dissolved in nematic solvents. 2.2.5 Dynamic Effects on Short Time Scales The focus of previous sections was on cases where spectroscopic parameters in condensed phases could be computed by an essentially static approach: the PCM was able to effectively reproduce the influence of the solvent on the EPR parameters; in some instances, the explicit introduction of some first-shell solvent molecules [33] also proved necessary. Despite the effectiveness of the approach described above, computation of reliable magnetic properties in solution calls for the consideration of true dynamic effects connected to the proper sampling of the many solute–solvent configurations energetically accessible to the system of interest: one could expect that the use of geometry optimized solute–solvent clusters for the computation of spectroscopic properties could lead to an overestimation of the solvent effects, since the thermal fluctuations of the system are being essentially neglected. When these subtle influences are of interest, molecular dynamics (MD) simulations represent the methods of choice for exploring the time evolution of liquid phase systems at finite temperatures. A detailed analysis of the many features and advantages of different MD approaches is clearly beyond the aim of the present section. Here we just want to stress the importance of a dynamic description of solute–solvent systems, when the spectroscopic computations aim at an accuracy quantitatively comparable with experimental data. Eventually, it must be said that we are concerned with the description of the evolution of the system on a short time scale (tens of picoseconds), in order to compute reliable and converged average values of experimental observables. An effective computational strategy involves two independent steps: first, MD simulations are run for sampling with one or more trajectories the general features of the solute–solvent configurational space; then, EPR observables are computed exploiting the discrete/continuum approach for supramolecular clusters, made by the solute and its closest solvent molecules, as averages over a suitable number of snapshots. It is customary to carry out the same steps also for the molecule in the gas phase, just to have a comparison term for quantifying solvent effects. The same approach has been validated also for predicting NMR and UV parameters of organic molecules in aqueous solution [35]. The a posteriori calculation of spectroscopic properties, compared to other on-the-fly approaches, allows us to exploit different electronic structure methods for the MD simulations and the calculation of EPR parameters. In this way, a more accurate treatment for the more demanding molecular parameters, of both first (hyperfine coupling constants) 160 Continuum Solvation Models in Chemical Physics and second (electronic g tensor shifts) order, could be achieved independently of structural sampling methods: first-principles, semiempirical force fields, as well as combined quantum mechanics/molecular mechanics approaches could be all exploited to the same extent, once the accuracy in reproducing reliable structures and statistics is proven. As case studies, let us consider the aqueous solutions of the derivatives 1, 2 and 3 shown in Figure 2.1, namely dimethyl-nitroxide (DMNO), the di-tert-butyl-nitroxide (DTBN) and the 2,2,5,5-tetramethyl-pyrroline-N -oxyl (PROXYL). In order to overcome the limitations of currently available empirical force field parameterizations, we performed Car–Parrinello (CP) Molecular Dynamic simulations [36]. In the framework of DFT, the Car–Parrinello method is well recognized as a powerful tool to investigate the dynamical behaviour of chemical systems. This method is based on an extended Lagrangian MD scheme, where the potential energy surface is evaluated at the DFT level and both the electronic and nuclear degrees of freedom are propagated as dynamical variables. Moreover, the implementation of such MD scheme with localized basis sets for expanding the electronic wavefunctions has provided the chance to perform effective and reliable simulations of liquid systems with more accurate hybrid density functionals and nonperiodic boundary conditions [37]. Here we present the results of the CPMD/QM/PCM approach for the three nitroxide derivatives sketched above: details on computational parameters can be found in specific papers [13]. Figure 2.8 DMNO–H2 O30 cluster and convergence test of nitrogen HCC (see Colour Plate section). The DMNO radical is not very stable in aqueous solutions, nevertheless it is a good model to test the effectiveness of the discrete/continuum approach, since it directly exposes the nitroxide oxygen atom to the solvent molecules, such that calculation of magnetic parameters could be carried out without considering other kind of solute– solvent interactions. Three DMNO–water clusters containing up to 30 solvent molecules were extracted from the CPMD trajectories and aN calculations were performed on these structures, with and without addition of bulk solvent effects by PCM: as shown in Figure 2.8, by including 2–5 explicit water molecules in the calculation, together with the PCM, it is possible to reproduce with a good accuracy full QM results. The perturbation of the solvent on the hyperfine coupling constants could be described by Properties and Spectroscopies 161 means of QM/PCM, provided that a couple of water molecules is explicitly included in the QM calculations. Thus proper account of the first two water molecules close to the nitroxide at a QM level is necessary and sufficient for the description of the short-range solvent–solute interactions, while the rest of the solution is acting on the solute in terms of electrostatic effects. To validate the approach that combines CPMD and QM/PCM calculations of EPR parameters, we focused on a stable nitroxide, DTBN, in aqueous solution, which many experimental data are available for. We performed first-principle MD simulations of the DTBN aqueous solution and, for comparison, in the gas phase. The results can be summarized in three main points: the effect of the solvent on the internal dynamics of the solute, the very flexible structure of the DTBN–water H-bonding network and the quantification of solvent effects onto molecular parameters. Magnetic parameters are quite sensitive to the configuration of the nitroxide backbone, and in the particular case of DTBN, the out-of-plane motion of the nitroxide moiety is strongly affected by the solvent medium. While the average structure in the gas phase is pyramidal, the behaviour of DTBN in solution presents the maximum probability of finding a planar configuration: this does not mean that the DTBN minimum in solution is planar, but that there is a significant flattening of the potential energy governing the out-of-plane motion and that the solute undergoes repeatedly an interconversion among pyramidal positions. The vibrational averaging effects of these large amplitude internal motions have been taken into account by computing the EPR parameters along the CPMD trajectories. The H-bonding network embedding the nitroxide moiety in aqueous solution presented a very interesting result: the dynamics of the system points out the presence of a variable number of H-bonds, from zero to two, with the highest probability of only one genuine H-bond. Such a feature of the DTBN–water interaction is actually system dependent, the high flexibility of the NO moiety and the steric repulsion of the tert-butyl groups decreases the energetically accessible space around the nitroxide oxygen. Table 2.4 lists all the aforementioned effects on the EPR spectroscopic observables. Thus, after proper averaging along the MD trajectories, the proposed discrete/continuum approach provided solvent shifts and absolute values in remarkable agreement with the experimental data of DTBN in aqueous solution [13]. Finally, it is worth noting the importance of the dynamical description of the very flexible hydrogen bond network embedding the nitroxide oxygen atom. Focusing of Table 2.4 EPR parameters of DTBN in aqueous solution: nitrogen isotropic HCC (aN in gauss) and isotropic g shift (giso in ppm) aN QCISD/DFT GIAO-DFT Dynamical effects Solvent effects TOTAL Experimental data giso 155 01 16 172 1717 3736 123 −475 3348 3241 162 Continuum Solvation Models in Chemical Physics PROXYL, a more rigid five-member ring nitroxide, from analysis of CPMD trajectories, the average number of water molecules H-bonded to the solute is close to two. As a matter of fact, in this case the substituents embedding the NO moiety are constrained in a configuration where methyl groups are never close to the nitroxide oxygen, and also the backbone of the nitroxide presents an average value of the CNC angle which is lower than in the case of the DTBN, thus providing evidence of a better exposure of the NO moiety to the solvent molecules in the case of the PROXYL radical. Nevertheless, the behaviour of the closed ring nitroxide in water could not be generalized to all protic solvents: a similar simulation of the PROXYL molecule in methanol solutions presented, on average, only one genuine solute–solvent H-bond, possibly because the H-bonded methanol molecule prevents an easy access to the NO moiety for other solvent molecules. Therefore, the solvation structure of prototypical spin probe molecules depends in a sensitive way on the nature of the solvent as well as on the chemical structure of the solute. The H-bonding picture arising from all these CPMD simulations is depicted by Figure 2.9. Thus, the observed differences between experimental EPR data, collected for spin probes dissolved in water or in methanol, could be mainly due to differences in the solvent network embedding the nitroxide molecule, rather than to the diverse bulk dielectric constants. (a) 1 Methanol (a) Water (b) (b) (b) P(n) (b) (b) 0 (b) (b) (a) 0 (b) 1 0 1 2 3 0 1 2 number of H-bonds Figure 2.9 Number of solute–solvent H-bonds along CPMD trajectories for (a) PROXYL and (b) DTBN. In conclusion, our analysis is directly concerned with relatively fast and local solvent motions and the results highlight the importance of careful computational modelling for the interpretation of experimental data on the behaviour of nitroxide spin probes in water and other protic solvents. 2.2.6 Dynamics and Line Shapes on Long Time Scales We shall consider, for purposes of illustration, the system p-(methylthio)-phenyl-nitronylnitroxide (MTPNN,7 in Figure 2.1) in toluene solution [38]. Principal values and orientations of magnetic and diffusion tensors have been taken from QM calculations, Properties and Spectroscopies 163 according to the computational approaches described in previous sections. Although at least two relevant internal degrees of freedom, i.e. dihedral angles, can be identified, between the SCH3 group and the phenyl group and between the phenyl group and the nitroxide group, we assume here that the motional regime for the first angle is fast enough to be practically negligible, while we may assume that the second angle is affected by localized librations around the planar conformation. To keep our example simple, we shall not consider explicitly the coupling with this relatively soft degree of freedom. Thus, we end up with the following magnetic Hamiltonian of the system, which includes Zeeman and hyperfine interaction for the unpaired electron and the two nitrogen nuclei Ĥ = Ĥe + ĤeN = e B0 · g · Ŝ + e Î1 · A1 · Ŝ + Î2 · A2 · Ŝ (2.45) Since the system is dissolved in an isotropic fluid, and no glassy phases will be considered, the motional regime assumed for the molecules is purely free diffusive. The only adjustable parameters, valid for the entire set of spectra are the reference translational diffusion coefficient, DT0 = 1 498 × 10−8 m2 s−1 , and an inhomogeneous broadening constant which has been taken equal to 4.7 G for T < 190 K 2 8 G for 190 K < T < 170 K and zero for T < 170 K. Inhomogeneous broadening is required in order to account for residual line width resulting from super-hyperfine coupling with hydrogen nuclei, which are not accounted for explicitly in the simplified Hamiltonian defined in Equation (2.33). Notice that it is feasible to determine coupling terms for all the hydrogen atoms on the basis of the evaluation of coupling constants resulting from the QM calculation, and to evaluate the inhomogeneous broadening constant and its weak temperature dependence via a partial averaging of an extended SLE which include super-hyperfine coupling. In Figure 2.10 we show a selection of results, in which experimental and calculated spectra are compared at 292 and 155 K. The results are quite satisfactory, especially when considering that no fitted parameters, but only calculated quantities (via QM and hydrodynamic models) have been employed. The overall satisfactory agreement of the spectral line shapes, particularly at low temperatures, is a convincing proof that the simplified dynamic modelling implemented in the SLE through the ˆ and the hydrodynamic calculation purely rotational stochastic diffusive operator , of the rotational diffusion tensor, is sufficient to describe the main slow relaxation processes. In our opinion, the above results show the potentialities of an integrated computational approach and the validity of the assumptions made in the specific application. This procedure has been applied here to a radical in a single phase, but with magnetic interactions more complex than those typical of a nitroxide spin probe. The success of this method when applied to more challenging systems can be foreseen, as it is based on the link between sophisticated QM calculations of molecular properties giving amazingly reliable magnetic parameters tailored for each environment of the probes, and refined stochastic models for their reorientational motions in any dynamical régime and orienting potential symmetry. 164 Continuum Solvation Models in Chemical Physics 292 K 155 K Figure 2.10 Experimental (continuous line) and calculated (dotted line) CW ESR spectra of MTPNN in toluene at 292 and 155 K. 2.2.7 Concluding Remarks The present contribution is devoted to the development and application of an integrated computational approach to the EPR spectra of organic radicals in solution. Using nitroxides as test cases we have shown how the magnetic properties are modulated by structural, environmental and dynamical effects. The use of methods able to provide accurate results for all these contributions is thus mandatory for a reliable calculation of magnetic parameters. The development of reliable density functionals coupled to effective discrete/continuum solvent methods and suitable dynamical approaches is allowing researchers to achieve an accuracy comparable with experimental measurements for phenomena dominated by short time dynamics. The situation is different for long time dynamical effects, such as Properties and Spectroscopies 165 line shapes. Here, only the integration of quantum mechanical and stochastic techniques could offer a viable route. The first examples of such an effort are indeed quite promising and suggest that further work will lead to exciting results for more complex situations. However, it is important to point out that the different effects determining the overall experimental observables are not always separable, often being mutually interrelated and strongly coupled. 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Soc., 128 (2006) 15865. 2.3 Continuum Solvation Approaches to Vibrational Properties Chiara Cappelli 2.3.1 Introduction Vibrational spectra of isolated molecules depend on the presence of certain chemical groups, and finer details extracted from the large wealth of information enclosed in the spectrum permit the better characterization of the molecule, its conformation, its chemical linkage, and the mutual interactions between atoms and the atomic charges, modulated by the intrinsic temperature. When the system is not isolated, the interpretation of the spectrum becomes more complex, as additional factors due to the interaction of the molecule with the surrounding have to be taken into account. This should be kept well in mind when developing any computational approach to vibrational spectra of molecules in a condensed phase. The direct comparison between calculated and experimental properties for systems in solution also requires the inclusion in the calculated data of the maximum possible number of effects which are believed to be present in the experimental sample. For this reason, a way of treating nonequilibrium, local field and specific solvent effects should be included in the model. The recent progress of computational quantum chemistry has made it possible to get realistic descriptions of vibrational frequencies for polyatomic molecules in solution. The first attempt in this direction was made by Rivail et al. [1] by exploiting a semiempirical QM molecular model coupled with a continuum description of the medium to compute vibrational frequency shifts for molecular solutes. An extension to ab initio QM methods, including the treatment of electron correlation effects and electrical and mechanical anharmonicities, was then proposed [2–4] in the framework of the Polarizable Continuum Model (PCM). Still within continuum solvation models, Wang et al. [5] have used an ab initio SCRF Onsager model to compute vibrational frequencies at different levels of the ab initio QM molecular theory, the G-COSMO model has been used by Stefanovich and Truong to calculate vibrational frequencies at the DFT level [6], and the multipole SCRF model, developed by the group of Rivail, has been extended to the calculation of frequency shifts at the HF, MP2 and DFT levels, including nonequilibrium effects [7]. More recently, the PCM has been amply extended to the treatment of vibrational spectroscopies, by taking into account not only solvent-induced vibrational frequency shifts, but also vibrational intensities in a unified and coherent formulation. Thus, models to treat IR [8], Raman [9], IR linear dichroism [10], VCD [11] and VROA [12] have been proposed and tested, by including in the formulation local field effects, as well as an incomplete solute–solvent regime (nonequilibrium) and, when necessary, by extending the model to the treatment of specific solute–solvent (or solute–solute) effects. 2.3.2 Classical Approach to Vibrational Spectroscopy within Continuum Solvation Models Solvent effects on molecular vibrational (IR and Raman) spectra have been studied for many years: the attention paid to this subject is due to the observation that environmental 168 Continuum Solvation Models in Chemical Physics factors may affect the frequency and the intensity of normal vibrational modes as well as the band shape. Models to describe frequency shifts have mostly been based on continuum solvation models (see Rao et al. [13] for a brief review). The most important steps were made in the studies of West and Edwards [14], Bauer and Magat [15], Kirkwood [16], Buckingham [17,18], Pullin [19] and Linder [20], all based on the Onsager model [21], which describes the solvated solute as a polarizable point dipole in a spherical cavity immersed in a continuum, infinite, homogeneous and isotropic dielectric medium. In particular, in the study of Bauer and Magat [15] the solvent-induced shift in frequency is given as: −1 =C 0 20 + 1 (2.46) where C is a constant depending on the solute [22]. Moving to IR intensities, special efforts have been made to investigate the relation between intensity values in gas Agas and liquid phase Asol , so to formulate a value of the ratio f = Asol /Agas for pure liquids [23–25] and systems in solution [17, 18, 26–29]. Almost all the classical models for solvent effects on IR intensities, such as those due to Buckingham [17, 18], Mecke [30], Polo and Wilson [23], Mirone [29], and Warner and Wolfsberg [31] are based on a continuum (Onsager) description of the solvent. Such classical approaches start from an expression for f of the type: f= Esol Egas 2 (2.47) where Esol and Egas are the vibrating electric fields acting on the molecule in the liquid and in the gas phase. Actually, Esol is the microscopic local electric field acting on the molecule, which is different from the macroscopic Maxwell field EM acting inside the liquid. In Onsager’s theory, the local field is written as a function of the Maxwell field and the electric dipole moment of the molecule, so that Esol is expressed as the sum of two terms: the term depending on EM is called the ‘cavity field’ and the other, which is related to the dipole moment, is the ‘reaction field’: Esol = 3 2 − 2 EM + 2 + 1 r 3 2 + 1 (2.48) where is the dielectric constant of the liquid. The electric dipole moment in Equation (2.48) can be written as: = perm + Esol (2.49) where perm is the permanent dipole moment of the isolated molecule and the Esol term is the field-induced dipole moment. As the re-orientation time of the molecules is greater than the vibrational period of the radiation field, it is possible to assume that only the induced moment contributes to the vibrating electric field at the absorption frequency. With this assumption and by using the Lorenz–Lorentz equation it is possible to derive Properties and Spectroscopies 169 an expression for Esol as a function of n and EM . In addition, within the IR range of frequencies it is reasonable to assume the dielectric constant of the solution to be equal to square of the solution refractive index n2s . With this assumption and by considering that, in order to have 2 the same probing intensity I both in solution and in vacuo, it must hold that EM /Egas = 1/ns , it is possible to derive the Polo–Wilson equation for pure liquids n = ns [23]: f= 1 n n2 + 2 3 2 (2.50) and the Mallard–Straley [27] and Person [28] equation for solutions: 2 1 n2 + 2 f= ns n2 /n2s + 2 (2.51) In Buckingham’s approach [17,18], it is assumed that the solution is composed of small solvent macroscopic spheres (small with respect to the radiation wavelength) comprising a single solute molecule and surrounded by pure solvent; each sphere is independent of the others (i.e. the solution is dilute). The ratio between the integrated absorption in solution and in gas phase can be written as: f∝ sol M /Q gas /Q 2 (2.52) sol where M is the dipole of the sphere averaged over all solvent configurations and gas is the dipole moment of the isolated molecule. It is possible to show that [17, 18]: 2 sol M 9n2s sol = n2s + 2 2n2s + 1 Q Q (2.53) where sol is the dipole moment of the solute molecule in a sphere very small relative to the macroscopic sphere. The factor in brackets arises from the oscillating dipole induced in the solvent portion between the microscopic and the macroscopic spheres. This part of the solvent interacts with the solute as a continuum. By expanding sol as a function of the dipole of the isolated molecule and the polarizability of the molecule, it is possible to obtain an expression for sol /Q as a function of , the solute refractive index n, the solution refractive index ns and [17, 18]. Note that the Buckingham approach accounts for nonequilibrium solvent effects (see below), described in terms of the optical dielectric constant opt A comparison between PCM calculated IR intensities and classical equations is reported in ref. [8]. Similarly to IR, classical theories have also been proposed in the literature for Raman intensities in solution [29, 32–38]. The starting point is again the definition of the ‘local field’ Esol acting on the molecule. In all cases the local field factor is defined as f = sc sc Ssol /Svac , with S sc being the scattering intensity. 170 Continuum Solvation Models in Chemical Physics The need for a local field correction in Raman spectra was first suggested by Woodward and George [39] who, however, made no attempt to present a quantitative expression for the magnitude of the effect. Starting from Onsager’s theory, Pivovarov derived an expression for the ratio between polarizability derivatives in solution and in vacuo (and then Raman intensities) [34, 35]: fP = eff /Q = /Q 3n2s 2 2 −2 s 2n2s + 1 1 − 2n 2 3 2n +1 r (2.54) s where eff is the effective polarizability of the molecule in the cavity (see below for discussion), the polarizability of the isolated molecule, ns the refractive index of the medium and r the radius of the (spherical) cavity. Still starting from the Onsager’s theory, Mirone and co-workers [29, 36, 38] proposed a relation for the ratio between Raman intensities in solution and in vacuo given by the following formula: ⎡ ⎤4 3opt ⎦ fM = ⎣ 2 −2 2opt + 1 1 − 2opt +1 r3 (2.55) opt where opt = n2s . By assuming that the ratio /r 3 can be approximated by using the Lorenz–Lorentz formula, Equation (2.55) becomes: ⎡ fM = ⎣ ⎤4 n +2 ⎦ n2 +2 2 (2.56) opt If it is applied to pure liquids, such an expression reduces to that proposed by Eckhardt and Wagner [37]. A comparison between PCM Raman intensities and classical theories is reported in ref.[9]. 2.3.3 Quantum Mechanical Models for Vibrational Spectroscopies of Systems in a Condensed Phase The strategy which is commonly followed in the QM calculation of vibrational spectra of systems in a condensed phase is to start from the theory developed for isolated systems and to supplement that theory with solvent specificities. By taking as a reference the calculation in vacuo, the presence of the solvent introduces several complications. In fact, besides the ‘direct’ effect of the solvent on the solute electronic distribution (which implies changes in the solute properties, i.e. dipole moment, polarizability and higher order responses), it should be taken into account that ‘indirect’ solvent effects exist, i.e. the solvent reaction field perturbs the molecular potential energy surface (PES). This implies that the molecular geometry of the solute (the PES minima) and vibrational frequencies (the PES curvature around minima in the harmonic approximation) are affected by the presence of a solvating environment. Also, the dynamics of the solvent molecules around the solute (the so-called ‘nonequilibrium effect’) has to be Properties and Spectroscopies 171 taken into account to gain a realistic picture of the system and, depending on the nature of the solute–solvent system, specific (solute–solvent or solute–solute) interactions can be present, which can markedly affect the calculated properties. Lastly, it should be considered that in the framework of continuum solvation models, the electric field acting on the molecule in the cavity is different from the Maxwell field in the dielectric: however, the response of the molecule to the external perturbation depends on the field locally acting on it (‘local field’ effects). This last effect modifies the solute response to external electric and magnetic fields (the radiation), i.e. vibrational intensities. In order to develop a reliable continuum model for vibrational properties of systems in a condensed phase, all such effects should be accurately modelled [40]. The development of quantum mechanical (QM) methods for the calculation of vibrational spectra (frequencies and intensities) of systems in a condensed phase follows the development of reliable and computationally affordable algorithms for the evaluation of (free)energy first and second derivatives with respect to nuclear coordinates and/or external electric or magnetic fields. This is why this subject is relatively new in the literature (see ref. [41] for a discussion and relevant references). In recent years a great effort has been made towards the development of analytic algorithms for the calculation of free energy derivatives within the framework of continuum solvation models (see ref. [41] and the contribution by Cossi and Rega in this book) and thus the applications of such models to vibrational (as well as other response) properties are increasing. Reaction Field Effects The quantities of interest in vibrational spectra are frequencies and intensities. Within the double harmonic approximation, vibrational frequencies and normal modes for solvated molecules are related, within the continuum approach, to free energy second derivatives with respect to nuclear coordinates calculated at the equilibrium nuclear configuration. The QM analogues for ‘vibrational intensities’, depend on the spectroscopy under study, but in any case derivative methods are needed. Also, because such derivatives are to be evaluated at the equilibrium geometry, a key point is the determination of that geometry on the solvated PES, which leads to the socalled ‘indirect solvent effects’, which still requires a viable method to calculate free energy gradients (and possibly hessians). The problem of the formulation of free energy derivatives within continuum solvation models is treated elsewhere in this book and for this reason it will not considered here. Instead, it is worth remarking in this context another implication of such a formulation, i.e. that a choice between a complete equilibrium scheme or the account for vibrational and/or electronic nonequilibrium solvent effects [42, 43] should be done (see below). The Local Field Problem In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical Eloc ) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and E . However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the ‘local field’ effect, is normally solved by resorting to the Onsager–Lorentz theory of dielectric polarization [21, 44]. In such an approach the macroscopic quantities are related to the microscopic electric response of 172 Continuum Solvation Models in Chemical Physics the liquid constituents as it is in the gas phase by using a simple multiplicative factor. In particular, it is assumed that [44] E loc = n2 + 2 +2 E Eloc = E 3 3 (2.57) A more general framework to treat local field effects in linear and nonlinear optical processes in solution has been pioneered, among others [45], by Wortmann and Bishop [46] using a classical Onsager reaction field model (see the contribution by the Cammi and Mennucci for more details). Such a model has not been extended to treat vibrational spectra. Still within a continuum solvation approach [22, 41], a unified treatment of the ‘local field’ problem has recently been formulated within PCM for (hyper)polarizabilities [47] and extended to several optical and spectroscopic properties, including IR, Raman, VCD and VROA spectra [8, 9, 11, 12]. The key differences between the PCM and the Onsager’s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent–solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the ‘local field’ relies on the assumption that the ‘effective’ field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of ‘effective’ molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E [8, 47, 48] (see also the contribution by Cammi and Mennucci). By analogy with the the Onsager’s theory, it is assumed that the response of the molecule to an external probing field can be expressed in terms of an ‘external dipole moment’ , sum of the molecular dipole moment and the dipole moment arising from the molecule-induced dielectric polarization. Following ref. [8] and ref. [47], a = −tr R ma + Na (2.58) where R is the density matrix and Na indicates the nuclear contribution to the ath a matrix in Equation (2.58) is defined starting from an component of [8]. The m additional charge distribution spread on the cavity surface (the external charge). Using the standard Boundary Element Method, this charge is discretized into a set of pointlike charges, q ex , placed on representative points, sl , on the cavity surface. Within this formalism: a = − m l q ex Vsl l Ea (2.59) Properties and Spectroscopies 173 where Vsl are potential integrals evaluated at the point sl . The qlex charges represent the component of the solvent polarization that is induced by the external field EM oscillating at the frequency of the radiation, and are computed by exploiting the optical dielectric constant of the medium. The ‘effective’ properties in solution with the cavity-field effects taken into account matrix. For example, the IR intensity are formulated in terms of , i.e. in terms of the m can be expressed as [8]: Asol = NA 3ns c2 + Qi 2 (2.60) The approach just sketched in terms of ‘effective’ properties has also been applied to other vibrational spectroscopies, such as Raman [9], IR linear dichroism [10], VCD [11] and VROA [12], as well as to (hyper)polarizabilities [47–49] and birefringences of systems in a condensed phase (see refs. [50, 51] and the contribution by Rizzo in this book). Solvation Regime The motions associated with the degrees of freedom of the solvent molecules involve different time scales. In particular, typical vibration times being of the order of 10−14 –10−12 s, it is clear that the orientational component of the solvent polarization cannot instantaneously readjust to follow the oscillating ‘solute’, so that a nonequilibrium solute–solvent system has to be considered. The solvent polarization can be formally decomposed into different contributions each related to the various degrees of freedom of the solvent molecules. In common practice such contributions are grouped into two terms only [41, 52]: one term accounts for all the motions which are slower than those involved in the physical phenomenon under examination (the ‘slow’ polarization), the other includes the faster contributions (the ‘fast’ polarization). The next assumption usually exploited is that only the slow motions are instantaneously equilibrated to the momentary molecule charge distribution whereas the fast cannot readjust, giving rise to a nonequilibrium solvent–solute system. This partition and the subsequent nonequilibrium approach were originally formulated and commonly applied to electronic processes (for example solute electronic transitions) as well as to the evaluation of solute response to external oscillating fields [41]. Such phenomena are discussed elsewhere in this book: suffice it to say that in these cases the fast term is connected to the polarization of the electron clouds and the slow contribution accounts for all the nuclear degrees of freedom of the solvent molecules. In the case of vibrations of solvated molecules the same two-term partition can be assumed, but in this case the ‘slow’ term will account for the contributions arising from the motions of the solvent molecules as a whole (translations and rotations), whereas the ‘fast’ term will take into account the internal molecular motions (electronic and vibrational) [42]. After a shift from a previously reached equilibrium solute–solvent system, the fast polarization is still in equilibrium with the new solute charge distribution but the slow polarization remains fixed to the value corresponding to the solute charge distribution of the initial state. 174 Continuum Solvation Models in Chemical Physics Such a scheme has been implemented within the PCM framework to treat nonequilibrium effects on IR frequencies and intensities [42], where as a further refinement it is assumed that the geometry of the molecular cavity does not follow the solute vibrational motion. In Table 2.5 a comparison between equilibrium (eq) and nonequilibrium (neq) IR intensity shifts (solvent–gas) is reported for some methylketones in a medium polarity solvent (1,2-dichloroethane) and in a polar solvent (acetonitrile). Data are taken from ref. [42]. Nonequilibrium shifts are in very good agreement with experimental measurements, whereas a pure equilibrium model fails in reproducing the solvent–induced shifts. Table 2.5 Comparison between equilibrium (eq) and nonequilibrium (neq) B3LYP/631G(d) intensity shifts km mol−1 with respect to the gas phase for dimethyl ketone (DMK), methyl ethyl ketone (MEK), sec-butyl methyl ketone (SBMK) and tert-butyl methyl ketone (TBMK). Experimental data from ref.[53] are also shown for comparison DMK eq 1,2-dce 92 acn 113 neq 49 49 MEK exp eq 46 ± 10 99 55 ± 7 121 neq 55 54 SBMK exp eq 55 ± 7 114 55 ± 7 152 neq 64 76 TBMK exp eq neq exp 64 ± 7 112 68 ± 6 137 61 59 68 ± 6 65 ± 7 The Raman effect can be seen, from a classical point of view, as the result of the modulation due to vibrational motions in the electric field-induced oscillating dipole moment. Such a modulation has the frequency of molecular vibrations, whereas the dipole moment oscillations have the frequency of the external electric field. Thus, the dynamic aspects of Raman scattering are to be described in terms of two time scales. One is connected to the vibrational motions of the nuclei, the other to the oscillation of the radiation electric field (which gives rise to oscillations in the solute electronic density). In the presence of a solvent medium, both the mentioned time scales give rise to nonequilibrium effects in the solvent response, being much faster than the time scale of the solvent inertial response. The dynamic (nonequilibrium) response of the solvent to the external field-induced oscillation in the solute electronic density (electronic nonequilibrium) has been formulated within the PCM in ref.[9], whereas ‘vibrational nonequilibrium’ effects (due to the dynamics of the solvent resulting from solute vibrational motions) have been formulated, still within the PCM, in ref.[43]. It should be noted that, even though vibrational nonequilibrium effects have been shown to give substantial corrections to IR absorption intensities of molecules in solution, these effects are in general negligible for Raman intensities [43]. Vibrational nonequilibrium effects have also been tested in the case of VCD [11], whereas electronic nonequilibrium effects have been formulated within the PCM for VROA spectra [12]. Specific Solute–Solute and Solute–Solvent Effects Continuum solvation models are generally focused on purely electrostatic effects; the solvent is a homogeneous continuous medium and its response is determined by its dielectric constant. Electrostatic effects usually constitute the dominating part of the solute – solvent interaction but in some cases explicit solute–solvent (or solute–solute) Properties and Spectroscopies 175 interactions should be taken into account to achieve a reliable and accurate estimate of the phenomenon. This requirement is particularly pressing when the phenomenon under study is dominated by the so-called first-solvation-shell effects, such as hydrogen bonding. In such cases there is within the continuum approach a kind of ambiguity in determining which part of the system constitutes the solute and which one the solvent, i.e. where the solute stops and the continuum begins. There are essentially two approaches used to go beyond the standard continuum approach. One is the so-called ‘supermolecule’ method, which is the most straightforward methodology to treat explicit solvation effects. In its basic formulation, it redefines the system as constituted by the solute molecule and a (small) number of explicitly treated solvent molecules. The ‘clusters’ thus defined are then treated quantum mechanically in vacuo. It is clear that the validity of such an approach is crucially determined by the number of the explicitly treated solvent molecules but also that the complexity of the system increases enormously as this number becomes larger. In addition, even for a small molecule and a small number of solvent molecules, it is likely that the PES would present a large number of local minima, whose contribution to the solvation should in principle be averaged. A second approach consists of the inclusion of a few explicit solvent molecules together with a continuum model able to take into account the bulk effect of the solvent. Such a methodology should, in principle, lower the number of solvent molecules to be explicitly treated to keep a given level of accuracy. The validity of the two approaches sketched above has been quite amply tested against the ability of reproducing various molecular properties of hydrogen-bonded systems (see elsewhere in this book) including vibrational spectroscopies [11, 54–56]. For example, in Figure 2.11 calculated versus experimental IR spectra of gallic acid in water solution are reported for different levels of treatment of the specific solute–solvent interaction [54]. The portion of the spectrum in the range 1200–1500 cm−1 is poorly reproduced (both frequencies and peak intensities) by the calculations on the two most stable conformers (A and B), either in the absence or in the presence of the continuum dielectric (Figure 2.11, top), thus showing that in this case the reduction of the effects of the aqueous environment to an average dielectric effect is not sufficient to explain the experimental behaviour (the treatment is even worse if the isolated conformers are considered). Turning to a mixed continuum–discrete approach, few differences are found between the spectra of the clusters with only one water molecule bound to the carbonyl group and the averaged A + B spectra (data not shown, see ref.[54] for details), showing that the pure continuum approach is able to reproduce well the solvent-induced polarization on the carbonyl group even in the absence of the explicit consideration of first-shell effects. In contrast, IR spectra markedly different from both those of the one-water clusters and that of the A + B system are obtained if two water molecules around the carbonyl are considered, either when the continuum solvent is considered or not (Figure 2.11, middle). The comparison between the spectrum of the two-water cluster and experimental findings shows an improvement in the overall description as a result of the introduction of the two water molecules, even though the intense band at about 1345 cm−1 is still not reproduced well. Such a band can be reproduced well (both frequencies and intensities) when model structures of gallic acid with all the potential hydrogen bond sites saturated by water molecules are considered (Figure 2.11, bottom). It should be noted that in this case the further inclusion of a continuum dielectric environment does not change the picture, thus 176 Continuum Solvation Models in Chemical Physics exp AB VAC AB IEF 1000 1100 1200 1300 1400 1500 1600 1700 1500 1600 1700 1500 1600 1700 ν (cm–1) exp A2w 1000 1100 1200 1300 1400 ν (cm–1) exp A8w + B8w 1000 1100 1200 1300 1400 ν (cm–1) Figure 2.11 B3LYP/6-311++G∗∗ versus experimental pH = 168 IR spectra of gallic acid in water solution. showing a substantial saturation of solvent effects when the clusters with eight water molecules are taken into account. A further example where specific effects, in this case solute–solute aggregation effects, are noticeable is the VCD spectra of −-3-butyn-2-ol in CCl4 solution at different Properties and Spectroscopies 177 concentrations [11] (see also the contribution by Stephens and Devlin). The simulation of the spectra of this system going from dilute to concentrated solutions is a challenging problem, mainly because of three issues: (1) the solute may exist in three different conformations, which are differently stabilized by the solvent; (2) possible modifications in the dielectric properties of the local environment surrounding the solute molecules due to changes in the concentration of the solution occur; (3) clusters made of two or more hydrogen-bonded molecules of the solute can exist. The three problems mentioned have been solved in ref. [11] by resorting to population-weighted spectra of all conformers and by computing the VCD spectra of the conformers and of the possible dimers in two different dielectric environments: CCl4 and a hypothetical dielectric medium with macroscopic characteristics of the pure alcohol, so to account in an approximate way for larger clusters. The results of such an approach are shown in Figure 2.12, where a simulation of spectra at different concentrations is reported in terms of a superposition of spectra of monomers and aggregates. Although the correct way of reproducing such a phenomenon would be to calculate thermodynamic constants for all possible aggregation equilibria and to use them to evaluate the concentration of each species, for a qualitative estimate it is sufficient to show VCD spectra resulting from a combination of the spectra of monomers and dimers, obtained by introducing three different arbitrary weights corresponding to 75:25, 50:50, and 25:75 percentages of monomers and dimers, respectively (see Figure 2.12). The reported spectra confirm not only that the observed spectra are always a superposition of different contributions but also that, by combining the effects of clustering with those 25–75 50–50 75–25 Δε Δε 0.858 M 0.308 M 11 12 15 0.103 M 16 13 14 1500 1400 1300 1200 1100 1000 ν (cm–1) 900 1600 1200 800 ν (cm–1) Figure 2.12 Calculated VCD spectra resulting from a combination of the spectra of (S)−-3-butyn-2-ol monomers in CCl4 and dimers in pure alcohol, obtained by introducing three different arbitrary weights corresponding to 75:25, 50:50, and 25:75 percentages of monomers and dimers, respectively. Experimental spectra at different concentrations in CCl 4 are also reported (right-hand panel). 178 Continuum Solvation Models in Chemical Physics induced by the dielectric environment, the trend observed in the experimental spectra at different concentrations can be correctly reproduced. References [1] J.-L. Rivail and D. Rinaldi, in B. 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Struct., 175 (1988) 313. [54] C. Cappelli, B. Mennucci and S. Monti, J. Phys. Chem., A 109 (2005) 1933. [55] C. Cappelli, S. Monti and A. Rizzo, Int. J. Quantum Chem., 104 (2005) 744. [56] C. Cappelli, B. Mennucci, C. O. da Silva and J. Tomasi, J. Chem. Phys., 112 (2000) 5382. 2.4 Vibrational Circular Dichroism Philip J. Stephens and Frank J. Devlin 2.4.1 Introduction Circular Dichroism (CD) is the differential absorption of left- and right-circularly polarized light: A = AL − AR (2.61) CD is exhibited by solutions of chiral molecules. In dilute solutions, Beer’s Law applies, when A = c (2.62) where is the difference in the extinction coefficients of left- and right-circularly polarized light, L − R ; c is the solute molarity; and l is the sample pathlength in centimeters. Chiral molecules exist in two forms, enantiomers, which are nonsuperposable mirror images. The CD of the two enantiomers, E1 and E2 , is equal in magnitude but opposite in sign, at all frequencies: E1 = − E2 (2.63) As a result, the CD of a chiral molecule can in principle be used to determine its enantiomeric form, referred to as its Absolute Configuration (AC). Since the discovery of CD by Cotton [1], the determination of the ACs of chiral compounds has been the predominant application of CD spectroscopy. Until the 1970s, all CD measurements were carried out within the near-infrared – visible–ultraviolet spectral range and the CD observed originated in electronic transitions, referred to as Electronic CD (ECD). Soon after World War II commercial CD instrumentation became available, measuring CD using modulation spectroscopy [2]. The earliest instruments used KDP electrooptic modulators (Pockels cells), which transform linearly polarized light into alternating left-and right-circularly polarized light. After transmission through a sample with CD, an oscillating light intensity results, whose magnitude is proportional to the CD. In order to use its CD to determine the AC of a chiral molecule, a theory is required which predicts the sign of the CD of a given enantiomer. The utilization of CD by organic chemists was greatly stimulated by the development of the Octant Rule, which predicts the CD of the n– ∗ electronic excitation of carbonyl functional groups [3]. Subsequently, socalled Sector Rules were developed for many other electronic chromophores, extending the applicability of CD [4]. In the early 1970s, the question was raised: can the CD of vibrational transitions be measured in the infrared (IR) spectral region? At USC, we had already built a CD instrument which functioned in the near-IR (down to ∼ 3300 cm−1 [5]. This instrument used Properties and Spectroscopies 181 the standard modulation spectroscopy technique, with the Pockels cell replaced by a quartz Photoelastic Modulator (PEM), a new type of phase modulator invented in the 1960s [6]. In 1973 we began the extension of this instrument to the fundamental IR region. By 1974, measurements of Vibrational CD (VCD) spectra had been successfully accomplished [7]. Critical to this success was the development by post-doc Dr Jack Cheng of a new PEM, whose optical element was amorphous ZnSe [8], whose IR transmission limit is ∼ 650 cm−1 . Over time, the lower frequency limit of our VCD instrument was extended, eventually, after the incorporation of a closed-cycle refrigerated detector system, permitting the use of detectors operating at near-liquid-helium temperatures, such as As-doped Si, reaching the 650 cm−1 ZnSe transmission limit [9]. To utilize the now-measurable VCD of chiral organic molecules for the determination of their ACs, a reliable method for predicting VCD spectra was required. Although methods called the Fixed Partial Charge method [10] and the Coupled-Oscillator method [11] had been proposed and implemented in the early 1970s, these methods were seriously flawed [12] and insufficiently reliable to provide an acceptable basis for AC determination. What was needed was a quantum mechanical theory, consistent with the state-of-the-art theories of vibrational absorption spectra and the magnetic properties of molecules. Such a theory was developed by Stephens during the late 1970s and early 1980s [12, 13], providing equations for VCD which have been the basis for all reliable applications of VCD spectroscopy ever since. By the early 1980s, it was clear that the most accurate predictions of vibrational absorption spectra were provided by ab initio quantum mechanical methods. From the beginning, therefore, the implementation of Stephens’ theory of VCD was carried out using ab initio methods. Initially, the Hartree–Fock (HF) methodology was employed [14]. A decade later, Density Functional Theory (DFT) had become the method of choice, having the best compromise of numerical accuracy and computational labor of any quantum mechanical methodology. As a result, the Stephens theory was implemented using DFT by Drs Jim Cheeseman and Mike Frisch at Gaussian Inc. [15] within the famous and widely used ab initio package called GAUSSIAN [16]. The enormously greater accuracy of DFT calculations of VCD spectra, as compared to HF calculations, resulted in an enormous surge in the utilization of VCD in determining the ACs of organic molecules. It also encouraged a number of companies, including Bruker, Jasco and Bomem, to produce Fourier Transform (FT) VCD instruments. Thus, with the commercial availability of both VCD instrumentation and ab initio DFT software for predicting VCD spectra, VCD spectroscopy has become easily accessible and usable. In this contribution, we summarize the Stephens theory of VCD (Section 2.4.2), discuss its implementation using ab initio methods, most importantly DFT (Section 2.4.3), discuss the determination of the ACs of chiral molecules using VCD (Section 2.4.4), and, finally, discuss future developments expected to enhance the prediction of VCD spectra (Section 2.4.5). 2.4.2 Theory We restrict our discussion to the case of isotropic dilute solutions of randomly oriented molecules, e.g. liquid solutions or amorphous solid solutions. (In practice, the vast majority of VCD experiments are carried out using liquids at room temperature.) 182 Continuum Solvation Models in Chemical Physics Semi-classical treatment of the interaction of molecules with electromagnetic waves leads to equations for and in terms of molecular properties: ¯ = 8 3 N g Dgk fgk gk 2 303 3000hc gk (2.64) = 32 3 N R f 2 303 3000hc gk g gk gk gk (2.65) Dgk = g el k2 Rgk = Im g el k • k mag g (2.66) (2.67) where g → k is amolecular excitation of frequency gk g is the fraction of molecules in state g, and f gk is a normalized line shape function (e.g. Lorentzian). Dgk and Rgk are the dipole strength and rotational strength of the excitation g → k. el and mag are the electric and magnetic dipole moment operators: el = − i eri + Z! e R! ≡ eel + nel (2.68) ! e Z! e ri × pi + mag = − × R! × P! ≡ emag + nmag 2mc 2M c ! i ! (2.69) Here, −e and Z! e ri , and R! pi , and P! , are the charge, position and momentum of electron i and nucleus " respectively. Equations (2.64) and (2.65) do not include the effects of the condensed-phase medium either on the molecular properties g Dgk Rgk and gk or on the electromagnetic fields of the radiation: ‘solvent effects’. In the case of vibrational transitions, g and k are vibrational levels of the ground electronic state, G. Within the Born–Oppenheimer (BO) approximation: g r R = #G r R Gg R (2.70) k r R = #G r R Gk R (2.71) where Hel r R #G r R = WG R #G r R WG R + Tn R Gv R = Ev Gv R (2.72) (2.73) r and R denote electronic and nuclear coordinates respectively. Hel is the adiabatic ‘electronic Hamiltonian’: Hel = Te + Vee + Ven + Vnn (2.74) comprising the electronic kinetic energy and the Coulombic interactions of electrons and nuclei. #G and WG are the wavefunction and energy of the ground electronic state. Gv Properties and Spectroscopies 183 and Ev are the wavefunction and energy of the vibrational level v arising from vibrational motion on the potential energy surface (PES) WG R. For simplicity, we restrict discussion now by assuming that only the lowest vibrational level is populated and that the PES, WG , is harmonic: WG = WG0 2 WG 1 1 + X! X! = WG0 + k Q2 2 !! X! X! o 2 i i i (2.75) where WG0 is the energy of G at equilibrium, R = R0 $ X! is the displacement of nucleus !! = 1 N along Cartesian axis = x y z $ Qi are normal coordinates, simultaneously diagonalizing the nuclear kinetic energy: Tn = 1 2 Q̇ 2 i i (2.76) The force constants, ki , determine the normal mode frequencies: i = 1 ki 2 (2.77) The vibrational states of this harmonic PES are of energy Ev1 v2 v3N = i 1 vi + h 2 i (2.78) vi = 0 1 2 For six modes, corresponding to translational and rotational motions, ki and i , are zero. Within the harmonic approximation (HA), electric dipole transition moments are g el k ≡ #G Gg el #G Gk = Gg #G el #G Gk (2.79) which, on expanding #G el #G ≡ Gel with respect to the normal coordinates Qi : Gel = Gel Gel Q + + Qi 0 i i (2.80) leads to nonzero transition moments from the vibrational ground state (all vi = 0 only for fundamental transitions involving one mode alone, i.e. to the states vi = 1 vj = 0j = i. The transition moment for the fundamental in mode i is 0 el 1i = Gel Qi 0 4i 1/2 (2.81) 184 Continuum Solvation Models in Chemical Physics Equation (2.81) can be rewritten in terms of derivatives of the molecular electric dipole moment Gel with respect to the Cartesian displacement coordinates, X! . With X! = S!i Qi (2.82) i equation (2.81), becomes 0 el 1 i = 4vi 1/2 ! S!i P (2.83) ! where ! P = G el X! (2.84) 0 ! The second-rank molecular tensors, P , are termed atomic polar tensors (APTs). Separating electronic and nuclear parts: ! ! ! P = E + N G ! #G el #G E = X! 0 (2.85) ! N = Z! e We can further write ! E =2 #G X! e 0 # el G (2.86) 0 The dipole strength of the fundamental excitation of mode i is then G 2 el Di = Qi 0 ! = S!i P S! i P! 4vi !! 4vi (2.87) The formulation of magnetic dipole transition moments is unfortunately less straightforward. Compare the electronic contributions to the electric and magnetic dipole moments of G: Gel Gmag e e = #G eel #G = #G emag #G (2.88) (2.89) Properties and Spectroscopies 185 Considering only nondegenerate electronic ground states (in practice very few chiral are exceptions) Gel and Gmag are qualitatively different because emolecules #G mag #G = 0 at all molecular geometries. That is, electrons make zero contribution to the adiabatic magnetic dipole moment. It follows that, in the case of magnetic dipole transition moments, the BO approximation leads to a nonphysical result. The treatment of magnetic dipole transition moments requires more accurate vibronic wavefunctions. The vibrational states g and k must be written. #E Ee cGgEe g = #G Gg + E=G k = #G Gk + (2.90) #E Ee cGkEe E=G allowing for the admixture of BO functions of excited electronic states E into the ground state. This in turn permits nonzero vibrational transition moments of emag to be obtained; simply put, electronic magnetic dipole transition moments are ‘stolen’ by mixing of BO states. The reader is referred to the literature for the details [12, 13]. The final result is that 1/2 ! (2.91) 0 mag 1 = 43 vi S!i M i ! where ! ! ! = I + J M ! #G #G ! I = X! 0 H 0 ! J = (2.92) i Z! e R0! 4c ! ! ! The tensors M are termed atomic axial tensors (AATs); I and J are the electronic and nuclear components. Here, #G /X! 0 is the same derivative which occurred ! already in Equation (2.86). The electronic AAT, I is the overlap integral with the derivative #G /H 0 . The latter is defined via H H = Hel + H H H H = − emag H (2.93) H H #G H = WG H #G H That is: #G H is the wavefunction of G in the presence of a uniform external magnetic field, H , approximating the perturbation by the linear magnetic dipole interaction H H . The rotational strength of the fundamental excitation of mode i is then ! Ri = 2 S!i P S! i M! (2.94) !! 186 Continuum Solvation Models in Chemical Physics 2.4.3 Ab Initio Implementation Within the HA, the prediction of a vibrational absorption spectrum amounts to the calculation of the harmonic normal mode frequencies, vi , and dipole strengths, Di . The frequencies are obtained from the harmonic force field (HFF). With respect to Cartesian displacement coordinates, this is the Hessian 2 WG /X! X! 0 . Diagonalization (after mass-weighting) yields the force constants ki ; the frequencies, i ; and the normal coordinates, Qi , i.e. the transformation matrices, S!i . The dipole strengths depend in addition on the APTs; these require calculation of #G /X! 0 . The prediction of a VCD spectrum amounts likewise to the calculation of the harmonic frequencies and rotational strengths, Ri . All of the quantities required in predicting the absorption spectrum are again needed; in addition, the AATs must be calculated. Since #G/X! 0 is already required for the APTs, the AATs require additionally only #G /H 0 . and VCD spectra requires (i) 2In sum: the prediction of both absorption WG /X! X! 0 ; (ii) #G /X! 0 and (iii) #G /H 0 . The prediction of the VCD spectrum requiresrelatively little more than is needed for the absorption spectrum: specifically, #G /H 0 . The accurate calculation of these molecular properties requires the use of ab initio methods, which have increased enormously in accuracy and efficiency in the last three decades. Ab initio methods have developed in two directions: first, the level of approximation has become increasingly sophisticated and, hence, accurate. The earliest ab initio calculations used the Hartree–Fock/self-consistent field (HF/SCF) methodology, which is the simplest to implement. Subsequently, such methods as Møller–Plesset perturbation theory, multi-configuration self-consistent field theory (MCSCF) and coupled-cluster (CC) theory have been developed and implemented. Relatively recently, density functional theory (DFT) has become the method of choice since it yields an accuracy much greater than that of HF/SCF while requiring relatively little additional computational effort. The second direction in which ab initio theory has progressed is that of derivative techniques [17]. Many molecular properties of interest – including, as shown above, the HFF, APTs, and AATs – can be expressed in terms of derivatives of energies and wavefunctions with respect to perturbations. Such derivatives can be evaluated using either numerical or analytical methods. For example, the energy gradients WG /X! 0 can be evaluated either by calculating WG at R0 and R0 + X! and using WG X! ≈ 0 WG R0 + X! − WG R0 X! (2.95) or by formulating an equation for WG /X! 0 and then carrying out direct evaluation. Similarly, a Hessian matrix can be obtained by finite differences of gradients or analytically. Analytical derivative methods are much more efficient. Much of the recent expansion in usage of ab initio quantum chemistry has resulted from advances in formulating and implementing analytical derivative techniques for an increasing diversity of molecular properties at an increasing number of theoretical levels. Properties and Spectroscopies 187 At the present time, the simultaneous calculation of HFFs, APTs and AATs using analytical derivative ab initio methods has been implemented in three program packages: CADPAC, DALTON and GAUSSIAN. The levels of implementation are: CADPAC HF/SCF$ DALTON HF/SCF and MCSCF$ GAUSSIAN HF/SCF and DFT. The accuracies of these methods are: HF/SCF < MCSCF DFT The computational effort is: HF/SCF < DFT MCSCF The ratio of accuracy to effort is: DFT HF/SCF > MCSCF Thus, DFT is currently the most cost-effective methodology available. An additional variable in ab initio calculations is the basis set. Two choices are to be made: (i) perturbation independent or perturbation dependent; (ii) size and composition. In calculating derivatives with respect to nuclear displacements, X! , one can adopt basis functions which either (a) are not or (b) are functions of nuclear position. The latter add computational complexity but vastly improve convergence of properties with increasing basis set size (i.e. decrease the errors associated with the use of basis sets of finite size.) Modern computational packages use only nuclear-position-dependent basis sets. In the same way, derivatives with respect to magnetic fields can use basis functions which either (a) are not or (b) are functions of magnetic field. The standard choice for the latter are socalled London orbitals or gauge-invariant atomic orbitals (GIAOs) [18]. The use of GIAOs vastly reduces basis set error and is increasingly de rigueur in computation of magnetic properties (e.g. NMR shielding tensors). In addition, very importantly, the use of GIAOs leads to origin-independent rotational strengths. With regard to the implementation of AATs in CADPAC, DALTON and GAUSSIAN, we should add that DALTON and GAUSSIAN use GIAOs, while CADPAC does not. With respect to basis set size we can simply note that (a) accuracy increases with increasing basis set size; (b) the rate of increase in accuracy is rapid at small sizes and less rapid at large sizes. Finally, in DFT calculations there is the question of the density functional. The accuracy of DFT calculations varies greatly with the choice of functional. The exact functional gives exact results. Very crude functionals give very inaccurate results. Functionals used in the recent past can be grouped into three classes; (a) local; (b) nonlocal/gradientcorrected; (c) hybrid. Overall, the relative accuracy is [19]: Local < nonlocal < hybrid 188 Continuum Solvation Models in Chemical Physics At this time, hybrid functionals are generally regarded as state of the art. There are many: the original is B3PW91 [20]; a popular-choice is B3LYP [21]. In order to evaluate the accuracy of DFT/GIAO calculations of VCD spectra, the conformationally rigid chiral molecules shown in Figure 2.13 have been studied [22]. A thorough study of the dependence of predicted VCD spectra on the choice of basis set and functional was carried out for methyl-oxirane 2 [22b,c]. Comparison to the experimental VCD spectrum (Figure 2.14) clearly shows (i) that the agreement of calculated and experimental VCD improves rapidly with increasing basis set size and (ii) that the hybrid functionals B3LYP, B3PW91, B3P86, and PBE1PBE, yield VCD spectra in best agreement with experiment. Quantitative comparisons of calculated and experimental rotational strengths [22b,c], the latter obtained via Lorentzian fitting of the experimental VCD spectrum, shows (i) the relative accuracies of eight basis sets to be: 3-21G 6-31G∗ ∼ 6-31G∗∗ ∼ cc-pVDZ TZ2P ∼ cc-pVTZ ∼ cc-pVQZ ∼ VD3P$ and (ii) the relative accuracies of eight functionals to be: BHandH < LSDA ∼ BHand HLYP BLYP ∼ B3LYP ∼ B3PW91 ∼ B3P86 ∼ PBE1PBE tBu O D O D S O O Me 1 2 3 O 4 Men O O 5 O O O O n = 0, 1, 2 8 7 6 9 N O O O 10 N 11 12 Figure 2.13 Conformationally rigid chiral molecules whose VCD has been studied. Less extensive studies on other molecules have subsequently confirmed the generality of these results, and have confirmed the conclusion that the optimum compromise of size and accuracy of the basis set is TZ2P. A recent study [22p] of the chiral alkane perhydrotriphenylene (PHTP) (12, Figure 2.13) further illustrates the accuracy of B3LYP/TZ2P Properties and Spectroscopies 189 and B3PW91/TZ2P VCD spectra. In Figure 2.15, the calculated and experimental VCD spectra are compared. In Figure 2.16, the calculated and experimental rotational strengths, the latter obtained via Lorentzian fitting, are compared. In the case of PHTP, the B3PW91 functional provides somewhat more accurate results than the B3LYP functional. In some molecules the reverse is true [22]. Thus, it is always wise to carry out VCD calculations with a range of hybrid functionals, in order to determine which is the optimum for the molecule under study. 3–21G Δε ∗ 103 6–31G* TZ2P 6 15 cc-pVTZ 18 16 14 8 10 17 10 15 9 13 6 11 10 18 14 8 7 12 16 17 9 13 11 –10 expt 7 600 1500 1400 1300 1200 1100 1000 900 Wavenumbers (cm–1) (a) Figure 2.14a Mid-IR VCD spectra of R-+-2. The experimental spectrum is in CCl4 solution. DFT/GIAO spectra are calculated using the B3LYP functional and a range of basis sets. Band shapes are Lorentzian = 4 cm−1 . Fundamentals are numbered. 190 Continuum Solvation Models in Chemical Physics LSDA Δ ε ∗ 103 BLYP B3PW91 6 15 18 B3LYP 14 16 17 15 10 18 16 8 10 9 13 6 11 10 14 8 7 12 17 9 13 –10 11 expt 7 600 1500 1400 1300 1200 Wavenumbers 1100 1000 900 (cm–1) Figure 2.14b Mid-IR VCD spectra of R-(+)-2. The experimental spectrum is as in 2.14a. DFT / GIAO spectra are calculated using the cc-pVTZ basis set and a range of functionals. Band shapes are Lorentzian = 4cm− 1. Fundamentals are numbered. 2.4.4 The Determination of Absolute Configuration Using DFT/GIAO Calculations of VCD Spectra In order to determine the Absolute Configuration (AC) of an enantiomer of a chiral molecule of defined specific rotation, its VCD spectrum is measured and compared with the predicted DFT/GIAO VCD spectra of the two enantiomers. Assuming that the Properties and Spectroscopies 191 75/74 105 78–76 B3LYP/TZ2P 61/60 73 64/62 83/82 95/94 81/80 104–100 43/42 51/50 69/68 52 71/70 59/58 45 56–53 92–89 105 78–76 75/74 93 67/66 73 87–85 B3PW91/TZ2P 61/60 64 81/80 71/70 84/83 104–100 43/42 50–47 95/94 44 51 59/58 45 99/98 56/55 Δ ε ∗ 103 92–89 93 25 54/53 67/66 105 expt: (+) –12 in CCl4 75/74 78–76 95/94 87–85 81/80 64 73 61/60 43/42 50–47 44 99/98 59/58 84/83 71/70 104–100 56/55 –25 93–89 1500 1400 51 45 54/53 67/66 1300 1200 Wavenumbers 1100 1000 Figure 2.15 Comparison of the B3LYP/TZ2P, B3PW91/TZ2P VCD spectra of S-12 and experimental VCD spectra of (+)-12. The bandshapes of the calculated spectra are Lorentzian = 40 cm−1 . The numbers define the fundamental modes of 12 contributing to the spectral bands. DFT/GIAO predicted spectra are accurate, the predicted spectrum of one enantiomer will be in excellent agreement with the experimental spectrum, while for the other enantiomer the agreement will be very poor. By way of illustration, the B3PW91/TZ2P VCD spectra of the R- and S- enantiomers of PHTP are compared with the experimental VCD spectrum of + – PHTP in Figure 2.17, and the calculated and experimental rotational strengths 192 Continuum Solvation Models in Chemical Physics 75 B3PW91/TZ2P 50 B3LYP/TZ2P 25 Rcalc 0 –25 –50 –75 –100 –100 –75 –50 –25 0 25 50 75 Rexpt Figure 2.16 Comparison of the B3LYP/TZ2P, B3PW91/TZ2P rotational strengths of S-12 and experimental rotational strengths of (+)-12. For bands assigned to multiple vibrational modes, calculated rotational strengths are the sums of the rotational strengths of contributing modes. The straight line, of slope +1, is the ‘line of perfect agreement’. are compared in Figure 2.18. The results unambiguously determine the AC of + – PHTP to be S [22p]. As shown by this example, the determination of the ACs of conformationally rigid molecules is straightforward. Many chiral organic molecules, however, are conformationally flexible and multiple conformations are in equilibrium at the temperature of the experimental VCD measurements. In such cases, Conformational Analysis must first be carried out, leading to the structures, relative energies and room temperature equilibrium populations of all conformers. Then, for those conformers which are significantly populated, DFT/GIAO spectra are calculated, weighted by the fractional populations, and summed, to give the conformationally averaged VCD spectrum. The conformationally averaged VCD spectra of the two enantiomers are then compared to the experimental VCD spectrum of a sample of defined specific rotation in order to determine its AC. Conformationally flexible molecules for which DFT/GIAO VCD spectra have been calculated [23] are shown in Figure 2.19. For molecules with limited numbers of dihedral angles with respect to which internal rotation can occur, the most reliable way to find their conformations is to carry out PES scans using DFT. For example, in the case of the cyclic sulfoxide 1-thiochroman4-one S-oxide (18, Figure 2.19), a B3LYP/6-31G∗ 2D PES scan with respect to the two dihedral angles C5C4C3C2 and C8C9SC1 (Figure 2.20) clearly shows that there are Properties and Spectroscopies 193 B3PW91/TZ2P S-isomer Δ ε ∗ 103 20 expt: (+)–12 0 –20 R-isomer 1600 1500 1400 1300 1200 1100 1000 900 800 Wavenumbers Figure 2.17 Comparison of B3PW91/TZ2P VCD spectra of R- and S-12 to the experimental VCD spectrum of +-12. two stable conformations, a and b [23e,f]. Optimization of the structures of these two conformations, starting from the lowest energy structures in the PES scan, leads to the equilibrium structures of these conformations, shown in Figure 2.21. The B3LYP/TZ2P VCD spectra of conformations a and b, together with the conformationally averaged spectrum and the experimental VCD spectrum are shown in Figure 2.22. Assignment of the experimental spectrum clearly shows VCD bands due to the individual conformations a and b, supporting the reliability of the conformational analysis. The good agreement of the predicted VCD spectrum of S-18 with the experimental VCD spectrum of + –18, leads to the unambiguous assignment of the AC of 18 as S-+ [23e,f]. 194 Continuum Solvation Models in Chemical Physics 75 R-configuration b 50 25 Rcalc 0 –25 –50 –75 –100 –100 –75 –50 –25 0 25 50 75 0 25 50 75 Rexpt 75 50 S-configuration a 25 Rcalc 0 –25 –50 –75 –100 –100 –75 –50 –25 Rexpt Figure 2.18 Comparison of B3PW91/TZ2P rotational strengths for R- and S-12 to the experimental rotational strengths of +-12. Properties and Spectroscopies SOMe O O O 195 O 15 14 13 16 O COOMe S S S O O 17 OCOMe 18 O 19 21 20 O O O X HO O OH OR S O R = Ac, tBu, SiMe3 X CH(OH)Me COOMe X = o-Br, p-Me, m-F X = H, Br 22 24 23 25 O O H O N N O P OEt O OEt N O OEt N H S COOMe O 28 27 26 Br H H3CO H O O H N H R1 H H O O H O 30 O OMe H H O H R2 29 O OMe N H3CO O O O R1 = H, Me R2 = Me, H 31 OCOMe Figure 2.19 Conformationally flexible molecules whose VCD have been studied. For much more flexible molecules than 18, DFT PES scans can be impractical. Currently, for such molecules conformational analysis is most efficiently carried out in two stages: first, Monte Carlo searching using a molecular mechanics force field (MMFF) determines the stable conformations predicted by the MMFF; second, these conformations are re-optimized using DFT. For example, conformational analysis of the 196 Continuum Solvation Models in Chemical Physics 105 120 225 135 150 180 8 210 3 3 7 6 2 4 4 6 5 1 3 210 7 2 8 10 9 2 76 8 2 4 5 8 2 10 9 3 2 5 4 5 150 9 10 S 4 8 7 7 6 135 150 165 7 6 5 8 135 150 3 6 9 10 O 120 5 5 8 105 4 8 1 9 165 2 4 76 8 2 8 6 5 5 3 180 7 7 3 4 3 4 1 3 6 9 10 O 9 10 b 2 4 1 165 195 8 76 5 5 180 8 6 3 3 4 5 7 6 255 7 4 5 a 3 240 9 10 5 4 4 5 1 225 225 6 9 210 8 6 2 10 195 195 7 5 C5C4C3C2 (deg) 165 4 8 135 180 195 210 225 240 255 C8C9SC1 (deg) Figure 2.20 The B3LYP/6-31G∗ PES of S-18. The dihedral angles C5C4C3C2 and C8C9SC1 were varied in 15 steps. Contours are shown at 1 kcal mol−1 intervals. Figure 2.21 The B3LYP/TZ2P structures of conformations a and b of S-18. The perspective demonstrates the near-planarity of the C2C3C4C5C6C7C8C9S moiety. oxazol-3-one, 26 (Figure 2.19), using this protocol predicts that the three conformations, a–c (Figure 2.23), are significantly populated at room temperature [23l,m]. Prediction of their VCD spectra at the B3PW91/TZ2P level, followed by conformational averaging, leads to the conformationally averaged VCD spectra of R-26 and S-26, shown in Figure 2.24, together with the experimental VCD spectrum of +-26. The agreement of calculated and experimental VCD spectra leads to the AC S-+ of 26, which is confirmed by the comparison of the calculated and experimental rotational strengths shown in Figure 2.25 [23l,m]. 2.4.5 Discussion As a result of the studies of the molecules in Figures 2.13 and 2.19, it is now clear that the VCD spectra predicted using Stephens’ equation for vibrational rotational strengths, Properties and Spectroscopies 197 37 b 29 34 31 40 42 25 35 22 24 33 41 30 32 23 36 28 21 39 39 a 30 29 36 41 42 40 28 22 27 35 38 21 25 24 33 23 31 34 Δ ε ∗ 103 29a 39a 37 36 41 a+b 29b 30a 25 31b 34b 42 40a 24 39b 33 31a 34a 21a 22 23b 23a 21b 37 39a 30a 20 29a 41 34b 40b 32a/31b expt 29b 25 40a 21a 24 42 –20 22 39b 36 37 23b 23a 34a 33 21b 31a 1500 1400 1300 1200 1100 1000 900 800 Wavenumbers Figure 2.22 Calculated and experimental VCD spectra of 18. Spectra of conformations a and b are calculated at the B3LYP/TZ2P level for S-18. Lorentzian band shapes are used = 40 cm−1 . The spectrum of the equilibrium mixture of a and b is obtained using populations calculated from the B3LYP/TZ2P energy difference of a and b. The numbers indicate fundamental vibrational modes. Where fundamentals of a and b are not resolved only the number is shown. implemented using DFT and GIAOs, together with an accurate basis set, such as TZ2P, and a state-of-the-art density functional, such as B3LYP or B3PW91, are in impressive agreement with experiment. Consequently, Absolute Configurations (ACs) determined by comparison of calculated and experimental VCD spectra are of excellent 198 Continuum Solvation Models in Chemical Physics (a) (b) (c) Figure 2.23 B3LYP/6-31G∗ structures of conformations a, b and c of S-26. reliability (as long as the basis set and functional used in the calculations are well chosen). We do recognize, however, that calculated and experimental VCD spectra and rotational strengths are not in perfect agreement. The differences can be attributed to both experimental and calculational errors. VCD instrumentation is notoriously susceptible to artifacts, pseudo-VCD signals which do not originate in the VCD of the sample [24]. The magnitudes of artifacts can be assessed by comparison of the VCD spectra of the two enantiomers, ∈ + and ∈ −, measured using the VCD spectrum of the racemate as baseline. In the absence of artifacts, ∈ + = − ∈ −, and therefore ∈ + + ∈ − = 0. Deviations from zero of the ‘sum spectrum’, ∈ + + ∈ −, define the magnitudes of artifacts [25]. Calculational errors can be attributed to: (1) the neglect of anharmonicity; (2) the neglect of solvent effects; (3) imperfection of the density functional; and (4) basis set error. The presence of anharmonicity is responsible for the overall shift of calculated frequencies from experimental frequencies (see Figures ??, 2.15, 2.17, 2.22 and 2.24). The magnitude of anharmonic corrections to vibrational rotational strengths has not been defined to date; the development of software permitting anharmonicity to be included in calculations of rotational strengths is urgently needed. Solvent effects can be expected to be significant also. Unfortunately, to date the experimental study of solvent effects on VCD spectra is limited to a single study of methyloxirane (2, Figure 2.13), so far unpublished [26]. The solvent dependence of the experimental rotational strengths, using the solvents CCl4 C6 H6 CH3 2 CO CH3 OH and CH3 CN, is shown in Figure 2.26. In this case, solvent effects are minor. Since the vast majority of VCD measurements have been made in CCl4 and CHCl3 (or CDCl3 ) Properties and Spectroscopies 199 84.8 % a + 13.0 % b + 2.2 % c S-config Δ ε ∗ 103 50 expt: (+)–26 –50 R-config 1700 1600 1500 1400 1300 1200 1100 1000 900 800 Wavenumbers Figure 2.24 Comparison of the conformationally averaged B3PW91/TZ2P VCD spectra of S-26 and R-26 to the experimental VCD spectrum of +-26. solutions, it seems likely that this finding is applicable to most VCD spectra of conformationally rigid molecules. Clearly, more experimental studies are needed to confirm this conclusion. From the theoretical standpoint, to date solvent effects have been incorporated in DFT VCD calculations using the Polarizable Continuum Model (PCM) [27]. So far, the quantitative reliability of PCM DFT VCD calculations in reproducing solvent effects on the experimental VCD spectra of conformationally rigid molecules has not been thoroughly investigated. Such studies are to be desired. (We note that a detailed study of TDDFT predictions of solvent effects on the optical rotations of conformationally rigid chiral molecules, using the PCM, found that the PCM was not reliable for chlorinated solvents, such as CCl4 and CHCl3 [28]. It seems quite possible that the same will be true for PCM VCD calculations.) 200 Continuum Solvation Models in Chemical Physics 200 R-configuration 150 100 Rcalc 50 0 –50 –100 –150 –200 –200 –150 –100 –50 0 50 100 150 200 50 100 150 200 Rexpt 200 S-configuration 150 100 Rcalc 50 0 –50 –100 –150 –200 –200 –150 –100 –50 0 Rexpt Figure 2.25 Comparison of calculated rotational strengths for R-26 and S-26 to the experimental rotational strengths of +-26. Properties and Spectroscopies 40 201 C6H6 (CH3)2CO 20 CH3OH R solvents CH3CN 0 –20 –40 –60 –60 –40 –20 0 20 40 R CCl4 Figure 2.26 Solvent dependence of the experimental rotational strengths of R-+-2. In the case of conformationally flexible molecules VCD spectra are also dependent on the fractional populations of the populated conformers, which are determined by their relative free energies. It is very likely that solvent effects on conformer free energies and populations can give rise to greater solvent effects on VCD spectra than the solvent effects on rotational strengths. This raises the question: how accurately can solvent effects on conformer relative free energies be predicted? Solvent effects on free energies can be calculated using DFT via the PCM. However, a thorough comparison of PCM/DFT predictions with experimental data has not yet been reported. Such studies are also to be desired. The errors in calculated rotational strengths due to imperfection of the density functional are difficult to evaluate, since there is no alternative method available which yields perfect predictions. Nevertheless, a huge amount of effort continues to be devoted to the improvement of functionals, and one can anticipate that in the near future such improvements will be available, and will permit the errors caused by the current state-of-the-art functionals to be defined. Because of the studies of the basis set dependence of DFT rotational strengths, the errors of many basis sets are well defined. As discussed above, TZ2P and larger basis sets (e.g. cc-pVTZ) are very good approximations to the complete basis set limit. For these basis sets, errors are negligible. Of course, for much smaller basis sets, such as 6-31G∗ , the opposite is true. Thus, significant improvements of calculated rotational strengths await the incorporation of anharmonicity and solvent effects and the development of superior functionals. In the meantime, it is clear that the current DFT/GIAO methodology is of very high 202 Continuum Solvation Models in Chemical Physics accuracy, and that ACs determined using DFT/GIAO calculations together with wellchosen functionals and basis sets are of high reliability. References [1] A. Cotton, Ann. Chim. Phys., 8 (1896) 347. [2] L. Velluz, M. Legrand and M. Grosjean, Optical Circular Dichroism, Academic Press, New York, 1965. [3] W. Moffitt, W. B. Woodward, A. Moscowitz, W. Klyne and C. Djerassi, J. Am. Chem. Soc., 83 (1961) 4013–4018. [4] P. Crabbé, ORD and CD in Chemistry and Biochemistry, Academic Press, New York, 1972. [5] G. A. Osborne, J. C. Cheng and P. J. Stephens, A near-infrared circular dichroism and magnetic circular dichroism instrument, Rev. Sci. Instrum., 44 (1973) 10–15. [6] (a) M. Billardon and J. Badoz, C. R. Acad. Sci. Paris, 262B (1966) 1672–1675; (b) J. C. Kemp, J. Opt. Soc. Am., 59 (1969) 950–954. [7] (a) L. A. Nafie, J. C. Cheng and P. J. Stephens, Vibrational circular dichroism of 2,2,2trifluoro-1-phenylethanol, J. Am Chem. Soc., 97 (1975) 3842; (b) L. A. Nafie, T. A. Keiderling and P. J. Stephens, Vibrational circular dichroism, J. Am. Chem. Soc., 98 (l976) 2715–2723; (c) P. J. Stephens and R. Clark, Vibrational circular dichroism: the experimental viewpoint, in S. F. Mason, (ed.), Optical Activity and Chiral Discrimination, Reidel, Dordrecht, l979, pp 263–287. [8] J. C. Cheng, L. A. Nafie, S. D. Allen and A. I. Braunstein, Appl. Opt., 15 (1976) 1960–1965. [9] F. Devlin and P. J. Stephens, Vibrational circular dichroism measurement in the frequency range of 800 to 650 cm−1 , Appl. Spectrosc., 41 (1987) 1142–1144. [10] J. A. Schellman, J. Chem. Phys., 58 (1973) 2882–2886; 60 (1974) 343–348. [11] G. Holzwarth and I. Chabay, J. Chem. Phys., 57 (1972) 1632–1635. [12] P. J. Stephens and M. A. Lowe, Vibrational circular dichroism, Annu. Rev. Phys. Chem., 36 (1985) 213–241. [13] (a) P. J. Stephens, Theory of vibrational circular dichroism, J. Phys. Chem., 89 (1985) 748–752; (b) P. J. Stephens, Gauge dependence of vibrational magnetic dipole transition moments and rotational strengths, J. Phys. Chem., 91 (1987) 1712–1715. [14] (a) M. A. Lowe, P. J. Stephens and G. A. Segal, The theory of vibrational circular dichroism: trans l,2-dideuteriocyclobutane and propylene oxide, Chem. Phys. Lett., 123 (1986) 108–116; (b) M. A. Lowe, G. A. Segal and P. J. Stephens, The theory of vibrational circular dichroism: trans-1,2-dideuteriocyclopropane, J. Am. Chem. Soc., 108 (1986) 248–256; (c) R. D. Amos, N. C. Handy, K. J. Jalkanen and P. J. Stephens, Efficient calculation of vibrational magnetic dipole transition moments and rotational strengths, Chem. Phys. Lett., 133 (1987) 21–26; (d) K. J. Jalkanen, P. J. Stephens, R. D. Amos and N. C. Handy, Theory of vibrational circular dichroism: trans-1(S), 2(S)-Dicyanocyclopropane, J. Am. Chem. Soc., 109 (1987) 7193–7194; (e) K. J. Jalkanen, P. J. Stephens, R. D. Amos and N. C. Handy, Basis set dependence of ab initio predictions of vibrational rotational strengths: NHDT, Chem Phys. Lett., 142 (1987) 153–158; (f) K. J. Jalkanen, P. J. Stephens, R. D. Amos and N. C. Handy, Gauge dependence of vibrational rotational strengths: NHDT, J. Phys. Chem., 92 (1988) 1781–1785; (g) K. J. Jalkanen, P. J. Stephens, R. D. Amos and N. C. Handy, Theory of vibrational circular dichroism: trans-2,3-dideuterio-oxirane, J. Am. Chem. Soc., 110 (1988) 2012–2013; (h) R. W. Kawiecki, F. Devlin, P. J. Stephens, R. D. Amos and N. C. Handy, Vibrational circular dichroism of propylene oxide, Chem. Phys. Lett., 145 (1988) 411–417; (i) R. D. Amos, K. J. Jalkanen and P. J. Stephens, Alternative formalism for the calculation of atomic polar tensors and atomic axial tensors, J. Phys. Chem., 92 (1988) 5571–5575; (j) K. J. Jalkanen, R. W. Kawiecki, P. J. Stephens and R. D. Amos, Basis set and gauge Properties and Spectroscopies [15] [16] [17] [18] [19] [20] [21] [22] 203 dependence of ab initio calculations of vibrational rotational strengths, J. Phys. Chem., 94 (1990) 7040–7055; (k) P. J. Stephens, K. J. Jalkanen and R. W. Kawiecki, Theory of vibrational rotational strengths: comparison of a priori theory and approximate models, J. Am. Chem. Soc., 112 (1990) 6518–6529; (l) R. Bursi, F. J. Devlin and P. J. Stephens, Vibrationally induced ring currents? The vibrational circular dichroism of methyl lactate, J. Am. Chem. Soc., 112 (1990) 9430–9432; (m) R. Bursi and P. J. Stephens, Ring current contributions to vibrational circular dichroism? Ab initio calculations for methyl glycolate-d1 and -d4 , J. Phys. Chem., 95 (1991) 6447–6454; (n) R. W. Kawiecki, F. J. Devlin, P. J. Stephens and R. D. Amos, Vibrational circular dichroism of propylene oxide, J. Phys. Chem., 95 (1991) 9817–9831. (a) J. R. Cheeseman, M. J. Frisch, F. J. Devlin and P. J. Stephens, Ab initio calculation of atomic axial tensors and vibrational rotational strengths using density functional theory, Chem. Phys. Lett., 252 (1996) 211–220; (b) P. J. Stephens, C. S. Ashvar, F. J. Devlin, J. R. Cheeseman and M. J. Frisch, Ab initio calculation of atomic axial tensors and vibrational rotational strengths using density functional theory, Mol. Phys., 89 (1996) 579–594. GAUSSIAN, Gaussian Inc., www.gaussian.com (a) R. D. Amos, Adv. Chem. Phys., 67 (1987) 99; (b) Y. Yamaguchi, Y. Osamura, J. D. Goddard, H. F. Schaefer, A New Dimension to Quantum Chemistry: Analytic Derivative Methods in Ab Initio Molecular Electronic Structure Theory, Oxford University Press, Oxford 1994. R. Ditchfield, Mol. Phys., 27 (1974) 789–807. J. W. Finley and P. J. Stephens, Density functional theory calculations of molecular structures and harmonic vibrational frequencies using hybrid density functionals, J. Mol. Struc. (Theochem.), 357 (1995) 225–235. A. D. Becke, J. Chem. Phys., 90 (1993) 1372, 5648. P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields, J. Phys. Chem., 98 (1994) 11623–11627. (a) J. R. Cheeseman, M. J. Frisch, F. J. Devlin and P. J. Stephens, Ab initio calculation of atomic axial tensors and vibrational rotational strengths using density functional theory, Chem. Phys. Lett., 252 (1996) 211–220; (b) P. J. Stephens, F. J. Devlin and A. Aamouche, Determination of the structures of chiral molecules using vibrational circular dichroism spectroscopy, in J. M. Hicks (ed.), Chirality: Physical Chemistry, ACS Symp. Ser., 810, (2002), Chapter 2, pp 18–33; (c) P. J. Stephens, Vibrational circular dichroism spectroscopy: a new tool for the stereochemical characterization of chiral molecules, in P. Bultinck, H. de Winter, W. Langenaecker and J. Tollenaere (eds), Computational Medicinal Chemistry for Drug Discovery, Marcel Dekker, New York, 2003, Chapter 26, pp 699–725; (d) C. S. Ashvar, F. J. Devlin and P. J. Stephens, Molecular Structure in Solution: An ab initio vibrational spectroscopy study of phenyloxirane, J. Am. Chem. Soc., 121 (1999) 2836–2849; (e) A. Aamouche, F. J. Devlin, P. J. Stephens, J. Drabowicz, B. Bujnicki and M. Mikolajczyk, Vibrational circular dichroism and absolute configuration of chiral sulfoxides: tert-butyl methyl sulfoxide, Chem. Eur. J., 6 (2000) 4479–4486; (f) F. J. Devlin, P. J. Stephens, J. R. Cheeseman and M. J. Frisch, Prediction of vibrational circular dichroism spectra using density functional theory: camphor and fenchone, J. Am. Chem. Soc., 118 (1996) 6327–6328; (g) F. J. Devlin, P. J. Stephens, J. R. Cheeseman and M. J. Frisch, Ab initio prediction of vibrational absorption and circular dichroism spectra of chiral natural products using density functional theory: camphor and fenchone, J. Phys. Chem., 101 (1997) 6322–6333; (h) F. J. Devlin, P. J. Stephens, J. R. Cheeseman and M. J. Frisch, Ab initio prediction of vibrational absorption and circular dichroism spectra of chiral natural products using density functional theory: -pinene, J. Phys. Chem., 101 (1997) 9912–9924; (i) P. J. Stephens, C. S. Ashvar, F. J. Devlin, J. R. Cheeseman 204 Continuum Solvation Models in Chemical Physics and M. J. Frisch, Ab initio calculation of atomic axial tensors and vibrational rotational strengths using density functional theory, Mol. Phys., 89 (1996) 579–594; (j) C. S. Ashvar, P. J. Stephens, T. Eggimann and H. Wieser, Vibrational circular dichroism spectroscopy of chiral pheromones: frontalin (1,5-dimethyl-6,8- dioxabicyclo [3.2.1] octane), Tetrahedron Asymmetry, 9 (1998) 1107–1110; (k) C. S. Ashvar, F. J. Devlin, P. J. Stephens, K. L. Bak, T. Eggimann and H. Wieser, Vibrational absorption and circular dichroism of mono- and di-methyl derivatives of 6,8- dioxabicyclo [3.2.1] octane, J. Phys. Chem. A, 102 (1998) 6842– 6857; (l) P. J. Stephens, D. M. McCann, F. J. Devlin, T. C. Flood, E. Butkus, S. Stoncius and J. R. Cheeseman, Determination of molecular structure using vibrational circular dichroism (VCD) spectroscopy: the keto-lactone product of Baeyer–Villiger oxidation of +-(1R,5S)bicyclo[3.3.1]nonane-2,7-dione, J. Org. Chem., 70 (2005) 3903–3913; (m) P. J. Stephens, D. M. McCann, F. J. Devlin and A. B. Smith, III, Determination of the absolute configurations of natural products via density functional theory calculations of optical rotation, electronic circular dichroism and vibrational circular dichroism: the cytotoxic sesquiterpene natural products quadrone, suberosenone, suberosanone and suberosenol A acetate, J. Nat. Prod., 69 (2006) 1055–1064; (n) A. Aamouche, F. J. Devlin and P. J. Stephens, Determination of absolute configuration using circular dichroism: Tröger’s base revisited using vibrational circular dichroism, J. Chem. Soc., Chem. Comm., (1999) 361–362; (o) A. Aamouche, F. J. Devlin and P. J. Stephens, Structure, vibrational absorption and circular dichroism spectra and absolute configuration of Tröger’s base, J. Am. Chem. Soc., 122 (2000) 2346–2354; (p) P. J. Stephens, F. J. Devlin, S. Schurch and J. Hulliger, Determination of the absolute configuration of chiral molecules via density functional theory calculations of vibrational circular dichroism and optical rotation: the chiral alkane D3 – anti-trans-anti-trans-anti-trans- perhydro triphenylene, Theor. Chem. Acc., in press. [23] (a) F. J. Devlin and P. J. Stephens, Conformational analysis using ab initio vibrational spectroscopy: 3-methyl-cyclohexanone, J. Am. Chem. Soc., 121 (1999) 7413–7414; (b) A. Aamouche, F. J. Devlin and P. J. Stephens, Conformations of chiral molecules in solution: ab initio vibrational absorption and circular dichroism studies of 4, 4a, 5, 6, 7, 8 – hexa hydro – 4a – methyl – 2(3H)naphthalenone, and 3, 4, 8, 8a, – tetra hydro – 8a – methyl – 1, 6(2H, 7H) – naphthalenedione, J. Am. Chem. Soc., 122 (2000) 7358–7367; (c) P. J. Stephens, A. Aamouche, F. J. Devlin, S. Superchi, M. I. Donnoli and C. Rosini, Determination of absolute configuration using vibrational circular dichroism spectroscopy: the chiral sulfoxide 1-(2-methylnaphthyl) methyl sulfoxide, J. Org. Chem., 66 (2001) 3671–3677; (d) F. J. Devlin, P. J. Stephens, P. Scafato, S. Superchi and C. Rosini, Determination of absolute configuration using vibrational circular dichroism spectroscopy: the chiral sulfoxide 1-thiochroman S-oxide, Tet. Asymm., 12 (2001) 1551–1558; (e) F. J. Devlin, P. J. Stephens, P. Scafato, S. Superchi and C. Rosini, Determination of absolute configuration using vibrational circular dichroism spectroscopy: the chiral sulfoxide 1-thiochromanone S-oxide, Chirality, 14 (2002) 400–406; (f) F. J. Devlin, P. J. Stephens, P. Scafato, S. Superchi and C. Rosini, Conformational analysis using IR and VCD spectroscopies: the chiral cyclic sulfoxides 1-thiochroman-4-one S-oxide, 1-thiaindan S-oxide and 1-thiochroman S-oxide, J. Phys. Chem. A, 106 (2002) 10510–10524; (g) F. J. Devlin, P. J. Stephens, C. Oesterle, K. B. Wiberg, J. R. Cheeseman and M. J. Frisch, Configurational and conformational analysis of chiral molecules using IR and VCD spectroscopies: spiropentylcarboxylic acid methyl ester and spiropentyl acetate, J. Org. Chem., 67 (2002) 8090–8096; (h) V. Cerè, F. Peri, S. Pollicino, A. Ricci, F. J. Devlin, P. J. Stephens, F. Gasparrini, R. Rompietti and C. Villani, Synthesis, chromatographic separation, VCD spectroscopy and ab initio DFT studies of chiral thiepane tetraols, J. Org. Chem., 70 (2005) 664–669; (i) F. J. Devlin. P. J. Stephens and P. Besse, Conformational rigidification via derivatization facilitates the determination of absolute configuration using chiroptical spectroscopy: chiral alcohols, J. Org. Chem., 70 (2005) 2980–2993; (j) F. J. Devlin, P. J. Stephens and P. Besse, Are the absolute configurations of 2-(1-hydroxyethyl)-chromen-4-one and Properties and Spectroscopies [24] [25] [26] [27] [28] 205 its 6-bromo derivative determined by X-ray crystallography correct? A vibrational circular dichroism (VCD) study of their acetate derivatives, Tet. Asymm., 16 (2005) 1557–1566; (k) F. J. Devlin, P. J. Stephens and O. Bortolini, Determination of absolute configuration using vibrational circular dichroism spectroscopy: phenyl glycidic acid derivatives obtained via asymmetric epoxidation using oxone and a keto bile acid, Tet. Asymm., 16 (2005) 2653–2663; (l) E. Carosati, G. Cruciani, A. Chiarini, R. Budriesi, P. Ioan, R. Spisani, D. Spinelli, B. Cosimelli, F. Fusi, M. Frosini, R. Matucci, F. Gasparrini, A. Ciogli, P. J. Stephens and F. J. Devlin, Calcium channel antagonists discovered by a multidisciplinary approach, J. Med. Chem., 49 (2006) 5206–5216; (m) P. J. Stephens, F. J. Devlin, F. Gasparrini and E. Corosati, Determination of the absolute configuration of an oxadiazol-3-one calcium channel blocker, via density functional theory calculations of its vibrational circular dichroism, electronic circular dichroism and optical rotation, J. Org. Chem., 72 (2007) 4707–4715; (n) S. Delarue-Cochin, J. J. Pan, A. Daureloup, F. Hendra, R. G. Angoh, D. Joseph, P. J. Stephens, C. Cavé, Asymmetric Michael reaction: novel efficient occurs to chiral beta-ketophosphonates, Tetrahedron Asymmetry, 18 (2007), 685–691; (o) P. J. Stephens, J. J. Pan, F. J. Devlin, M. Urbanová and J. Hájíček, Determination of the absolute configurations of natural products via density functional theory calculations of vibrational circular dichroism, electronic circular dichroism and optical rotation: the schizozygane alkaloid schizozygine, J. Org. Chem., 72 (2007) 2508–2524; (p) P. J. Stephens, J. J. Pan, F. J. Devlin, M. Urbanová and J. Hájíček, Determination of the absolute configurations of natural products via density functional theory calculations of vibrational circular dichroism, electronic circular dichroism and optical rotation: the isoschizozygane alkaloids isoschizogaline and isoschizogamine, Chirality, on-line, doi: 10.1002/chir.20466; (q) P. J. Stephens, J. J. Pan, F. J. Devlin, K. Krohn and T. Kurtán, Determination of the absolute configurations of natural products via density functional theory calculations of vibrational circular dichroism, electronic circular dichroism and optical rotation: the iridoids plumericin and iso-plumericin, J. Org. Chem., 72 (2007) 3521–3536; (r) K. Krohn, M. H. Sohrab, D. Gehle, S. K. Dey, N. Nahar, M. Mosihuzzaman, N. Sultana, R. Andersson, P. J. Stephens and J. J. Pan, Prismatomerin, a new iridoid from Prismatomeris tetrandra (Rubiaceae). Structure elucidation determination of absolute configuration and cytotoxicity, J. Nat. Prod. 70 (2007) 1339–1343 P. J. Stephens and R. Clark, Vibrational circular dichroism: the experimental viewpoint, in S. F. Mason, (ed.), Optical Activity and Chiral Discrimination, Reidel, Dordrecht, l979, p. 263–287. (a) F. J. Devlin, P. J. Stephens, J. R. Cheeseman and M. J. Frisch, Ab initio prediction of vibrational absorption and circular dichroism spectra of chiral natural products using density functional theory: -pinene, J. Phys. Chem., 101 (1997) 9912–9924; (b) C. S. Ashvar, F. J. Devlin and P. J. Stephens, Molecular structure in solution: an ab initio vibrational spectroscopy study of phenyloxirane, J. Am. Chem. Soc., 121 (1999) 2836–2849; (c) A. Aamouche, F. J. Devlin and P. J. Stephens, Structure, vibrational absorption and circular dichroism spectra and absolute configuration of Tröger’s base, J. Am. Chem. Soc., 122 (2000) 2346–2354; (d) P. J. Stephens, F. J. Devlin, F. Gasparrini and E. Corosati, Determination of the absolute configuration of an oxadiazol-3-one calcium channel blocker, via density functional theory calculations of its vibrational circular dichroism, electronic circular dichroism and optical rotation, J. Org. Chem., 72 (2007) 4707–4715. F. J. Devlin and P. J. Stephens, unpublished results. C. Cappelli, S. Corni, B. Mennucci, R. Cammi and J. Tomasi, Vibrational circular dichroism within the polarizable continuum model: a theoretical evidence of conformation effects and hydrogen bonding for (S)-(-)-3-butyn-2-ol in CCl4 solution, J. Phys. Chem. A, 106 (2002) 12331–12339. See also section 2.3, 167–179, by C. Cappelli in this book. B. Mennucci, J. Tomasi, R. Cammi, J. R. Cheeseman, M. J. Frisch, F. J. Devlin, S. Gabriel and P. J. Stephens, Polarizable Continuum Model (PCM) calculations of solvent effects on optical rotations of chiral molecules, J. Phys. Chem. A, 106 (2002) 6102–6113. 2.5 Solvent Effects on Natural Optical Activity Magdalena Pecul and Kenneth Ruud 2.5.1 Introduction Natural optical activity, manifested as optical rotation (OR) in the transparent region and as electronic circular dichroism (CD) in absorption processes, is the lowest-order optical phenomenon associated with chirality, and is as such widely investigated [1]. Electronic circular dichroism has many applications in conformational analysis (especially of proteins) [2–4], and both OR and CD can be applied to determine the absolute configuration of chiral molecules [5, 6]. These fields have a long history [7] and for the case of CD, empirical schemes such as the octant rule have enabled the use of the method for establishing absolute configurations of chiral molecules [7,8]. Such empirical rules have been quite successful for CD (although the exceptions to them are quite numerous) [9], but less so in the case of OR [10]. The development of ab initio methods that can be used to calculate OR and CD directly have therefore led to a breakthrough in the field of determining absolute configurations of molecules, extending the applicability of OR and CD further. Ab initio calculations can also be useful in application of OR and CD (in particular the latter) for conformational analysis, since both these properties are sensitive to conformational changes. It is well known from experiment that both optical rotation and optical rotatory strength (the CD intensity) can vary dramatically with a change of solvent [11, 12], and even changes in the sign of a rotatory strength for a given electronic transition (or bands of transitions) are not uncommon [13]. Similar sign changes have also been observed in the case of optical rotation, even for rigid molecules such as methyloxirane [11, 12] where the solvation process does not involve significant conformational changes. Even greater solvent-induced changes of the optical rotation are observed for flexible molecules as a result of the changes in the conformational equilibria induced by the solvent, since different conformations may have very different optical rotation. The solvent effects on the rotatory strengths also vary for individual electronic transitions in a system, making a comparison of experimental spectra and theoretical results obtained for isolated molecules in some cases difficult. Without the possibility to estimate solvent effects on natural optical activity, the rigorous interpretation of experimental spectra is restricted to data collected in the gas phase. We note here that until very recently it was not even possible to measure OR in the gas phase, but the pioneering work of Müller et al. [11] has opened new possibilities for experimental investigations of solvent effects on optical rotation. Theoretical methods capable of accounting for solvent effects on OR and CD parameters are therefore important in order for these fields of research to continue to grow. Although there were early theoretical studies of OR [14,15] and in particular of CD [16– 21] it is only during the last 10–15 years that the field has grown significantly. This is partly due to the advent of computers powerful enough to allow routine calculations on chiral molecules, but also due to the implementation of efficient, gauge-origin independent methods [22–25] for the calculation of these properties, although largely limited to studies Properties and Spectroscopies 207 of isolated molecules. The calculations of natural optical activity in liquid phase, the subject of this review, have been started only in the last 2–3 years. The structure of this contribution is as follows. After a brief summary of the theory of optical activity, with particular emphasis on the computational challenges induced by the presence of the magnetic dipole operator, we will focus on theoretical studies of solvent effects on these properties, which to a large extent has been done using various polarizable dielectric continuum models. Our purpose is not to give an exhaustive review of all theoretical studies of solvent effects on natural optical activity; but rather to focus on a few representative studies in order to illustrate the importance of the solvent effects and the accuracy that can be expected from different theoretical methods. 2.5.2 Molecular Theory of Optical Activity Optical Rotation When plane-polarized light passes through a sample of chiral molecules with an excess of one enantiomer, its plane of polarization is rotated. This phenomenon, called optical rotation, is usually described quantitatively by the specific optical rotation , defined as = V ml (2.96) where is the rotation of the original polarization plane for incident light of frequency l is the optical path length, and m and V are the mass and volume of the chiral sample, respectively. The specific optical rotation is related to the trace of the Rosenfeld tensor through [26] = 288 × 10−30 2 NA a40 2 M (2.97) where = 13 Tr , and where the Rosenfeld tensor can be written in terms of the mixed electric dipole–magnetic dipole polarizability G = −−1 G (2.98) In these equations, NA is Avogadro’s number, a0 is the Bohr radius, M is the molar mass of the molecule in g mol−1 the frequency of the light in atomic units, and is −1 expressed in atomic units. The units of are deg cm3 g dm−1 . Most measurements of the optical rotation are carried out at a single frequency, usually corresponding to the sodium D-line. However, studies of the variation of the optical rotation with the frequency of the incident light are also known, and are referred to as optical rotatory dispersion (ORD) [7]. Historically, this was an important method for the determination of excitation energies in chiral molecules, but was later superseded by CD. We note that the calculation of ORD through regions of electronic absorption requires special care [27, 28]. 208 Continuum Solvation Models in Chemical Physics The mixed electric dipole–magnetic dipole polarizability G introduced in Equation (2.98) can be written as a sum-over-states expression [29] G = −2 n=0 Im 0 ˆ nnm̂ 0 2n0 − 2 (2.99) where ˆ and m̂ are the electric and magnetic dipole moment operators, respectively, 0 and n the reference and excited state, respectively, and n0 is the energy of the transition between these states. We note from this expression that G vanishes for static fields = 0. For oriented samples, the rotation of the plane-polarized light becomes a tensor – that is, the optical rotation becomes directionally dependent – and includes a contribution from the electric dipole–electric quadrupole polarizability tensor, which is traceless and thus vanishes for freely rotating molecules [30]. The term arising from these quadrupolar interactions can be expressed as [30] −A = − n=0 n0 0 ˆ nnQ̂ 0 2n0 − 2 (2.100) where is the Levi–Civita symbol, and Q̂ the electric quadrupole moment operator. Using response theory [31], the mixed electric dipole–magnetic dipole polarizability can be expressed as G = Im ˆ $ m̂ (2.101) which is equivalent to the sum-over-states expression in Equation (2.99) for exact wavefunctions. Within the same formalism, the mixed electric dipole–electric quadrupole polarizability can be expressed as A = −Re ˆ $ Q̂ (2.102) The magnetic dipole operator m̂ is proportional to the angular momentum operator l̂ , whereas the electric dipole operator ˆ can be expressed in length or in velocity form – that is, by the position operator r̂ or by the momentum operator p̂ . It is customary to refer to the expression in Equation (2.99) (or Equation (2.101)) as being the expression for the optical rotation in the length gauge when it involves the position operator, and in the velocity gauge when it involves the momentum operator. We note that the lengthgauge formulation is origin dependent for approximate wave functions, in contrast to the origin-independent velocity gauge formulation. The latter, however, suffers from slower basis set convergence. The optical rotation in the velocity gauge can be obtained by using the relation i 0 p n = no 0 r n (2.103) Properties and Spectroscopies 209 and the optical rotation is then in the velocity gauge given by G = 1 p̂ $ m̂ (2.104) We note that the relation in Equation (2.103) is only valid for variational wavefunctions in the limit of a complete basis set, and therefore the length and velocity gauges in general give different results. Whereas the velocity gauge in general gives somewhat slower basis set convergence than the length gauge, the results obtained with the velocity gauge are origin independent. It has been shown [32] that much improved basis set convergence can be obtained for the optical rotation in the velocity gauge by subtracting the static limit from the electric dipole–magnetic dipole polarizability p̂ $ l̂ → p̂ $ l̂ − p̂ $ l̂ (2.105) 0 This ensures basis set convergence comparable with that of the optical rotation in the length-gauge formulation, while not affecting origin independence. The lack of origin independence in the length-gauge formulation has been solved by using London Atomic Orbitals (LAOs) [22–24, 33]. The use of LAOs ensures that the optical rotation (or optical rotatory strengths) is independent of the choice of magnetic gauge origin for variational wavefunctions also in finite basis sets. The LAOs are defined as [34] 1 B = exp − i B × RNO · r r N (2.106) 2 where rN is a Gaussian atomic orbital centred on nucleus N and RNO is the position of nucleus N relative to the gauge origin O. The complex phase factor thus shifts the global gauge origin O to the best local gauge origin for each basis function, namely to the nucleus to which the basis function is attached. In addition to providing origin-independent results, the LAOs also lead to somewhat faster basis set convergence, although this effect is in general less pronounced for the optical rotation than it is for properties such as magnetizabilities [35]. The consequences of the explicit dependence of LAOs on the magnetic induction B are discussed in ref. [22]. Reference [24], where the Hartree–Fock formulation is described, provides the expression for the angular momentum operator in terms of the LAOs and a discussion of gauge-origin independence of the rotatory strength in the length gauge formulation when LAOs are used. The behaviour of the exact operator is examined in ref. [36]. We note that a formulation of the LAOs for time-dependent electromagnetic fields has recently been presented by Krykunov and Autschbach [37]. Electronic Circular Dichroism Circular dichroism is the differential absorption of left- and right-circularly polarized light by a sample with excess of one enantiomer. The effect is usually expressed as the difference between the molar extinction coefficients for left- and right-circularly polarized light (L ! and R !) ! = L ! − R ! (2.107) 210 Continuum Solvation Models in Chemical Physics ! is related to the rotatory strength n R of the transition between the ground state 0 and the nth excited state through the equation 1 n 16 2 NA a0 e2 ! = √ R exp 7 3c × 10 ln 10 me n n " ! − !n0 − n 2 # (2.108) where we have assumed a Gaussian band shape with half-width !n ! is the wavelength of the incident light and !n0 is the wavelength for the electronic transition. The rotatory strength n R was derived from quantum mechanical theory by Rosenfeld [26], and was shown for isotropic samples to be given as the product of the electric dipole and magnetic dipole transition moments, which in atomic units can be written as n " # 3 R = Im 0 ˆ n n m̂ 0 − 0 ˆ n n m̂ 0 4 (2.109) For oriented samples, there is also a contribution from interactions with the electronic quadrupole moment [36] n 3 RQ = n0 0 ˆ n n Q̂ 0 4 (2.110) This contribution is purely anisotropic, and thus vanishes upon orientational averaging and does not contribute in the case of isotropic samples such as a liquid. Using the formalism of response theory [24, 31], the scalar rotatory strength for a transition from the ground state 0 to an excited state n can be evaluated as the residue of the linear response function. In the velocity gauge formulation, n R is given by the equation $ 1 1 R = Tr 0p̂n · nL̂0 = 2n0 2n0 lim − n0 p̂$ L̂ n v % →n0 (2.111) whereas it in the length gauge formulation is given as R =− n r i i 0r̂n · nL̂0 = − Tr 2 2 $ % lim − n0 r̂$ L̂ →n0 (2.112) As was the case for the optical rotation, the length-gauge formulation is origin dependent for finite basis set calculations, but we note that origin-independent results can be obtained using London atomic orbitals [24, 25]. Optical Activity of Solvated Molecules OR and CD have for a long time been recognized as being very sensitive to the molecular environment. This hampers the comparison between theory and experiment, since the calculations are usually carried out for a single isolated molecule, whereas the measurements are usually conducted in liquid phase. Thus, attempts to account for solvent effects were undertaken at an early stage of theoretical modelling of natural optical activity. Properties and Spectroscopies 211 Solvent effects on the optical rotation are traditionally accounted for using the Lorentz effective field approximation [38], in which the optical rotation is multiplied by a local field factor LF = s + 2 3 (2.113) where s is the frequency-dependent dielectric constant of the solvent. This relation, which results in an increase of the optical rotation with increasing solvent polarity for all solvated molecules, has been shown many times not to describe properly the actual effects [39, 40], and more sophisticated models are required. At a more detailed level, we note that the solvent effects on the optical rotation have the same origins as solvent effects on the energy itself, as described in detail in other contributions to this book. Most other studies of solvent effects on natural optical activity have focused on the electrostatic contributions. These contributions can be partitioned into direct effects arising from the influence of the dielectric environment on the electronic density of the solute, and into indirect effects arising from the relaxation of the nuclear structure in the solvent. For conformationally flexible molecules, we may also consider a third possible solvent effect due to the changes in the conformational equilibria when going from the gas phase to solution. The electrostatic effects can be accounted for by means of a polarizable continuum model (PCM), where the solute molecule, treated quantum mechanically, is placed in a cavity in the solvent which is modelled as a dielectric continuum, characterized only by its dielectric constant. Computational techniques based on the PCM have been developed independently by several groups. They differ mainly in the cavity shape, and in the way the charge interaction with the medium is calculated. The cavity is defined as a sphere, an ellipsoid or a more complicated shape following the surface of the molecule. To compute the electrostatic component of the solvation free energy this model requires the solution of a classical electrostatic Poisson problem. Nowadays, the most popular method of solving this problem is a PCM developed primarily by the group of Tomasi and coworkers [41–43]. In this approach, the cavity is made from spheres centred on nuclei in the solute molecule, and the cavity surface is divided into a number of small surface elements (see the contribution by Pomelli), where the reaction field is modelled by distributing charges onto the surface elements, i.e. by creating apparent surface charges [44–47]. The electrostatic part of the solvent–solute interaction represented by the charge density spread over the cavity surface (apparent surface charges, ASC) gives rise to an operator to be added to the Hamiltonian of the isolated system in order to obtain the final effective Hamiltonian and the related free energy functional. ASC–PCM calculations [42, 43] can be carried out in different ways. The most widespread approach is the IEF–PCM method (Integral Equation Formalism) of Cancès et al. [46], which uses a molecule-shaped cavity to define the boundary between the solute and the solvent. Another approach is the COSMO method (COnductorlike Screening MOdel) due to Klamt and co-workers [48–50], in which the surrounding medium is modelled as a conductor instead of a dielectric. Apart from the ASC–PCM method developed by the Pisa group, there are several other PCM-based methods: the MPE (multipole expansion method) of the Nancy group 212 Continuum Solvation Models in Chemical Physics [51, 52] and of Mikkelsen and co-workers [53, 54] with a spherical cavity, and the GBA (generalized Born approximation) [48, 55–57] and others. Calculations of OR and CD are getting increasingly widespread. This is due to the development of computational protocols for calculating these properties which are made available in popular quantum chemical program packages. Calculations of optical rotation and optical rotatory strengths can be performed for example using the freeware program package DALTON [58] (for density functional theory DFT, single- and multireference self-consistent field (MCSCF) wave functions, coupled cluster theory (CC), and secondorder polarization propagator theory (SOPPA)) or commercial program packages such as Gaussian03 [59] (using DFT or Hartree–Fock (HF)), Amsterdam Density Functional program (ADF) [60] for DFT, or Turbomole [61, 62] (for Hartree–Fock or DFT). In some of these programs, solvent effects can be calculated using for instance MPE with a spherical cavity [53, 54], IEF–PCM [41–43], or the COSMO model [48–50]. Other solvent models based on polarizable continuum concepts are available in other programs. Still, the majority of theoretical investigations of natural optical activity are done on molecules in the gas phase, and the consequences and effects of a solvent on natural optical activity are not yet fully understood, in particular for solvents that may display strong specific interactions with the solute. 2.5.3 Calculations of Solvent Effects on Natural Optical Activity Optical Rotation Solvent effects on the optical rotation of several conformationally rigid chiral organic molecules (fenchone, camphor, - and -pinene, camphorquinone, verbenone and methyloxirane) have been studied by Mennucci et al. [39] using the IEF–PCM combined with DFT. The solvent effects were found to be substantial. For the solvents under investigation, the results obtained using the PCM were in most cases found to be in good agreement with experiment. However, the solvents benzene, chloroform and carbon tetrachloride showed disagreement with experiment, and it was concluded that for these solvents other interactions than the purely electrostatic ones play a more important role. The excellent agreement obtained for the wide range of solvents studied – ranging in polarity from cyclohexane to acetonitrile – suggests that in these cases PCM represents a suitable level of approximation for the study of solvent effects on the optical rotation, superior to the Lorenz effective field approximation. Solvent effects on the optical rotation have also been performed by the same group for 6,8-dioxabicyclo[3.2.1]octanes [40] using IEF–PCM. It was demonstrated that the Lorentz effective field approximation does not properly account for the solvent effects in this case. In contrast to this, DFT calculations combined with the IEF–PCM lead to a mean absolute deviation in the calculated optical rotations when compared to experiment −1 −1 of 12 6 deg cm3 g dm−1 , to be compared with 16 6 deg cm3 g dm−1 when PCM is not used. However, this finding may be fortuitous, since only one conformation was taken into account for each molecule, although we note that other conformations were shown to lie significantly higher in energy. The indirect influence of the solvent on the optical rotation due to the change in the conformational equilibrium upon solvation was studied by Polavarapu et al. [63] for R-epichlorhydrin. No solvent model was used, and the conformer populations in Properties and Spectroscopies 213 solution were obtained from the experimental IR spectra. The purpose of this study was to investigate the origins of the observed sign change of the specific rotation of R-epichlorhydrin in CCl4 compared with that in more polar solvents. The authors found that by using the optical rotation calculated at the DFT/B3LYP level for gasphase structures of different conformers of (R)-epichlorhydrin combined with conformer populations obtained from the IR spectra, one can reproduce the experimentally observed solvent dependence of the optical rotation quite successfully. Historically, optical rotation has been a property strongly associated with carbohydrates, and the IEF–PCM/DFT model for calculating optical rotation has been applied to study the OR of glucose [64]. The geometric parameters of eight conformers of glucose were optimized in the gas phase, and then transferred (without reoptimization) into the dielectric continuum model of an aqueous solution. It was found that the difference between the natural optical rotation of glucose in the gas phase (calculated as a Boltzmann average) and in aqueous solution primarily arises from the influence of the solvent on the conformer population statistics, whereas the direct effects on the optical rotation of the individual conformers were found to be much less significant. However, no account was taken of geometry relaxation effects or specific interactions such as hydrogen bonds in the calculations, which may change the picture dramatically. The authors did obtain, despite the limitations inherent in their computational model, good agreement with the experimental optical rotation for glucose in aqueous solution, which indicates that the effects mentioned above are either small or that a very fortunate cancelation of errors takes place for this model system. IEF–PCM calculations including all three contributions from the solvent (direct, through geometry changes and through changes in conformer population) have been carried out by Marchesan et al. [65] for paraconic acid and by Coriani et al. [66] for -butyrolactones. The objective was to investigate whether DFT calculations combined with the PCM are capable of correctly assigning the absolute configuration of highly flexible molecules. The results for paraconic acid indicate that the sign reversal of the optical rotation in going from vacuum to methanol solution is mainly due to changes in the conformer populations. However, the results are very sensitive to the computational method chosen, and the agreement with experiment was found to be much better when geometric parameters and energies obtained with Møller–Plesset second-order perturbation theory (MP2) were used instead of the DFT results. The calculations for the -butyrolactones family (of which paraconic acid is a precursor) were carried out for isolated molecules and for molecules in methanol as modelled by IEF–PCM. The solvent effects were strong, and it was found that the use of IEF–PCM is essential in order to bring the computed optical rotation into close agreement with experiment. The signs of the calculated optical rotations were in all cases found to be in agreement with experiment, and the authors therefore concluded that DFT/PCM is an appropriate method for the determination of the absolute configuration of this class of molecules. Less optimistic conclusions about the performance of the DFT/PCM scheme were drawn in a study of solvent effect on the optical rotation of (S)--methylbenzylamine [67]. The authors compared the optical rotation of this amine measured in 39 different solvents (whenever possible extrapolated to infinite dilusion) with the results obtained by means of IEF-PCM with the B3LYP functional and the aug-cc-pVDZ basis set. They observed substantial discrepancies for many of the hydrogen-bond forming solvents (which is not 214 Continuum Solvation Models in Chemical Physics surprising), but also for some solvents with low polarity (most noticeably for carbon tetrachloride). The latter fact is probably due to dispersion effects not accounted for by the PCM, and demonstrates the limitations of the method well. This study therefore largely corroborates the findings of the study by Mennucci et al. [39]. The PCM/DFT model failed to predict the intrinsic rotation (i.e. the specific rotation extrapolated to infinite dilution) of R-3-methylcyclopentanone dissolved in carbon tetrachloride, methanol and acetonitrile [68]. This molecule has been investigated because it exists in both an equatorial and an axial form, allowing researchers to investigate the interplay of solvent and conformational effects. The conformer populations used in the Boltzmann averaging were derived from IR absorption and VCD spectra. The deviation of the calculated optical rotation from experiment was found actually to be larger when IEF–PCM was used to account for direct solvent effects (and geometry relaxation) on the optical rotation than when the gas-phase values were used. The calculations of OR employing the PCM are not limited to DFT. Coupled cluster methods (CC2 and CCSD) combined with a PCM using a spherical cavity [69] has been developed by Kongsted et al. [70] and used to model solvent effects on the optical rotatory dispersion of methyloxirane. The results for the wavelength of 589.3 nm were compared with experimental studies [11, 12] and with IEF–PCM results (combined with DFT) of Mennucci et al. [39]. From the comparison it appears that the approach of Mennucci et al. [39] is somewhat more successful in modelling the solvent effects than the method of Kongsted et al. [70], although the remarkably large difference of the optical rotation of methyloxirane in gas phase and in cyclohexane solution [11] is not reproduced by either approach. The study of the importance of solvents on optical rotation was given a significant boost by the development of a cavity ring-down spectrometer capable of measuring optical rotation of molecules in the gas phase for a wide frequency range [11]. In this work, it was demonstrated that the optical rotation of S-methyloxirane in the gas phase actually is positive, in contrast to the sign observed in most solvents, and also in contrast to most of the theoretical data that had been obtained for the isolated molecule at the time. This finding led to a substantial theoretical effort to reproduce the experimental observations, including both electron correlation [70,71], vibrational corrections [72] and solvent effects [72,73], and we note in particular the recent study of optical rotation using the quantum mechanics/molecular mechanics (QM/MM) approach [74]. These studies have demonstrated the sensitivity of the optical rotation to the choice of computational method, and care has in general to be exercised in using theoretical predictions of optical rotations of less than about 30 in magnitude for determining the absolute configuration of even rigid molecules. Vaccaro and co-workers have later presented other experimental studies of optical rotations of molecules in the gas phase [74,75]. Electronic Circular Dichroism Ab initio calculations of solvent effects on ECD spectra are less abundant than those on OR. An ab initio study of the solvent effects on the ECD spectra were carried out by Pecul et al. [76] using the IEF–PCM method [44, 45, 47] at the DFT/B3LYP level using LAOs. The rotatory strengths were shown to be strongly influenced by a change of solvent, and for certain transitions in molecules such as methyloxirane, even Properties and Spectroscopies 215 the sign of the rotatory strength changed. This is at first glance somewhat surprising considering that methyloxirane is a fairly rigid molecule, and thus does not change its conformation upon a change of solvent. However, this sensitivity of the ECD spectrum of methyloxirane to solvent effects could be anticipated considering the strong solvent effects observed experimentally for the optical rotation [12]. For flexible molecules, even greater solvent effects can be anticipated. In ref. [76], calculations of the CD spectra of chiral bicycloketones in several organic solvents were also performed, and the initial results showed promising agreement with experiment for low-lying valence transitions. Transitions to diffuse states (Rydberg transitions) were found to be more difficult, though it is not obvious whether this is due to limitations in the solvent model or inherent limitations in the DFT functional used for the study of diffuse excited states. Another approach for calculating solvent effects on ECD spectra based on a dielectric continuum model was presented by Kongsted et al. [69], who used the coupled cluster method coupled with the MPE approach to model the influence of a solvent on the rotatory strength tensors of formaldehyde (a nonchiral molecule that exhibits optical activity only in oriented samples). Both the length and velocity gauge formulations were employed. As in ref. [76], the presence of the dielectric continuum was found to change the sign of the optical rotatory strengths of some of the transitions. Reaction field theory with a spherical cavity, as proposed by Karlström [77, 78], has been applied to the calculation of the ECD spectrum of a rigid cyclic diamide, diazabicyclo[2,2,2]octane-3,6-dione, in an aqueous environment [79]. In this case, the complete active space self-consistent field (CASSCF) and multiconfigurational secondorder perturbation theory (CASPT2) methods were used. The qualitative shape of the solution-phase spectrum was reproduced by these reaction field calculations, although this was also approximately achieved by calculations on an isolated molecule. Another system investigated using continuum models is 1-R-phenylethanol, for which the effect of the aqueous solution has been calculated by Macleod et al. [9] by means of the configuration interaction singles (CIS) method and DFT. In this case, both the IEF-PCM method and a supermolecular model (using small singly and doubly hydrated clusters) were used to model the effects of the aqueous environment on the CD spectrum of 1R-phenylethanol. The results obtained were, however, still at variance with experiment. The best (although still not perfect) agreement with experiment was obtained when calculations were performed on the averaged structures of solvated 1-R-phenylethanol obtained from molecular dynamics simulations. The CD spectrum of 1-R-phenylethanol was further investigated by the same group, who also carried out calculations for 1-R-phenylethylamine and its protonated cation [80] using the CIS method (DFT was found to be less reliable, especially for 1R-phenylethylamine). The influence of the solvent was accounted for by two methods: (1) using rigid hydrated clusters containing from one to three water molecules; (2) by carrying out molecular dynamics simulations in an aqueous ensemble, taking representative snapshots of geometries which then were used to calculate the CD spectra. The CD calculations were carried out for 1-R-phenylethylamine with the water molecules removed, so only indirect (through changes in the geometry) solvent influence were accounted for. The results were compared with experimental CD spectra collected in aqueous solution and in nonpolar solvents. The authors observed that solvent-induced changes in the geometry are the primary sources for the differences between CD spectra 216 Continuum Solvation Models in Chemical Physics of 1-R-phenylethanol and 1-R-phenylethylamine in polar and nonpolar solvents, since only in this case did they obtain satisfactory agreement with experiment. The COSMO solvent model has been used to simulate the influence of water on the electronic spectrum of N -methylacetamide [81], and the results was compared with the results of molecular dynamics simulations (where the electronic spectrum were calculated as an average over 90 snapshots from MD simulations). Most of the hydration effects were found to come from the first solvation shell hydrogen-bonded water molecules, and the continuum model does not properly account for these effects. The rotatory strengths were not calculated directly in ref. [81]. However, the results were used to model ECD spectra of peptides via the coupled oscillator model, with satisfactory result. 2.5.4 Perspectives The accurate and effective modelling of solvent effects is one of the most important challenges facing quantum chemistry in the years to come. Solvent effects on OR and CD are here of particular importance, since they are atypically strong, and sign reversals are not uncommon. This strong dependence on the inclusion of solvent effects makes it imperative to include these effects in the models in order for ab initio studies of these properties to have predictive powers. If theoretical predictions are to be compared with experimental results in order to extract information such as absolute configuration or conformational composition of a given compound, solvent effects have to be accounted for. The PCM has been shown to be quite successful in some cases in the modelling of solvent effects on optical rotation in polar solvents which do not form hydrogen bonds. However, in other cases the PCM fails to reproduce the experimentally observed effect. These failures can in many cases be explained by the presence of specific interactions such as hydrogen bonds, or by the dominance of dispersion effects in the solute–solvent interactions, neither of which is accounted for in PCMs. However, in some instances the reason for the failure of the PCM in reproducing solvent effect on OR is not obvious. Calculations of solvent effects on CD spectra have so far been less frequent than on OR and thus no general conclusions can yet be drawn, but it seems that the performance of the PCM for CD is even less stable than for OR. It should be recalled that the calculation of solvent effects on optical activity presents some unique problems. A chiral solute induces a chiral structure of the surrounding solvent, even when the individual solvent molecules are achiral. 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It is also a powerful analytical tool for determining absolute configurations, and for investigating conformational equilibria of chiral molecules and of their interaction in the liquid phase. The ability to probe the solution structure of molecules dissolved in water has made ROA a method of choice for some aspects of the solution structure of biopolymers as solution structures cannot be investigated by X-ray crystallography, and as the NMR time scale can be too slow to distinguish structures which interconvert. First measurements for identifying optical activity in Raman scattering were undertaken soon after the Raman effect itself was discovered but proved unsuccessful [1, 2]. The fact that the measurement of solutions of biomolecules is at present the most important application of ROA is remarkable in the face of the experimental difficulties which haunted these early attempts to observe it, even for pure, chiral liquids, the most favorable experimental situation. Early theoretical treatments [3,4] of optically active scattering by molecules did little to arouse renewed interest in a measurement of ROA. The decisive cross-terms between the electric dipole–electric dipole polarizability and the optical activity tensor were missed, and effects predicted for the optical activity tensor alone were too small to be practically useful. It was only after the proper cross-terms were identified [5] that new interest in the measurement of ROA accrued, and that the existence of the phenomenon was finally proved [6, 7]. It was not long before the first measurement of whole ROA spectra was demonstrated [8]. The technological advance which made the measurement of ROA possible in the early 1970s was the invention of the argon ion laser, which also turned ordinary Raman spectroscopy into a general analytical tool. Further progress depended on the development of optical multichannel detection and holographic grating technology [9–12], and the solution of the decisive offset problem [13, 14]. Being able to measure a phenomenon and being able to understand the measured results are two sides of a coin. Although a number of conceptually interesting models for optically active Rayleigh and Raman scattering [15–20] were developed early on, actually predicting even a small segment of a not yet measured ROA spectrum remained elusive. An empirical rule [8], derived from one of the first ROA measurements, could be confirmed [21], but further rules of general usefulness for assigning the absolute configuration of small molecules did not evolve. Relative configurations could, on a case by case basis, be determined by a comparison of ROA spectra. Once the ROA measurement of biopolymers became possible, due to the development of backscattering [22, 23], the power of empirical rules relating observed ROA to molecular structure became evident [24, 25]. In the case of biopolymers, the absolute Properties and Spectroscopies 221 configuration of individual building blocks is generally known and it is the secondary and tertiary structures which became the goals of a spectroscopic investigation [26]. While the interpretation of the ROA spectra of biopolymers remains at present empirical and based on the comparison of the ROA for known structures, notably through the use of pattern recognition techniques [27–29], there has also been a recent attempt to improve the theoretical understanding by the direct ab initio computation of the ROA of a decapeptide [30]. The situation is more favourable for smaller chiral molecules. Computational advances in quantum chemistry have been to the interpretation of ROA what the development of laser technology and of electro-optics was to its measurement. Over a period of about two decades [31–39] the ab initio computation of ROA spectra has matured from a somewhat haphazard exercise to a reliable tool for determining absolute configurations, and solution conformations. We will focus in this chapter on the basic formalism of Raman and ROA scattering, and on the understanding of ab initio computed vibrations, electronic tensors, and Raman and ROA scattering cross-sections. The usefulness of decomposing ab initio computed data will be demonstrated in the context of their comparison with the measured spectra of +-P-1,4-dimethylenespiropentane [40] which exhibits an unusual dependence on the solvent environment. 2.6.2 Basic Theoretical Expressions Circular Sum and Difference Scattering Cross-sections The molecular measure for Raman scattering is the scattering cross-section , where is defined as the rate at which photons are removed from an incident beam of light by scattering into a solid angle of 4, relative to the rate at which photons cross a unit area perpendicular to their direction of propagation [41]. thus has the dimension of m2 per molecule. For ROA, the measure is the difference scattering cross-section for leftand right-circularly polarized light [42, 43]. The sign convention in optical activity for molecular properties is the value measured for left- minus that for right-circular light. Measured ROA spectra are, for historical reasons, displayed as the scattered intensity of right-circular (R) minus that of left-circular (L) light [44, 45]. A potential sign confusion between measured and computed data is avoided by representing computed data not as , but as −. Though the integral scattering cross-sections and for scattering into a solid angle of 4 are the definitive molecular measure [34] and easiest to calculate [46] due to their lack of quadrupole contributions [42], they are rarely measured in Raman spectroscopy. Most spectra are recorded for a particular scattering direction, or rather for a cone of scattered light about the direction of observation. The theoretical measures which then allow for the comparison with experimental data are the circular sum and circular difference differential scattering cross-sections per unit of solid angle , namely d%/d and d%/d, where % is the scattering angle. In addition to the choice of the scattering direction, there are three basic polarization schemes [47] for measuring ROA, namely ICP [6] (incident circular polarization), SCP [48] (scattered circular polarization), and DCP [49] (dual circular polarization). All have been experimentally demonstrated. ICP, where the polarization of the exciting light 222 Continuum Solvation Models in Chemical Physics is modulated between right- and left-circular and the variation of the intensity of the scattered light is measured, is historically the oldest method. In SCP the polarization of the exciting light is kept unchanged and the difference in the intensity of the right- and left-circular scattered light is detected, and in DCP the circular polarization of the incident light is modulated and the content of circular scattered light analysed. If the modulation and detection are in phase, i.e. right-circular light is detected when right-circular light is used for irradiating the sample, the designation DCPI is used, while DCPII stands for out-of-phase modulation and detection [50]. In the off-resonance case, DCPII vanishes and only DCPI is thus of interest here. A further important distinction arises for SCP and ICP ROA, if they are not measured in a collinear scattering geometry, by the possibility to choose the incident light (SCP) or scattered light (ICP) either naturally (n), or linearly polarized oriented parallel or perpendicular ⊥ to the scattering plane. For right angle measurements, parallel polarization leads to depolarized Raman and ROA spectra and perpendicular polarization to polarized ones. As ROA is done on isotropic samples, the information which ideally results from an appropriately chosen set of measurements is rotational invariants of the scattering tensor. In practice, though, these invariants are rarely separately determined [11, 51] because the measurements done with different scattering angles and polarization schemes are not necessarily directly comparable, owing to instrumental limitations. The scattering arrangements of practical interest are SCP and DCPI backward and forward scattering, and polarized and depolarized ICP right angle scattering. Formulae for the general scattering angle and polarization dependence of all Raman and ROA scattering arrangements are available [52–55] but they are of importance mainly to experimentalists intent on extracting the invariants of the various parts of the scattering tensor from measured data. We give only the basic SCP formulae here. The scattering cross-sections of all other scattering arrangements are expressible through them in the off-resonance case [45]. We also note that ICP and SCP have the same invariant combination. The rotational invariants of the scattering tensor, namely a2 and 2 for ordinary Raman scattering, and aG 2G , and 2A for ROA scattering, are explained in the following section. If we drop the explicit mention of the molecular states between which the molecule transits during the scattering process, then the scattering cross-sections for the off-resonance situation can be written as: ⊥ d%SCP = 2K45a2 + 72 d d%SCP = 2K 62 + cos2 %45a2 + 2 d 4K 45aG2 + 72G + 2A + cos %45aG2 − 52G − 32A d c 4K 2 6G − 22A + cos %45aG2 − 52G − 32A − d%SCP = 4 + cos2 %45aG2 + 2G + 32A d −⊥ d%SCP = −n SCP = 8K 180aG2 + 402G 3c (2.114) (2.115) (2.116) (2.117) (2.118) Properties and Spectroscopies 223 where e.g. −⊥ d%SCP = − ⊥ dL %SCP −⊥ dR %SCP . The cross-sections for natural polarization follow as averages of the polarized and depolarized ones. K depends on the circular frequency 0 of the exciting and p of the scattered light. For in units of m2 its value is given by K = Kp = 1 0 2 0 3p 90 4 (2.119) where 0 is the permeability of the vacuum. Mechanical and Electrical Harmonic Approximation The tensors which enter theoretical expressions are transition tensors Tf ←i for a transition between an initial state i and a final state f . The Placzek polarizability theory for vibrational Raman scattering [56], which we use here, is valid in the far from resonance limit. i and f are then vibrational states. If we assume that they differ for normal mode p, then the transition tensors can be written as e e & T T e Tf ←i ≈< f T i >≈ < f Qp i >≈ (2.120) Qp 0 Qp 0 400c˜p with ˜p in units of cm−1 . We notice that terms independent of Qp on the right-hand side of Equation (2.120) vanish because of the orthogonality of the vibrational wavefunctions f and i. For the explicit form of the integral < f Qp i > it is assumed that Qp is a normal mode and the vibration p therefore harmonic, and that f ← i is a fundamental transition. If the transition starts from a level other than the level np = 0, where np is the vibrational quantum number, then the right-hand side of Equation (2.120) must be multiplied by √ np + 1 for an upward and np for a downward transition. The form (2.120) for Tf ←i further implies the electrical harmonic approximation by assuming that derivatives of T e higher than the first one vanish. The derivatives of the electronic tensor T e with respect to the normal coordinate Qp can be expressed in terms of the derivatives with respect to the Cartesian displacements xi of the nuclei : e T e T e T e xi T x = = L = · Lx (2.121) Qp 0 i xi 0 Qp 0 i xi 0 ip x 0 p as one has xi = Lxip Qp (2.122) with i where m is the mass of nucleus . m Lxip 2 = 1 (2.123) 224 Continuum Solvation Models in Chemical Physics Ordinary Raman scattering is determined by derivatives of the electric dipole–electric dipole tensor e , and ROA by derivatives of cross-products of this tensor with the e imaginary part G of the electric dipole–magnetic dipole tensor (the optical activity tensor) and the tensor Ae which results from the double contraction of the third rank electric dipole–electric quadrupole tensor Ae with the third rank antisymmetric unit tensor of Levi–Civita. The electronic property tensors have the form: e = 2 jn Re< nj ˆ >< jn ˆ > j=n 2jn − 20 2 0 Im< nj ˆ >< jm̂n > j=n 2jn − 20 (2.125) 2 jn ˆ > Re &< nj ˆ >< jn j=n 2jn − 20 (2.126) Ge = − Ae = (2.124) The summation in Equations (2.124)–(2.126) extends over the electronic states j of the system, which in the absence of the perturbation by the radiation field is assumed to ˆ the be in the stationary state n. ˆ is the electric dipole, m̂ the magnetic dipole, and electric quadrupole operator. Rotational Averages and their Decomposition The expressions for Raman and ROA intensities depend on products of the various transition tensors. For ROA, isotropic samples are of interest and rotational averages of the products of these tensors are therefore required. The rotational averages can be expressed by double contractions of the isotropic (is), the anisotropic (anis), and the antisymmetric (a) part, which for a second rank tensor T are defined as is = T T (2.127) 1 anis T = T + T − T 2 1 a T = T − T 2 (2.128) (2.129) where T= 1 T 3 (2.130) T = T is + T anis + T a (2.131) For the double contraction of two tensors T1 and T2 it holds that T1 :T2 = T1is & T2is + T1anis & T2anis + T1a & T2a T1is & T2anis = T1is & T2a = T1ansis & T2a =0 (2.132) (2.133) Properties and Spectroscopies 225 with the three independent invariants of the product T1 T2 having the form: T1is & T2is = 3T1 T2 = T1anis & T2anis = 1 T T 3 1 2 (2.134) 1 1 T1 T2 + T1 T2 − T1 T2 3 2 T1a & T2a = (2.135) 1 T T − T1 T2 2 1 2 (2.136) The convention for summing over indices specified by the double dot product is as in ref. [57] and defined by AB & CD = A · CB · D (2.137) where A B C, and D are vectors and AB and CD dyads formed from them. We note that Equation (2.137) is not the only definition of the double dot product found in the literature [58], and that differences in the definitions must be observed for the antisymmetric invariant. If the form (2.120) of Tf ←i is used with the Cartesian derivatives (2.121), then the double contraction of two transition tensors for the states i and f of normal mode p becomes: T1 & T2 f ←i = T1 T2 f ←i ≈< f Qp i > 2 e e T1 T2 i j xi 0 xj Lxip Lxjp 0 (2.138) with analogous expressions for the isotropic, anisotropic, and antisymmetric invariant defined by Equations (2.134)–(2.136). The conventional invariants used in Raman and ROA spectroscopy carry additional factors and can be written as [42, 43]: 1 a2f ←i = isf←i & isf←i ≈< f Qp i >2 Lxp · V a2 · Lxp 3 3 2f ←i = anis & anis ≈< f Qp i >2 Lxp · V 2 · Lxp 2 f ←i f ←i 1 aGf ←i = isf←i & Gf ←i ≈< f Qp i >2 Lxp · V aG · Lxp 3 3 2Gf ←i = anis & Gf ←i ≈< f Qp i >2 Lxp · V 2G · Lxp 2 f ←i 2Af ←i = 0 anis &A ≈< f Qp i >2 Lxp · V 2A · Lxp 2 f ←i f ←i (2.139) (2.140) (2.141) (2.142) (2.143) 226 Continuum Solvation Models in Chemical Physics The right-hand sides of Equations (2.139)–(2.143) are of the form < f Qp i >2 Jp , with Jp = Lxp · V · Lxp . The vectors Lxp and the tensors V are given by ⎞ ⎛ Lx1p V11 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ x ⎟ ⎜ V1 L V = Lxp = ⎜ ⎜ ⎜ p ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎝ ⎝ ⎠ VN 1 LxNp ⎛ V12 V2 VN 2 V1 V VN ⎞ V1N ⎟ ⎟ ⎟ ⎟ VN ⎟ ⎟ ⎟ ⎟ ⎠ VNN (2.144) The expressions for the local tensors V are: e 1 e V a = 9 x 0 x 0 e G 1 e V aG = 9 x 0 x 0 $ e e e e # 3 V 2 = + x 0 x 0 x 0 x 0 4 e 1 e − 2 x 0 x 0 $ e e e e # G G 3 + V 2G = 4 x x x x 0 0 0 0 e 1 e G − 2 x 0 x 0 e e e e # A A 0 1 2 + V A = 2 2 x 0 x 0 x 0 x 0 2 (2.145) (2.146) (2.147) (2.148) (2.149) where the products on the right-hand side are dyads. In view of Equation (2.133), we have specified in Equations (2.139)–(2.143) the isotropic and anisotropic part only for f ←i . As its antisymmetric part vanishes outside resonance, antisymmetric invariants do not occur in ordinary Raman and ROA scattering. The expressions for the tensors V have been kept general and the symmetric nature of e has not been used to simplify them. We further note that the tensor Af ←i does not give rise to an isotropic invariant as it is traceless because Ae is symmetric in the second and third indices. 2.6.3 Interpretation of Raman and ROA Spectra Computed and measured Raman and ROA spectra contain a wealth of detailed information. Substantial portions of the vibrational spectra measured for polyatomic molecules Properties and Spectroscopies 227 have in the past been considered ‘fingerprint’ regions, implying that the pattern of observed vibrational absorption and Raman scattering intensities were characteristic of a molecule’s structure but little understood. While the recent ab initio computations of vibrational spectra have been highly successful, the numerical comparison of computed and measured data has tended to lead more to knowledge on individual molecules than to understanding and insight. We have chosen in the preceding section a form for the equations of the scattering cross–sections that permits inferring patterns, characteristic of specific structural elements, from computed results. Decomposition of Vibrational Motions The separation into a vibrational and an electronic part is implied by the Placzek polarizability theory. The further analysis of vibrational motions has in the past typically been accomplished by calculating the vibrational energy distribution in valence coordinates. For the large-scale skeletal motions often important in ROA, and for relating Raman and ROA scattering cross–sections to the vibrational motions of structural parts of an entity, a different approach is needed. Our starting point is the decomposition of the normal modes of a larger system into those of independently computed fragments [12]. An exact decomposition is possible if the number of the nuclei of the fragments equals those of the supersystem, and provided all normal modes are considered, which means rotations and translations must be included in the treatment. In order to avoid the otherwise ubiquitous mass factors, it is convenient to use the matrix L which gives the transformation between the mass-weighted excursions of the nuclei and the normal modes Qp , rather than Lx . The elements of the two √ matrices are related by Lip = m Lxip [59]. A normal mode LSp of the system S can be written as linear combination of the normal modes LAr LBr LCr · · · of the independent subunits A B C · · · with the numbers NA NB NC · · · of nuclei: LSp = 3NA r A A crp Lr + 3NB B B crp Lr + r 3NC C C crp Lr + · · · (2.150) r B A particular coefficient, e.g. cqp , follows as B cqp = LBq · LSp (2.151) where LBq is the vector 0 LBq 0 0 · · · of the same dimension as LSp , with zeros at the positions of LAr LCr , etc., and where the arrangement of the nuclei of the subunits A B C · · · is supposed to match that of S. The fraction with which LBq is contained in S B Lp then follows as the square of the coefficient cqp . This fraction can also be obtained more elegantly by double contracting the dyads LSp LSp and LBq LBq [12]: B B cqp cqp = LBq LBq & LSp LSp (2.152) B which depend on The use of dyads avoids passing through individual coefficients cqp arbitrary phase factors with which L vectors can be multiplied. Primes will not be further 228 Continuum Solvation Models in Chemical Physics specified for L vectors. Where vectors occur in contractions, they will be assumed to be defined in the same dimensional space. We may consider, as a limiting case, the nuclei of a molecule as its fragments. The normal modes of a nucleus are its translations in three orthogonal directions. As Equation (2.122) remains valid if displacements are replaced by velocities, we can define three normalized vectors Lx Ly , and Lz . Contracting them with LSp yields three coefficients cqp , with q = x y, and z. Their values correspond to those of Lxp Lyp , and Lzp which result from a normal mode analysis of the molecule. The example bridges the gap between decomposing normal modes of the system S and comparing nuclear motions, either on the same fragment of a molecule, or on similar A A fragments of different molecules. We can define the overlap Op2p 1 of two normal modes p and p on the similar fragments A1 and A2 of two molecules 1 and 2 as the double A A contraction of the dyads LAp 1 LAp 1 and Lp2 Lp2 : A A A A Op2p 1 = Lp2 Lp2 & LAp 1 LAp 1 (2.153) A A Op2p 1 varies between 0 and 1 and depends on the fraction of the normal modes located on the fragments. By renormalizing the normal modes on these fragments one obtains A A a measure of the similarity Sp2p 1 of the shape of the motions independent of their actual size: A A Op2p 1 A A Sp2p 1 = A A A A Lp2 Lp2 Lp 1 Lp 1 (2.154) In order for Equations (2.153) and (2.154) to yield meaningful results, the fragments A1 and A2 have to be aligned. This can be done by a quaternion rotation chosen so that the sum of the squares of the mass-weighted distances between the nuclei one wants to superpose is minimal [12,60]. In a normal mode analysis, the Eckart–Sayvetz conditions are observed for the whole of a system and they are not, therefore, in general satisfied for computed nuclear motions on a fragment only. The dyads in the above expressions will thus contain translational tA rA rA and rotational components. If LtA q Lq and Lq Lq are the dyads for the translational and rotational normal modes of the fragment A, respectively, then the dyad corresponding to the local vibrational component in normal mode p is given by LpvA LpvA = LpA LpA − trans q rot LqtA LqtA & LpA LpA LqtA LqtA − LqrA LqrA & LpA LpA LqrA LqrA (2.155) q where q runs over the translational and rotational modes of the fragment as indicated. By substituting appropriate terms of Equation (2.155) into Equations (2.153) and (2.154), one obtains the overlap and the similarity separately for the translational, rotational, and vibrational components of the nuclear motions of two fragments for the normal modes p and p . In comparing the results of a computation with measured data, one must be aware that a normal mode analysis supposes a harmonic force field. In a normal mode the ratio of Properties and Spectroscopies 229 all displacement coordinates is constant in time, which implies for nondegenerate modes nuclear motions along straight lines. Computed directions therefore represent tangents to the actual (classical) trajectories of the nuclei at the equilibrium position, and comparing normal modes amounts to comparing the directions of these tangents, with the relative size of nuclear excursions based on the assumption of rectilinear motion. We note that such a comparison can remain meaningful even where the computation of vibrational absorption and scattering intensities based on normal modes might no longer be so. Group Coupling Matrices and Group Contribution Patterns The form of Lxp and V , Equation (2.144), makes it evident that the invariants If ←i of products of transition tensors can be written in the frame of the polarizability theory as sums over mono- and dinuclear terms: If ←i ≈< f Qp i >2 Jp =< f Qp i >2 Jp =< f Qp i >2 Lxp · V · Lxp (2.156) Each set of values Jp for normal mode p forms a N × N matrix, where N is the number of nuclei. A diagonal term Jp represents the contribution which atom makes to Jp , and the sum Jp + Jp the contribution due to the coupled motion of the pair of nuclei and . The graphical representation as full and empty circles, depending on the sign, in a matrix, in upper triangular form, maps the way nuclear motion creates Raman and ROA intensity in the vibrating molecule [42]. Matrices for individual nuclei are helpful for comparing the patterns for various invariants, for assessing the influence of computational parameters, and for studying changes due to the interaction of a molecule with its environment. The bewildering amount of information they contain, particularly for ROA, in the form of cancelling positive and negative terms, limits their usefulness for understanding actual spectra. Better insight is often gained by collecting nuclei into groups, and by representing the contributions due to these groups, and to their interactions, as group coupling matrices [42]. The meaningful choice of groups depends on the particular normal mode the ROA of which one wants to analyse. An example is given in a subsequent section. A different approach for extracting relevant information is to define quasi-atomic quantities the sum of which yields the value of Jp [43]. This can be done even though a decomposition into transferable additive terms, a long standing pipe dream in optical activity, is not possible. To this end, the dinuclear terms in Equation (2.156) have to be split between two atoms, in proportion to the motion of their nuclei and the size of the gradients of the electronic tensors. For an atom with nucleus in its molecular environment, a quasi-atomic contribution Jp can then be defined as Jp = Jp + J r p + Jp r p (2.157) with a meaningful choice of the coefficients r p and r p discussed elsewhere [42]. As with group coupling matrices, a clearer picture often emerges by adding the values Jp of a group of nuclei, such as those of a methyl or a phenyl group. 230 Continuum Solvation Models in Chemical Physics ROA of Clusters The decomposition of the vibrational motions of a larger entity into the normal modes of fragments opens up the possibility for decomposing Raman and ROA scattering crosssections into a part due to the noninteracting subsystems, and a contribution due to their interaction. In order to keep the notation simple, we will consider a cluster of two subunits only, but the approach is general and extensible to an arbitrary number. Examples for two unit systems would be two temporarily aligned molecules in the condensed phase, or a hydrogen-bonded dimer of two carboxylic acid units. With the combined system S consisting of the two subunits A and B LSp is given by the first two sums on the right-hand side of Equation (2.150). The tensor V S , Equation (2.144), AA AB can be written in the form of blocks V AA V AB etc. of the local tensors V V given by Equations (2.145)–(2.149), AA AB V V V = V BA V BB S (2.158) where the notation V AA and V BB implies that the tensors are computed for A and B as parts of the cluster. For A and B as independent, noninteracting units one can likewise write VS = A V 0 0 VB (2.159) A particular invariant specified by Equations (2.139)–(2.143) can be written for cluster S in the form S xS JpS = LxS p · V · Lp = 3NA A A A xA xA A xA crp csp LxA r · V · Ls + Lr · V · Ls rs + 3NA 3NB r + 3NB A B AB xB BA crp csp LxA · LxB · LxA r ·V s + Ls · V r (2.160) s B B B xB xB B xB crp csp LxB r · V · Ls + Lr · V · Ls rs where V A = V AA − V A and V B = V BB − V B are the changes in the tensors V A and V B , respectively, upon cluster formation. The significance of the terms in Equation (2.160) can best be understood by looking at a situation where A and B are identical chiral units in a C2 symmetric arrangement. If their interaction is weak, then the vibrations of the cluster S will occur in pairs of a symmetric and an antisymmetric linear combination of the two monomer modes q xS degenerate in the absence of interaction. For such a pair LxS p+ and Lp− due to the monomer xA xB modes Lq and Lq one can write A xA B xB LxS p± = cqp Lq ± cqp Lq (2.161) Properties and Spectroscopies 231 √ A A B A with cqp = cqp = cqp = cqp− = 1/ 2. Equation (2.160) then takes the form + S xS A A xA A xA B B xB B xB JpS± = LxS p± · V · Lp± = cqp cqp Lq · V · Lq + cqp cqp Lq · V · Lq A A xA B B xB B xB + cqp cqp Lq · V A · LxA q + cqp cqp Lq · V · Lq A B AB xB BA ± cqp cqp LxA · LxB · LxA q ·V q + Lq · V q (2.162) The first two terms after the equality sign are the parts which stem from the Raman or ROA scattering of the isolated, noninteracting subunits A and B, the terms with V A and V B reflect the change of the electronic tensors of the subunits when A and B interact in the cluster, and the last term is due to the tensors V AB and V BA , which optically couple the vibrational motions of the subunits. We notice that V AB and V BA do not decrease with the distance between the subunits A and B, though their size will vary as a result of the distance dependence of m and . Even for an infinite distance, they will lead to a nonzero term in Equation (2.162), a consequence of neglecting in the derivation of Equations (2.114)–(2.118) the dimension of the system considered in comparison with the wavelength of the light. Computed ROA due to the interaction of A and B vanishes despite this for an infinite distance, because the vibrations p+ and p− are then degenerate, and the sum of their ROA due to V AB and V BA cancels. Equation (2.162) permits an ab initio interpretation, for Raman optical activity, of the two-group model originally developed for Rayleigh optical activity [15–18, 20]. One might ask what difference there is between the approach taken in this section and the decomposition into group coupling matrices discussed earlier. Group coupling matrices depend simultaneously on the nuclear motions and on the electron distribution, and they do not, therefore, yield the separate insight into the vibrational part and the electronic tensor part which Equations (2.160) and (2.162) provide. They do not, on the other hand, require the separate computation of individual groups, something which Equation (2.160) implies. We will show in the following section that a qualitative understanding can also be gained through Equation (2.162) without a computation of individual fragments, by considering their known group vibrations, and that this information can be related to that provided by group coupling matrices. 2.6.4 +-P-1,4-Dimethylenespiropentane Theoretical ROA in the C=C Stretching Region In +-P-1,4-dimethylenespiropentane [40], the two local C=C stretching motions can couple in phase and out of phase. One of the two molecular vibrations which results from their coupling transforms like the symmetric representation of the point group C2 of the molecule, the other like the antisymmetric one. The symmetric mode is expected to occur at lower energy and should give rise to an intense, fairly polarized band in the Raman spectrum, and the antisymmetric mode at higher energy, with a less intense and completely depolarized Raman band. Apart of their coupling, the nuclear motions are expected to be mostly confined to the two achiral C=CH2 fragments. If we equate these fragments with the subunits A and B in Equation (2.162), then V A and V B will vanish for aG 2G , and 2A . If we further neglect V A and V B in 232 Continuum Solvation Models in Chemical Physics comparison with V AB and V BA , which is reasonable as the direct environment of A and xB B is achiral, and if we associate the vibrations LxA q and Lq with the two localized C=C stretching motions, then the SCP backscattering ROA of the coupled motions follows from Equations (2.116) and (2.117) with % = as 4K A B AB < f Qp± i >2 cqp cqp 12 LxA 2G ) · LxB q ·V q c xA AB 2 xB BA 2 xA xB + Lq · V G ) · Lq + 4 Lq · V A ) · Lq BA + LxB 2A ) · LxA q ·V q −dSCP± = ± (2.163) One thus expects two ROA bands of the same size and of opposite sign in the 1650–1850 cm−1 region. The ab initio computed Raman and ROA spectra [61] shown in Figure 2.27 confirm this qualitative reasoning. In addition, they predict that the ROA couplet due to the coupled C=C stretching vibrations should be the largest feature by far in the ROA spectrum of +-P-1,4dimethylenespiropentane, with the exception of the lowest frequency vibration predicted to occur outside the presently measurable range. Higher quality computed data [38] give the same result. Measured Data and Influence of Solvent Environment The measured [61] liquid phase ROA spectrum of +-P-1,4-dimethylenespiropentane is included in Figure 2.27. It does not confirm the calculated gas phase data for the C=C stretching region. The predicted dominant ROA couplet is absent, and four small ROA bands are found instead. Figure 2.28 shows the C=C stretching region on an extended scale, including Raman and polarization data. Instead of one computed intense polarized Raman band, there are two polarized bands of comparable intensity, in addition to a much weaker, close to depolarized band at higher energy. The usual culprit for bands which cannot be accounted for by vibrational states computed within the harmonic approximation is Fermi resonance [64]. The occurrence of two comparably strong, polarized bands in the 1650–1850 cm−1 range can easily be explained by it. There are several vibrational states of species A, due to overtone and combination frequencies, which have an appropriate energy for interacting with the fundamental of the symmetric combination of the two C=C stretching motions. The larger width of the higher energy band, and the fact that a small ROA couplet is associated with it rather than a single ROA band, point to multiple Fermi resonances. The small, only slightly polarized band at higher energy must then be due to the antisymmetric coupled C=C stretching vibration. The fact that it is not completely depolarized, which would imply a degree of circularity of 57 in Figure 2.27 [41], appears to be due to overlap with the larger polarized band at its low energy side. The lack of the dominant computed ROA couplet is more difficult to understand. Calculations for the gas phase, with basis sets known to reproduce experimental data well [37], invariably lead to a couplet of substantial size. Fermi resonance should conserve ROA intensity the same way as it conserves Raman intensity. A mutual compensation of the ROA intensities of the in-phase and out-of-phase vibrations by mixing cannot occur. Properties and Spectroscopies 233 Figure 2.27 Computed Raman and ROA backscattering spectra and measured ROA backscattering spectrum of +-P-1,4-dimethylenespiropentane. From bottom to top: computed Raman, computed ROA, measured ROA, computed degree of circularity for backscattering. Computational parameters: vibrational modes, density functional theory with B3LYP/aug-cc-pVTZ as implemented in Gaussian [62]; electronic tensors, time-dependent Hartree–Fock with aug-cc-pVDZ as implemented in DALTON [63]. Isotropic and anisotropic bandwidths for computed spectra: 3.5 and 10 cm−1 , respectively, convoluted with the instrumental line shape. Experimental spectrum: exposure time, 40 min.; laser power at sample, 150 mW; exciting wavelength, 532 nm; sample size, 35 l; resolution, 7 cm−1 . The number of electrons is per column on the CCD detector with a spectral width of 24 cm−1 . A reduction of the couplet’s size through the interaction of vibrational states in the condensed phase is a possibility. If this were so, then replacing a molecule’s identical neighbours by a different kind should increase the size of the couplet. The spectra +-P-1,4-dimethylenespiropentane recorded in trideuterioacetonitrile are also shown in Figure 2.28 and prove that this is not so. A direct interaction of vibrational states in the liquid phase can thus be ruled out as the cause for the small size of the ROA observed in the 1650–1850 cm−1 region. 234 Continuum Solvation Models in Chemical Physics Figure 2.28 Comparison of the Raman and ROA bands of +-P-1,4-dimethylenespiropentane (a) in substance and (b) as a 20 % by volume solution in trideuterioacetonitrile measured in backscattering for the 1650 to 1830 cm−1 region. From bottom to top: Raman, ROA, degree of circularity. The relative scattering intensities in substance and trideuterioacetonitrile solution were normalized so that the largest peak in the measured Raman spectra, vibration 8 at 609 cm−1 , has the same height. The experimental parameters are as in Figure 2.27. Numerous calculations [61] of the electronic tensors with different basis sets have shown, on the other hand, that the computed size of the couplet depends critically on the presence or absence of diffuse basis functions with valence angular momentum numbers. It is the diffuse part of the electron distribution of a molecule which is primarily affected by nonspecific interactions in the condensed phase. This suggests that the absence of a sizable couplet in the condensed phase, in substance as well as in trideuterioacetonitrile, is the result of the change of the electron distribution of +-P1,4-dimethylenespiropentane by nonspecific interactions. Figure 2.29 shows, by means of group coupling matrices, the effect which the presence of diffuse functions has on computed electronic tensors. The basis set rDPS includes such functions and yields a large couplet while rDP lacks them [37] and leads to negligible ROA in the 1650–1850 cm−1 region. The elements which stem from V AB and V BA show the strongest dependence on the presence of diffuse functions. Group coupling matrices for other basis sets display a similar behaviour. We conclude this section on +-P-1,4-dimethylenespiropentane, and this contribution on the understanding of ROA, by pointing out the difference in the relative Properties and Spectroscopies 235 Figure 2.29 The A symmetric (30) and B symmetric (31) coupled C =C stretching vibrations with their ROA group coupling matrices as implemented in VOAView [65]. The volume of the bicoloured spheres is proportional to the vibrational energy and the direction of motion indicated by the colours. The five groups in the group coupling matrices are the the four carbon atoms as indicated, with the fifth group being the remainder of the molecule. Computational parameters: vibrations: as in Fig. 1; electronic tensors: as in Fig. 1 with basis sets as indicated (see Colour Plate section). height of the two principle Fermi resonance Raman bands in the 1650–1850 cm−1 region in substance and in trideuterioacetonitrile, and the pronounced change in the shape of the higher energy band. Other regions of the Raman and ROA spectra exhibit likewise a dependence on the solvent environment which is unusual for a nonpolar, rigid hydrocarbon molecule devoid of conformational degrees of freedom. While the ab initio computation of vibrational spectra has advanced in leaps and bounds over the past decade, such experimental data are a stark reminder of the fact that much ground still needs to be covered for the reliable modelling of observed vibrational spectra. For isolated molecules, mechanical, and possibly electrical, anharmonicity will have to be taken into account. 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Cuony, Calcul de l’activité optique vibrationnelle Raman, PhD Thesis, University of Fribourg, 1981. [54] L. Hecht and L. A. Nafie, Mol. Phys., 72 (1991) 441. [55] L. D. Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, Cambridge, 2004. [56] G. Placzek, Rayleigh-Streuung und Raman Effekt, in “Handbuck der Radiologic”, E. Maux (ed)., Akademische Verlagsgesellschaft, Leipzig, 1934; p. 205. [57] H. Goldstein, Classical Mechanics, Addison-Weseley, New York, 1980. [58] L. Rosenfeld, Theory of Electrons, Dover Publications, New York, 1965. [59] E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. [60] G. R. Kneller, Mol. Simul., 7 (1991) 113–119. [61] W. Hug, J. Haesler, S. Kozhushko and A. de Meijere, Chem. Phys. Chem., 8 (2007) 1161. [62] M. J. Frisch et al. Gaussian 03, Revision C.02, Gaussian, Inc., Wallingford, CT, 2004. [63] DALTON, a molecular electronic structure program, release 1.1. http://www.kjemi.uio.no/ software/dalton/dalton.html, 2000. [64] E. Fermi, Z. Phys., 71 (1931) 250. [65] J. Haesler, Construction of a new forward and backward scattering Raman and Raman optical activity spectrometer and graphical analysis of measured and calculated spectra for (R)2 H1 2 H2 2 H3 -neopentane, PhD Thesis, University of Fribourg, 2006. 2.7 Macroscopic Nonlinear Optical Properties from Cavity Models Roberto Cammi and Benedetta Mennucci 2.7.1 Introduction The increasing efforts devoted to investigations of linear and nonlinear optical (NLO) properties of solvated molecules and liquids follow the success of modern quantum chemical tools in the prediction of the same properties for isolated systems. The approach, which is generally adopted in the modelling of solvated systems, consists in applying the same methodologies developed so far for the isolated systems, with the additional introduction of solvent-dependent features as described in other contributions to this book. Among them, we cite the fact that the presence of the solvating environment modifies the geometry and the electronic density of the molecule. Also, nonequilibrium solvent effects in response to the external perturbation (connected to motions of solvent molecules around the solute) as well as, in some cases, specific aggregation effects can be relevant. However, even when all these effects are included in the solvation model, the calculated quantities are still microscopic and cannot be directly compared with their macroscopic manifestation, i.e. the macroscopic susceptibilities determined experimentally. Historically, the way of making the connection between solution measurements and the theoretical molecular properties which govern NLO processes (polarizabilities and hyperpolarizabilities) has been to introduce local field factors, often of the Onsager– Lorentz form. This has been done by both theoreticians and experimentalists [1]. A more general framework to treat local field effects in linear and nonlinear optical processes in solution has been pioneered, among others, by Wortmann and Bishop [2]. Still, by using a classical Onsager reaction field model, one can introduce the solvent effects in two steps. First, the solute polarizability is defined by taking into account all the effects caused by the static reaction field induced by the solute dipole. Secondly, an ‘effective polarizability’ is defined to include the effects due to the difference between the local field acting on solute molecules and the macroscopic optical field (Maxwell field) in the medium. These effective properties represent the main result of the theoretical formulation of NLO phenomena for solvated systems [2–6] as they describe the response of the solute in terms of the macroscopic field in the surrounding medium and thus they may be directly related to the macroscopic properties determined from experiments. The approach which will be reviewed here has been formulated within the framework of the quantum mechanical polarizable continuum model (PCM) [7]. Within this method, the ‘effective properties’ are introduced to connect the outcome of the quantum mechanical calculations on the solvated molecules to the outcome of the corresponding NLO experiment [8]. The correspondence between the QM–PCM approach and the semiclassical approach will also be discussed in order to show similarities and differences between the two approaches. Another aspect of the NLO properties in condensed phase which will be considered here is that concerning the evaluation of susceptibilities of pure liquids. In a typical Properties and Spectroscopies 239 nonlinear optical experiment the presence of a large number of chromophores perturbed by the optical radiation at a fixed (fundamental) frequency produces a macroscopic polarization density at the output frequency which in turn acts as a source of an additional perturbing field. The analysis of this effect which is usually done in term of classical local field factors [9] is here reformulated within the PCM framework using the same approach introduced for dilute solutions [10]. 2.7.2 Macroscopic Susceptibilities and Molecular Effective Polarizabilities The response of a medium to a macroscopic field Et generated by the superposition of a static and an optical component Et = E0 + E cost is represented by the dielectric polarization vector (dipole moment per unit of volume) Pt: Pt = P0 + P cost + P2 cos2t + (2.164) where each Fourier amplitude can be rewritten as a power series with respect to the applied macroscopic (or Maxwell) field in the medium: 1 P0 = 0 + 1 0$ 0 · E0 + 2 0$ 0 0 & E0 E0 + 2 0$ − & E E + 2 (2.165) P = 1 −$ · E + 2 2 −$ 0 & E E0 + 3 3 −$ 0 0 E E0 E0 + (2.166) 1 3 P2 = 2 −2$ & E E + 3 −2$ 0 E E E0 + 2 2 (2.167) The tensorial coefficients are now the nth-order macroscopic susceptibilities n . They are tensors of rank n + 1 with 3n+1 components. The prefactors in Equations (2.165)– (2.167) result from trigonometric identities and from intrinsic permutation symmetries. Many of the different susceptibilities in Equations (2.165)–(2.167) correspond to important experiments in linear and nonlinear optics. 0 describes a possible zero-order (permanent) polarization of the medium; 1 0$ 0 is the first-order static susceptibility which is related to the permittivity at zero frequency, 0, while 1 −$ is the linear optical susceptibility related to the refractive index n at frequency . Turning to nonlinear effects, the Pockels susceptibility 2 −$ 0 and the Kerr susceptibility 3 −$ 0 0 describe the change of the refractive index induced by an externally applied static field. The susceptibility 2 −2$ describes frequency doubling usually called second harmonic generation (SHG) and 3 −2$ 0 describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). Further specifications are required if we consider as a macroscopic sample a liquid solution of different molecular components, each at a concentration cJ ; if we assume 240 Continuum Solvation Models in Chemical Physics that the effects of the single components are additive, the global measured response becomes [11–13]: n = n 'J cJ (2.168) J n where 'J are the nth-order molar polarizabilities of the constituent J . The values of the n single 'J can be extracted from measurements of n at different concentrations. In order to relate these molar quantities to properties of the single molecule we can apply arguments of statistical classical mechanics. At moderate intensity, the electric field gives rise to a dipole density by electronic and atomic translation (or deformation) effects and by rotation (or orientation) effects. We recall that the rotation effects are counteracted by the thermal movement of the molecules and thus they are strongly dependent on the temperature T whereas the translation effects are only slight dependent on T because they are intramolecular phenomena. The general expression to be used to define the Fourier amplitudes (2.165)–(2.167) is: P = N ¯ where N is the number of particle per volume unit and the frequency adopts all values involved in the NLO process under consideration. ¯ is the average dipole moment of a single molecule of the species J at a temperature T and in the presence of the macroscopic field Et. Its value can be computed from the energy W of the dipole in the field; this energy is dependent on that part of the electric field (called the directing field) tending to direct the dipoles, namely [14]: . 2 . 2 ¯ = 0 a ka exp−W/kT sin % d% d( 0 . 2 . 2 exp−W/kT sin % d% d( 0 0 (2.169) where Einstein summation and the Boltzmann law are assumed, % and ( are the usual spherical coordinates that define the molecule orientation with respect to X,Y and Z k is the cosine of the angle between the molecular axis a and the laboratory axis Z, and the bar indicates an average over a statistical distribution of molecular orientations. In Equation (2.169) is defined as the value of the electric dipole moment of the single molecule considered as a function of the field acting upon that (also called the internal field). Within the framework of continuum solvation models it is possible to expand both the energy W and the dipole in terms of the applied Maxwell field instead of the directing and the internal components. This is obtained by introducing effective polarizability and hyperpolarizabilities [2–4]. Here, the term effective indicates that the related molecular property (from now on represented by a tilde) has been modified by the combination of the two different environment effects represented in terms of ‘cavity’ and ‘reaction’ fields [1, 15] (see also Section 2.7.4). Within this formalism the dipole becomes: = 0 + cost + 2 cos2t + 3 cos3t + (2.170) Properties and Spectroscopies 241 where the Fourier amplitudes may be expanded as power series with respect to the field amplitudes: 1˜ 1˜ 0 0 & E0 E0 + 0$ ˜ 0 · E0 + 0$ − & E E + 0 = 0 + 0$ (2.171) 2 4 1 ˜ ˜ ˜ · E + −$ 0 & E E0 + −$ = −$ 0 0 E E0 E0 + (2.172) 2 1˜ 1 ˜ & E E + −2$ (2.173) 2 = −2$ 0 E E E0 + 4 4 In parallel the the field-dependent part of the free energy of the molecule in the presence of the Maxwell field is: 1 W = −∗ · E0 − ∗ & E0 E0 + 2 (2.174) where the quantities with the ‘star’ correspond to derivatives of the free energy of the system with respect to the static components of the Maxwell field [1, 15] (see Section 2.7.3, The Orientational Energy). It is now possible to give the operative equation relating the macroscopic (or molar) properties ' n to the microscopic (or effective properties); its general form is: n ¯ Z n (2.175) 'ZZ = N EZ EZ E→0 where we have introduced the Z space-fixed axes of the laboratory. For example, for the first-order (both static and frequency-dependent) we obtain [13, 16]: ∗ · 0 (2.176) + ˜ is 0$ 0 ' 1 0$ 0 = N 3kT ' 1 −$ = N ˜ is −$ and for the third-order EFISHG process [13]: ˜ −2$ · ∗ 3 + ˜ s −2$ 0 ' −2$ 0 = N 15kT (2.177) (2.178) In Equations (2.176)–(2.178) ˜ is is 13 of the trace of the effective polarizability and in Equation (2.178) ˜ s −2$ 0 is the ‘scalar part’ of the third-order polarizability. Analogous relations hold for the other NLO properties; their expressions can be found, for example, in ref. [15]. n As the molar polarizabilities 'J represent an easily available ‘experimental’ set of data, the expressions above become important for the theoretical evaluation of molecular response properties; in fact they represent the most direct quantities to compare with the computed results obtained applying a given model for the solvent effects. 242 Continuum Solvation Models in Chemical Physics By extending the concept of effective polarizabilities to pure liquids, a further issue has to be introduced. The optical radiation at the fundamental frequency produces in the liquid a macroscopic polarization density Pn which acts as source of an additional perturbing field at the output frequency n. The response of each molecule of the liquid to such a field can be represented in terms of a new effective polarizability −n$ ¨ n at the output frequency. The introduction of this additional field has been controversial. Wortmann and Bishop [2] in their seminal paper on the effective properties excluded the possibility of a cavity field with a frequency different from that of the Maxwell field. However, soon later Munn et al.[9] gave a convincing argument in favour of a cavity field with the same frequency as that of the nonlinear polarization. They introduced the term ‘cascading’ to indicate this effect. Starting from that analysis focused on molecular crystals, more recently a parallel analysis has also been given for liquids for which the term ‘output wave effect’ has been used. The introduction of such a new term is here exemplified for the EFISHG process for which the resulting expression for the induced dipole becomes: 1˜ 2 2 ¨ 2PEFISH + −2$ & E E EFISH = −2$ 4 1 ˜ + −2$ 0 E E E0 + 4 (2.179) 3 2 PEFISH = EFISH E E E0 (2.180) where In Equations (2.178) the effective quantities indicated with a tilde have the same meaning described above. As a result the definition of the molar polarizability for the third-order EFISHG process given in Equation (2.177) for a solution has to be modified as follows: 3 'liq −2$ 0 = ' 3 −2$ 0 1 − −2$ ¨ 2 (2.181) where ' 3 has exactly the same definition as in Equation (2.178). The denominator of Equation (2.181) represents the effect of the polarization density as a source field at the output wave frequency, and it depends on the numeral density of the liquid and on the effective polarizability −2$ ¨ 2 here through its average value defined as the trace of the corresponding tensor. In the following sections we shall present how both the ‘effective’ and the ‘star’ molecular properties appearing in Equations (2.176)–(2.178) can be evaluated within the framework of the PCM continuum model. 2.7.3 Effective Polarizabilities Within the PCM Formalism The theory of PCM calculation of the effective polarizabilities is based on a timedependent response theory that describes the interaction between the molecular solutes and the Maxwell electric field. We will review the method in three separate sections, the Properties and Spectroscopies 243 first concerning the electronic component of the (hyper)polarizabilities, the second the orientational free energy W , and the last the vibrational component of the same effective response properties. The electronic component The theory of the PCM has been extensively treated in other parts of the present book. Here we just report the main conclusions as necessary for a better understanding of the present formulation. In the presence of a Maxwell field Et the electronic Hamiltonian of the solute can be written as Ĥ = Ĥ 0 + V̂MS + V̂ t + V̂ " t (2.182) where Ĥ 0 is the Hamiltonian of the isolated system and V̂MS is the electrostatic interaction between the solute and the apparent charges representing the polarization (or reaction field) of the solvent. In the PCM these charges (placed on the cavity surface) are determined by the solvent permittivity, the shape of the cavity, the topology of the surface and the electrostatic potential induced by the solute on the same surface. The last two time-dependent terms represent the interaction of the solute with the Maxwell field in the medium and the interaction with a uniform nonlinear polarization, respectively; we note that the second of these terms appears only when a pure liquid is considered. The first time-dependent perturbation V̂ t can be represented as [4] V̂ t = ˆ E eit + e−it + E0 ex q k it q0ex k 0 −it + V̂k E e + e + E E E0 k (2.183) where ˆ and V̂ indicate the electronic dipole moment operator of the solute and the electronic electrostatic potential at the cavity surface, respectively. In Equation (2.183) new surface charges, q ex , have been introduced; these charges can be described as the response of the solvent to the external field (static or oscillating) when the volume representing the molecular cavity has been created in the bulk of the solvent. We note that the effects of q ex in the limit of a spherical cavity coincide with that of the cavity field factors historically introduced to take into account the changes induced by the solvent molecules on the average macroscopic field at each local position inside the medium: more details on this equivalence will be given in Section 2.7.4. The last time-dependent perturbation V̂ t of Equation (2.182) appears only when a pure liquid is under scrutiny. It represents the interaction of the selected molecule (the ‘solute’) with the uniform nonlinear polarization density Pn produced by the other equivalent molecules [10]: V̂ t = k V̂k q̈ex k n int P e + e−int Pn In Equation (2.183) the additional surface charges q̈ex have been introduced; they correspond to the charges representing the electrostatic potential produced by the uniform 244 Continuum Solvation Models in Chemical Physics polarization density Pn . They are linearly proportional to the normal component of the polarization density at the cavity surface, i.e.: ex q̈n = A Pn · n where A is the diagonal matrix collecting the areas of the tesserae and n is the outward pointing vector at the cavity surface. We note that the effects of q̈ex in the limit of a spherical cavity coincide with that of the historical Lorentz approximation for the evaluation of the electric field produced inside a spherical cavity by a uniform polarization density [1]. Approximate solutions of the time-dependent Schrödinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree–Fock or Kohn–Sham equation: F C − i SC = SC t (2.184) with the proper orthonormality condition; S, C and represent the overlap, the MO coefficient, and the orbital energy matrices, respectively. In Equation (2.184) the prime on the Fock matrix indicates that terms accounting for the solvent effects are included, i.e.: F = F0 R + m · E eit + e−it + E0 + m̃ · E + m̃ · E e 0 0 it +e −it (2.185) + m̈ · P e n nt +e −int (2.186) where F0 R represents the Fock matrix for the molecule in the absence of the Maxwell field but in the presence of the solvent reaction field and R is the one-electron density matrix. The matrices m m̃ and m̃0 , are the matrices containing the dipole integral and the dipole due to the apparent charge q ex induced by the external oscillating and static field, respectively, namely m̃x = − Vk k qxex k Ex with x = 0 (2.187) where Vk is the matrix containing the solute potential integrals computed on the surface cavity. In a parallel framework, the matrix m̈ which represents the effects of the uniform non-linear polarization density Pn becomes: m̈ = − k Vk ex q̈n k = − Vk a k nk n P k (2.188) where ak is the area of the tessera k and nk is the outward pointing vector at the cavity tessera k. Properties and Spectroscopies 245 The solution of the time-dependent HF or KS Equation (2.184) can be obtained within a time-dependent coupled HF or KS approaches (TDHF or TDDFT) by expanding all the involved matrices (F, R, C and ) in powers of the field components. It has to be noted that the solvent-induced matrices present in F0 R depend on the frequency-dependent nature of the field as they depend on the density matrix R and as they are determined by the value of the solvent dielectric permittivity at the resulting frequency. By applying standard iterative procedures, all the perturbed density matrices can be analytically computed and thus also the electronic component of the effective properties (2.171)–(2.173), namely we have: b ˜ el ab −1 $ 1 = −tr ma R 1 bc ˜ el abc − $ 1 2 = −tr ma R 1 2 el − $ 1 2 3 = −tr ma Rbcd 1 2 3 ˜ abcd (2.189) (2.190) (2.191) with a,b,c indicating the Cartesian coordinates of the applied field and = )i . A similar scheme can be exploited to compute the additional polarizability − ¨ $ ; in this case the time-dependent problem to solve is determined by the Fock operator in which the external perturbation is the polarization P and thus the dipole-like operator to be included is m̈ only. The resulting polarizability is now: ¨ ab − $ = −tr ma Rb As shown by Equations (2.189)–(2.191) the procedure briefly sketched above allows us to take into account all the effects of the solvent, both those intrinsic, i.e. due to the reaction potential, and the others related to the presence of the external field, in a compact and self-consistent form. In this way no a posteriori corrections, such as those usually introduced by cavity factors, are required, but the computed properties can be used as they are and introduced in the expressions linking the microscopic to the macroscopic. Let us now turn to consider the two further contributions necessary to obtain the complete description, starting from the definition of the angle-dependent energy W (2.174) in the presence of the solvent effects. The Orientational Energy In Equation (2.174) we have shown that the field-dependent part of the free energy can be written in terms of the dipole ∗ (and, at higher order, the polarizability ∗ ). Classically, this expression can be obtained by expanding the Boltzmann potential energy in terms of the field (here appearing only through its static components); in the framework of the PCM description of solvation such energy has to be replaced by the free energy analogue, i.e.: 1 GE0 = G0 + WE = G0 + ∗ · E0 + ∗ & E0 E0 + 2 (2.192) where G0 is the free energy of the solvated system in the absence of the field whereas ∗ and ∗ are the gradient and the Hessian of G with respect to the field components, respectively. 246 Continuum Solvation Models in Chemical Physics Both the components of the gradient and of the Hessian have to be computed at E0 = 0; in the framework of the coupled HF, or KS, approach described above, they can be expressed in terms of the unperturbed density matrix, and of its derivative with respect to the static field, respectively, i.e.: ∗a = ∗ab = G E0 E0 =0 2 G Ea0 Eb0 = −tr R0 ma + m̃a0 = −tr Rb ma + m̃a0 (2.193) (2.194) E0 =0 where ma and m̃a0 , are the matrices introduced in Equation (2.185). These expressions are the PCM results for the evaluation of the orientational averaging required in Equation (2.169). The Vibrational Component The detailed treatment of the nuclear effects on the electric (hyper)polarizabilities has been addressed by Bishop et al. [18] using a perturbational treatment. According to this derivation, the vibrational contribution to the (hyper)polarizabilities should contain two distinct effects [19], the ‘curvature’ related to the field dependence of the vibrational frequencies (i.e. the changes in the potential energy surface in the presence of the external field) and including the zero-point vibrational correction, and the ‘nuclear relaxation’ arising from the field-induced nuclear relaxation (i.e. the modification of the equilibrium geometry in the presence of the external field). In the following analysis, however, only the former nuclear relaxation will be considered, and only in the static limit; vibrational effects in the presence of frequency-dependent fields are in fact usually small and here they will be completely neglected. In the limit of static fields, the nuclear relaxation contribution (from now on just ‘vibrational’) to the polarizabilities can be computed in the double harmonic approximation, i.e. assuming that the expansions of both the potential energy and the electronic properties with respect to the normal coordinates can be limited to the quadratic and the linear terms, respectively (i.e. assuming both mechanical and electric harmonicity). As shown in ref.[20], the double harmonic procedure can be reformulated within the PCM so as to obtain the analogues of the classical expressions in terms of summations of derivatives of dipoles and polarizabilities with respect to normal coordinates but with all the properties computed in the presence of the solvent (i.e. exploiting effective properties), namely we obtain: ˜ vab = 3N −6 i ˜ vabc = a Qi 0 ∗b Qi / 2i (2.195) 0 3N −6 ∗ ∗ c ab ac b + Qi 0 Qi 0 Qi 0 Qi 0 i ∗ / bc a 2i + Qi 0 Qi 0 (2.196) Properties and Spectroscopies 247 where i = 2i is the circular frequency associated with the normal coordinate Qi for the solvated molecule and each partial derivative is evaluated at the proper equilibrium geometry. We in fact recall that equilibrium geometry as well as vibrational frequencies, force constants and normal modes are computed in the presence of the solvent interactions as derivatives of the free energy functional with respect to the nuclear coordinates. The derivatives of the star-quantities in Equations (2.195) and (2.196) can be obtained including the contributions due to the external charge q ex in the expansion of G with respect to the field to be used in the derivation of the PCM double-harmonic scheme, exactly as we have done in the previous section to evaluate the orientational averaging; namely: ∗ a = −tr Ri ma + m̃a0 + R0 m̃ai (2.197) Qi 0 ∗ ab = −tr Rbi ma + m̃a0 + Rb m̃ai (2.198) Qi 0 where m̃ai represents the derivative with respect to the normal coordinate i (and thus the nuclei coordinates) of the so-called external component of the solvent reaction. The matrix m̃a0 in fact, contrary to the dipole matrix ma , depends on the nuclei geometry through the form of the molecular cavity, and as a consequence its variations with respect to the nuclei motions should be included. 2.7.4 Effective Polarizabilities in the Semiclassical Models In this section we compare the PCM formulation of the effective polarizabilities with the semiclassical Onsager–Wortmann–Bishop model [2] (from now on indicated as OWB). The OWB model describes the solute as a classical polarizable point dipole located in a spherical or ellipsoidal cavity in an isotropic and homogeneous dielectric medium representing the solvent. In the presence of a macroscopic Maxwell field E, the solute experiences an internal (or local) field Ei , given by a superposition of a cavity field EC and a reaction field ER . In terms of Fourier components Ei EC ER of the fields we have Ei = EC + ER (2.199) The cavity and the reaction fields are related to the Maxwell field in the medium and to the total (permanent+induced) dipole moment of the molecule at the frequency by EC = f C E ER = f R (2.200) where f C and f R are the cavity and reaction field factor, respectively. Expressions for the factors f C and f R have been proposed in the literature for spherical and ellipsoidal cavities and their physical meaning is immediate; in the first case we have: f C = f R 3 2 + 1 1 2 − 1 = 3 a 2 + 1 (2.201) 248 Continuum Solvation Models in Chemical Physics where a is the radius of the sphere and the solvent permittivity at frequency . Once the definition of the internal field is known, the component of the induced dipole moment of the solute is given by = sol −$ Ei (2.202) where sol −$ is the ‘solute polarizability’, i.e. the polarizability of the solute in the presence of the solvent interactions, namely R0 sol + aa −$ = aa −$ + aab −$ 0Eb (2.203) and −$ and −$ 0 are, respectively, the first- and second-order polarizability of the isolated molecular solute. In other words, sol −$ describes the linear response of the solute to a probing optical field in presence of its own static reaction field ER0 . The effective polarizabilities of the OWB solute are finally obtained in terms of the cavity and reaction field factors; for example, for the linear effective polarizability we obtain R sol ˜ aa −$ = f C Faa aa −$ (2.204) R where the factor Faa represents the coupling between the induced components of the dipole and its environment, namely R Faa = 1 1 − f R sol aa −$ (2.205) Similarly, we have effective second- and third-order effective polarizabilities as R C R C sol R ˜ aaa − $ 1 2 = Faa Faa 1 faa 1 Faa 2 faa 2 aaa − $ 1 2 R C R C R R1 C1 ˜ aaaa − $ 1 2 3 = Faa faa Faa Faa 2 faa 2 Faa 3 faa 3 (2.206) sol aaaa − $ 1 2 3 The OWB equations obtained in this semiclassical scheme analyse the effective polarizabilities in term of solvent effects on the polarizabilities of the isolated molecules. Three main effects arise due to (a) a contribution from the static reaction field which results in a solute polarizability, different from that of the isolated molecules, (b) a coupling between the induced dipole moments and the dielectric medium, represented by the reaction field factors F R , (c) the boundary of the cavity which modifies the cavity field with respect the macroscopic field in the medium (the Maxwell field) and this effect is represented by the cavity field factors f C . All these effects are considered in a more consistent and general way in the PCM framework, where the coupling between the induced electronic charge distribution (not limited to the dipolar component but described by the QM wavefunction) and the external medium is represented by the reaction potential produced by the apparent charges, while the boundary effect on the Maxwell field is represented by the matrices m̃a . Properties and Spectroscopies 249 The so-called solute polarizabilities of the OWB scheme can be obtained by considering the perturbed Fock matrices in which both the reaction and the boundary effects are neglected. This corresponds to performing a response calculation of the polarizabilities of the solute in presence of a fixed static reaction field. All the effects due to the coupling of the induced charge distribution with the external medium can be recovered by introducing in the perturbed Fock matrices the corresponding solvent reaction terms. It is worth recalling that almost all the quantum mechanical continuum methods proposed for the evaluation of the solvent effects on polarizabilities have been limited to considering these two effects only. Finally, by introducing into the perturbed Fock matrices, the perturbation corresponding to the cavity-induced modification of the Maxwell field, m̃a , we also have a direct evaluation of the cavity field effect obtaining a coherent description of the effective polarizabilities. A parallel analysis may be performed for the vibrational contribution to the effective polarizabilities. In the OWB approach, the equivalents of ‘star’ dipole and polarizability involved in the vibrational contributions (2.194) and (2.195), are expressed in terms of the cavity and reaction field factors effects as ∗a = f C0 F R0 a ∗aa 0$ 0 = f f F aa 0$ 0 C0 C0 R0 (2.207) (2.208) where a and aa are the dipole and polarizability components of the isolated molecule. The corresponding PCM expressions (2.193) and (2.194) show that the same physical effects are considered: the static cavity field effects are explicitly represented by the matrices m̃0 , while the static reaction field effects are implicit in the coupled perturbed HF (or KS) equations which determine the derivative of the density matrix. 2.7.5 Conclusions We have reviewed the quantum mechanical approach to the determination of NLO macroscopic properties of systems in the condensed phase using the Polarizable Continuum Model. This approach is based on the introduction of molecular effective polarizabilities, i.e. molecular properties which have been modified by the combination of the two different environment effects represented in terms of ‘cavity’ and ‘reaction’ fields. In terms of these properties the outcome of quantum mechanical calculations can be directly compared with the outcome of the experimental measurements of the various NLO processes. The explicit expressions reported here refer to the first-order refractometric measurements and to the third-order EFISH processes, but the PCM methodology maps all the other NLO processes such as the electro-optical Kerr effect (OKE), intensitydependent refractive index (IDRI), and others. More recently, the approach has been extended to the case of linear birefringences such as the Cotton–Mouton [21] and the Kerr effects [22] (see also the contribution to this book specifically devoted to birefringences). We have also shown that this approach is not limited to the case of a single solute molecule in a infinite solution, but it can be extended to the case of a pure liquid. In this 250 Continuum Solvation Models in Chemical Physics case the further effect of the outcoming polarization field resulting in the NLO processes must be taken into account. Finally, we remark that the problem of the calculation of molecular quantities directly comparable with the outcome of experiments in the liquid phase is not limited to the realm of the NLO processes. All experiments involving the interaction of light with molecules in condensed matter are plagued by this problem. The methodology reviewed here has been applied (with appropriate modifications) to various spectroscopies, IR [23], Raman [24], Surface Enhanced Raman Scattering (SERS) [25], vibrational circular dichroism (VCD) [26] and linear dichroism [27] with equal reliability, and other extensions will come. References [1] (a) C. J. F. Böttcher, Theory of Electric Polarization, Vol.I., Elsevier, Amsterdam, 1973; (b) C. J. F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol.II, Elsevier, Amsterdam, 1978. [2] R. Wortmann and D. M. Bishop, J. Chem. Phys., 108 (1998) 1001. [3] (a) Y. Luo, P. Norman and H. Ågren, J. Chem. Phys. 13, (1998) 21; (b) P. Norman, P. Macak, Y. 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Macak and H. Agren, J. Chem. Phys., 114 (2001) 3105. [10] R. Cammi, L. Frediani, B. Mennucci and J. Tomasi, J. Mol. Struct. (THEOCHEM), 633 (2003) 209. [11] (a) W. Liptay, J. Becker, D. Wehning, W. Lang and O. Burkhard, Z. Naturforsch. A, 37 (1982) 1396; (b) W. Liptay, D. Wehning, J. Becker, and T. Rehm, Z. Naturforsch. A, 37 (1982) 1369. [12] K. D. Singer and A. F. Garito, J. Chem. Phys., 75 (1981) 3572. [13] R. Wortmann, P. Krämer, C. Glania, S. Lebus and N. Detzer, Chem. Phys., 173 (1993) 99. [14] D. M. Bishop, Rev. Mod. Phys., 62 (1990) 343. [15] J. J. Wolfe and R. Wortmann, Adv. Phys., Org. Chem., 32 (1999) 121. [16] (a) W. Liptay, R. Wortmann, H. Schaffrin, O. Burkhard, W. Reitinger and N. Detzer, Chem. Phys., 120 (1988) 429; (b) R. Wortmann, K. Elich, S. Lebus, W. Liptay, P. Borowicz and A. Grabowska, J. Phys. Chem., 96 (1992) 9724. [17] R. Cammi and J. Tomasi, Int. J. Quantum Chem., 60, (1996) 297. [18] D. M. Bishop, J. M. Luis and B. Kirtman, J. Chem. Phys. 108 (1998) 10013; (b) D. M. Bishop, Adv. Chem. Phys., 104 (1998) 1. Properties and Spectroscopies 251 [19] (a) J. Martí, J. L. Andrés, J. Bertán and M. Duran, Mol. Phys., 80 (1993) 625; (b) J. Martí, D. M. Bishop, J. Chem. Phys., 99 (1993) 3860; (c) Bishop and B. Kirtman, J. Chem. Phys., 95 (1991) 2646; (d) J. Chem. Phys., 97 (1992) 5255. [20] R. Cammi, B. Mennucci and J. Tomasi, J. Am. Chem. Soc., 34 (1998) 8834. [21] C. Cappelli, A. Rizzo, B. Mennucci, J. Tomasi, R. Cammi, G. L. J. A. Rikken, R. Mathevet and C. Rizzo, J. Chem. Phys., 118 (2003) 10712. [22] C. Cappelli, B. Mennucci, R. Cammi and A. Rizzo, J. Phys. Chem. B, 109 (2005) 18706. [23] R. Cammi, C. Cappelli, S. Corni and J. Tomasi, J. Phys. Chem. A, 104 (2000) 9874. [24] S. Corni, C. Cappelli, R. Cammi and J. Tomasi, J. Phys. Chem. A, 105 (2001) 8310. [25] (a) S. Corni and J. Tomasi, J. Chem. Phys., 114 (2001) 3739; (b) S. Corni and J. Tomasi, J. Chem. Phys. Lett., 342 (2001) 135. [26] C. Cappelli, S. Corni, B. Mennucci, R. Cammi and J. Tomasi, J. Phys. Chem. A, 106 (2002) 12331. [27] C. Cappelli, S. Corni, B. Mennucci, J. Tomasi and R. Cammi, J. Quantum Chem., 104 (2005) 716. 2.8 Birefringences in Liquids Antonio Rizzo 2.8.1 Introduction The term birefringence indicates an anisotropy of some kind in the real part of refractive index n exhibited by a beam of electromagnetic radiation after it traverses a medium. The best known example of birefringence, natural optical activity (NOA), was discussed in another contribution to this book by Pecul and Ruud. The birefringences discussed in this contribution are observed when radiation interacts with molecules in external electromagnetic fields. We focus here in particular on the computational aspects of the study of some linear birefringences in condensed phases. General references for this section are the books by Böttcher and Bordewijk [1], Barron [2] and Raab and De Lange [3]. Specific reviews discuss the aspects related to theory and experiment for some classical birefringences as Kerr and Cotton–Mouton effects in condensed phases [4–7]. Birefringences and their computational study with the models and techniques developed in recent years within analytical response theory [8, 9] are discussed, mainly with reference to the gas phase, in some review work involving the author [10–13]. The absorptive counterparts of birefringences are dichroisms, due to the appearance of anisotropies in the complex part of the refractive index n, see also the contributions by Ruud and Pecul and by Stephens and Devlin. A linear birefringence involves the occurrence of an anisotropy between the components of the refractive index associated with linearly polarized monochromatic light whose polarization vector is directed along two perpendicular optical axes. In the examples discussed here the principal optical axis lies parallel to an external applied field, and nlin = n − n⊥ (2.209) An external electric field yields the Kerr effect (KE); a magnetic field is responsible for the Cotton–Mouton effect (CME); an electric field gradient induces the Buckingham effect (BE). Linear birefringences can be seen in isotropic fluids, and they can involve static and optical fields. Where linear birefringences occur, the changes in the polarization state of the electromagnetic beam result in an ellipticity * which is proportional to nlin . If ! is the wavelength (corresponding to a circular frequency ) and l is the path length *≈ lnlin ! (2.210) * is the observable related to linear birefringences. One of the mechanisms responsible for the emergence of linear birefringences is the temperature T dependent orientational effect the fields have on the molecules of the sample, through the interaction with their permanent multipoles. On the other hand this fact alone would not explain the occurrence of birefringences also for atoms or molecules with spherical symmetry. Electronic rearrangements, involving high order responses to Properties and Spectroscopies 253 the radiation and external fields, yield a T -independent contribution, usually small albeit seldom negligible even in systems of low spatial symmetry. The general expression for nlin T as a function of molecular properties and of the parameters characterizing the electromagnetic radiation can be expressed as (see Table 2.6) nlin T = w1 F mW T (2.211) where F includes the field dependence, m W T is identified as the molecular ‘constant’ for that particular birefringence and w1 is a combination of fundamental constants, characteristic of the given process. Table 2.6 shows that the KE and CME are quadratic in the electric and magnetic induction field strengths, respectively. BE is linear in the strength of the electric field gradient. m W T can be written as m W T 0 1 A A2 = w2 A0 + 1 + 2 2 +··· kT kT (2.212) Table 2.6 See Equation (2.211). 0 is the vacuum permittivity, the relative permittivity of the medium, Vm the molar volume. E, B and E are the strengths of the electric, magnetic induction, and electric field gradient fields, respectively W F w1 KE K E2 27 1 n2 + 22 + 22 2Vm n 9 9 CME C B2 27 1 n2 + 22 2Vm 40 n 9 BE Q E 3 1 n2 + 22 2 + 3 2Vm n 9 5 where k is the Boltzmann constant. In Table 2.7 the constant w2 and the parameters An n = 0 1 2, are given for the linear birefringences discussed here, assuming that Table 2.7 See Equation (2.212). NA is Avogadro’s number. See text for other definitions KE CME BE w2 A0 A1 A2 NA 540 K4 1 K1 + K3 5 1 K 5 2 1 Q 15 bEQC F EQC 2NA 27 2NA 450 254 Continuum Solvation Models in Chemical Physics no permanent magnetic moments are present in the sample. The missing definitions are (Einstein implicit summation over repeated indices, is the alternating tensor) 1 1 −$ 0 0 + −$ 0 0 15 10 1 + −$ 0 0 10 K1 = −$ 0$ 0 − 3iso iso 0 (2.214) K2 = −$ − iso (2.215) K4 = K = − 2 4 K3 = −$ 0 3 1 * = * − * 3 Q = 3 −$ − −$ (2.213) (2.216) (2.217) (2.218) EQC bEQC =B −$ 0 − BEQC −$ 0+ − 5 J −$ 0 EQC F EQC = + −$ (2.219) (2.220) Among the molecular properties introduced above are the permanent electric dipole moment and traceless electric quadrupole moment + , the electric dipole polarizability −$ iso = 13 −$ , the magnetizability , the dc Kerr first electric dipole hyperpolarizability −$ 0 and the dc Kerr second electric-dipole hyperpolarizability −$ 0 0. The more exotic mixed hypersusceptibilities are defined, with the formalism of modern response theory [9] para dia −$ 0 + * −$ 0 0 * = * dia dia * −$ 0 ∝ ˆ $ ˆ ˆ 0 para −$ 0 0 * ∝ ˆ $ ˆ m̂ m̂ 00 (2.221) (2.222) (2.223) ˆ $ ˆ +̂ 0 B −$ 0 ∝ (2.224) B −$ 0 ∝ ˆ $ +̂ ˆ 0 (2.225) J −$ 0 ∝ ˆ $ m̂ ˆ 0 (2.226) where the electric dipole , ˆ the traceless +̂ electric quadrupole, the magnetic dipole m̂ and the diamagnetic susceptibility ˆ dia operators appear. The superscript ‘EQC’ in the entries of Table 2.7 related to BE indicates that the origin-dependent quantities to which they are associated refer to the so–called effective quadrupolar centre [14], REQC , a particular frequency-dependent vector in the coordinate space defined with respect to a given choice of origin of the coordinates, ‘or’. Properties and Spectroscopies 255 REQC is a null vector for nondipolar molecules. For dipolar systems with dipole moment aligned along the z direction REQC = 0 0 REQC , with (frequency z dependence of the response functions omitted for sake of brevity) = REQC z or 5 or or G or − 5 Gyx + Aor xzx + Ayzy + Azzz xy 3 − 25 Mixed + 23 yy − 25 Mixed + 2zz xx yy 2 xx (2.227) Above or or (2.228) ˆ $ m̂ G −$ ∝ or (2.229) 1 ˆ $ Mixed ˆ p −$ ∝ (2.230) ˆ $ +̂ A −$ ∝ or The origin with respect to which the electric quadrupole and magnetic dipole operators are defined is indicated by the superscript. ˆ p is the component of the velocity operator. The connection between the quadrupole moment referred to ‘or’ – for example the centre of nuclear masses – and the EQC is EQC EQC or +xx = +yy = +xx + z REQC z EQC +zz = or +zz − 2z REQC z (2.231) (2.232) All the linear and nonlinear optical properties introduced above are therefore expressed in terms of linear, quadratic and cubic response functions. They can be computed with high efficiency using analytical response theory [9] with a variety of electronic structure models [8]. 2.8.2 Birefringences in Liquids and Solutions Birefringences are mostly observed in condensed phases, especially pure liquids or solutions, since the strong enhancement of the effects allows for reduced dimensions (much shorter optical paths) of the experimental apparatus. Nowadays measurements of linear birefringences can be carried out on liquid samples with desktop–size instruments. Such measurements may yield information on the molecular properties, molecular multipoles and their polarizabilities. In some instances, for example KE, CME and BE, measurements (in particular of their temperature dependence) have been carried out simultaneously on some systems. From the combination of data, information on electric dipole polarizabilities, dipole and quadrupole moments, magnetizabilities and higher order properties were then obtained. For measurements in solution of linear birefringences, a ‘specific’ birefringence constant s W l T =m W l T/Ml is usually defined for component l [6, 7]. Ml is the molar weight. In a multicomponent solution [4] sW solution T = l xl s W l T + l m xl xm s W lm T + · · · (2.233) 256 Continuum Solvation Models in Chemical Physics which assumes an additive scheme for the various contributions, and where xl is the molar fraction of l and s W lm T accounts for the contribution arising from the interaction of components l and m. For a two-components mixture, where we identify a solute (sol, whose birefringence we are interested in) and a solvent, SOL, it is often assumed that sW solution T ≈ xSOL s W SOL T + xsol s W sol T (2.234) and an infinite-dilution constant for the solute is defined as sol T =s W SOL T + s W s W solution T xsol (2.235) Measurements of the linear birefringences of Table 2.6 imply the determination of n , the density and of the constant s W solution T for the solution. Also, s W SOL T is assumed to be known. An extrapolation to infinite dilution is then made, according to Equation (2.235), often under the further assumption that all parameters of the solution depend linearly on xsol . From the point of view of theory, the formulae of Table 2.6 are equally applicable to both gas and condensed phase samples, as they include the local field factors, which account for local modifications to the Maxwell fields due to bulk interactions within the Onsager–Lorentz model. For gases, n = ≈ 1 is an excellent approximation. The easiest approach to condensed phases maintains this approximation, where calculations of the molecular first-order and response properties are performed for the isolated molecule, while accounting for the effect of intermolecular interactions through the number density N = Na /Vm , and therefore by taking appropriate values of Vm . This rough, often at best qualitative, approach is somewhat relaxed by employing expansions of the birefringence constant with the density, that is in inverse powers of Vm . This introduces the appropriate virial coefficients [15,16] m W T = AW T + BW T + CW T2 + · · · (2.236) where AW T is the constant for noninteracting species whereas BW T CW T · · · take into account two–, three– etc. body interactions, and are the so– called second, third,· · · virial coefficients. Their ab initio calculation involves the detailed knowledge of intermolecular potentials and interaction-induced properties, a far from trivial task even with nowadays huge progress in the field. Nevertheless, second Kerr virial coefficients BK T have been determined, at varying levels of approximations, for systems of different complexity, from closed shell atoms [17] to molecules [18–20]. The corresponding quantities for CME [21], BC T, and BE [22], BQ T, have been computed for helium. We are unaware of calculations of CW T (or higher) for any of the birefringences discussed in this section. A nowadays more easily applicable framework to treat local field effects in optical processes involving pure liquids or solutions has been discussed at length elsewhere in this book, and it consists in resorting to dielectric continuum solvation models. In the last pages of this section some application of such models the study of birefringences in condensed phases will be briefly discussed. Properties and Spectroscopies 257 A dielectric continuum model was adopted in the computational study of the CME of liquid water [23, 24]. A single molecule of water was placed in a spherical cavity surrounded by the homogeneous polarizable dielectric. The electric dipole polarizability and the magnetizability, see Equation (2.218), were computed using an electron– correlated wavefunction model – multiconfigurational self consistent field, MCSCF – and a basis set of London Atomic Orbitals (LAOs, also known as Gauge Including Atomic Orbitals, GIAOs [25]). The latter ensure origin independence of magnetic properties. The components of the hypermagnetizability, cf. Equations (2.221)–(2.223), yielding the anisotropy, Equation (2.217), were approximated by their infinite wavelength limit. They were obtained by a finite (electric) field approach, since [26] 4 B E B E E 2 E * 0$ 0 0 0 = − = B B E E E E (2.237) where B E indicates the molecular energy, which, like the magnetizability , in an equilibrium dielectric continuum model depends on the fields and on the dielectric constant . Origin-independent magnetizabilities were then computed analytically for different electric field strengths. Second derivatives were obtained numerically. This model accounts only partially for the specific structure of liquid water, and to refine it, calculations within supermolecular and semicontinuum models were also performed. In these cases, the properties were computed for a cluster of five water molecules, simulating the inclusion of a first solvation shell. In the semicontinuum model, the cluster was immersed in the dielectric continuum. Because of the (prohibitive for the times) size of the cluster, it was possible to obtain only an uncorrelated result. On the other hand, a nonequilibrium solvation model was used in computing the orientational contribution of Equation (2.218). Finally, to determine m C T, an extensive property, a differential shell method was employed. Table 2.8 summarizes the results [23, 24]. Going from the gas to the liquid, the Table 2.8 Cotton–Mouton constant m C T 1020 G−2 cm3 mol −1 ) for liquid water. 0 and Q in atomic units. T = 29315 K = 6328 nm. aug-cc-pVTZ basis set. See refs. [23, 24] for further details Phase Wavefunction Gas SCF MCSCF SCF MCSCF SCF SCF SCF SCF Exp.c Liquid Liquid a b c Solvent model 0 Continuum Continuum Supermolecule Semicontinuum Semicontinuuma Semicontinuumab 236 241 384 397 −36 01 01 01 Nonequilibrium solvation model for the electric dipole polarizability. Result corrected for local field effects. Ref.[27], mean value for T between 283.15 and 293.15 K. Q 00291 00394 000847 000816 −09719 −09890 −08579 −13620 m C T 97 101 147 152 −276 −267 −231 −367 −11815 258 Continuum Solvation Models in Chemical Physics response of the system is remarkable. The orientational term, surprisingly ineffective for water vapour and even less important when computed within a pure dielectric continuum solvation model, largely dominates in the supermolecular and semicontinuum approaches, where the * contribution becomes negligible at standard temperatures. The effect of electron correlation, cf. MCSCF and SCF results, is quite limited. Neglecting short-range interactions between the water molecules in the liquid yields an estimate of the effect of opposite sign with respect to experiment. Apparently, specific interactions in the liquid are so influential that they reverse the direction of the ellipticity passing from the gas to the liquid phase. This occurs through a strong change of character of the effect, which is dominated by the electronic rearrangement mechanism in the gas phase, and is essentially all of the Langevin type in the liquid. The missing factor of three in the computed CME constant (≈ −37 G−2 cm3 mol−1 versus experiment −118 ± 15 G−2 cm3 mol−1 ) is attributable mainly to the lack of averaging of the dynamic structures as a result of the adoption of a fixed solvation shell arrangement in the cluster. It is nowadays commonly accepted that continuum solvation models alone are inadequate for the description of hydrogen-bonded liquids. Supermolecular and semicontinuum approximations may be costly, besides imposing the need for differential shell techniques to recover extensive observables for the solvated molecule, a procedure assuming additivity of the effects of the different shells. Molecular dynamics, coupled to quantum mechanical methods of sufficient sophistication to provide good high order optical properties, may provide in the long run relief in this field. Meanwhile, for the treatment of local field effects for high order optical properties of systems where specific interactions are not important, a good performance can be obtained with models where local field effects are accounted for through the definition of effective polarizabilities [1]. For birefringences, effective molecular response properties, embedding the response of the solute to the Maxwell fields, are introduced within the quantum mechanical polarizabile continuum model (PCM) [28], in the so-called integral equation formulation (IEF) [29], see elsewhere in this book. Again specializing to linear birefringences, it is convenient to define ‘effective’ constants m W T which are obtained from those given in Table 2.7 multiplied by the local field factors originally included in w1 , cf. Table 2.6 n2 + 22 + 2 9 3 2 2 n + 2 m C T = m C T 9 2 n + 22 2 + 3 m Q T = m Q T 9 5 mK T = m K T (2.238) (2.239) (2.240) Equation (2.211) is therefore formally rewritten as nlin = w1 Fm W T (2.241) with the local field factors displaced from w1 to w2 . Their role is then taken care of in the effective constants by the effective molecular properties. These are defined by a Properties and Spectroscopies 259 perturbative expansion of the molecular multipoles in terms of the Maxwell field of the medium, and represent the solvent–modified response of the solute to the macroscopic external fields (for more details see the contribution by Cammi and Mennucci). For electric properties in an optical Maxwell field, for example, the defining equations are 1 = −$ EMax + −$ 0EMax EMax 0+ 2 1 −$ 0 0EMax EMax 0EMax 0 + · · · + 6 (2.242) The quantity on the left is the Fourier component of the dipole moment induced by the optical field EMax . These equations can be generalized to mixed frequency-dependent electric dipole, electric quadrupole, magnetic dipole properties, and similar equations can , be written for the Fourier components of the permanent electric quadrupole, + . For static Maxwell fields similar expansions yield effective and magnetic dipole, m (starred) properties, defined as derivatives of the electrostatic free energies. Upon the introduction of effective properties, the effective constants m W T assume exactly the form valid for dilute gases, that is that given in Table 2.7, where ‘tilde’ and ‘star’ properties are employed for dynamic and static response properties, respectively. This approach has recently been employed for studies of Kerr [30], Cotton–Mouton [31] and Buckingham [32] linear birefringences of pure liquids and solutions. In the calculation of the frequency-dependent electric properties a nonequilibrium solute–solvent regime was employed. Magnetizabilities, quadrupole moments and *, the latter again in its infinite wavelength limit, *0, were obtained in the equilibrium solvation regime. Electric and magnetic effective properties were computed in a coupled perturbed approach. The study of the CME of furan, thiophene and selenophene combined experiment and theory [31]. Effective electric dipole polarizabilities and magnetizabilities were computed in the gas phase, for the pure liquids and for solutions involving a selection of common solvents. A DFT/B3LYP wavefunction model was adopted, and properties were obtained using a Coupled Perturbed Kohn–Sham approach. *0 was obtained using a finite electric field technique applied to the effective magnetizability, see above. For magnetic properties a continuous set of gauge transformations (CSGT) formalism, ensuring origin invariance, was employed. The results were compared with experiment, where the m C T of the solution was obtained by extrapolating to infinite dilution measurement made at different low concentrations. Tables 2.9 and 2.10 summarize the findings for pure liquids and solutions, respectively. The agreement between theory and experiment in Table 2.9 is quite excellent for furan, less so for its homologues. For the latter, experiment highlights an early tendency of the Cotton–Mouton constant to deviate from a linear dependence on the concentration of the solute as the latter increases. This indicates some degree of aggregation, not reproduced by the calculations. Table 2.10 shows the trend of the observable as the polarity of the solvent increases. Theory and experiment are in quite satisfactory agreement, albeit in some instances the former underestimates the effect. The investigation proved that cavity field effects on the response properties are important for the individual tensor components, whereas their influence is quenched on the averages Q and * nl T, 260 Continuum Solvation Models in Chemical Physics Table 2.9 DFT/B3LYP/d-aug-cc-pVDZ and experimental results (atomic units) for the CME of pure liquid furan, thiophene and selenophene. =632.8 nm, T=293.15 K. Experimental geometries Q 109 × nl Ta 109 × nl Texpa a nl T = NA B 2 solvent 4n0 Vm Furan Thiophene 42833 −1984 153 142 85685 9658 259 193 + 1 15kT Q Selenophene 94680 21697 238 197 Table 2.10 DFT/B3LYP and experimental results for the CME of furan, thiophene and selenophene in solution (atomic units). =632.8 nm, T = 293.15 K. d-aug-cc-pVDZ for furan and thiophene, aug-cc-pVDZ for selenophene. Experimental geometries. See also Table 2.9 Q Furan Thiophene Selenophene 109 × nl T 109 × nl Texp Q 109 × nl T 109 × nl Texp Q 109 × nl T 109 × nl Texp Acetone Cyclohexane CCl4 41007 −2328 150 141 43308 −1910 102 098 43584 −1928 113 110 79739 10399 292 263 84067 9306 199 192 84658 9425 219 86882 23656 319 281 91775 21123 218 202 92211 21355 239 definition in Table 2.9, changes by ca. 2–3 %, but for the cases analysed in ref. [31] the same effect, but with opposite sign, is observed when molecular geometries are relaxed, and re–optimization in the presence of the dielectric is carried out. In the study of the KE for a selection of pure liquids [30] the concept of effective polarizabilities was extended to introduce the contribution of the output wave. Radiation at a frequency induces a macroscopic nonlinear polarization density P NL at the same frequency, the output wave, generating an additional perturbing field. The molecules of the liquid respond with an additional effective polarizability −$ ¨ , whereby Equation (2.242) becomes P NL + = ¨ −$ P −$ EMax + 1 + −$ 0EMax EMax 0+ 2 Properties and Spectroscopies 1 + −$ 0 0EMax EMax 0EMax 0 + · · · 6 261 (2.243) With the introduction of −$ ¨ the effective molar Kerr constant of pure liquids can be written as pl T = mK m K T (2.244) 1 − −$ ¨ pl T is the Kerr constant to be compared with experiment, and including therefore the effect of the polarization density P NL as a source field, whereas m K T is given by Equation (2.212), see Table 2.6, where the appropriate ‘tilde’ and ‘star’ quantities are employed in Equations (2.213)–(2.216). Details on the computational procedure for −$ ¨ are given in ref. [30]. Again, nonequilibrium calculations were performed using a DFT/B3LYP model, and employing for all systems (see below) experimental gas phase geometries and an augcc-pVDZ basis set. The linear response functions ∗iso 0 −$ and ¨ −$ were obtained analytically, whereas the higher order polarizabilities, −$ 0 and −$ 0 0, were determined via a finite electric field technique. The results published in ref. [30] are shown in Table 2.11. The Table allows for a comparison between gas phase results, m K vac T, results obtained by including reaction and cavity field effects, m K T, and those – directly comparable to experiment – further taking into account the effect of the output wave, pl m K T. The increase in the Kerr constant in going from gas to condensed phase is in general noticeable, in some cases huge, see e.g. nitrobenzene. The inclusion of the effect of the output wave leads to a general increase in the constant, with a detailed analysis – not discussed here – showing that the enhancement afforded by the PCM–IEF model is slightly lower than that predictable by a classical Onsager model. In ref. [30] it was also mK −1 Table 2.11 Kerr constant 10−26 V −2 m5 mol of pure liquids. = 6328 nm T = 29815 K except where noted otherwise. Experimental gas phase geometries. Experimental densities, see Equation (2.244). See text for definitions and ref. [30] for further details and pertinent references vac T mK Benzene F-benzenea Pyridine Acetone Benzonitrile Acrylonitrile Nitrobenzene Acetonitrile a T = 29315 K. 169 12 830 125 m KT) 271 305 711 556 pl T mK 361 35 91 69 253 937 674 741 273 662 1001 347 910 534 145 182 mK exp 268 ± 004 44.1 125 79 ± 2 99 ± 5 2020 440 2364 ± 142 2546 ± 51 277 262 Continuum Solvation Models in Chemical Physics 2 term (vanishing for the shown that the mixed dipole polarizability proportional to the K nonpolar benzene ring) is dominating the constant (in both the gas and the condensed 4 , is, as expected in phases) whereas the T -independent contribution, proportional to K particular for a dipolar system, almost always markedly smaller than the others (but see acetone as a notable exception). The agreement between theory and experiment is quite good, albeit the former consistently underestimates the effect for polar systems. This is particularly evident for nitrobenzene, which experiment places as the system with the strongest effect among the selection, while calculations place it second after benzonitrile. This mismatch is the likely consequence of aggregation effects. As for the study of the CME of furan and its homologues, the dependence of the results of the PCM– IEF calculations on the only external parameter of the calculation, the cavity radius, and on the re-optimization of the geometry in the dielectric continuum, as estimated on the most challenging system among those considered (nitrobenzene) is found to be small. Very recently, the study of linear birefringences has been extended to BE [32], where, as for ref. [31], furan and its homologues were investigated, in this case in solutions of cyclohexane. The latter were the subject of an experimental analysis by Dennis et al. [33]. In ref. [32] advantage is taken of the recent development of frequency-dependent quadratic response in the nonequilibrium PCM solvation regime [34]. In Table 2.12 we concentrate only on the comparison between theory and experiment for the quadrupole moment at the centre of nuclear masses (CM – measured in microwave Zeeman experiments), for the quadrupole moment at the EQC – the direct observable in BE measurements, and for two ‘indirect’ observables, the EQC itself and the combination Table 2.12 BE of furan, thiophene and selenophene in cyclohexane. Comparison of theory (PCM/DFT/B3LYP both for the geometry optimization and for the properties, aug-cc-pVTZ basis set) and experiment. See text for definitions and ref. [32] for further details. In particular, A + 5/G indicates the numerator of the right-hand side of Equation (2.227). Atomic units. =632.8 nm. CM stands for the centre of nuclear masses, chosen as reference origin in the calculations Theory Experiment Furan A + 5 G EQC Rz CM zz EQC zz 470794 17546 −43403 −48587 −9169 ± 6877 5669 ± 5669 −4525 ± 02898 −6063 ± 1204 Thiophene A + 5 G EQC Rz CM zz EQC zz −495375 −07551 −53143 −51353 −9169 ± 1948 3779 ± 9449 −6174 ± 1627 −7132 ± 1204 Selenophene A + 5 G EQC Rz CM zz EQC zz −1185218 −14289 −58735 −56046 1261 ± 4355 −5669 ± 1701 −6397 ± 2073 −5639 ± 156 Properties and Spectroscopies 263 of linear response properties at the numerator of Equation (2.227). The large error bars associated with the experiment), especially for the indirect observables, allow for a nice agreement between theory and experiment, albeit some of the trends apparently arising EQC from the experimental work are not confirmed by calculations. For instance, +zz is estimated to be greater, in absolute value, for selenophene than for thiophene, whereas the centres of the experimental distribution would indicate the reverse. References [1] C. J. F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol. II, Elsevier, Amsterdam, 1978. [2] L. D. Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, Cambridge, 1982. [3] R. E. Raab and O. L. De Lange, Multipole Theory in Electromagnetism. Classical, Quantum and Symmetry Aspects, with Applications, International Series of Monographs in Physics, 128. Oxford Science Publications, Clarendon Press, Oxford, 2005. [4] J. H. Williams, Adv. Chem. Phys., 85 (1993) 361. [5] A. D. Buckingham, in C. O’Konski (ed.), Molecular Electro-Optics, Marcel Dekker, New York, 1976. [6] C. G. LeFévre and R. J. W. LeFévre, Rev. Pure Appl. Chem., 5 (1955) 261. [7] C. G. LeFévre and R. J. W. LeFévre, in A. Weissberger and B. W. Rossiter (eds), Physical Methods of Chemistry, Part 3c, Wiley-Interscience, New York, 1972. [8] T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory, John Wiley and Sons, Ltd, Chichester, 1999. [9] J. Olsen and P. Jørgensen, in D. R. Yarkony (ed.), Modern Electronic Structure Theory, World Scientific, New York, 1995, p. 857. [10] C. Rizzo, A. Rizzo and D. M. Bishop, Int. Rev. Phys. Chem., 16 (1997) 81. [11] S. Coriani, A. Halkier and A. Rizzo, in G. Pandalai (ed.), Recent Research Developments in Chemical Physics, Vol. 2, Transworld Scientific, Kerala, India, 2001, p. 1. [12] A. Rizzo and S. Coriani, Adv. Quantum Chem., 50 (2005) 143. [13] O. Christiansen, S. Coriani, J. Gauss, C. Hättig, P. Jørgensen, F. Pawłowski and A. Rizzo, in M. G. Papadopoulos, A. J. Sadlej and J. Leszczynski (eds), Non-Linear Optical Properties of Matter: From Molecules to Condensed Phases, Series: Challenges and Advances in Computational Chemistry and Physics, Vol. 1, Springer, 2006, p. 51 [14] A. D. Buckingham and H. C. Longuet-Higgins, Mol. Phys., 14 (1968) 63. [15] A. D. Buckingham and J. A. Pople, Faraday Soc. Disc., 22 (1956) 17. [16] A. D. Buckingham, Proc. Phys. Soc. A, 68 (1955) 910. [17] A. Rizzo, S. Coriani, D. Marchesan, J. López Cacheiro, B. Fernández and C. Hättig, Mol. Phys., 104 (2006) 305. [18] D. A. Dunmur and N. E. Jessup, Mol. Phys., 37 (1979) 697. [19] V. W. Couling and C. Graham, Mol. Phys., 93 (1998) 31. [20] V. W. Couling, B. H. Halliburton, R. I. Keir and G. L. D. Ritchie, J. Phys. Chem., A, 105 (2001) 4365. [21] A. Rizzo, K. Ruud and D. M. Bishop, Mol. Phys., 100 (2002) 799. [22] D. Marchesan, S. Coriani and A. Rizzo, Mol. Phys., 101 (2003) 1851. [23] K. Ruud, T. Helgaker, A. Rizzo, S. Coriani and K. V. Mikkelsen, J. Chem. Phys., 106 (1997) 894. [24] K. Ruud, H. Ågren, P. Dahle, T. Helgaker, A. Rizzo, S. Coriani, H. Koch, K. O. Sylvester-Hvid and K. V. Mikkelsen, J. Chem. Phys., 108 (1998) 599. 264 Continuum Solvation Models in Chemical Physics [25] F. London, J. Phys. Radium, 8 (1937) 397. [26] A. Rizzo, T. Helgaker, K. Ruud, A. Barszczewicz, M. Jaszuński and P. Jørgensen, J. Chem. Phys., 102 (1995) 8953. [27] J. H. Williams and J. Torbet, J. Phys. Chem., 96 (1992) 10477. [28] S. Miertus, E. Scrocco and J. Tomasi, Chem. Phys., 55 (1981) 117. [29] J. Tomasi, B. Mennucci and R. Cammi, Chem. Rev., 105 (2005) 2999. [30] C. Cappelli, B. Mennucci, R. Cammi and A. Rizzo, J. Phys. Chem. B, 109 (2005) 18706. [31] C. Cappelli, B. Mennucci, J. Tomasi, R. Cammi, A. Rizzo, G. L. J. A. Rikken, R. Mathevet and C. Rizzo, J. Chem. Phys., 118 (2003) 10712. [32] A. Rizzo, L. Frediani and K. Ruud, J. Chem. Phys., submitted. [33] G. R. Dennis, I. R. Gentle, G. L. D. Ritchie and C. G.Andrieu, J. Chem. Soc., Faraday Trans. 2, 79 (1983) 539. [34] L. Frediani, H. Ågren, L. Ferrighi and K. Ruud, J. Chem. Phys., 123 (2005) 144117. 2.9 Anisotropic Fluids Alberta Ferrarini 2.9.1 Introduction Anisotropic fluids, of which nematic liquid crystals are the most representative and simplest example, are characterized by an anisotropic dielectric permittivity. The nematic phase has Dh symmetry, and in a laboratory frame with the Z axis parallel to the C symmetry axis (the director) the permittivity tensor has the form: ⎞ ⎛ ⊥ ⎜ ⊥ ⎟ ⎟ =⎜ ⎝ ⎠ with and ⊥ denoting the longitudinal and transverse component, respectively [1]. The mean permittivity, < >= 2⊥ + /3, and the dielectric anisotropy, = − ⊥ , are often used as independent parameters, in place of the parallel and perpendicular components. The magnitude of the dielectric anisotropy depends upon the polarity of the constituting molecules and the degree of orientational order. Figure 2.30 shows the temperature dependence of the permittivity tensor for two nematics, with > 0 and < 0; the different sign originates from the presence of longitudinal and transversal dipoles in the constituting molecules. In the figure we can recognize a typical behaviour: vanishes in the isotropic liquid, has a discontinuity at the first-order nematic-isotropic transition, and in the nematic phase increases with decreasing temperature (i.e. increasing order) [1, 4]. Hereafter, we shall generally refer to the static permittivity; however, most considerations could be extended to the optical permittivity. In the case of nematics 2 this is also an axially symmetric tensor, with ⊥ = neo , and an anisotropy which is always small (generally smaller than 1) and positive. As an example, in Figure 2.30 NC 20 15 C5H11 C2H5O OC6H10 ε ε⊥ ⏐⏐ ε 10 NC 10 ⊥ 2 ne ε 2 0 no 280 (a) ⏐⏐ 290 300 T/K 310 330 (b) 340 350 360 370 T/K Figure 2.30 Permittivity of two nematics with opposite dielectric anisotropy, as a function of temperature [1, 2]. In (a) the square of the refractive index =589 nm) is also shown [3]. 266 Continuum Solvation Models in Chemical Physics the temperature dependences of the squares of the ordinary ne and extraordinary ne refractive indices are also shown for the nematogen 5CB.1 Solvation in anisotropic fluids has been subjected to few experimental and theoretical investigations [5]. However, the role of electrostatic interactions has been a matter of debate for a long time, in connection with the origin of orientational order. Solutes have been widely used to probe the molecular order in liquid crystals, mostly with the NMR technique (for a comprehensive review see ref. [6] and references cited therein). Intensive investigation was stimulated from the finding that dideuterium preferentially aligns its axis with the director in the nematic ZLI1132, and perpendicular to the director in EBBA [7]. When EBBA is added to ZLI1132, the degree of alignment of dideuterium with the director decreases, and goes through zero at a given composition, which has been denoted the ‘magic mixture’. This orientational behaviour has been ascribed to differences in electrostatic solute–solvent interactions in ZLI1132 and EBBA; such interactions would drive the alignment of small solutes, such as molecular hydrogen and methane, in nematics. In contrast, in the case of bulkier molecules, the orientational order is dominated by the short-range steric and dispersion interactions. These molecules have a preferential alignment even in the ‘magic mixture’; however, their order changes in opposite ways upon increasing the concentration of either EBBA or ZLI1132, and electrostatic solute–solvent interactions have been claimed to be responsible for the observed changes [6]. Recently, an explanation in terms of the dielectric anisotropy of the solvent has been proposed, in the framework of the polarizable continuum model [8, 9]. Indeed, ZLI1132 and EBBA have opposite dielectric anisotropy (ZLI1132 13 and EBBA −0 3), and the presence of compounds with opposite values is a feature shared by all the ‘magic mixtures’ which have been found after the original one [6]. Within the dielectric continuum model, the electrostatic interactions between a probe and the surrounding molecules are described in terms of the interaction between the charges contained in the molecular cavity, and the electrostatic potential these changes experience, as a result of the polarization of the environment (the so-called reaction field). A simple expression is obtained for the case of an electric dipole, 0 , homogeneously distributed within a spherical cavity of radius a embedded in an anisotropic medium [10–12], by generalizing the Onsager model [13]. For the dipole parallel (perpendicular) to the director, the reaction field is parallel (perpendicular) to the dipole, and can be calculated as [10]: R E ⊥ = 0 a2 ⊥ − 2 ⊥ + A ⊥ 1 − A ⊥ A ⊥ (2.245) 1 − A ⊥ As customary, acronyms will be used for nematic solvents throughout the text: 5CB, (4-n-pentyl)-4 -cyanobiphenyl; EBBA, N -(p-ethoxybenzylidene)-p -butylanyline; ZLI1132, eutectic mixture of 1,4-(trans-4 -n-alkylcyclohexyl)-cyanobenzene and 1,4-(trans-4 -n-pentylcyclohexyl)-cyanobiphenyl. 1 Properties and Spectroscopies −6 −6.2 W (kJ/mol) −6.1 W (kJ/mol) 267 ⊥ || −6.1 || ⊥ −6.2 −6.3 0 5 10 15 −6 Δε −4 −2 0 Δε (a) (b) Figure 2.31 Electrostatic free energy for a dipole = 565 D in a spherical cavity of radius a=5.15 Å in a uniaxial dielectric, as a function of the dielectric anisotropy . The average permittivity is (a) < > = 11 and (b) < > = 95. Solid and dashed lines refer to the dipole parallel and perpendicular to the director, respectively. with 2 a3 ds A ⊥ = √ 2 2 ⊥ s + a / ⊥ R 0 with 2 a2 R2 = s + a2 / s + ⊥ (2.246) Figure 2.31 displays the electrostatic free energy, W = − 21 0 E R [10], as a function of the dielectric anisotropy, calculated for a solute with dipole moment 0 = 5 65 D, cavity radius a = 5 15 Å, and permittivity values corresponding to the two nematics considered in Figure 2.30. We can see that the free energy is lower when the dipole is along the direction of higher permittivity. The change in sign of the free energy difference, W = W − W⊥ , in the two nematics can be correlated with the opposite signs of changes in order parameters, experimentally observed for a given molecule, when dissolved in media with > 0 and < 0 [6]. We can also see that the free energy difference W increases with the magnitude of the dielectric anisotropy, but it remains a small fraction of the entire free energy: with the molecular parameters used, which should be appropriate for a molecule such as coumarin, an energy difference less than 0 1 kJ mol−1 is predicted even for polar nematics with high . The model of a dipole in a spherical cavity can only provide qualitative insights into the behaviour of real molecules; moreover, it cannot explain the effect of electrostatic interactions in the case of apolar molecules. More accurate predictions require a more detailed representation of the molecular charge distribution and of the cavity shape; this is enabled by the theoretical and computational tools nowadays available. In the following, the application of these tools to anisotropic liquids will be presented. First, the theoretical background will be briefly recalled, stressing those issues which are peculiar to anisotropic fluids. Since most of the developments for liquid crystals have been worked out in the classical context, explicit reference to classical methods will be made; however, translation into the quantum mechanical framework can easily be performed. Then, the main results obtained for nematics will be summarized, with some illustrative 268 Continuum Solvation Models in Chemical Physics examples. We shall focus on the effects of the dielectric anisotropy on the orientational distribution of molecules in anisotropic fluids and on average properties which depend on such a distribution. The effects of the dielectric anisotropy on magnetic tensors, whose calculation is discussed in more details in the contribution by Sadlej and Pecul, will be only mentioned here. 2.9.2 Theoretical Background The Integral Equation Formalism (IEF) The IEF methodology [14–16] enables us to express the electrostatic potential experienced within a cavity of arbitrary shape and containing an arbitrary charge distribution, 0 , embedded in an anisotropic dielectric, as the sum of two contributions: one arises from 0 , and the other from a charge density, R , spread on the boundary of the cavity. The latter is obtained by solving an integral equation derived from the Poisson equation, which with suitable boundary conditions defines the electrostatic problem inside CI and outside CE the cavity, through the Green functions of the differential operators involved (see also the contribution by Cancès): GI r r = 1 40 r − r GE r r = r ∈ CI r ∈ CI 1 40 det r − r · −1 · r − r (2.247) r ∈ C I r ∈ CI (2.248) The integral equation reads: A R = B(0 (2.249) where (0 is the Coulomb potential generated in a vacuum by the molecular charge density 0 : (0 r = 2 GI r r 0 r dr (2.250) A and B are integral operators, defined on the molecular surface S, which depend on the shape of the cavity boundary, on the dielectric permittivity of the surrounding dielectric and on the charge distribution within the cavity: I I A = − De Si + Se − Di 2 2 I B = − De + 0 Se i 2 (2.251) (2.252) Properties and Spectroscopies 269 Here i = /r · sr, with sr being an outward pointing unit vector, normal to the surface in r I is the identity operator, and Se Si De Di are integral operators, defined as: 2 Si hr = GI r r hr dr 2 S G r r hr dr Di hr = 0 sr I r S2 r r ∈ S Se hr = GE r r hr dr 2 S GI r r De hr = 0 sr · · r dr r S The IEF methodology was originally proposed in the quantum mechanical context [15–17]; in this case the charge distribution 0 comprises all the nuclear and electronic charges. Within a classical approach, the charge distribution can be identified with a set of atomic charges, qK0 , located at the nuclear positions, r K [8]; so, for a molecule with Na atoms we can write: 0 r = Na qK0 r − r K (2.253) K=1 These are intended as charges in the isolated molecule, in the absence of external fields; the molecular polarizability can be included [9], as explained in the following. Classical approach: inclusion of the molecular polarizability Let us describe the charge distribution induced in a molecule by an external field, in terms of a set of mutually interacting dipoles centred at the nuclear positions. The dipole induced at the J th site by the reaction field can then be expressed as: ind J =− Na M−1 JK · K=1 2 S tK r R r dr (2.254) Here the dot indicates contraction over Cartesian coordinates, tK is a component of the vector tK r = r − r K /40 r − r K 3 , and M is the 3 × 3 block, corresponding to the Kth and J th atoms, of the so-called relay matrix, which gives an atomic representation of the molecular polarizability. The supermatrix M has dimension 3Na × 3Na , and is defined as: ⎡ −1 1 ⎢T ⎢ 21 M=⎢ ⎢ ⎣ T N1 T 12 T 1N −1 2 T 2N T N2 −1 N ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (2.255) where J = J I3 , with I3 being the 3 × 3 unit tensor, J the atomic polarizability of the J th center and T JK the dipole field tensor [10]. A modified form of this tensor has 270 Continuum Solvation Models in Chemical Physics been proposed by Thole, with a distance-dependent attenuation to avoid divergence of the polarizability when dipoles lie too close to each other [17]. If the molecular polarizability is taken into account, Equation (2.249) becomes: A R = B(0 + (Rind (2.256) where (Rind r is the Coulomb potential generated by the induced dipoles: (Rind = Na tJ r · M−1 JK · K=1 2 S tK r R r dr (2.257) Given the linear relationship between (Rind and R , Equation (2.256) is conveniently rewritten in the form: A + Aind R = B(0 (2.258) with Aind R = −B Na tJ r · M−1 JK · JK=1 2 S tK r R r dr (2.259) Presence of an applied electric field The IEF formalism can be extended to take into account the presence of an applied field, E [18]. In this case, under linear conditions, Equation (2.259) becomes: A + Aind R+C = B(0 − gC (2.260) gC = E · r (2.261) with For E = 0 the surface charge density, R+C , obtained by solving Equation (2.260), contains a contribution which accounts for the cavity field, i.e. the electric field generated in the cavity by the applied field. Evaluation of physical observables The usefulness or the IEF approach relies on the possibility of expressing physical properties, which depend on the interaction between the molecular charges and the reaction/cavity field, as surface integrals of the general form [8, 9, 18]: 2 (2.262) f = vrr dr S where is the charge density on the surface of the cavity, defined in Equations (2.249), (2.258) or (2.260). The surface integral can then be rewritten as: 2 f = vrA + Aind −1 ur dr (2.263) S Properties and Spectroscopies 271 where u v are functions which depend on the specific observable. Examples are reported in Table 2.13. Table 2.13 Explicit form of the functions appearing in Equation (2.263) for the calculation of the dipole induced at the Jth site by the reaction and cavity fields, ind J , and of the electrostatic free energy, W . This free energy includes a contribution from the reaction field, WR , and one from the cavity field, WE . In the absence of an applied field gC vanishes, and only the reaction field contribution to the induced dipole and to the electrostatic free energy remains f v W ind J − u 0 M JK · tK K 5 − 12 B0 − gC −B0 − gC If = 0 the reaction/cavity fields, and then the molecular property f , depend on the molecular orientation. Such a dependence affects the physical observables, which are obtained by averaging over the orientational distribution. Considering in general a tensorial property, we can express the average value as: fIJLAB = 2 fIJLAB p d (2.264) where are the Euler angles defining the molecular orientation, and I J denote the Cartesian components in the laboratory frame. The function p is the orientational distribution function, which is a constant in isotropic fluids. Equation (2.24) is appropriate for rigid molecules, but could be generalized to the case of flexible systems, at the cost of introducing the internal degrees of freedom as additional variables. It is usually convenient to express tensorial properties in the molecular frame; so Equation (2.264) can be rewritten as: fIJLAB = 2 fijMOL RiI RjJ p d (2.265) ij where R is the Euler rotation matrix for the transformation from the laboratory to the molecular frame, and the labels i j are used for Cartesian components in the molecular frame. Here the possiblity of an intrinsic orientation dependence of the molecular property has been taken into account. Dealing with orientational properties, it is customary to use the spherical tensor notation [19]. The relation between observable in the laboratory frame and property in the molecular frame becomes: Lm fLAB = 2 n Ln fMOL DL∗ mn p d (2.266) 272 Continuum Solvation Models in Chemical Physics where DLmn are Wigner rotation matrices [19]. Also the intrinsic orientation dependence of the molecular property can be expressed in terms of Wigner rotation matrices, according to the following expansion: Ln Jpq∗ J Ln fMOL = fMOL Dpq (2.267) Jpq with Ln Jpq = fMOL 2L + 1 2 Ln∗ fMOL DJpq d 8 2 (2.268) Using Equations (2.267) and (2.268), Equation (2.266) can be rewritten as: Ln Jpq∗ 2 L∗ Lm = fMOL Dmn DJpq p d fLAB Jpqn = Jpqn Ln Jpq∗ fMOL J CL J J $ m p m + pCL J J $ n q n + qDJm+pn+q (2.269) In the second line, the symbols C are Clebsch–Gordan coefficients [19], whereas DLmn are the so-called order parameters, in the ireducible spherical notation; these are defined as: 2 DLmn = DLmn p d (2.270) The symmetry of the system can be exploited to restrict the expansion to subsets of Wigner functions. For instance, in the absence of applied fields, p in nematics is invariant for rotation around the director and for reflection with respect to a plane perpendicular to the director. It follows that, if the laboratory Z axis is taken parallel to the director, only Wigner matrices DLmn with m = 0 and even L values can contribute. In the presence of a longitudinal electric field, reflection is no longer a symmetry operation, and also odd L values must be included. The nonvanishing order parameters thus depend upon the symmetry of the system; their magnitude quantifies the degree of order. It follows from Equation (2.269) that tensorial properties which are washed out in the isotropic phase, where all order parameters vanish, can yield measurable quantities in the nematic phase. In fact, Equation (2.269), or its simplified version in the case of an angular-independent property in the molecular frame, Ln Lm fLAB = fMOL DLmn (2.269–b) Ln is used to obtain order parameters from experimental data. The orientational order of molecules in nematics is often described in terms of the so-called Saupe matrix [1], whose elements are defined as: Sij = 2 3R R − iZ jZ ij p d 2 (2.271) Properties and Spectroscopies 273 where i j are molecular axes, Z is a laboratory axis parallel to the director, and ij is the Kronecker symbol. A simple relationship exists between the Saupe matrix and the second rank order parameters D20n [20]. The Saupe matrix is symmetric and traceless. The order parameter Sii can range from -0.5 and 1, for complete alignment perpendicular and parallel to the director, respectively, and vanishes in the absence of orientational order. Orientational Distribution Function The orientational distribution p can be expressed as p = exp −U/kB T = 62 exp −U/kB T d (2.272) where U is the orienting potential. Even in the absence of an applied field, in the nematic phase there is a contribution to this potential, which derives from the anisotropy of the intermolecular interactions. This many-body effect cannot be expressed in a simple way; therefore a number or approximations are generally taken. Adopting the usual distinction between electrostatic and short-range interactions of steric and dispersion origin, we can express the nematic potential, in the absence of an applied field, in the following form: U = Usr + WR (2.273) The electrostatic part, WR , can be evaluated with the reaction field model. The shortrange term, Usr , could in principle be derived from the pair interactions between molecules [21–23]. This kind of approach, which can be very cumbersome, may be necessary in some cases, e.g. for a thorough analysis of the thermodynamic properties of liquid crystals. However, a lower level of detail can be sufficient to predict orientational order parameters. Very effective approaches have been developed, in the sense that they are capable of providing a good account of the anisotropy of short-range intermolecular interactions, at low computational cost [6, 22]. These are phenomenological models, essentially in the spirit of the popular Maier–Saupe theory [24], wherein the mean-field potential is parameterized in terms of the anisometry of the molecular surface. They rely on the physical insight that the anisotropy of steric and dispersion interactions reflects the molecular shape. 2.9.3 Applications Electrostatic Free Energy and Order Parameters The electrostatic free energy of a solute in a nematic solvent can be calculated as a function of the molecular orientation, according to Equation (2.263). It is strongly affected by the shape of the molecular cavity and the distribution of charges within this. Coumarin, which was taken as an example in the introduction, can be considered again to illustrate this point. For coumarin in a nematic solvent with = 11 and = 15, the difference between the maximum of the electrostatic free energy, which corresponds to the dipole perpendicular to the director, and the minimum, for the parallel orientation, is predicted to be about 2 kJ mol−1 . This is about 20 times as great as the free energy difference obtained, under the same conditions, for a dipole in a sphere of radius 5.15 Å, as estimated from 274 Continuum Solvation Models in Chemical Physics the volume of the molecular shaped cavity (see Figure 2.31); a reduction of the radius to less than 2 Å would be required to get a free energy difference of about 2 kJ mol−1 with this model. Energy differences of the order of a few kJ mol−1 are small, but can be sufficient to affect the orientational distribution of molecules in nematics, and thus their order parameters. However, as explained in the previous section, these depend on the anisotropy of all the intermolecular interactions experienced by a molecule in the liquid crystal phase. It follows that, to disentangle the electrostatic contribution, a reliable model for the shortrange interactions is also needed. A well tested approach, which can be easily integrated with the dielectric continuum model, is the so-called ‘surface tensor’ method [25]. Here it is assumed that each element of the surface of a molecule has a preference to lie parallel to the director. In analogy with the Rapini–Papoular expression for the anchoring energy of nematics to the surface of macroscopic bodies [26], the contribution of each element of the molecular surface, dS, to the mean-field potential, is expressed in the form: dUsr = P2 cos n · sdS, where P2 is the second Legendre polynomial, and n, s are unit vectors parallel to the director and to the surface normal, respectively. The parameter defines the orienting strength of the medium; it can be related to the order parameters and the reduced temperature, and in calamitic nematics2 it takes positive values, increasing with the degree of order. It follows that, for a model particle with the shape of a prolate ellipsoid, the minimum of the mean-field potential Usr corresponds to the configuration with the long molecular axis parallel to the nematic director; in the case of a molecule, the alignment tendency can be evaluated on the basis of a realistic representation of its shape. The angular dependence of the nematic potential, Equation (2.273), can then be expressed as: U = L −T Ln∗ L2 + WRLn∗ DL0n (2.274) L=24 n=−L Here WRLn are the coefficients of the expansion of the electrostatic free energy, which can be obtained from the free energy WR , according to Equation (2.269). T 2n are the irreducible spherical components of the (second rank) surface tensor, which describe the anisometry of the molecular shape, and can be calculated in the form of integrals over the molecular surface [25]. Given the nematic potential U, the distribution function p can be calculated according to Equation (2.272), and then the Cartesian order parameters with Equation (2.271). Recently, the surface tensor model has been used together with the dielectric continuum model to calculate the orientational order parameters of solutes in nematic solvents [8, 9, 27]. Figure 2.32 shows the theoretical results for anthracene and anthraquinone in nematic solvents with different dielectric anisotropy. Considering only the surface tensor contribution, positive Szz and Sxx and negative Syy are obtained, with Szz > Sxx > Syy . This corresponds to what could be expected on the basis of the molecular shape: the long axis (z) is preferentially aligned with the director, and the normal to the 2 Calamitics are the most common nematics, wherein molecules can be approximated as rods, which preferentially align their long axis in the same direction. Properties and Spectroscopies 0.21 0.15 0.20 0.14 –0.12 z 0.6 –0.45 –0.24 0.4 Δε < 0 0.2 Δε > 0 –0.4 –0.3 SZZ –SXX SZZ –SXX –0.13 –0.13 0.6 0.64 –0.11 0.4 Δε < 0 0.2 –0.2 –0.4 SYY (a) 0.15 Y z –0.54 0.16 0.8 Y 0.8 275 (b) –0.3 Δε > 0 –0.2 SYY Figure 2.32 Order parameters calculated for (a) anthracene and (b) anthraquinone dissolved in nematics with different dielectric anisotropy [9]. For the case > 0, the results obtained in the absence of induction effects are also shown (dotted line). The temperature dependence of the dielectric anisotropy is taken into account, with the values NI = 102 NI = 8 and NI = 52 NI = −05 for the two cases at the nematic–isotropic transition. Atomic charges in absolute value greater than 0.1 are shown (in e units). The y axis is perpendicular to the molecular plane. aromatic plane (y axis) tends to lie perpendicular to the director. Although both solutes are apolar, the inclusion in the mean field of the electrostatic contribution produces nonnegligible effects. As we can see in Figure 2.32, the curve representing Szz − Sxx as a function of Syy is shifted either upwards or downwards, depending on the dielectric anisotropy of the nematic solvent. In the case of anthraquinone, an increase in the alignment of the z over the x axis is predicted for > 0; the opposite is predicted for anthracene. The values used in the calculations were chosen so as to make possible a comparison with the available experimental data [28–30]; this is the reason for the low value taken for negative , and then for the small effects found in this case. A clear agreement between experimental and theoretical results is obtained; the dependence of order parameters on the dielectric anisotropy of the solvent is sligthly underestimated by the polarizable continuum model, but the experimental trend is well reproduced. So, for example, for a given Syy value, a higher Szz − Sxx difference is predicted and measured for anthraquinone than for anthracene, when dissolved in nematics with high and positive dielectric anisotropy; the opposite would be obtained in the absence of electrostatic interactions. Dielectric Permittivity The principal components of the dielectric permittivity of nematics are related to the average dipole moment of the constituting molecules as: ⇑⊥ = 1 + N ⇑⊥ E V0 E (2.275) with ⇑⊥ E = 0⇑⊥ E + ind ⇑⊥ E (2.276) 276 Continuum Solvation Models in Chemical Physics where the labels 0 and ind are used for permanent and induced dipoles, the symbols ⊥ denote components in the laboratory frame, and the subscript E indicates that the average is over the orientational distribution in the nematic phase in the presence of the applied field, pE . This function is associated with the orientational potential: UE = U + WE (2.277) where U is the nematic mean field, Equation (2.273), and WE accounts for the interaction between the molecule and the applied field. For sufficiently weak applied fields the power expansion of the exponential, exp−WE /kB T, can be truncated at the linear term, and the orientational distribution function can be approximated as W pE ≈ p 1 − E (2.278) kB T where p is the nematic distribution function, corresponding to the nematic potential U. Thus, the averages in Equation (2.276) can be expressed as: ⇑⊥ E ≈ 0⇑⊥ + ind ⇑⊥ − 1 WE 0⇑⊥ + WE ind ⇑⊥ kB T (2.279) where the averages on the right-hand side are performed over the distribution function p. Thus the dielectric permittivity can be calculated as: N ⇑⊥ ≈ 1 + V0 7 ind ⇑⊥ E ! 1 WE 0 − + ind ⊥ kB T E ⊥ (2.280) The nematic mean-field U , the molecule–field interaction potential, WE , and the induced dipole moment, ind , are evaluated at different orientations using Equation (2.263), and then the coefficients of their expansion on a basis of Wigner rotation matrices can be calculated, according to Equation (2.268). The permittivity is obtained by a selfconsistency procedure, because the energy WE and the induced dipole moment ind , as well as the reaction field contribution to the nematic distribution function p, themselves depend on the dielectric permittivity. Figure 2.33 shows the dielectric permittivity of 5CB calculated in this way, using the surface tensor model for the nonelectrostatic contribution to the nematic meanfield [18, 31]. Only the all-trans conformer was taken, because only a small dependence on the conformation was found. The good agreement with the experimental data has a twofold origin: the modelling of electrostatic effects is responsible for the average permittivity, while the model for short-range nonelectrostatic interactions has the main responsibility for the behaviour of the dielectric anisotropy. For comparison, the results obtained using the Maier–Meier theory [4] are also shown; this is a generalization of the Onsager model [13] to uniaxial media. The same dipole moment used for the calculations with the molecular shaped cavity was assumed, and the radius a was taken to be 3.9 Å, a value derived from the density of the system. Improvement of the predictions, when the sphere is replaced by a molecular shaped Properties and Spectroscopies 277 Figure 2.33 Dielectric permittivity of 5CB. Experimental data [2] (solid line), and theoretical results obtained with the IEF method (filled diamonds) and with the Maier–Meier theory [4] (open diamonds). cavity, was found as a general result even for other polar isotropic solvents, e.g. a set of mono-substituted benzenes [18]. However, in none of the cases examined were the dramatic effects obtained for nematic solvents [31], and in particular for 5CB, observed; this result can be traced back to the strong shape anisometry of the mesogenic molecules. Calculation of Magnetic Tensors The magnetic tensors for a molecule embedded in an anisotropic medium can be calculated on the basis of an effective Hamiltonian which, in addition to the Hamiltonian of the isolated molecule, contains a contribution describing the response function of the reaction potential [32]. The general aspects are presented in the contribution by Sadlej and Pecul. With this methodology, the NMR shielding tensors of solutes [33, 34] and the A, g tensors of radical spin-probes [35] in nematic solvents were calculated. In all cases it was found that the presence of a moderately polar solvent (nematics with average permittivity ranging from about 5 to about 10 were considered) has nonnegligible effects on the magnetic tensors. In contrast, these tensors are scarcely affected by the dielectric anisotropy of the solvent, i.e. they are practically independent of the molecular orientation. The angular dependence was found to be lower than 1 ppm over about 30 ppm for the shielding tensor of 13 C nuclei; stronger effects were found only for more environmentally sensitive nuclei, such as 15 N and 17 O, especially if located in strongly polar molecules. A negligible dependence on the molecular orientation was also predicted in the case of the g and A tensors. 278 Continuum Solvation Models in Chemical Physics Order parameters are usually derived from the measured spectral splittings through relationships such as Equation (2.269–b); thus the availability of good estimates of the magnetic tensors is an essential requirement to obtain accurate order parameters. The theoretical results suggest that the magnetic tensors obtained from calculations in a solvent, introduced by the polarizable continuum model, should definitely be a better choice than the tensors derived, as customary, from solid-state data or from calculations for molecules in a vacuum. 2.9.4 Conclusions When considering the dielectric properties of anisotropic fluids, two main differences from the case of isotropic liquids arise. Firstly, the dielectric permittivity is anisotropic, and this can induce an orientation dependence in the properties involving electrostatic intermolecular interactions. Secondly, the orientational distribution of molecules is anisotropic, and this has to be taken into account when relating physical observables to molecular properties. Both issues introduce some more difficulty into the study of anisotropic systems; however, they can be very valuable, being behind the emergence of effects, undetectable in isotropic liquids, which can give new insights into the microscopic origin of the dielectric behaviour of fluids. Theoretical tools which make it possible to deal with both aspects have been worked out; the state of the art has been summarized in this contribution. Using the IEF methodology, it is possible to calculate the reaction and cavity fields within a molecular cavity of arbitrary shape, supporting an arbitrary distribution of charges, embedded in an anisotropic medium. This approach is suitable for quantum mechanical, as well as classical calculations. In principle, the dielectric anisotropy introduces an angular dependence of the reaction and cavity fields. However, this dependence can be more or less relevant, according to the property under investigation. For instance magnetic tensors, and in particular nuclear shielding tensors, although strongly dependent on the average dielectric permittivity of the environment, are only slightly affected by the the dielectric anisotropy, so their angular dependence is very weak. In contrast, the angular dependence of the electrostatic free energy is sufficient to affect the degree of orientational order of molecules in anisotropic media. The theoretical estimates of this dependence are strongly influenced by the choice of the cavity shape and the description of the charge distribution, which does not need to be polar. Recently, the limits of the model of the dipole in a sphere, deriving from the neglect of the dipolar correlations, were pointed out in relation to the interpretation of Stokes shifts in nematics [5]. The results summarized here clearly show that, when comparing dielectric continuum models with experimental data, the features of the molecular cavity and of the charge distribution cannot be ignored. This is particularly important for mesogenic molecules, which are characterized by a strongly anisometric shape, and for solutes used to probe the orientational order in liquid crystals, whose shape is always far from spherical. The IEF methodology, combined with an effective mean-field modelling of short-range orienting interactions, can also provide good predictions of the permittivity of nematics and its temperature dependence. Indeed, a realistic account of the cavity shape allows us to calculate the permittivity purely on the basis of the structure of the constituting Properties and Spectroscopies 279 molecules, without the need of introducing an adjustable parameter, i.e. the cavity radius, which changes in an unpredictable way from system to system. Appendix. Computational Methods In this appendix the numerical methodology adopted for the calculation of electrostatic free energy, order parameters and dielectric permittivity, reported in the text, are summarized. Lm in Equation (2.266) requires integration over Evaluation of the average values fLAB the Euler angles = . The Gauss quadrature algorithm has been used, with Legendre polynomyals for cos and and Chebyshev polynomyals for cos cos [36]. Ln The values of the tensorial components fMOL have to be calculated for a given set of orientations; for each value, Equation (2.263) is used, which is conveniently transformed into a set of algebraic equations, corresponding to the discretization over a set of tesserae Ln defined on the surface of the cavity. The component fMOL is then calculated with the biconjugated gradient algorithm, without the need for full matrix inversion [9]. For all the examples reported here a smoothed cavity surface was considered, defined by the rolling sphere algorithm [37], in the implementation by Sanner et al. [38]; the same van der Waals radii [39, 40] where used in all cases, along with a rolling sphere of radius 3 Å. The same surface was assumed for the electrostatic problem and for the surface tensor calculation. Ab initio geometry optimization was performed [41], and atomic charges were calculated using the Merz–Kollman–Singh procedure [42]. The Thole model [17] was used for the polarizability, with the parameterization proposed by van Dujnen et al. [43]. For the prediction of the dielectric permittivity, which is very sensitive to the value of the dipole moment, the calculated charges were scaled to reproduce the experimental dipole. References [1] G. Vertogen and W. H. de Jeu, The Physics of Liquid Crystals, Fundamentals, Springer, Berlin, 1988. [2] A. Bogi and S. 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Barone, Accurate prediction of electron-paramagnetic-resonance tensors for spin probes dissolved in liquid crystals, J. Chem. Phys. 123 (2005) 194909. [36] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Wetterling, Numerical Recipes, Cambridge University, Cambridge, 1988. [37] F. M. Richards, Areas, volumes, packing and protein structure, Annu. Rev. Biophys. Bioeng., 6 (1997) 151–176; M. L. Connolly, Analytical molecular surface calculation, J. Appl. Crystallogr., 16 (1983) 548–558. [38] M. F. Sanner, A. J. Olson and J. C. Spehner, Reduced surface: An efficient way to compute molecular surfaces, Biopolymers, 38 (1996) 305–320. [39] A. Bondi, Van der Waals volumes and radii, J. Phys. Chem., (1964) 441–451. [40] D. R. Lide (ed.), Handbook of Chemistry and Physics, CRC, Boca Raton, FL, 1996–1997. [41] M. J. Frisch et al., Gaussian 98, Gaussian Inc., Pittsburgh PA, 1998. [42] B. H. Besler, K. M. Merz and P. A. Kollman, Atomic charges derived from semiempirical methods, J. Comput. Chem., 11 (1990) 431–439; U. C. Singh, P. A. Kollman, An approach to computing electrostatic charges for molecules, J. Comput. Chem., 5 (1984) 129–145. [43] P.Th. van Duijnen and M. Swart, Molecular and atomic polarizabilities: Thole’s model revisited, J. Phys. Chem. A, 102 (1998) 2399–2407. 2.10 Homogeneous and Heterogeneous Solvent Models for Nonlinear Optical Properties Hans Ågren and Kurt V. Mikkelsen 2.10.1 Introduction This contribution provides a theoretical background of homogeneous and heterogeneous solvation models for calculations of nonlinear optical molecular properties at the level of correlated electronic structure methods. It is based on the work presented by SylvesterHvid and co-workers [1–6] and Jørgensen et al. [7–9]. Methods investigating the effects of the solvation on the electronic wave function of solvated molecules have been considered since the 1970s [1, 2, 10–42]. The primary focus of these methods has been on the utilization of uncorrelated electronic structure descriptions of the solvated molecule [10– 20, 23–27]. More recently, methods involving correlated electronic structure descriptions of the electronic wavefunction of the solvated molecule have been given, for instance at the second-order Møller–Plesset (MP2) level [34, 36], the multiconfigurational selfconsistent reaction field (MCSCRF) level [21, 28] and the coupled-cluster self-consistent reaction field (CCSCRF) level [42]. Calculations of time-dependent electromagnetic properties of molecules at the correlated electronic structure level are conveniently carried out by the utilization of modern response theory [43–51]. The transition of modern response theory for gas phase molecular systems to solvated molecules has been established [1–6] and these methods include the use of correlated electronic wavefunctions. These methods, reviewed here, have given rise a large number of computational approaches for calculating electric and magnetic molecular properties of solvated molecules. Heterogeneous dielectric media models have included the developments of Jørgensen et al. [7–9] (reviewed here) and Corni and Tomasi [52, 53]. Generally, the number of methods for determining frequency-dependent molecular electronic properties, such as the polarizability or first- and second hyperpolarizability tensors of heterogeneously solvated molecules, is very limited. The present contribution considers general electronic states of solvated molecules and is not limited to closed shell molecular compounds. For closed shell molecular systems, methods utilizing closed shell coupled-cluster electronic structure and closed shell density density functional theory for the electronic structure of the solvated system have appeared in the literature [54–67]. The electronic structure of molecular systems interacting with the outer solvent is given by a correlated electronic structure wavefunction, which, coupled to solvent response theory, enables the calculations of molecular properties of solvated molecules such as: • • • • • frequency-dependent second hyperpolarizabilities ; three-photon absorption; two-photon absorption between excited states; frequency-dependent polarizabilities of excited states; frequency-dependent first hyperpolarizability tensors; Properties and Spectroscopies 283 • two-photon matrix elements; • frequency-dependent polarizabilities; • excitation and de-excitation energies along with their corresponding transition moments. This contribution contains a section on the energy functional and the response theory of a solvated molecule interacting with a solvent given by either a homogeneous or a heterogeneous dielectric media model. The energy functionals for the homogeneous and heterogeneous dielectric media are covered in this section and generally the basic idea is to consider a large system divided into two subsystems where one of the subsystems is of principal interest and is described by quantum mechanics, whereas the other subsystem is treated by a much coarser method such as classical electrostatics [1, 2, 10–39]. For these methods, the interactions between the quantum subsystem and the outer solvent is given by the induced polarization in the outer solvent and the electric field due to the charge distribution of the solvated quantum mechanical subsystem [10–26]. The coupling between the quantum mechanical and the classical subsystems is accomplished by an effective interaction operator, which provides a modified quantum mechanical equation for finding the quantum mechanical electronic wavefunction of the solvated molecule[2, 23–25, 27, 32, 34–39]. 2.10.2 Response Theory The time evolution of the molecular state is derived by demanding that the Ehrenfest equation d d 0̃T̃† 0̃ = 0̃ T̃† 0̃ − i0̃T̃† H0̃ dt dt (2.281) is fulfilled for the following set of operators: T = q† R† q R (2.282) defined according to † R†n = n0qpq = Epq = a†p aq (2.283) The function of these operators is to determine effectively the time evolution of the multiconfigurational self-consistent field (MCSCF) state 0̃ = ei,t eiSt 0 (2.284) and for mathematical convenience appear as their time transformed counterparts in Equation (2.281): † † −i,t q̃pq = ei,t qpq e q̃pq = ei,t qpq e−i,t R̃†n = ei,t eiSt R†n e−i,t e−iSt R̃n = ei,t eiSt Rn e−i,t e−iSt (2.285) 284 Continuum Solvation Models in Chemical Physics The propagator ei,t is given by ,t = ,pq tEpq + ,∗qp tEqp (2.286) qp and the propagator eiSt is written as St = Sn tR†n + Sn∗ tRn (2.287) n We are able to write the amplitudes as a vector ⎛ ⎞ ⎜S⎟ ⎟ =⎜ ⎝ ∗ ⎠ S∗ which, following a transformation to the more general operator basis [69] {O}, provides a partition into an orbital and a configurational part ,t + St = T = O (2.288) Oj = Oo j + Oc j (2.289) Finally, we are able to rewrite the corresponding amplitude of ,t + St on this basis ,t = j Ooj St = j j Ocj (2.290) j and the Ehrenfest equation on this basis becomes d d (2.291) 0̃Õ† 0̃ = 0̃ Õ† 0̃ − i0̃Õ† H0̃ dt dt 0 1 with the time transformed operators, Õ , defined in analogy with Equation (2.285). Since the molecular system is interacting with a time-dependent perturbation due to a time-dependent electromagnetic field, the Hamiltonian of the total system is given as H = H0 + Wsol + Vt (2.292) with the condition that Vt → − = 0 and for this limit the generalized Brillouin condition is given by 0! H0 + Wsol 0 = 0 where ! designates variational parameters in orbital and configuration space. (2.293) Properties and Spectroscopies 285 2.10.3 Homogeneous Dielectric Medium Model We consider a solvated molecular system interacting with a time-dependent electromagnetic field, where the molecular system is in either an equilibrium or a nonequilibrium solvation state. The solute molecule is placed within a spherical cavity in a linear, homogeneous and isotropic dielectric medium with the molecular charge distribution obtained by using the multiconfigurational self-consistent field (MCSCF) procedure. The charge distribution induces a polarization field in the dielectric medium. The induced polarization field is partitioned into optical and inertial polarization vectors and is coupled self-consistently to the MCSCF procedure. In the sudden processes, only the optical polarization field is allowed to equilibrate with the solute state and the inertial polarization field remains unchanged reflecting the initial charge distribution. This gives a nonequilibrium solvent configuration. We model the nonequilibrium solvation for a molecular state using the following interaction operator between the outer dielectric medium and the molecular system[2] W̃sol = n 2 gl op Tlm + lm n 2 gl st op Tlm + T̃g st op (2.294) lm n is where the nuclear charge moment operators Tlm n Tlm = Zg tlm Rg (2.295) g and Zg is the nuclear charge on atom g Rg its position and tlm the real spherical harmonics. The reaction field factor is l + 1 − 1 1 gl = − R−2l+1 g = gl st − gl op cav 2 l + l + 1 l st op (2.296) where Rcav is the cavity radius and l the order of the multipole expansion. The third term is given by a sum a b c T̃g st op = gsol + g̃sol + gsol (2.297) where: a gsol = −2 n e gl op Tlm Tlm (2.298) lm b g̃sol = 2 e e gl op Tlm 0̃Tlm 0̃ (2.299) lm c gsol = −2 e gl st op Tlm i Tlm (2.300) lm All terms involving the electronic charge moment operators are defined as; e Tlm = pq lm tpq Epq = pq (p tlm r(q Epq (2.301) 286 Continuum Solvation Models in Chemical Physics and we denote the creation and annihilation operators for an electron in spin-orbital (p as a†p and ap . Having a summation over the spin quantum number , we have that the spin-free operator Epq is represented as Epq = a†p aq (2.302) c n e The expectation value appearing in gsol is Tlm = Tlm − Tlm . We observe from Equation (2.297) that: a • The term gsol is due to the optical polarization, induced by the nuclear charge distribution, interacting with the solute electronic charge distribution. • The corresponding polarization interaction due to the electronic charge distribution is given by b the term g̃sol . This operator describes the instantaneous coupling between the molecular state and the optical polarization state in the dielectric medium. c • The third term, gsol , describes the interaction of the inertial polarization vector with the electronic charge distribution and the inertial polarization vector is due to the nuclear charge distribution e and the initial solute electronic charge distribution Tlm i . The evolution of the electronic wavefunction of the molecular system is determined by the appropriate Ehrenfest equation d d 0̃Õ† 0̃ = 0̃ Õ† 0̃ − i0̃Õ† H0 + Vt + W̃sol 0̃ dt dt (2.303) We observe that the first two terms of W̃sol do not contribute to Equation (2.303) and therefore we only consider the electronic part of W̃sol 1 0 a c b a+c W̃ = gsol + gsol + g̃sol = gsol + Alm 0̃Blm 0̃ (2.304) lm e e with Alm = 2gl op Tlm and Blm = Tlm and we note that for the limit t → − W̃ reduces to: a+c W = gsol + Alm 0Blm 0 (2.305) lm At this point we have that Equation (2.303) is given by d d a+c 0̃Õ† 0̃ =0̃ Õ† 0̃ − i0̃Õ† H0 + Vt0̃ − i0̃Õ† gsol 0̃ dt dt − i 0̃Õ† Alm 0̃0̃Blm 0̃ (2.306) lm The vacuum part of Equation (2.306) has been derived for all but the solvent contribution by Olsen et al. [69,72,73,76]. The matrix representation of the solvent modifications to the response equations is found by expanding the last two terms. Generally, the solvent contributions have the following structure 0̃Õ† A0̃B0̃0̃ = 0̃Õ† A0̃0̃B0̃ (2.307) Properties and Spectroscopies 287 where A and B are time-independent operators. Additionally, we expand 0̃B0̃ a normal expectation value and the matrix representation of Equation (2.306) becomes: n+1 in Sjl1 l2 ln ˙ l1 t n=1 n 8 l t = − n 0 18 n+1 tn+1 in+1 Ejl1 l2 ln + Vjl1 l2 ln l t n=0 =2 − =1 7 i n+1 n+1 Cjl1 l2 ln n=0 n 8 + n - lmn−k+1 lmk Ajlk+1 ln Bl1 l2 lk k=0 lm l t (2.308) =1 n+1 tn+1 where we utilize the definitions of Sjl1 l2 ln and Vjl1 l2 ln from ref. [68]. Furthermore, we have lmn Bl1 l2 ln = n k n 8 8 −1n 0 Ôcl Ôol Blm 0 =1 k=0 k!n − k! =k+1 (2.309) and n+1 Xjl1 l2 ln = n k n 8 8 −1n † 0Ocj Ôcl Ôol X0 =1 k=0 k!n − k! =k+1 + 0 k 8 † Ôcl Ooj =1 n 8 (2.310) Ôol X0 =k+1 where X is to be replaced by either a+c n+1 H0 → Ejl1 l2 ln gsol n+1 lmn+1 → Cjl1 l2 ln Alm → Ajl1 l2 ln (2.311) We obtain the terms for the solvent modifications of the quadratic response functions, 3 denoted Wjkl , by collecting all terms for n = 2 in Equation (2.308) 1 0 lm2 lm1 3 3 lm1 lm2 lm3 Wjl1 l2 l1 l2 = iCjl1 l2 l1 l2 + i Bl1 l2 Aj + Bl1 Ajl1 + Blm0 Ajl1 l2 l1 l2 lm (2.312) The contributions, due to the interactions with the solvent, to the cubic response theory are obtained by adding the third-order solvent contribution to the corresponding vacuum equations and we have from Equation (2.308) 4 O3 −i0̃Õ† W̃ 0̃ = iGjh1 h2 h3 h1 h2 h3 0 lm3 lm1 lm2 lm2 lm1 lm3 +i Dh1 h2 h3 Cj + Dh1 h2 Cjh2 + Dh1 Cjh1 h2 lm 1 lm4 +Dlm0 Cjh1 h2 h3 h1 h2 h3 (2.313) 288 Continuum Solvation Models in Chemical Physics 2.10.4 Heterogeneous Dielectric Medium Model The basic outline of the heterogeneous dielectric media method is to divide the total system into two subsystems. The solvated molecule is encapsulated in a cavity C which is given by the surfaces )m and )l . The cavity is surrounded by a heterogeneous environment given by two part Sm and Sl . The two dielectric media are in contact with the cavity through the surfaces )m and )l . Each of the two dielectric media is taken to be a linear, homogeneous and isotropic dielectric medium and is characterized by a scalar, optical, inertial or static dielectric constant. As an illustration, we consider two dielectric media, Sm and Sl , characterized by the dielectric constants m and l , respectively, with the following spatial positions: 0 1 VSm = r = r % (% − $ 2 2 0 1 VSl = r = r % (r ≥ R % ∈ − $ 2 2 (2.314) The heterogeneous dielectric model presented is represented by a hemispherical cavity, C, having the radius R. The cavity is placed such that it is on the surface of Sm and embedded in Sl and the volume is given by 0 1 VC = r = r % (r ≤ R % ∈ − $ 2 2 (2.315) We describe the molecular system of interest, M, by a correlated electronic structure wavefunction and the distribution of the molecular system is given by M . The charge distribution of the molecular system within the cavity induces polarization charges in the surrounding environment. The interactions between the molecular system and the dielectric media are represented by a reaction field and a polarization potential, Upol . The polarization energy is given by an integral over the interactions between the induced reaction field and the molecular charge distribution at a point r inside the cavity Epol = 12 dr M rUpol r 2 (2.316) Therefore it is necessary to solve simultaneously the quantum mechanical problem (the Schrödinger equation) and the classical electrostatic problem (the Poisson and Laplace equations) using the boundary conditions as defined by the physical problem. For the physical problem presented, where the coordinates of the electons and nuclei are given as • Nel electrons q ≡ q1 qNel and • M nuclei Q ≡ Q1 QM we have that the Hamiltonian for the molecular system coupled to the two dielectric media is given by HM q$ Qq$ Q = EQq$ Q (2.317) Properties and Spectroscopies 289 where HM q$ Q = HM0 q$ Q + Wpol (2.318) It is clearly seen that the Hamiltonian consists of two terms: • the Hamiltonian, HM0 , for the isolated molecular system in vacuum, and • the interaction operator, Wpol , that includes the interactions between the molecular system and the two dielectric media. Furthermore, the interaction operator, Wpol , depends on two terms: • the induced potential, Upol , in the two dielectric media and • the molecular charge distribution m r where Qi denotes the partial charge on the ith nucleus i. at position R m r = M i Qi r − R (2.319) i=1 The molecular system interacts with the outer environment and the total electrostatic potential, -, is the sum of • a potential arising from the molecular charge distribution, UM , • and a potential arising from the induced charges in the two dielectric media, Upol , that is -r = UM r + Upol r r ∈ VC (2.320) The induced potential, Upol and the total electrostatic potential - within the cavity are determined by solving the Poisson and Laplace equations with the appropriate boundary conditions, i.e., . 2 -r = −4M r r ∈ VC =0 r ∈ VS1 =0 r ∈ VS2 (2.321) and the condition that the molecular charge distribution is nonzero within the cavity, C, i.e. r = M r for r ∈ VC and zero outside the cavity. The method of image charges determines how a charge qi located at ri = ri %i (i in C leads to the following contributions to the induced potential: • One charge qi1 = −qi at ri1 = ri − %i (i in the metal surface, S2 , due to the perfect conductor–vacuum interface. • A second charge qi2 = −qi a − 1/ + 1ri at ri2 = a2 /ri %i (i due to the hemispherical environment between the vacuum and the dielectric medium. • A third qi3 = −qi2 at ri3 = a2 /ri − %i (i due to the perfect conductor–dielectric medium. 290 Continuum Solvation Models in Chemical Physics The induced potential is determined as i Upol r = 3 qij j r − rij (2.322) and in the case of a molecular system represented by N partial charges we find the total induced potential to be Upol r = N i Upol r (2.323) i=1 Therefore, introducing the electronic wavefunction for the molecular system, the interaction operator is given by the following expression Wpol = 1 k Upol rj (p qj (q Epq 2 j k pq (2.324) where we have used the following terms: • (p and (q represent the molecular orbitals p and q. • The excitation operator, Epq , is defined in Equation (2.302). • The operators a†p and ap are the creation and annihilation operators for an electron in the spin orbital (p . It is observed that the interaction operator is given by the product of two-electron operators. Next, we present the fundamental equations for determining the time-dependent electromagnetic properties of a molecular system interacting with a heterogeneous dielectric medium. For the heterogeneous dielectric media model we utilize the representation that is given in Equation (2.234), which makes it possible to rewrite the contributions due to the two dielectric media as k −i T†t Wpol = −i (p qj (q 0t T†t Epq 0t Upol rj j k (2.325) pq We obtain the contributions to the linear, quadratic and cubic response equations by expanding Ot and T†t followed by collecting terms for a given order of the expansion: −i T†t Wpol = G0 T†t + G1 T†t + G2 T†t + G3 T†t + (2.326) and we identify the terms containing ,t and St that are related to an orbital and a configurational parts following the procedure from the previous section. 2.10.5 Sample Application: Solvent Effects on Two-photon Absorption The homogeneous and hetergeneous models for solvent effects are generic for nonlinear optical properties and cover a great number of processes emerging from strong matter– light interaction. These are of considerable fundamental interest, and also possess Properties and Spectroscopies 291 in several cases an inherent practical applicability as diagnostic tools. One such nonlinear process is two-photon absorption, which we here use for sample applications illustrating the solvent theory for nonlinear optical processess outlined in the foregoing. Although the formal theoretical description of two-photon absorption (TPA) has been well established for a long time (see excellent summary of Mahr [70, 71]), the application of the theory was traditionally mostly addressed in the context of the physics of atoms and small molecules. The applicability of TPA has greatly developed owing to improvements in experimental conditions, and multiphoton – and in particular twophoton – absorption spectra can today be found for a wide selection of systems. This interest in TPA stems to a large extent from the various, potential, technical applications that can be realized in the fields of medical therapy and photonics [74, 75, 77]. For most applications of TPA it is highly desirable to employ molecules with a very large TPA cross-section, possibly coupled with a certain transparency to the traditional one-photon absorption (OPA) process. This demand has pushed forward the research on the design of such molecules: synthesis [79–83], spectroscopy [84–89] and theoretical works [90–97] have been published. Building on the knowledge originally developed for hyperpolarizabilities [98] the research on this topic has shifted rapidly from dipolar structures [99,100] towards quadrupolar [89,95,101,102] and octupolar [103, 104] structures. Organometallic systems such as porphyrines have been investigated because of the possibility to fine tune their response by functionalization[105–107]. Systems of increased the dimensionality have been of particular interest [108–111]. Concomitant to the large effort to establish useful structure-to-properties relationships, considerable effort has now been put to investigate the environmental effects on TPA[112–114]. For example, the solvent effect has been studied for a small linear push–pull chromophore using a self-consistent reaction field (homogeneous solvation) method employing a spherical cavity and an internal force field (IFF) method[112]; in another study the polarizable continuum model has been employed to calculate the relevant quantities to obtain the TPA cross-section in the limit of a two-state model[113]; Woo et al. made a critical study of experimental comparison of TPA cross-sections in different solvents[114]. The TPA cross-section for an excitation from the ground state 00 to a final state 0f encountered in experimental measurements is defined in terms of the normalized line shape function, g + , and the TPA transition amplitude tensor, T f [115, 116]: TPA = 2e4 g + T f 2 c2 (2.327) 2f , between the The cartesian components of the TPA transition amplitude tensor, Tab ground state 00 and the excited state 0f of an isolated molecular system are defined as 0 ˆ a kk ˆ b kk ˆ b f 0 ˆ a f 2f + a b = x y z (2.328) Tab = k − f /2 k − f /2 k>0 where we assume that the frequency of the incoming radiation is equal to half of the excitation energy from the ground state 00 to the excited state 0f , i.e. = f /2. In the 292 Continuum Solvation Models in Chemical Physics above equation, ˆ a and ˆ b are the cartesian components of the dipole moment operator ˆ k and f are the frequencies of excitation from 00 to 0k and 0f , respectively. The straightforward application of this formula is limited, both since it involves explicit summation over excited states of a molecular system and since it requires computations of matrix elements of the dipole moment operator ˆ between excited states. In response theory, this problem is overcome in that the TPA transition amplitude is extracted from a single residue of the quadratic response function 2f lim C − f ˆ a$ ˆ b ˆ c f /2C = −Tab f ˆ c 0 C →f (2.329) Heterogeneous Dielectric Medium Response for Two-photon Absorption In this example the two-photon processes for molecules interacting with dielectric media are briefly considered using the carbon monoxide molecule; for original investigations see refs. [7, 8]. The CO molecule is typically placed in the half spherical cavity having the surfaces in contact with a semi-infinite dielectric medium (representing a perfect conductor) and embedded in a solvent given by a dielectric medium. The perfect conductor is confined to the half space z < 0 and the molecular axis of CO is perpendicular to the perfect conductor. All the calculations in this example were been performed in the Cs symmetry group with an aug-cc-pVDZ basis set and a complete active space consisting of ten electrons. Generally, the transition energies relative to the transition energy for the molecule in vacuum are shifted: • The third transition energies were red shifted for both symmetries. • For both symmetries the second and fourth transition energies were blue shifted. • The energy of the first state was reduced for the 1 A0 symmetry but increased for the 1 A00 symmetry. • Of the four transitions, it is clear that the transition energy of the fourth state was more sensitive to the distance and dielectric constants than the others. We consider the effects of the heterogeneous dielectric media on the two-photon transition matrix elements by defining pol vac Tab = Tab − Tab (2.330) where we have used the following terms: • The two-photon transition matrix element of the molecule in the heterogeneous dielectric media pol is Tab . vac • Tab is the two-photon transition matrix element of the molecule in vacuum. The calculational results indicate that: • The transition moments to the first two states were zero. • The transition moments to the third state were less than those for vacuum. • The transition moments to the fourth state were greater than those for vacuum. Properties and Spectroscopies 293 Generally, the decrease of the transition moments to the third state was strongly related to the general blue shifting of the intermediate states. The increase of the transition moments to the fourth state was dominated by how the intermediate states were red shifted. Additionally, we have observed that the relative magnitude of the shift of the transition moments increased as a function of the dielectric constant of the heterogeneous dielectric media. Homogeneous Dielectric Medium Response for Two-photon Absorption A recent implementation by Frediani et al. [117] introduced the possibility to study solvent effects on specific classes of TPA dyes (Figure 2.34). This derivation is based on the polarizable continuum model (PCM) for homogeneous solvation coupled to quadratic response theory and it generalizes the earlier one for linear response given by Cammi et al. [118]. As the PCM model has been thoroughly reviewed in other contributions to this book we rest here by stating that the key expressions contain solute–solvent interaction terms depending on the solute potential and the nuclear inertial and dynamic apparent surface charges. This generalization made it possible for Frediani et al. [117] to perform the first TPA cross-section calculations of solvated molecules. Their attention was focussed on trans-stilbene as the -backbone as it is one of the most common building blocks in the design of TPA chromophores. The bare backbone was employed together with its di-functionalized derivatives with the electron donating (D) amino-group and the electron accepting (A) nitro-group (see Figure 2.34). TSB (a) NH2 H2N DATSB (b) NO2 H2N NATSB (c) NO2 O2N DNTSB (d) Figure 2.34 Structures and corresponding labels of the molecules: (a) trans-stilbene (TSB), (b) 4-amino-4 -amino-trans-stilbene (DATSB), (c) 4-nitro-4 amino-trans-stilbene (NATSB), (d) 4-nitro-4 -nitro-trans-stilbene (DNTSB). From ref. [117]. Reprinted from L. Frediani et al., J. Chem Phys., 123, 144 117. Copyright (2005), with permission from American Institute of Physics. 294 Continuum Solvation Models in Chemical Physics TSB 1100 Cross section (GM) 1000 900 B3LYP 800 BLYP 700 SVWN 600 500 400 300 200 100 0 H2 O O 2 H1 h. 2 H1 sp H2 C6 C6 O Ga H2 O 2 H1 h. 2 H1 sp 31Ag H2 C6 C6 O Ga H2 O 2 H1 h. 2 H1 sp H2 C6 C6 Ga 21Ag 41Ag Figure 2.35 PCM calculations of TPA cross-sections for the first three 1 Ag excited states of TSB in the different environments with three functionals employed. From Ref. [117]. Reprinted from L. Frediani et al., J. Chem Phys., 123, 144 117. Copyright (2005), with permission from American Institute of Physics. For these four molecules (, D––D, D––A and A––A), they obtained the TPA cross-sections in vacuum, in a nonpolar solvent, cyclohexane, and in a polar solvent, water. The results are reported in the Figures 2.35–2.37. In all figures the data are ordered as follows: each set of excited states is grouped together, within each set of excited states the environment goes from gas phase, to cyclohexane and to water, for each root and solvent the first three columns are the results obtained with gas phase geometry, whereas the second three are obtained with the appropriate solvent-optimized geometry. Each set of three columns with different colours corresponds to the three functionals employed (B3LYP, BLYP and SVWN). It was shown that the solvent effect is generally significant and that it therefore needs to be taken into account properly. For nonpolar structures such as the bare backbone of TSB such an effect has been found to follow closely the refraction index of the medium though deviations may occur as a result of the nature of the excited states involved. Such deviations are more prominent when polar groups are attached to the backbone and become quite large for the dipolar structure of NATSB. It was shown by Frediani et al. [117] that to enhance the solvent effect it is more important to have a solvent with a high refractive index and that the static polarity of the solvent plays a minor role for the nondipolar structures which are known as the most promising ones. Another source of solvent dependence can also be found in how the electronic structure of the Properties and Spectroscopies 295 DATSB 1600 Cross section (GM) 1400 1200 B3LYP BLYP SVWN 1000 800 600 400 200 0 H2 O O H2 2 H1 C6 2 H1 C6 h. sp Ga O H2 sp O H2 2 H1 C6 2 H1 h. C6 Ga 21Ag 31Ag Figure 2.36 PCM calculations of TPA cross-sections for the first two 1 Ag excited states of DATSB in the different environments with the three functionals employed. From Ref. [117]. Reprinted from L. Frediani et al., J. Chem Phys., 123, 144 117. Copyright (2005), with permission from American Institute of Physics. molecule changes upon excitation. Such variations are found in charge transfer systems such as NATSB. Any comparison with experiment should here also acknowledge the fact that crosssection results obtained are very dependent on pulse lengths [119]. The usefulness of the results rests on the understanding they can give on how the molecular structure and the environment affect such a property, something that can be used to devise interesting structure–environment combinations which can be exploited for applications. Acknowledgments We thank Dr. Luca Frediani for providing data and figures [117] to the last section of this review. H. Å. acknowledges the Swedish Science Research Council for support. K.V.M. thanks Statens Naturvidenskabelige Forskningsråd, Statens Tekniske Videnskabelige Forskningsråd, the Danish Center for Scientific Computing. 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The original, macroscopic notion of a surface is simply the boundary between two different regions of space displaying different properties (e.g., density, dielectric constant). However, microscopic investigations suggest that the interface should be considered as a peculiar environment itself instead of a mere boundary dividing two regions. Surfaces and interfaces play a key role in numerous processes and applications covering the entire spectrum of pure and applied sciences. It is therefore important to investigate their nature from a microscopic point of view. A common method to investigate microscopically a specific environment is to study the properties of a probe located in that environment. Surfaces and interfaces in this respect present the additional complication that the environment is nearly two-dimensional; thus technical issues such as the weakness of a response signal with respect to the bulk media defining the interface must be addressed. For instance, in order to determine the electronic absorption spectrum of a surfactant it is not possible to acquire it directly, because the interface response is ‘masked’ by the absorption in bulk solution. One could instead exploit Surface Second Harmonic Generation (SSHG) since Second Harmonic Generation (SHG) is interface specific due to the absence of inversion symmetry at the interface. From the intensity of the SHG signal the absorption spectrum of the probe at the interface is then indirectly recovered. Quantum Mechanical (QM) calculations, however, do not suffer from this limitation since the properties of a single molecule are directly obtained. It is thus very appealing to perform QM calculations on molecules located at the interface, since the interesting properties could then be obtained directly and eventually compared with the experimental results acquired indirectly. In order to reproduce the environmental effect, Continuum Models (CMs) of solvation have been applied, in particular for the case of gas–liquid, liquid–liquid, gas–solid and liquid–solid interfaces. The first two environments and the last two require different treatments since, whereas a gas and a liquid could be handled within the same continuum description, the modelling of solid surfaces requires a different strategy. In addition, when the solid participating in the interface is a metal, a new class of phenomena (known as Surface Enhanced, SE) can take place. Such phenomena, consisting in the amplification of molecular electromagnetic properties at the interface, depend on the dielectric response of the metal, and are thus naturally treated within CMs. 2.11.2 Gas–Liquid and Liquid–Liquid Interfaces The simplest model to deal with a surface (liquid–liquid interface or liquid–gas surface) is the classical view of two media sharing a boundary. In the spirit of CMs, the problem consists in placing a solute in between the two media or in proximity of the interface. The first attempt to deal with such an environment for an ion species is due to Onsager Properties and Spectroscopies 301 and Samras [1]. Since then, the theoretical modelling of interfaces has attracted much attention. In recent years the majority of such studies have made use of simulation techniques as demonstrated by recent reviews on the subject [2–4] which the reader is referred to for further details. Continuum models have been used less frequently and in most cases both the environment and the solute are modelled with molecular details discarded [5–7]. Here, instead, we will focus on such models where the solute is treated ab initio and the solvent is described through one or more ‘perturbation’ terms in the Hamiltonian of the solute. In the following, the modelling of solute–solvent interactions at the interface will be discussed: in Section Electrostatic and in Section Nonelectrostatic Interactions (cavitation, dispersion, and repulsion). Applications of the methods developed to the modelling of molecular properties at liquid surfaces will be described in the Section Energetics and Properties at Liquid–Gas and Liquid–Liquid Interface. Electrostatics For an infinite planar surface, the electrostatic energy of a point charge at a distance r from a planar surface can be obtained analytically [8]. It can be shown that, for two media with permittivities 1 > 2 , a point charge located in medium 1 is repelled from the interface, whereas in medium 2 it is attracted to the interface, thus providing the driving force for the phase transfer of a charged species from the low permittivity medium to the high permittivity one. However, it must be kept in mind that, because of the image-charge potential [8], the model yields a divergent energy for a charge approaching the interface. Although this behaviour is acceptable at the macroscopic level, it would certainly not be valid microscopically: modifications both in the structure of the interface and in the molecular solute would eventually occur in response. The divergence is less severe either when the charge is described by a volumetric density (as the electronic density) or when point charges (pointwise nuclei) are embedded in a cavity and thus are prevented from approaching the surface too closely. The sharp dielectric surface was implemented for the Polarizable Continuum Model (PCM) for the first time by Bonaccorsi et al. [9] and further developed by Hoshi et al. [10] The only requirements for the employment of the PCM is a knowledge of the constitutive parameters of the system: geometry of the dielectrics and corresponding dielectric constants. The same model has been subsequently revisited in 2000 [11] supplemented with the modelling of nonelectrostatic interactions (see later). The sharp planar interface between two infinite dielectrics M1 and M2 with dielectric constants 1 and 2 could also be modelled through the Integral Equation Formalism (IEF) (see also the contribution by Cancès) since the expression for the Green’s function Gr r is a consequence of the availability of the analytical solution for the point charge problem [8]. In particular, if r r and r are respectively a generic point, the unit charge position and the image charge position: Gr r = ⎧ 1 1 −2 1 ⎪ ⎨ 1 r−r + 1 +2 1 r−r 2 1 ⎪ 1 +2 r−r ⎩ if r r ∈ M1 if r ∈ M1 r ∈ M2 (2.331) 302 Continuum Solvation Models in Chemical Physics The advantage of the IEF–PCM formulation in this respect is that the surface between the two dielectrics is ‘hidden’ within the Green’s function and only the explicit description of the molecular cavity is needed as for bulk IEF–PCM. Although the IEF–PCM implementation was able to overcome several numerical problems as a result of the explicit description of the dielectric interface, the BEM discretization technique for those tesserae in close proximity of the sharp interface could still potentially lead to unphysical divergences. Such problems are overcome through a more physically meaningful modelling of the interface with a smoothly varying dielectric permittivity z along the direction perpendicular to the interface, instead of a discontinuity from one bulk value to the other. This problem was originally analysed in the 1970s [12–14]: a numerical strategy to obtain the potential of a point charge located at a diffuse interface was given [14] and an analytic solution was also obtained for particular choices of the profile [12, 13]. As for a sharp interface, the knowledge of the potential of a point charge is equivalent to obtaining the Green’s function of the problem; therefore it can be included in the IEF–PCM formalism simply by providing the new Green’s function. This approach has recently been developed [15] for a permittivity profile z of arbitrary shape. Although a numerical integration is required, the Green’s function expression can be formally written as [15]: Gr r = 1 + Gim r r Dr r r − r (2.332) in order to highlight the subdivision into a direct interaction term mediated by an ‘effective’ dielectric constant Dr r and an image-like term Gim corresponding to the second term in the first expression of Equations (2.331). Nonelectrostatic Interactions Electrostatics is certainly the most important interaction between a dielectric medium and a molecular species. Therefore, it has also been investigated extensively for interfaces as shown in the previous section. Nonelectrostatic forces are often neglected in the bulk solution since their contribution to the solvation energy is often limited because of reciprocal cancellation and their effect on molecular properties is small [16] (repulsion and particularly dispersion) or zero (the present understanding of cavitation is strictly empirical). As a consequence, much less effort has been devoted to model these interactions, compared with electrostatics. This picture changes when interfaces are considered. Although the magnitude of those contributions will not be significantly affected, their variations through the interface span different length scales for the different interactions. Therefore the picture of a predominant electrostatic contribution could change at the interface. This has in fact been observed [11, 17, 18]. The first attempt to describe nonelectrostatic interactions at the interface has been done by extending the semiempirical models [19, 20] of such interactions to a sharp interface. In order to illustrate how semiempirical models are extended to the interface let us consider the dispersion interaction. In bulk solution dispersion is modelled through a sum of interatomic interactions between the solute atoms and the solvent atoms. The assumption of a uniform distribution of the solvent atoms allows us to obtain a closed Properties and Spectroscopies 303 form that can be integrated over the solvent volume [19]. The extension to a sharp interface has been done by splitting the integral over the volumes occupied by the two solvents and using the appropriate parameters for each solvent. The same approach has been followed for repulsion and cavitation.1 The semiempirical model cannot be employed if one is interested in the modelling of molecular properties since the direct effect of the interaction on the wavefunction or electronic density is then required. A quantitative model for repulsion and dispersion interactions has been derived by Amovilli and Mennucci [21] based on the theory of weak interactions [22]. Cavitation is strictly empirical in this context since it does not depend on the molecule but only on the cavity shape and on the environment: it will have an indirect effect on properties only by contributing to the determination of the preferred molecule–interface orientation. Following the work of Amovilli and Mennucci [21] a model for repulsion interactions at diffuse interfaces has been developed. Since the repulsion energy depends on the solvent density it is then natural to replace the constant density with a positiondependent density z. The first attempt made use of z in the final expression for the repulsion energy [17]. Such a model has subsequently been improved by a derivation of a new repulsion expression [18]. The extension of the quantitative model for dispersive forces is currently under development [23] following the same idea as for repulsive forces [18], namely by deriving an expression for a solvent with position-dependent macroscopic properties. Additional complications arise here owing to the dependence of the PCM equations on the dispersion frequency . For homogeneous dielectrics the problem has been overcome by neglecting autopolarization effects and thus obtaining a closed form for the dispersion energy [21]. In case of a position-dependent dielectric constant, further simplifications will have to be considered in order to obtain a manageable expression. The cavitation expression may also be extended to diffuse interfaces, by weighting each contribution coming from the tesserae with an appropriate position-dependent function. Energetics and Properties at Liquid–Gas and Liquid–Liquid Interfaces The main objection to the use of CMs to describe the solvent effect of an interfacial environment is that such a model neglects the specific effects arising from the interface, thus preventing a faithful description. It is therefore important to test the model and to compare the results obtained with those from other theoretical methods (e.g. simulations) and experiments. The early studies of interfaces with PCM [24, 25] were promising but could not be compared with other results since direct experimental investigations [26] and simulations [27–29] were performed a few years later. In 2000, the original model was revisited and extended to compute all four fundamental interactions [11] (electrostatics, dispersion, repulsion and cavitation) and a satisfactory agreement with experiments [26] and simulations [27–29] was obtained. The need for a more flexible model was met by developing the formalism for diffuse interfaces [15, 17, 18]. First, the electrostatic solvation for a diffuse interface has been 1 Cavitation is given as an integration over the cavity surface instead of a volume integration. 304 Continuum Solvation Models in Chemical Physics investigated [15] by showing the energetics of phase transfer both for a simple interface and for a model membrane. The model has also been applied to the investigation of the excitation energy of N N -diethyl-p-nitroaniline (DEPNA) [15] at the water surface. The lack of any other interaction prevented in that case a comparison with experimental results beyond an approximate estimate of the interface polarity [30]. More recently, the electrostatic model for diffuse interfaces has also been employed to model the Excitation Energy Transfer process at the water surface [31]. Because of their crucial role in the comparison between theoretical modelling and experiments, nonelectrostatic interactions have recently been reconsidered for diffuse interfaces. In particular, the introduction of repulsion proved to be the key to obtaining agreement between the CM and experimental findings. In particular, it allowed the surfactant behaviour of halides [17] and N3− [18] to be modelled. The fully quantum mechanical model is not yet available because of the lack of dispersion, which is currently under development [23]. Nevertheless, the results obtained so far are encouraging. In particular CMs offer from this point of view a significant advantage: the same molecular system is investigated in three very different environments, namely gas phase, interface and bulk solution, with the very same method. We remark that no other method has at present the same capability: when comparisons among the three environments are drawn, the use of different models thus introduces a dishomogeneity in the results obtained whose impact is not easy to evaluate [32]. (a) (b) Figure 2.38 Interfacial solvation: (a) a solvated molecule embedded in a cavity lying on top of a metal surface; (b) a solvated molecule at the diffuse interface characterized by position-dependent properties, e.g. permittivity z. 2.11.3 Gas–Solid and Liquid–Solid Interfaces In this section, we shall focus on the use of CMs to study molecules at the interface between a solid and a fluid (gas or liquid). In particular, we reserve the term ‘continuum models’ to approaches that consider both the solid and the fluid as structureless continuum bodies characterized by their dielectric response, and treat the molecule at some microscopic level. Properties and Spectroscopies 305 The use of a continuous description for a solid surface may appear less justified than for a liquid. In fact, ordinary liquids are homogeneous and uniform media on the average, as a result of thermal motions. In contrast, the spatial arrangement of the atoms in a solid does not average to a continuum distribution in time.2 Thus, considering molecule–solid systems in the light of the general theory for weakly interacting systems [22], the rigid atomic structure of solids implies that the coulombic charge–charge interaction term does not average to zero as it does for liquids, and the dielectric constant should not be uniform and isotropic. However, there is at least a class of solids for which, at the first order, the equilibrium charge distribution is relatively simple and destructured, and the dielectric response is close to be homogeneous: the inorganic metals. In fact, the charge density and the response of metals mainly depend on the weakly bound conduction electrons, which behave, at the zeroth order, as a confined homogeneous system. Looking back in the literature at the applications of CMs to systems involving solid interfaces, metals are by far the most common systems. However, CMs have also been applied to other solids, e.g. ref. [33]. Early Continuum Models for Metals: Image and van der Waals Interactions CMs for metals have an old history, related in the beginning to the calculation of the interaction energy between an atom or ion and a metal surface. When a real chemical bond is established between the molecule and the metal (chemisorption), interaction energies are mainly determined by the bond strength, and a description based only on CMs is precluded. Heterogeneous catalysis is thus beyond the scope of a simple CM. However, when the molecule is only physisorbed, the interaction is dominated by the polarization induced by the molecule in the metal and the dispersion interaction between them.3 These two components of the interaction energy were, in particular, studied via CMs. The simplest model that can be adopted is to consider the metal as a perfect conductor with a planar, infinite surface. As for polarization, the interaction of an ion (or of an electron) with metal described in such a way, gives rise to the image potential [8, 34, 35] introduced in Section 2.11.2. Green’s function for the vacuum space outside the metal is obtained from Equation (2.331), taking 1 = 1 and 2 = met → . Obvious questions regarding the modelling of a metallic surface by a conductor plate are the position of the fictious conductor surface (z0 ) with respect to the real metal atomic lattice and the minimum distance between the ion and the surface for which the image charge approach works. Starting from the 1970s, several studies addressed these issues [36–42] by comparing the simple conductor-based model with quantum mechanical (QM) approaches to the metal response. One of the most frequently used QM approaches was the jellium model [43], in which conduction electrons are considered quantum mechanically while the cores of the metal atoms are replaced by a homogeneous positive density of charge (called jellium). Summarizing the results of such efforts, the image approach seems to be a good approximation at a distance of around 2–2.5 Å from z0 , and z0 is around 0.2–1 Å outside the edge of the jellium background. 2 However, even simple liquids have short-range order around the solute, i.e., in the region that is most important in determining solute–solvent interactions, and this order does not prevent solvation continuum models from being quite successful. Obviously, we are not discussing here the interaction between a polar molecule and a charged metal surface, which is clearly dominated by the Coulombic interaction between them. 3 306 Continuum Solvation Models in Chemical Physics Although quite obvious, it is important to remark that a perfect conductor-like model completely neglects the chemical nature of the metal (i.e., all the metals behave in the same way). This is different to what happens for solvents, where even the simplest models include at least one solvent-specific parameter, the static dielectric constant. However, the chemical nature of the metal is relevant for another aspect of the static dielectric response that is neglected by the conductor model: the nonlocal effects. They will be discussed in the following Section, The Response Properties of a Molecule Close to a Metal Specimen: Surface Enhanced Phenomena and Related Continuum Models. The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44]. Although these results overestimate the dispersion energy, the correct 1/d3 dependence was recovered (d is the metal–molecule distance). Later studies [45–47] extended the work of LennardJones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule–metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability embedded in a dielectric medium with permittivity sol (note that no cavity is built around the molecule) reads: Edisp = − 2 i met i − sol i d 4d3 0 sol i sol i + met i (2.333) The chemical nature of the metal appears in Equation (2.333) via the frequency-dependent permittivity met , evaluated at the imaginary frequency i. For simple metals, the Drude form is often reasonable: D = 1 − 2p + i/0 (2.334) where p is the plasma frequency (which depends on the density of the conduction electrons in the metal) and 0 a relaxation time. The use of CMs to calculate metal–adsorbate interaction energies has been discontinued, being superseded by the full quantum mechanical treatment of the system (i.e., molecule and metal) made possible by the developments in Density Functional Theory (DFT). An exception is the recent work by Okuno and Mashiko [49] but the discrepancy with experimental results appears to be marked. Concerning the use of DFT to treat metal–molecule interactions, we remark that present exchange-correlation functionals give rise to difficulties in properly treating dispersion interactions, and the extension of the works on CMs in this direction (e.g., improving the description of the solid response, by including surface and nonlocal effects) seems a promising field. The Response Properties of a Molecule Close to a Metal Specimen: Surface Enhanced Phenomena and Related Continuum Models The evolution of CMs to describe metal–molecule interactions has been greatly fostered by the flourishing of experimental studies of phenomena related to the electromagnetic Properties and Spectroscopies 307 response properties of a molecule in close proximity of metal surfaces in the 1970s. We refer in particular to experiments measuring radiative and nonradiative luminescence lifetimes of chromophores at various distances from a flat metal surface [50–52] and to the discover of SERS, Surface Enhanced Raman Scattering [53–57]. Recently, the introduction of SNOM (Scanning Near-field Optical Microscopy) has motivated other works on the subject [58]. The exact microscopic quantities that must be calculated depend on the precise kind of measurement one aims to reproduce or predict. However, a central quantity is the frequency-dependent polarizability of the overall system. For instance at the first order and out of molecular resonance, the derivative of with respect to the normal coordinate of the vibration under study determines the SERS intensity; for luminescence (and SNOM), nonradiative lifetimes are related to the imaginary part of the poles of , while radiative lifetimes depend on the residues of such quantity. Since the CM literature on the response properties of molecules close to metals is quite extensive, we shall not give an analytical report of the various works on the subject. Instead, we shall give a few classification criteria for the models, describing a few of them as notable examples. In particular, we can group models as follows: • the shape of the metal specimen considered in the model (e.g., planar surface, spherical particle and so on); • the inclusion of electromagnetic retardation effects; • the level of description of the nonlocal character of the metal response; • the inclusion of the specific surface effects in the metal dielectric response; • the model used for the molecule; Shape of the metal specimen The shape of the metal specimen considered is obviously related to the kind of system to be modelled. For SERS and the others SE phenomena, the presence of curved surfaces, with a curvature radius on the nanometric scale, is fundamental for the enhancement. Thus, spheres, ellipsoids, ensembles of spheres, spheres close to planar metal surfaces and planar metal surfaces with random roughness have been considered. We refer to the review by Metiu [57], which describes most of these analytically solvable models. More recently, the modelling of the electric field acting on the point molecule has moved to more realistic shapes (including fractal metal specimen) [59] which require numerical methods to be tackled. The aim of these approaches is usually to calculate the total electric field around the metal particle, and the molecule does not even appear explicitly in the calculations. Interested readers are referred to some recent reviews on the subject [60] (see also Chapters 2 and 5 of ref. [56]). Retardation effects When the size of the metal specimen is much smaller (approximately a factor of 10) than the wavelength of the incident or scattered light in the system, the description of the electromagnetic fields for molecules at metal interfaces can be simplified and based on time-varying electrostatic potentials that satisfy the Poisson equation (quasi-static approximation). In contrast, if the long-wavelength condition is not fulfilled, Maxwell equations cannot be simplified and should be solved, thus taking into account retardation effects. Large nanoparticles and aggregates of nanoparticles are clear examples of such systems, but we should not forget that planar surfaces are, by definition, larger than 308 Continuum Solvation Models in Chemical Physics any wavelength.4 Thus, if phenomena involving light coming into or out of a system involving a planar interface are of interest, electrodynamic and retardation effects should be included at some point. A complete electrodynamic treatment of the emission of a classical oscillating point dipole in front of a planar metallic surface has been given by Chance et al. ([61] and references cited therein). By exploiting this treatment, the authors calculated radiative and nonradiative decay rates for electronically excited molecules in proximity of a metal surface. An interesting point in their analysis is that it is possible to partition the radiative and nonradiative molecular decay times into two contributions: one that reduces to the usual image form when the molecule is close (i.e., metal– molecule distance wavelength) to the metal surface; the other that takes into account the behaviour of the long-range contribution of the dipole-emitted field. This partition suggests that the first term can be conveniently replaced by the results of a quasistatic model that describe in more detail the molecules and its interaction with the metal, using the second term as a electrodynamic correction. This is the approach taken in ref. [62], where radiative and non-radiative decay times of byacetil close to a silver surface were studied with a quasistatic, quantum mechanical CM. Nonlocal effects In the language of reciprocal space, ‘nonlocal metal response’ refers to the dependence of the metal dielectric constant on the wavevector k of the various plane waves into which any probing electric fields can be decomposed. Such an effect is often mentioned in reports on SERS, but it is usually neglected. One of the oldest papers addressing the importance of nonlocal effects on the polarizability of an adsorbed molecule is the article by Antoniewicz, who studied the static polarizability of a polarizable point dipole close to a linearized Thomas–Fermi metal [63]. The static dielectric constant TF k of such a model metal can be written as: TF k = 1 + 2 kTF k2 (2.335) 2 is a quantity proportional to the electronic density of the metal under study. where kTF Thus, in contrast to the conductor model, this model is reminiscent of the chemical nature of the metal, even for static phenomena. A pioneering paper on the effects of nonlocality in the framework of CMs for the metal is that by Ford and Weber [64]. They considered a molecule (represented as a polarizable point dipole or as a polarizable dielectric sphere) close to a planar metallic surface, and they describe the metallic response via a modified Lindhard–Mermin dielectric function (we refer to the original paper for the expression of this LM k function). Within this model, they calculate the molecular radiative and nonradiative lifetimes (see ref. [61]). Fuchs and Barrera[65] studied the same problem, but limited the nonlocal level of description to the hydrodynamic dielectric constant, which is valid in the region of small k h k = 1 − 4 2p + i/0 − 2 k2 Clearly, an infinite planar surfaces is a model of a macroscopically large, although not infinite, surface (2.336) Properties and Spectroscopies 309 is related to the Fermi velocity vF of the electrons in the metal. The approach by Fuchs and Barrera (called the SemiClassical Infinite Barrier, SCIB, model) was later extended to spherical metallic nanoparticle as well [66–68]. We remark, however, that the hydrodynamic constant does not properly take into account excitations of electron–hole pairs in the metal bulk. This effect is important for a molecule close (tens of Å) to the metal surface and even has qualitative effects: while calculations based on the hydrodynamic constant predict a decrease of the metal quenching effects with respect to the local constant, the Lindhard–Mermin treatment predicts an increase of quenching, caused by energy transfer to electron–hole pairs having a short lifetime [62]. Surface effects on the metal response The use of a bulk-like dielectric constant, such as those in Equations (2.334)–(2.336), neglects the specific contribution given by the surface to the dielectric response of the metal specimen. For metal particles, such a contribution is often introduced in the model by considering the surface as an additional source of scattering for the metal conduction electrons, which consequently affects the relaxation time 0 [69]. Experiments indicate that the precise chemical nature of the surface also plays a role [70]. The presence of a surface affects the nonlocal part of the metal response as well, giving rise to surfaceassisted excitations of electron–hole pairs. The consequences of these excitations appear to be important for short molecule–metal distances [71]. It is worth remarking that, when the size of the metal particle becomes very small (2–3 nm), the electron behaviour is affected by the confinement, and the metal response deviates from that of the bulk (quantum size effects) [70]. The level of description of the molecule The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64]. To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72]. They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability eff to demonstrate that the effect of the image potential on eff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal–molecule system in a molecularly shaped cavity [62, 73–78]. In particular, the molecule was treated at the Hartree–Fock, DFT or ZINDO level, while for the metal different models have been explored: for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. Another quantum mechanical CM for heterogeneous system has been proposed by Jørgensen et al. [79]. They considered a perfect conductor behaviour at all frequencies for 310 Continuum Solvation Models in Chemical Physics the metal, described as a planar infinite surface. The solvent is modelled by a dielectric in which an hemispherical cavity is built to host the molecule. The latter has been considered at the Multi-Configurational Self-Consistent Field (MCSCF) level. The (quasi-) electrostatic interactions in the system have been treated by using the image method and approximating the molecule as a set of point charges. Nonlinear optical response properties up to the cubic order were calculated. Although this model can give useful information on the static (hyper-)polarizability of the molecule, considering the metal as a perfect conductor at all frequencies precludes a correct description of dynamic response properties, and thus of the related physical quantities (e.g., luminescence lifetimes or SE phenomena). Models for Heterogeneous Electron Transfer Reactions Another research field in which CMs for metal–liquid systems has been used is the modelling of electrochemical electron transfer (ET) reactions (see also the contribution by Newton). In particular, CMs have been used to address calculations of the heterogeneous electron transfer reorganization energy. For example, Marcus [80] considered a spherical cavity for the molecule and a planar perfect conductor for the metallic electrode. The problem of calculating reorganization energy with analytically solvable CMs has been also tackled by others [81–83]. In these studies the solute is treated as a classical density of charge (a point charge, a point dipole, a conducting sphere upon which a surface density of charge is spread). Recently, reorganization energies in the presence of an electrode have been calculated via a quantum mechanical continuum model akin to PCM [84]. In particular, the system considered was an electron transfer protein (Azurin) supported on a gold electrode in the presence of a scanning tunnelling microscopy tip and immersed in water. The active site of the protein was treated quantum mechanically at the DFT level, while the rest of the protein was considered as a complex-shaped dielectric. The tip and electrode were considered as perfect conductors. The solution of the Poisson problem for such a complex system was based on the IEF approach. Finally, another refinement of the model used in ref. [84] was the introduction of a nonlocal dielectric constant for the protein medium. 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Mikkelsen, Theor. Chem. Acc., 111 (2004) 122; (b) T. Hansen, T. B. Pedersen and K. V. Mikkelsen, Chem. Phys. Lett., 405 (2005) 118. 3 Chemical Reactivity in the Ground and the Excited State 3.1 First and Second Derivatives of the Free Energy in Solution Maurizio Cossi and Nadia Rega 3.1.1 Free Energy Derivatives in Solution In other contributions to this book (see, for example, Tomasi et al. and Orozco et al.) it is shown how the molecular (free) energy and electronic properties are influenced by the solvent, modelled as a polarizable continuum. The environment also affects the geometrical structure of the dissolved molecules: more precisely, the shape of the molecular potential energy surfaces (PES) depends, sometimes critically, on the interactions with the surrounding medium, so that new minima can appear which do not exist in the gas phase, the existing minima can be shifted to new positions, and the PES slope around a minimum (and hence the vibrational parameters) can change. These phenomena are sometimes referred to as ‘indirect solvent effects’, to be distinguished from the ‘direct’ ones, i.e. the effects of the solute electronic polarization at fixed geometry. The solvent-induced geometrical distortions can be very important to understand the properties and the reactivity of systems in solution: for example, SN 2 reactions on halomethanes, which proceed by a simple activated mechanism in the gas phase, are described by a double well mechanism in polar solutions, where stable precursor complexes appear [1]. In addition, for some systems important regions of the conformational space become accessible only if solute–solvent interactions are taken into account. An outstanding example is provided by (poly)peptides whose conformations are often described in terms of the two backbone dihedral angles and (see Figure 3.1). Continuum Solvation Models in Chemical Physics: Theory and Applications © 2007 John Wiley & Sons, Ltd Edited by B. Mennucci and R. Cammi 314 Continuum Solvation Models in Chemical Physics OH O H C CH3 N C N H CH2 ψ φ H CH3 C O Figure 3.1 Dihedral angles and for tyrosyl dipeptide analogue (-acetylamino-tyrosylN-methylamide, TDA). In vacuo most peptides are constrained to quasi-planar conformations 0 180 , while Polarizable Continuum Model (PCM) calculations show that in aqueous solution another stable structure appears for −60 −60 : this is noteworthy because such angles are typical of -helix conformations of polypeptides, which is particularly favoured by the solvent [2]. This feature is illustrated in Figure 3.2, where Ramachandran maps (i.e. plots of the energy versus and ) are reported both in vacuo and in aqueous solution. To account for indirect solvent effects, solvation models must allow for geometry optimizations and frequency calculations including the solute–solvent interactions. Indeed, many ab initio continuum solvation models and in particular those belonging to the family of the PCM [3] provide analytical first and second derivatives of the free energy with respect to the nuclear coordinates [4, 5]. In the following we shall present in detail the formalism for the derivatives in the PCM and Conductor–PCM (CPCM) [6] models. When one solute atom moves, the solute–solvent interface (the ‘solute cavity’) changes. We have seen that the cavity is described in terms of ‘tesserae’, i.e. small elements on the surface of the spheres that form the cavity: the first derivatives of all the tesserae geometrical elements (position, shape and size) can be computed analytically with respect 00 3.0 0 00 60.00 8.00 0 8.0 60.00 4.00 0 6. 120.00 6.00 4.00 2.00 4.0 2. 8.00 120.00 180.00 8.00 180.00 4.00 6.00 8.00 4.00 4.00 –60.00 8.0 8.00 6. 00 –60.00 0.00 3.00 ψ ψ 0.00 4.0 0 0 –120.00 –120.00 4.0 0 6.0 0 00 0 –180.00 –180.00 –120.00 –60.00 (a) 8. 8.0 0.00 φ 60.00 120.00 180.00 –180.00 –180.00 –120.00 –60.00 (b) 0.00 φ 60.00 120.00 180.00 Figure 3.2 Ramachandran maps for TDA (a) in vacuo and (b) in aqueous solution. Chemical Reactivity in the Ground and the Excited State 315 to any nuclear displacement [7], and this is the basic step in the calculation of PCM contributions to energy gradients. Nonelectrostatic Contributions to Gradients As already illustrated in the contribution by Tomasi, different terms contribute to the free energy of a molecule in solution: in the PCM formalism it is customary to write them as G = Gel + Gcav + Gdisp–rep (3.1) separating the electrostatic, cavitation and dispersion–repulsion contributions (for an exhaustive analysis of the physical grounds and the algorithms proposed for the last two terms, the reader is addressed to the fundamental review by Tomasi and Persico [8]). The cavitation energy (related to the work needed to dig in the continuum a cavity large enough to accommodate the solute) is expressed in the PCM as a function of the cavity exposed surface: spheres Gcav = I AI HS G RI 4R2I (3.2) where RI and AI are the radius and the exposed surface of the Ith sphere forming the cavity, respectively (so that AI /4R2I is the fraction of the Ith surface actually in contact with the solvent), and GHS RI is the cavitation energy for a spherical cavity of radius RI according to Pierotti’s hard sphere formalism [9]. Since the sphere radii depend on the chemical nature of the solute atoms, the only term in Equation (3.2) affected by the nuclear motions is AI . Then the cavitation energy derivative with respect to a nuclear Cartesian coordinate x is spheres AxI HS Gcav = Gxcav = G RI x 4R2I I AxI = tesserae axi (3.3) (3.4) i∈I where ai is the area of ith tessera, and the last sum is extended to the tesserae belonging to sphere I. Here and in the following we adopt the short-hand symbol x to indicate differentiation with respect to x; the gradients of tesserae area, axi , are computed as indicated in ref. [7]. The dispersion and repulsion contributions have been modelled and computed with a variety of approaches [3,8]. The most diffused PCM version adopts the procedure developed by Floris and Tomasi [10], based on atom–atom interaction parameters, proposed by Caillet and Claverie from crystallographic data [11]: Gdisp–rep = solute solvent a s s tesserae dis rep ai as rai + as rai (3.5) i where a runs over solute atoms and s over solvent atoms, s is the number density of solvent atom s rai is the distance between solute atom a and tessera i. Interaction 316 Continuum Solvation Models in Chemical Physics parameters as are determined by assuming that dispersion energy depends on interatomic distance as r −6 , and repulsion energy as an exponential (Buckingham model) [7, 10]. Also in this case, the energy gradients are easily written in terms of the geometrical disrep x derivatives: axi from the area appearing in Equation (3.5) and as (which includes the derivatives of other geometrical parameters) as indicated in ref. [7]. Among the alternative definitions of the dispersion–repulsion energy, we mention the quantum mechanical approach presented in ref. [12], which has the merit of including this part of the nonelectrostatic contribution in the molecular Hamiltonian (like the electrostatic term), so that the solvent affects not only the free energy but also the electronic distribution. In this approach, however, the dispersion part is highly expensive except for very small systems, and it is not routinely used in any computational package: the analytical derivatives of quantum mechanical Gdisp–rep can be derived but they have not been implemented until now. Electrostatic Contribution to Gradients In most cases, and particularly for medium to high polarity solvents, the electrostatic interactions provide the largest contribution to the free energy in solution: as a consequence, this contribution to first and second energy derivatives in solution is also the most important and significant. If the self-consistent field (SCF) energy in the gas phase is written as E 0 = E 0 + VNN (3.6) where E is the Hartree–Fock (HF) or density functional theory (DFT) functional of the SCF electronic density 0 and VNN is the nuclear repulsion term, the free energy in solution can be written as Gel = E + VNN + 1 tesserae Vi qi 2 i (3.7) where the same functional as in the gas phase is used but with the polarized density , and an additional term appears: Vi is the electrostatic potential generated by the solute and qi the solvation charge in tessera i. Differentiating Equation (3.7) with respect to the nuclear position x, one obtains x Gxel = E x + VNN + x 1 tesserae 1 tesserae V i qi + Vi qix 2 i 2 i (3.8) The first term on the r.h.s. should be expanded as E x + E x , where E x collects all the terms which do not depend on x . If the molecular electronic density, or wavefunction, is determined variationally (for example in Hartree–Fock or density functional calculations) the term E x can be replaced by an expression not containing x : then in these methods there is no need to compute the derivative of [13]; this very useful Chemical Reactivity in the Ground and the Excited State 317 result holds in the gas phase as well as in solution. Then in variational methods the free energy gradient in solution reduces to x Gxel = E x + VNN + x 1 tesserae 1 tesserae V i qi + Vi qix 2 i 2 i (3.9) the first two terms on the r.h.s. are computed with the same algorithms as used for molecules in the gas phase: the only new quantities required for the gradients in solution are those involving the solvation charges. The derivatives of the solute potential on tesserae, Vix , can be calculated by standard algorithms, provided by most ab initio computational packages. On the other hand, the solvation charges are defined in terms of the solute potential on tesserae through the general expression (holding both for PCM and for CPCM): qi = tesserae Qij Vj (3.10) j where the matrix elements Qij are in turn defined as a function of tesserae geometrical elements (position and area). The charge derivatives can be computed as qix = tesserae Qij x j Vj + tesserae Qij Vjx (3.11) j x where we again find the derivatives of the solute potential, and the term Qij again depending on a combination of derivatives of geometrical elements (tesserae position and area), computed as in ref. [7]. Equations (3.9) and (3.11) can be combined to obtain a very efficient and fast definition of analytical derivatives, especially when the Q matrix is symmetric (as in CPCM) or can be symmetrized [4, 5]. A completely different approach has also been proposed to compute Gel /x [14, 15]: instead of finding the derivatives of Equation (3.7), one can differentiate the basic PCM electrostatic equations and then find the solutions to the new equations. By the repeated application of the divergence theorem, this procedure leads to the following expression for the free energy gradients: x Gxel = E x + VNN + tesserae Vix qi + Ux q1 · · · qi · · · (3.12) i where Ux is a function of the solvation charges and of the tesserae geometrical elements. Expression (3.12) is approximated for cavities formed by interlocking spheres (even if the approximation is often very good), so it is not used for PCM gradients (Equation (3.9) providing more reliable results at the same cost), but it is useful for computing second derivatives, as we shall see. In general, we can conclude that gradient calculations, and hence geometry optimizations, can be performed in solution with the same accuracy and reliability as in the gas phase, using this kind of continuum model. The remaining problems, which sometimes 318 Continuum Solvation Models in Chemical Physics make the optimizations in solution more lenghty and difficult to converge, are related to the higher ‘roughness’ of the PES, especially when the shape of the cavity changes markedly during the optimization: in these cases, the convergence can be facilitated by simplifying the shape of the cavity, e.g. by reducing the number of spheres, including some atomic group in the same sphere, etc. It is noteworthy that such problems usually arise at the end of the optimization runs, when the system has come close to a minimum and the very small discontinuities in the PES due to the numerical evaluation of solvation charges on the PCM cavity become more visible: sometimes it is sufficient to increase the convergence thresholds slightly to obtain a reasonable approximation of the converged structures. Electrostatic Contribution to Hessian Free energy second derivatives are mainly used to analyse the nature of stationary points on the PES, and to compute harmonic force constants and vibrational frequencies: to perform such calculations in solution, one needs analytical expressions for Gel second derivatives with respect to nuclear displacements (the alternative of using numerical differentiation of gradients is far too much expensive except for very small molecules). The straightforward differentiation of Equation (3.9) with respect to a second coordinate y leads to xy xy x y Gxy el = E + E + VNN + + xy x y 1 tesserae 1 tesserae V i qi + Vi qi 2 i 2 i y x 1 tesserae 1 tesserae Vi q i + Vi qixy 2 i 2 i (3.13) which contains some terms already found (first derivatives of potential and solvation charges), and some new quantities. Before analysing them, we observe that the last term involves the charge second derivatives, qixy , which could be obtained by differentiating Equation (3.11) but would require the second derivatives of tesserae geometrical elements (through Qxy ij ). Such derivatives have been developed formally, but lead to extremely complex expressions which have not yet been implemented [4]. In the practice then, one resorts to Equation (3.12) whose differentiation leads to xy xy x y Gxy el = E + E + VNN + tesserae i Vixy qi + tesserae i Vix qiy + Ux y (3.14) where the last term now contains only first derivatives of solvation charges and tesserae elements. As for the other quantities appearing in Equation (3.14), the potential second xy (the nuclear contriderivatives, Vixy , are routinely computed by most programs, and VNN xy bution) and E (containing all the terms which do not depend on density derivatives) are the same as in the gas phase. Special care must be taken with the term involving y , i.e. the electronic density derivatives: such quantities are obtained, as in the gas phase, through a so-called coupled perturbed SCF procedure, in which the Schrödinger equation is iteratively solved for any Chemical Reactivity in the Ground and the Excited State 319 y . During this procedure, the PCM contribution is expressed in terms of ‘perturbed’ potentials and solvation charges, i.e. potentials obtained by contracting the atomic basis on the density derivative y instead of the usual density , and charges computed through Equation (3.10) using the perturbed potentials [16, 17]. The above procedures allow for the analytical calculation of free energy second derivatives, and can be used for testing the optimization results or computing harmonic frequencies: as an example of application, we report in Figure 3.3 the vibrational spectra computed at the DFT level for two DNA bases in vacuo and in aqueous solution. Reliability of the Hessian in Solution We have noted above that, since the analytical double differentiation of the cavity elements is not feasible presently, an approximate expression is used as the starting point for the Hessian calculation. The reader may wonder if using different expressions, i.e. Equations (3.9) and (3.12), for the gradient in first and second derivatives leads to an inconsistent Hessian: in fact, however, the differences in the gradients obtained with the two approaches are very small in almost all cases (with the possible exception of negatively charged molecules, for which a nonnegligible difference sometimes arises). In general, we can safely assume that the same stationary points are found by both the procedures, so that the Hessian computed as indicated above is coherent with the optimizations using Equation (3.9). Another, somehow more delicate point concerns nonelectrostatic terms: as shown above, their gradients depend on cavity geometrical derivatives, so that the nonelectrostatic contributions to the Hessian would require cavity second derivatives and cannot be computed analytically. In some cases, especially for flexible molecules in nonpolar environments, these contributions are not negligible if compared with the electrostatic ones, and the Hessian can actually be inconsistent with the gradients. This problem can be overcome either by optimizing the structures without the contribution of nonelectrostatic terms (finding less accurate stationary points, but still suitable for vibrational analysis), or by computing the nonelectrostatic contributions to second derivatives numerically. In fact the calculation of cavitation, dispersion and repulsion energies and gradients is fast enough to be repeated for all the nuclear displacements. Gradients in Post-Hartree–Fock Calculations Many post-HF procedures, both variational and perturbative, have been extended to allow for the calculation of nuclear gradients (and thus for geometry optimizations) including solvent effects through the PCM model. Presently this is possible for MCSCF [18, 19], RASSCF [20] and MP2 [21]. It is evident that PCM is not the only continuum method used to include solvent effects in SCF and post-HF calculations [22–24], but it has the great advantage of its realistic cavity, which mimics the real shape of the solute. We have seen that the complex shape of the PCM cavity does not hinder the calculation of analytical first (and for some methods also second) derivatives, so that this is usually considered the most reliable and powerful approach for the ab initio study of reactivity in solution. A related issue concerns time-dependent (TD) DFT calculations, whose popularity for the study of excited state energy and reactivity is rapidly increasing as a result of the favourable combination of efficiency and reliability. Though TD DFT cannot be considered a post-HF method, strictly speaking, nonetheless it provides some 320 Continuum Solvation Models in Chemical Physics Absorbance / arbitrary units Adenine 4000 3500 3000 PBE0/6–311 G(d,p) CPCM(water)/PBE0/6–311G(d,p) 2000 1500 1000 500 Wavenumber / cm–1 (a) Absorbance / arbitrary units Cytosine 4000 3500 3000 PBE0/6–311 G(d,p) CPCM(water)/PBE0/6–311G(d,p) 2000 1500 1000 500 Wavenumber / cm–1 (b) Figure 3.3 IR spectra in vacuo (full curves) and in aqueous solution (dashed curves) computed using the PCM at the DFT level, with a hybrid GGA functional. Chemical Reactivity in the Ground and the Excited State 321 information usually sought in MCSCF or MP2 calculations. PCM solvent effects have been included in TD DFT calculations some years ago [17], and recently the expressions for analytical first derivatives have been published [25]: TD DFT geometry optimizations in solution are expected to be available soon, at least in some computational packages. It is also noteworthy that, even for methods having an intrinsic high formal complexity, the PCM corrections are always quite ‘simple’, thanks to the electrostatic nature of the solute–solvent interactions, which allows us to write the PCM-related operators in analogy to Coulomb operator (and allows us, for instance, to have the same PCM contribution in HF and DFT). References [1] M. Cossi, C. Adamo and V. Barone, Solvent effects on an SN2 reaction profile, Chem. Phys. Lett., 297 (1998) 1. [2] C. Adamo, V. Dillet and V. Barone, Solvent effects on the conformational behavior of model peptides. A comparison between different continuum models, Chem. Phys. Lett., 263 (1996) 113. [3] J. Tomasi, B. Mennucci and R. Cammi, Quantum mechanical continuum solvation models, Chem. Rev., 105 (2005) 2999. [4] M. Cossi, G. Scalmani, N. Rega and V. Barone, New developments in the polarizable continuum model for quantum mechanical and classical calculations on molecules in solution, J. Chem. Phys., 117 (2002) 43. [5] M. Cossi, N. Rega, G. Scalmani and V. Barone, Energies, structures, and electronic properties of molecules in solution with the C-PCM solvation model, J. Comput. Chem., 24 (2003) 669. [6] V. Barone, and M. Cossi, Quantum calculation of molecular energies and energy gradients in solution by a conductor solvent model, J. Phys. Chem. A, 102 (1998) 1995. [7] M. Cossi, B. Mennucci and R. Cammi, Analytical first derivatives of molecular surfaces with respect to nuclear coordinates, J. Comput. Chem., 17 (1996) 57. [8] J. Tomasi and M. Persico, Molecular interactions in solution: An overview of methods based on continuous distributions of the solvent, Chem. Rev., 94 (1994) 2027. [9] R. A. Pierotti, A scaled particle theory of aqueous and nonaqueous solutions, Chem. Rev., 76 (1976) 717. [10] F. M. Floris, J. Tomasi and J. L. Pascual-Ahuir, Dispersion and repulsion contributions to the solvation energy: Refinements to a simple computational model in the continuum approximation, J. Comput. Chem., 12 (1991) 784. [11] J. Caillet and P. Claverie, Theoretical evaluations of the intermolecular interaction energy of a crystal: application to the analysis of crystal geometry, Acta Crystallogr. B, 34 (1978) 3266. [12] C. Amovilli and B. Mennucci, Self-consistent-field calculation of Pauli repulsion and dispersion contributions to the solvation free energy in the polarizable continuum model, J. Phys. Chem. B, 101 (1997) 1051. [13] R. McWeeny, Methods of Molecular Quantum Mechanics, 2nd edn, Academic Press, London, 1992. [14] E. Cancès and B. Mennucci, Analytical derivatives for geometry optimization in solvation continuum models. I. Theory, J. Chem. Phys., 109 (1998) 249. [15] E. Cancès, B. Mennucci, and J. Tomasi, Analytical derivatives for geometry optimization in solvation continuum models. II. Numerical applications, J. Chem. Phys., 109 (1998) 260. [16] B. Mennucci, R. Cammi and J. Tomasi, Analytical free energy second derivatives with respect to nuclear coordinates: complete formulation for electrostatic continuum solvation models, 322 Continuum Solvation Models in Chemical Physics J. Chem. Phys., 110 (1999) 6858. [17] M. Cossi and V. Barone, Time-dependent density functional theory for molecules in liquid solutions, J. Chem. Phys., 115 (2001) 4708. [18] M. Cossi, V Barone and M. A. Robb, A direct procedure for the evaluation of solvent effects in MC–SCF calculations, J. Chem. Phys., 111 (1999) 5295. [19] L. Frediani, R. Cammi, C. S. Pomelli, J. Tomasi and K. J. Ruud, New developments in the symmetry-adapted algorithm of the polarizable continuum model, J. Comput. Chem., 25 (2004) 375. [20] G. Karlstrom, R. Lindh, P.-A. Malmqvist, B. Roos, U. Ryde, V. Veryazov, P.-O. Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady and L. Seijo, MOLCAS: a program package for computational chemistry, Comput. Mater. Sci., 28 (2003) 222. [21] R. Cammi, B. Mennucci and J. Tomasi, Second-order Möller–Plesset analytical derivatives for the polarizable continuum model using the relaxed density approach, J. Phys. Chem. A, 103 (1999) 9100. [22] V. Dillet, D. Rinaldi, J. Bertran, and J. L. Rivail, Analytical energy derivatives for a realistic continuum model of solvation: application to the analysis of solvent effects on reaction paths, J. Chem. Phys., 104 (1996) 9437. [23] O. Christiansen and K. V. Mikkelsen, Coupled cluster response theory for solvated molecules in equilibrium and nonequilibrium solvation, J. Chem. Phys., 110 (1999) 8348. [24] A. Schäfer, A. Klamt, D. Sattel, J. C. W. Lohrenz and F. Eckert, COSMO implementation in TURBOMOLE: extension of an efficient quantum chemical code towards liquid systems, Phys. Chem. Chem. Phys., 2 (2000) 2187. [25] G. Scalmani, M. J. Frisch, B. Mennucci, J. Tomasi, R. Cammi and V. Barone, Geometries and properties of excited states in the gas phase and in solution. Theory and application of a time-dependent DFT polarizable continuum model, J. Chem. Phys., 124 (2006) 094107. 3.2 Solvent Effects in Chemical Equilibria Ignacio Soteras, Damián Blanco, Oscar Huertas, Axel Bidon-Chanal and F. Javier Luque 3.2.1 Introduction The passage from the theoretical study of compounds isolated in the gas phase to molecular systems in solution is a challenging goal in theoretical chemistry. The main limitation comes from the enormous number of molecules that must be considered in a dynamic way to represent the assembly of chemical (solute, solvent) entities which constitute the solution state. Such a complexity has given rise to a wide variety of computational approaches, which involve different treatments for the description of the solute and solvent molecules. These methods rely on (i) the construction of physical functions, (ii) the computer simulation of classical liquids, where any property of the system is obtained from an ensemble of configurations representative of the solute–solvent system, (iii) a supermolecular description of the solution, which provides limited but detailed information about specific solute–solvent interactions, (iv) quantum mechanical treatments of the solute combined with statistically averaged descriptions of discrete solvent molecules, and (v) continuum models, where attention is mainly paid to one component of the system, the solute, whereas the solvent is treated in a very simplified way as a polarizable medium. Combined strategies using the different methods mentioned above for the solvent molecules in conjunction with classical or quantum mechanical treatments of the solute have given rise to a plethora of methods, which are under continuous progress. A comprehensive analysis of the evolution experienced by these methods has been given in several reviews [1–9] which provide a critical evaluation of their suitability for the study of chemical and biochemical systems. Here we give an overview of the current status and perspectives of theoretical treatments of solvent effects based on continuum solvation models where the solute is treated quantum mechanically. It is worth noting that our aim is not to give a detailed description of the physical and mathematical formalisms that underlie the different quantum mechanical self-consistent reaction field (QM–SCRF) models, since these issues have been covered in other contributions to the book. Rather, our goal is to illustrate the features that have contributed to make QM–SCRF continuum methods successful and to discuss their reliability for the study of chemical reactivity in solution. 3.2.2 The Free Energy of Solvation Of particular relevance to understanding the influence of solvation on chemical processes is the concept of the free energy of solvation, Gsol , which can be defined as the reversible work required to transfer the solute from the ideal gas phase to solution at a given temperature, pressure and chemical composition [10]. This definition is well suited for molecular formulations of the solvation problem, because it permits Gsol to be related to the difference in the reversible works necessary to ‘build up’ the solute both in solution and in the gas phase. 324 Continuum Solvation Models in Chemical Physics To determine the coupling work between solute and solvent, it is convenient to decompose Gsol into separate, more manageable terms, which typically involve the separation between electrostatic and nonelectrostatic contributions. The former accounts for the work required to assemble the charge distribution of the solute in solution, while the latter is typically used to account for dispersion and repulsion interactions between solute and solvent molecules, as well as for cavitation, i.e. the work required to create the cavity that accommodates the solute. This partitioning scheme has proven to be extremely valuable in the framework of QM–SCRF continuum models [11], where the solvent is treated as a continuum polarizable medium characterized by suitable macroscopic properties. By means of a detailed parameterization of both electrostatic and nonelectrostatic contributions, Gsol can be predicted with remarkable accuracy (less than 1 kcal mol−1 for water and generally around 05 kcal mol−1 for nonaqueous solvents) using the latest versions of QM–SCRF models, such as the multipolar expansion (ME) method developed at Nancy [12, 13], the generalized Born (GB) SMx solvation models from Cramer and Truhlar [14–17], the Polarizable Continuum model (PCM) from Tomasi and coworkers [18, 19], the Miertus–Scrocco– Tomasi (MST) version developed at Barcelona [20–23], or the conductor-like screening model from Klamt and coworkers [24–26]. Though convenient from a practical point of view, partitioning of the solvation free energy is not strictly rigorous because it neglects the mutual coupling between the components of the solute–solvent interaction potential. Since only the total free energy of solvation is experimentally measurable, the development of theoretical formalisms that consider explicitly the mutual coupling between the different contributions appears to be the only valuable approach to determine the reliability of the partitioning scheme [27–30]. It can be assumed that the most important fraction of the solvent reorganization effects arises from cavitation and electrostatics, especially for polar solvents. Since cavitation contains the largest portion of the repulsion term, the coupling between repulsion and electrostatics is expected to be small. It can also be considered that dispersion is weakly coupled to electrostatics, at least for neutral solutes and polar solvents. Support to these assumptions has recently come from the analysis of the coupling between electrostatic and dispersion–repulsion contributions to the solvation of a series of neutral solutes in different solvents [31]. It has been found that the explicit inclusion of both electrostatic and dispersion–repulsion forces have little effect on both the electrostatic component of the solvation free energy and the induced dipole moment, as can be noted from inspection of the data reported in Table 3.1. These results therefore support the separate calculation of electrostatic and dispersion–repulsion components of the solvation free energy, as generally adopted in QM–SCRF continuum models. 3.2.3 The Solute Cavity in Continuum Models A key issue in any continuum model is the definition of the solute–solvent interface, since it largely modulates the electrostatic contribution to the solvation free energy. Generally, cavities are built up from the intersection of atom-centred spheres, whose size is determined from fixed standard atomic radii [32–36]. However, other strategies have been proposed, such as the use of variable atomic radii, whose values depend Chemical Reactivity in the Ground and the Excited State 325 Table 3.1 Electrostatic contribution Gele kcal mol −1 to the solvation free energy and dipole moment (; Debye) for a series of representative neutral compounds in water determined from QM–SCRF B3LYP/ aug-cc-pVDZ calculations with and without coupling between electrostatic and dispersion–repulsion components Compound Without CH3 CONH2 CH3 COOH CH3 NH2 CH3 OH C6 H5 CONH2 C6 H5 COOH Gele −104 −78 −43 −52 −100 −76 With −103 −76 −42 −51 −98 −74 Without With 517 228 178 206 500 273 519 228 178 206 503 276 on the chemical environment of atoms in the molecule or on the charge distribution of the solute [37–43]. Other approaches have attempted to relate the atomic radii to the analysis of radial distribution functions obtained from discrete simulation of diluted systems [44, 45], or the choice of a given density isocontour [46–48]. Finally, though most methods use a common cavity for any solvent, solvent-adapted cavities have also been considered [20–22]. The choice of the cavity is even more delicate for ionic compounds, whose electrostatic field strongly perturbs the solvent, thus making the solvent molecules in the first hydration shells exhibit properties different from those of the bulk solvent. Several strategies have been used to account for these effects in continuum models, such as the reduction of the solute cavity [40, 43], or the use of electron density contour different for charged and neutral species [49, 50]. An alternative attempt consists in the addition of an arbitrary number of solvent molecules to the solute, while the rest of the solvent is treated as a continuum dielectric [51–59]. This approach, however, raises problems such as the increase in the computational task, the need to obtain appropriate averages of thermally accessible configurations or the treatment of the librational motions arising from weak interactions between solute and solvent molecules. Accordingly, it is convenient to limit the number of solvent molecules treated explicitly in cluster–continuum computations. For instance, the SM6 solvation model adds an explicit water molecule in the case of ionic species having highly concentrated regions of charge density [17]. In particular, this model has been shown to improve the agreement between calculated and experimental pKa values compared with the results obtained without the explicit inclusion of a water molecule [60]. Keeping in mind the intrinsic features associated with the definition of the cavity in the most popular QM–SCRF methods, it can be questioned what is the influence of the fine details of the cavity definition on the computed solvation free energies. This question has been investigated in a recent study by Takano and Houk [61], who have examined the dependence of the solvation free energies estimated for a series of 70 compounds, including neutral and charged species, on both the choice of the cavity and the level of theory used in computations within the framework of the conductor-like polarizable continuum model (CPCM). The mean absolute deviation (MAD) between calculated 326 Continuum Solvation Models in Chemical Physics and experimental values was found to vary from 2.6 to 81 kcal mol−1 depending on the choice of the cavity. The best agreement was found when CPCM computations were performed using the UAKS and UAHF(G03) cavities, which use the united atom topological model [40] with radii optimized for the HF/6–31G(d) and PBE0/6–31G(d) levels of theory. In contrast, lower differences were found between the results obtained from CPCM(UAKS) computations carried out at both HF/6–31+G(d) and B3LYP/6– 31+G(d) levels of theory, since the MADs varied from 2.6 to 33 kcal mol−1 for the same set of molecules. In addition, further extension of the basis set was found to have little influence on the mean absolute deviations between calculated and experimental aqueous solvation free energies. In line with the preceding studies, it might also be questioned whether there are fundamental differences in the electrostatic response provided by different QM–SCRF formalisms. This question has recently been addressed in a study that has compared the electrostatic response provided by three methods: the GB model as implemented in the SM5.42R method, the ME used in the QM–SCRF method developed at Nancy, and the PCM model as implemented in the MST method [62]. The results show that all the methods closely agree in reflecting the change in the electrostatic response obtained for different solvents. For the same cavity definition, both ME and PCM formalisms yield very similar values for the electrostatic component of the solvation free energy and the induced dipole moment. Adoption of the same cavity definition in the GB model, however, is inadequate, and the agreement with the ME and PCM results is recovered when a smaller cavity is used in GB computations, thus reflecting the intrinsic features of the SM5.42R method, which would yield cavities smaller than those used in the MST(PCM) model. Overall, it can be concluded that there must be a close correspondence between the use of a QM–SCRF continuum model and the specific details of the underlying parameterization in order to obtain an accurate description of solvation effects. 3.2.4 Solvent Effects on Electronic Distribution The reactivity of a given chemical species is intimately related to the charge distribution, which in turn is modulated by the interactions between solute and solvent molecules. For a given nuclear configuration, the transfer of the solute from the gas phase to a condensed phase changes the electron distribution of the solute. This is illustrated by the increase in the dipole moment of water upon condensation, which changes from 1.855 D for an isolated water molecule [63] to 2.4–2.6 D in the condensed phase [64]. In general, the increase in the dipole is estimated to be 20–30 % of the gas-phase values upon solvation in water [62, 65]. However, even in less polar solvents such as octanol, chloroform and carbon tetrachloride, the solvent-induced electronic polarization is not negligible, as indicated by changes in the dipole moment close to 16 %, 10 % and 6 %, respectively, according to MST solvation computations [22]. Besides the changes in the dipole moment, the electron density redistribution induced upon solvation can also be examined from the changes in the molecular electrostatic potential [66–68], or the variation in the molecular volume [69]. These studies have identified specific trends for the solvation of neutral, polar solutes and charged species. In the former case, solvation in polar solvents is accompanied by an electron shift from the Chemical Reactivity in the Ground and the Excited State 327 neighbourhood of electropositive atoms to the vicinities of electronegative ones, which results in an increase in the molecular dipole. For cations, the charge redistribution is less sensitive to the polarizing effect of the solvent, but in general it reflects an electron transfer from hydrogen atoms to the heteroatom in polar groups. In contrast, for anions there is a tendency to concentrate the electron charge in regions close to the molecule. These findings indicate the complexity of the solvent polarization effect on the solute charge distribution, which shows differential trends depending on the nature of the solute. In conjunction with the free energy of solvation, the analysis of the solvent-induced changes in the solute’s electron density should be valuable to shed light on the influence of solvation on the chemical reactivity of solutes. 3.2.5 Solvation and Chemical Reactivity This section will focus on the application of QM–SCRF continuum methods to chemical processes in solution. For brevity, however, we will limit the discussion to two kinds of chemical processes. Firstly, we will examine selected examples of tautomeric equilibria, which are well known to be highly sensitive to solvation effects. Secondly, we will move on the analysis of selected chemical reactions involving formation and breaking of bonds, whose description constitutes a challenge for any QM–SCRF continuum model. Tautomerism Tautomerism is an extremely solvent-dependent chemical process which affects the chemical properties of molecules. A well known example is the keto–enol equilibrium of -diketones, in which the enol form is the most populated species in apolar solvents, whereas the keto species is the most stable tautomer in aqueous solution [70]. Another classical example is the solvent influence on the keto–enol tautomerism of 4-pyridone, where the population ratio between the keto and enol tautomers changes by a factor of 104 upon its transfer from the gas phase to an aqueous solution [71]. Theoretical and experimental data show that polar solvents generally displace the tautomeric equilibrium so as to increase the population of the most polar tautomer, and this effect can be large enough to change the intrinsic tautomeric preference. This effect has been recently demonstrated in the case of the N9-methylated form of isoguanine [72], a nucleobase formed by oxidative stress of adenine. A large number of tautomeric forms are a priori available to isoguanine. In the gas phase, calculations performed at high level of quantum mechanical theory indicates that isoguanine mainly populates two amino-enol forms (AEc and AEt in Figure 3.4). In aqueous solution, however, the most favourable tautomeric forms correspond to the amino-oxo species (AO1 and AO3 in Figure 3.4), which are found to be around 1 kcal mol−1 preferred relative to the amino-enol tautomers AEc and AEt. These findings are in agreement with the available experimental data, which indicates that the amino-enol tautomers are the only detectable tautomers in dioxane, but that the amino-oxo species predominate in aqueous solution [73]. On the other hand, understanding the tautomeric dependence of isoguanine on the surrounding environment is critical to realize the mutagenic properties of this compound [72]. Another example comes from the benzo-fused derivatives of nucleic acid bases, which have been conceived of as potential building blocks to develop modified DNA duplexes with nanotechnological applications [74]. These benzo-fused bases consist of a benzene unit inserted between the six- and five-membered rings of purines, and of the same 328 Continuum Solvation Models in Chemical Physics H N H H N N O H N H N N N O N N H N O N AEt (0.0; 0.7 ) AEc (– 0.2; 1.2 ) H H N N N CH3 CH3 H H N H N N N O N CH3 N CH3 H AO1 (7.2; 0.0 ) AO3 (7.2; 0.4 ) Figure 3.4 Representation of the main tautomeric forms of isoguanine in the gas phase and in aqueous solution. The relative stabilities kcal mol −1 in the gas phase and in water are given in plain text and in italics, respectively. unit added to the six-membered rings of pirimidines. The tautomeric preferences of benzoadenine and benzothymine are little affected by solvation, because the canonicallike amino (benzoadenine) and dioxo (benzothymine) forms are populated both in the gas phase and in aqueous solution. However, a more complex tautomeric scenario is found for benzoguanine and benzocytosine [75]. Benzoguanine, which is predicted to exist mainly in the canonical-like amino-oxo forms AO19 and AO17 in the gas phase, might be found in up to four different species in aqueous solution (see Figure 3.5). In contrast, hydration changes the tautomeric preferences of benzocytosine for the oxo-imino forms OIc3 and OIt3 in the gas phase to the canonical-like oxo-amino OA1 form. O O H N H2N N N N N H2N H N N N N O O H H N H2N H H AO19 (0.0; 0.0 ) AO17 (0.5; 0.7 ) AO39 (8.5; 0.9 ) H NH2 H N O N H OA1 (0.0; 0.0 ) H2N N N O N H N H OIc3 (–0.2; 6.1 ) O N N H AO37 (7.2; 0.1 ) N N H N N H N N H OIt3 (–0.5; 4.9 ) Figure 3.5 Representation of the main tautomeric forms of benzoguanine and benzocytosine in the gas phase and in aqueous solution. The relative stabilities kcal mol −1 in the gas phase and in water are given in plain text and in italics, respectively. These two examples suffice to illustrate the dramatic effect of solvation on the tautomeric preferences. For our purposes here, it also worth noting the close similarity in the relative free energies of solvation determined from the QM–SCRF MST Chemical Reactivity in the Ground and the Excited State 329 model and the corresponding values obtained from Monte Carlo–Free Energy Pertubation (MC–FEP) computations. Thus, for a total set of 68 tautomers of the natural nucleic acid bases and their benzo-fused derivatives, there is a close agreement between the relative solvation free energies predicted from the two techniques Gsol MC-FEP = 0988 Gsol MST r = 098 F = 16073, which gives confidence in the suitability of continuum solvation models to predict the solvent influence on tautomeric equilibria. Reactive Processes Chemical reactivity is influenced by solvation in different ways. As noted before, the solvent modulates the intrinsic characteristics of the reactants, which are related to polarization of its charge distribution. In addition, the interaction between solute and solvent molecules gives rise to a differential stabilization of reactants, products and transition states. The interaction of solvent molecules can affect both the equilibrium and kinetics of a chemical reaction, especially when there are large differences in the polarities of the reactants, transition state, or products. Classical examples that illustrate this solvent effect are the SN 2 reaction, in which water molecules induce large changes in the kinetic and thermodynamic characteristics of the reaction, and the nucleophilic attack of an R−O− group on a carbonyl centre, which is very exothermic and occurs without an activation barrier in the gas phase but is clearly endothermic with a notable activation barrier in aqueous solution [76–79]. The potential impact of QM–SCRF methods on the study of the influence of solvation on chemical processes has been highlighted in recent studies. As an example, we limit ourselves to quote here a recent study [80] by Tondo and Pliego, who have used the PCM method to study the SN 2 reaction between acetate ions with ethyl halides. The study of the chemical reaction in dimethylsulfoxide (DMSO) was performed by scaling the cavity by a factor of 1.35, while that used in water was scaled by a factor of 1.10 according to the results reported in previous studies [81]. The calculated activation free energies for the reaction with ethyl chloride, ethyl bromide and ethyl iodide are 24.9, 20.0 and 185 kcal mol−1 in DMSO, which are in very good agreement with the experimental values of 22.3, 20.0 and 166 kcal mol−1 . In aqueous solution, the free energy barriers of 26.9, 23.1 and 221 kcal mol−1 are also agree with the estimated experimental values of 26.1, 25.2 and 247 kcal mol−1 , respectively. Again, the protic to dipolar solvent rate acceleration is correctly predicted, especially for ethyl chloride and ethyl bromide. Other studies have also shown that finely parameterized QM–SCRF continuum models are valuable to provide an accurate description of solvent effects on the energetics and kinetics of chemical reactions in solution. For instance, we quote here the work by Lui and Cooksy [82], who have examined the proton transfer reaction of -8a-(hydrodioxy)tocopherone and -8a-(methyl-dioxy)tocopherone to produce 1-benzopyrylium, the subsequent hydrolysis to 2H-1-benzopyran-6(8aH)-one, and the terminating rearrangement of 8a-hydroxytocopherone to 2,5-cyclohexadiene-1,4-dione from BP86 computations combined with the COSMO–RS model. The unimolecular rearrangement is found to be the rate-limiting step, and the predicted reaction rate constant 0056 min−1 is close to the experimental value 0046 min−1 . Furthermore, García et al. have used the PCM model to study the cycloaddition reaction between 9-hydroxymethylanthracene and N -methylmaleimide [83]. They have shown that the reaction rate decreases as the solvent polarity is increased, and that such an effect can be related to the formation of a hydrogen 330 Continuum Solvation Models in Chemical Physics bond between the two reactants in the transition state. As a final example, we mention the study of solvation in DMSO on the Witting reaction by Seth and co-workers [84]. By using the COSMO model, the authors found that the inclusion of solvent effects causes a change in the structure of the intermediate formed in the reaction from the oxaphosphetane structure to the dipolar betaine one, which illustrates the crucial role of solvation in cases where charge separation occur through the formations of ions or a dipolar structure. Here we further examine the suitability of QM–SCRF methods in two chemical reactions: the base-catalysed hydrolysis of methyl acetate in water, and the steric retardation of SN 2 reactions of chloride with ethyl and neopentyl chlorides in water. In the two cases the influence of the solvent is examined by using the MST version of the PCM model (see ref. [85] for a detailed description). Table 3.2 reports the calculated and experimental Gibbs free energies determined for the stationary points of the base-catalysed hydrolysis of methyl acetate in water. In the reaction, the approach of the hydroxide ion to methyl acetate leads to the formation of a tetrahedral intermediate through a transition state (TS1), and the final products (methanol and acetate ion) are formed passing through a second transition state (TS2), which are shown in Figure 3.6. The free energies were determined by adding the free energy differences determined in the gas phase at the MP2/6–31++G(d,p)//HF/6–31+G(d) level to the hydration free energies determined by using the B3LYP/6–31+G(d) version of the MST(IEF) model [86]. Table 3.2 Calculated and experimental Gibbs free energies for the stationary points of the base-catalysed hydrolysis of methyl acetate Reactants TS1 Intermediate TS2 Products a (a) MST Exptl.a 00 209 148 226 −183 00 185 100 174 −144 Taken from ref. [87]. (b) (c) Figure 3.6 Representation of the transition states TS1 (a) and TS2 (c), and the intermediate (b) formed in the base-catalysed hydrolysis of methyl acetate. Chemical Reactivity in the Ground and the Excited State 331 Despite the simplicity of QM–SCRF models, the results shown in Table 3.2 show reasonable agreement with the experimental data. The MST values slightly overestimate the stability of the products. With regard to the transition states, the Gibbs free energies overestimate the experimental values by around 2 and 5 kcal mol−1 . The greater difference obtained for the second transition state however, is not unexpected, as several studies have pointed out the potential involvement of an explicit water molecule [88, 89]. From a qualitative point of view, the theoretical results reflect the relevant role played by hydration in modulating the energetics of the final products, as well as in modulating the kinetics of the reaction. In fact, for the reaction of hydroxide anion with ethyl acetate, Pliego and Riveros have predicted a rate enhancement by a factor of 435 in the reaction when going from water to DMSO, where the activation free energy is predicted to drop to around 14 kcal mol−1 [90]. As a second example, we have determined the influence of solvation on the steric retardation of SN 2 reactions of chloride with ethyl and neopentyl chlorides in water, which has recently been studied by Vayner and coworkers [91]. In their study solvent effects were examined by means of QM–MM Monte Carlo simulations as well as with the CPCM model. Solvation causes a large increase in the activation energies of these reactions, but has a very small differential effect on the ethyl and neopentyl substrates. Nevertheless, a quantitative difference was found between the stability of the transition states determined using discrete and continuum treatments of solvation, since the activation free energies for ethyl chloride and neopentyl chloride amount to 23.9 and 304 kcal mol−1 according to MC–FEP simulations, but to 38.4 and 476 kcal mol−1 from CPCM computations. Table 3.3 reports the calculated activation free energies determined by using the MST(IEF) continuum model (at the B3LYP/6–31+G(d) level) for the transition states formed in the SN 2 reactions of chloride with ethyl and neopentyl chlorides in water (see Figure 3.7). Even though the retardation energy caused upon replacement of ethyl by neopentyl is reasonable well described at all the levels of theory, the MST values show close agreement with the MC–FEP ones, as indicated by a difference between the respective activation free energies of around 3 kcal mol−1 , which is smaller than that reported in ref. [91] from CPCM computations (see above). Table 3.3 Calculated activation free energies for the transition states formed in the SN 2 reaction of chloride anion with ethyl and neopentyl chloride in water determined from MST, CPCM and MC–FEP computations MST TS (ethyl chloride) TS (neopentyl chloride) Retardation energy a 263 341 78 MC–FEPa 239 304 65 Taken from ref. [91]. Overall, the preceding examples suffice to illustrate the capability of QM–SCRF computations to reflect the differential influence exerted by solvation on the relative stability of the different species formed in chemical reactions, and hence on the 332 Continuum Solvation Models in Chemical Physics (a) (b) Figure 3.7 Representation of the transition states formed in the SN 2 reaction of chloride anion with ethyl (a) and neopentyl (b) chloride. modulation exerted by the solvent on both the energetics and kinetics of chemical reactions. 3.2.6 Concluding Remarks During the last decade theoretical chemistry has widened the range of phenomena that can be examined in condensed phase. The development of elaborate QM–SCRF continuum models has contributed decisively to this marked change, partly due to the efforts devoted to the detailed parameterization of the different contributions to the solvation free energy. 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The solvent often plays a central role in determining the kinetics and outcome of liquid-phase chemical reactions, and the present chapter describes theoretical and computational methods that may be used to understand such effects in terms of continuum solvation models. How does the solvent influence a chemical reaction rate? There are three ways [1, 2]. The first is by affecting the attainment of equilibrium in the phase space (space of coordinates and momenta of all the atoms) or quantum state space of reactants. The second is by affecting the probability that reactants with a given distribution in phase space or quantum state space will reach the dynamical bottleneck of a chemical reaction, which is the variational transition state. The third is by affecting the probability that a system, having reached the dynamical bottleneck, will proceed to products. We will consider these three factors next. Reactant Equilibrium In a fixed-temperature gas, molecular collisions populate the various states of the reactants. In the absence of chemical reaction, the populations of these states would come into thermal equilibrium, as governed by Boltzmann statistics (or, in a quantal system, by Fermi–Dirac and Bose–Einstein statistics). However, when reactions occur, the most reactive states (usually the highest-energy ones) may react rapidly. In this case a steadystate distribution is set up in which the rate of population of these states by molecular collisions (and possibly back reaction) is balanced by their depletion by reactions. The resulting nonequilibrium distribution can be quite different from the equilibrium one, especially for unimolecular reactions in low-pressure gases [3,4]. It is often assumed that such nonequilibrium effects are always unimportant in liquids, but this is not necessarily true [5–7]. As a consequence, increasing the strength of solute–solvent couplings may promote a more equilibrated reactant state distribution and thereby change the reaction rate [4]. That is the exceptional case, though, and usually there is no observable nonequilibrium effect [8, 9]. Furthermore, such effects are not usually treated by continuum methods (the subject of this book); hence we will not discuss them in detail in this chapter. Rate of Attainment of the Transition State We define a generalized transition state as a hypersurface (usually one just says surface) in phase space separating reactants from products; it is sometimes called a dividing surface. Like any hypersurface, such a dividing surface has one less degree of freedom than the space in which it resides. For any dividing surface that we might define, the reactants must pass through the dividing surface at least once to reach products. One could imagine dividing surfaces that are crossed many times in the course of a typical Chemical Reactivity in the Ground and the Excited State 339 reactive event, but these are not very interesting. If one could find a dividing surface such that when reactants reach it, they almost surely go on to products without recrossing the surface, one could calculate a reasonably accurate rate constant by simply calculating the one-way rate of systems passing through the dividing surface. This is transition state theory [10], and the dividing surface is called the transition state. Since a system with an equilibrium distribution in reactant state space will evolve, by Liouville’s theorem, into a system with an equilibrium (Boltzmann) distribution in any other part of phase space, including the dividing surface [11, 12], the calculation can be further simplified into a calculation of the equilibrium one-way flux through the dividing surface. Notice that an equilibrium flux is independent of the nature of the dynamics that gets the system to the transition state dividing surface; it is even independent of the details of the potential energy surface in the vast expanse of phase space between reactants and the transition state. Calculating reaction rates gets even easier though when one recognizes [13, 14] that the equilibrium rate constant for passing through a dividing surface toward products may be written as k= kB T ‡o −G‡o RT K e h (3.15) where kB is Boltzmann’s constant, T is temperature, h is Planck’s constant, K ‡o is unity for a unimolecular reaction and the reciprocal of the concentration in the standard state for a bimolecular reaction, G‡o is defined by G‡o = G‡o − GRo (3.16) where GRo is the standard-state free energy of reactants, and G‡o is a new quantity that has exactly the same mathematical form as a standard-state free energy but for a system localized in the transition state by having one degree of freedom missing. The degree of freedom that is missing is the coordinate normal to the dividing surface. G‡o is called the standard-state free energy of the transition state, and G‡o is called the standard-state free energy of activation (or sometimes it may be called the standard-state quasithermodynamic free energy of activation). Now the calculation of an equilibrium one-way flux is reduced to the calculation of the difference of two free energies (technically, only GRo is a free energy; G‡o is a quasithermodynamic quantity, not a true free energy, because one degree of freedom is missing). Since the transition state is missing one degree of freedom, that degree of freedom must be treated as separable. The missing degree of freedom is usually called the reaction coordinate, and one could say that the separability of the reaction coordinate is the fundamental assumption of transition state theory. (If the reaction coordinate were globally separable there would be no recrossing.) The separability approximation usually breaks down most strongly when tunneling is important, and nonseparability effects can be included by including a multidimensional tunneling contribution as discussed in Section 3.3.2. Even when the separability assumption does not break down for the true variational transition state, i.e., for the true reaction coordinate (any trial transition state, being a dividing surface, defines a trial reaction coordinate as the coordinate normal to it, and vice versa), in practice we are limited to reaction coordinates that depend in a manageable way on only a manageable number of coordinates or that are defined by a simple model, and thus there may be nonseparability (and 340 Continuum Solvation Models in Chemical Physics hence recrossing) for calculations with practical reaction coordinates. New methods for identifying complex reaction coordinates and reaction coordinates in complex systems are under development [15–17]. The reader will have noticed that at key points in the above discussion we used classical mechanical concepts such as the flux through a dividing surface in phase space. Even the definition of the transition state (as a phase-space dividing surface) is classical. In fact there is no unique way to extend the definition of the transition state to quantum mechanics. However, if we write the free energies in terms of statistical mechanical quantities such as partition functions, there are well-known ways to replace classical mechanical partition functions by quantal ones. We will return to this issue and other practical issues involved in evaluating and improving Equation (3.15) in Section 3.3.2. However first we need to introduce another concept that will be important in computing free energies in liquid-phase solution, namely the concept of potential of mean force (PMF). The PMF is a statistical mechanical quantity that corresponds to the free energy for a system in which one or more coordinate is ‘nailed down,’ that is constrained to a constant value. For example a two-dimensional PMF corresponds to fixing two coordinates; they could be the x and y Cartesian coordinates of one of the atoms or – more likely in applications – they might be some functions of the internal coordinates such as the distance from atom A to atom B and the distance from atom B to atom C. (Note that distances are nonlinear functions of the atomic Cartesian coordinates; we call them curvilinear coordinates, whereas linear functions of atomic Cartesians are called rectilinear coordinates.) In general, if H is the Hamiltonian of the system, i.e., its total energy, the free energy G is defined by e−G/ RT = e−H / RT (3.17) where < > denotes a Boltzmann average over all phase space at temperature T . We label the set of constrained coordinates of a PMF calculation as R, the set of other coordinates as r, and the set of all conjugate momenta as p. Then Equation (3.17) becomes e−G/ RT = e−H / RT Rrp (3.18) Alternatively we can carry this out in two steps e−WR/ RT = e−H / RT rp e−G/ RT = e−WR/ RT R (3.19) (3.20) This defines WR as the PMF of coordinates R. For applications we need to specify whether a constant of integration is added to G or W to set their zero of energy [18]. The standard-state solvation free energy GoS of a solute corresponds to a statistical average over its coordinates, which may be called R, and the coordinates of the solvent, which may be called r. By analogy to the PMF we may define a constrained standard-state free energy of solvation as GoS R = WR − VR (3.21) Chemical Reactivity in the Ground and the Excited State 341 where WR is the liquid-phase PMF of the solute molecule, and VR is the gasphase potential energy of that molecule. Again we need to be careful about additive constants that account for standard states and zeroes of energy. According to the Born– Oppenheimer approximation [19], VR is given by the electronic energy of the molecule with fixed nuclear coordinates R. (The electronic energy is defined [19] to include the internuclear Coulomb repulsion.) Equation (3.21) shows that the potential of the mean force is an effective potential energy surface created by the solute–solvent interaction. The PMF may be calculated by an explicit treatment of the entire solute–solvent system by molecular dynamics or Monte Carlo methods, or it may be calculated by an implicit treatment of the solvent, such as by a continuum model, which is the subject of this book. A third possibility (discussed at length in Section 3.3.3) is that some solvent molecules are explicit or discrete and others are implicit and represented as a continuous medium. Such a mixed discrete– continuum model may be considered as a special case of a continuum model in which the solute and explicit solvent molecules form a supermolecule or cluster that is embedded in a continuum. In this contribution we will emphasize continuum models (including cluster–continuum models). Recrossing Transition state theory, as explained above, assumes an equilibrium distribution in reactant state space and no recrossing of the transition state. In a classical mechanical world, we could always find a transition state that is not recrossed, but – except very close to threshold [20] – the resulting dividing surface would typically be so convoluted that it would be impossible to use. A better strategy [21] is to find the best (but not perfect) dividing surface from among a sequence of practical dividing surfaces. Then one corrects the approximate transition state theory rate expression for recrossing by multiplying by a transmission coefficient. The best transition state is the one that minimizes the amount of recrossing, which corresponds to minimizing the one-way flux [22]. This best transition state is called the variational transition state, and its use to calculate reaction rates is called variational transition state theory [21–25]. Often when we say ‘transition state,’ it is shorthand for ‘variational transition state’ or ‘best transition state’ or ‘dynamical bottleneck’ although the phrase may also be used to refer to trial transition states. A new issue arises when one makes a solute–solvent separation. If the solvent enters the theory only in that VR is replaced by WR, the treatment is called equilibrium solvation. In such a treatment only the coordinates in the set R can enter into the definition of the transition state. This limits the quality of the dynamical bottleneck that one can define; depending on the system, this limitation may cause small quantitative errors or larger more qualitative ones, even possibly missing the most essential part of a reaction coordinate (in a solvent-driven reaction). Going beyond the equilibrium solvation approximation is called nonequilibrium solvation or solvent friction [4, 26–28]. This is discussed further in Section 3.3.2. Application Areas Continuum solvation models have been applied to many chemical processes in the liquid phase. Determining absolute free energies of activation is important because it allows one to predict the time scale on which a chemical process can take place. In addition, 342 Continuum Solvation Models in Chemical Physics the absolute barrier height is critical for determining mechanisms of chemical processes whose reaction pathway is not well known from experiment. There are four important problem areas where the study of chemical reactions in solution can be very useful. The first class of problems involves calculating the absolute free energy of activation or rate constant. Our ability to predict absolute reaction rates is critical in order to determine if a reaction can take place or not. A high free energy of activation indicates that a reaction is slow or does not occur or that the true reaction pathway was not found. In other cases, there is a clear experimental observation of the reaction, but the calculated barrier of the supposed mechanism is very high. In this case, if we assume that a sufficiently reliable level of electronic structure theory was used to calculate the barrier, it means that we do not know the real mechanism and a more detailed investigation should be done. This leads us to the second problem: elucidating the reaction mechanism. Although for some chemical processes the reaction mechanism is known, for many reactions the true mechanism is not known at all. In other cases, there are doubts about the real mechanism. These problems are discussed further in Section 3.3.4. The third problem is the competition between parallel pathways. The relative rate constants determine the product ratio. Thus, our ability to calculate relative rate constants allows us to predict which product is generated in a chemical reaction. An interesting application is to predict the products of a reaction of synthetic interest, which may result from a competition such as that between SN 2 and E2 process. This is discussed in Section 3.3.5. Another example of the need to calculate relative reaction rates is kinetic isotope effects, i.e., the relative rates of reaction of different isotopomers or isotopologs. Kinetic isotope effects are often used by experimentalists to elucidate reaction mechanisms and to gain an understanding of the nature of the transition state (i.e., of the dynamical bottleneck). They are discussed in Section 3.3.4. The fourth aspect is related to development of more efficient reaction media or catalysts. The dream of a chemist is to be able to induce a chemical transformation to take place quickly, efficiently, selectively, and with good specificity. Computational studies of reactions in solution allow us to understand the factors that influence reactivity and enable us to design new catalysts or solvent media with better properties. Because continuum models are fast, easy to use, and often reliable, they may be chosen for theoretical studies aimed at the development of catalysts or chemical processes. Examples of applications will be presented later in this contribution. However, first we will discuss transition state theory for liquid-phase reactions and parametrization of continuum models for reactive problems, because these theoretical constructs are required for applications to chemical reactions in the liquid phase. 3.3.2 Transition State Theory The most useful theoretical framework for studying chemical reactions in solution is transition state theory. Building on the material presented in the introduction, we will begin by presenting a general theory called the equilibrium solvation path (ESP) theory of reactions in a liquid. We then present an approximation to ESP theory called separable equilibrium solvation (SES). Finally we present a more complete theory, still based on an implicit treatment of solvent, called nonequilibrium solvation (NES). All three Chemical Reactivity in the Ground and the Excited State 343 theories assume reactant equilibrium, as discussed in the introduction. (Simple methods for including nonequilibrium reactant effects are reviewed elsewhere [29].) In discussing the ESP and the SES theories, R is always the set of atomic coordinates of an N -atom solute, although (as mentioned above) one may, if desired, include one or a few solvent molecules as part of the ‘solute’ (the so called ‘supermolecule’ approach). Equilibrium Solvation Path (ESP) In ESP theory [30–32] we treat the system by the same methods that we would use in the gas phase except that in the nontunneling part of the calculation we replace V (R) by W (R), and in the tunneling part we approximate V (R) by W (R) or a function of W (R). Next we review what that entails. In particular we will review the application of variational transition state theory [21–25] with optimized multidimensional tunneling [33, 34] to liquid-phase reactions for the case [31, 32] in which W (R) is calculated from V (R) by WR = VR + GoS R (3.22) and the constrained standard-state free energy of solvation is obtained by a continuum solvation model. Notice that we follow the usual practice of not indicating explicitly that W (R) depends on the standard state, although it does. The conventional definition of the transition state is a hyperplane passing through the saddle point of V (R) and orthogonal to the imaginary-frequency normal mode [10,13,14, 35,36]. This definition can also be applied using a saddle point of W (R) and normal mode analysis of W (R) instead of those for V (R). Call this saddle point geometry R‡ . Note (from the definition of W (R) given above) that a saddle point of W (R) corresponds to an average over an ensemble of solvent configurations at solute geometry R‡ . This does not correspond to a saddle point of the potential energy of the entire (solute+solvent) system. For that reason it is a misnomer to call the theory based on W (R) conventional transition state theory. In fact the conventional idea of calculating the rate constant using a dividing surface that passes through a saddle point of the entire system is not suitable for reactions in liquids because there are an uncountable number of saddle points, most of which differ only in the conformation of some far-away solvent molecules (by conformation here we mean not just intramolecular conformation but also hydrogenbonding and noncovalent-packing conformations of interacting solvent molecules). Thus, in the early days of transition state theory, the theory was generalized [37, 38] to liquidphase reactions by stating it in thermodynamic or statistical thermodynamic language. This obviated the need to define clearly the transition state for liquid reactions, and this task was only taken up more recently by using the concept of the PMF. In the next step, one finds the minimum (free) energy path (MEP) starting at R‡ and follows it toward both reactants and products. The progress variable s that measures the signed distance along the path from the saddle point is called the reaction coordinate, although that name is also used (see above) for the missing degree of freedom in the transition state, and the two coordinates are not always the same (a possible point of confusion, but both usages of ‘reaction coordinate’ are so well established that there can be no turning back). This path is defined as the path of steepest descents, which in general depends on the coordinate system in which it is computed [39]. In this chapter, 344 Continuum Solvation Models in Chemical Physics when we refer to the MEP we always mean the one computed in mass-scaled or massweighted rectilinear coordinates that diagonalize the classical mechanical kinetic energy; such coordinate systems are called isoinertial, and the MEP is the same in any such coordinate system [40]. The reasons for choosing an isoinertial coordinate system are that it promotes the local separability of the reaction coordinate, and it allows intuition about the motion of N atoms in three dimensions as if it were the motion of a single mass point in 3N dimensions [39–42]. The MEP is sometimes called the intrinsic reaction path or intrinsic reaction coordinate [43]. One next defines a sequence of dividing surfaces, which are trial transition states (these are sometimes called generalized transition states to denote that they are not conventional transition states at saddle points, but we will drop this semantic distinction here). Usually one uses a one-parameter sequence of dividing surfaces, either hyperplanes in a rectilinear coordinate system [21, 40, 44, 45] or curved dividing surfaces defined in valence internal coordinates [46, 47]. The parameter is the value of s at which the transition state, locally orthogonal to the MEP, intersects the MEP, and one optimizes this value to find the variational transition state location, which is called s∗ . The deviation of s∗ from the location along the MEP where potential energy is a maximum is called a variational effect. More generally one can variationally optimize not only the value of the reaction coordinate but also the orientation of the dividing surface; this approach can even be applied without computing the MEP [48, 49]. One can also use dynamically optimized reaction paths that pass through a sequence of variationally optimized multi-parameter dividing surfaces [50]. For each dividing surface one calculates the free energy of the transition state by standard statistical mechanical procedures in terms of partition functions by treating the transition state as a molecule with one degree of freedom missing [35,36]. The standardstate free energy of activation is then obtained from Equation (3.16). An alternative procedure for calculating G‡o would be to calculate a one-dimensional PMF, in particular Ws or Wz, where z is the distance along an arbitrary reaction path, and note that G‡o = W z∗ + Wcurv z∗ (3.23) where Wcurv z is a term, often but not always negligible, that vanishes when the missing degree of freedom is rectilinear [18]. We shall not pursue this here. Since we replaced the classical partition functions in GRo and G‡o by quantum mechanical ones, we have included quantum effects on all degrees of freedom of the reactants and all but the missing degree of freedom at the transition state. One then includes quantum effects on the remaining degree of freedom by a transmission coefficient , thereby replacing Equation (3.15) by k= kB T ‡o −G‡o RT K e h (3.24) Note that both and G‡o depend on temperature. The transmission coefficient is sometimes called the tunneling transmission coefficient because tunneling is the main quantum effect on the reaction coordinate. Chemical Reactivity in the Ground and the Excited State 345 The derivation of transition state theory as the flux through a dividing surface assumes that the system can be in the transition state only when it has positive classical mechanical kinetic energy there [14]. Tunneling is the phenomenon by which a particle passes through a barrier which it cannot pass through with positive classical mechanical kinetic energy. (Other definitions of tunneling are possible and sometimes preferable, but will not be used here.) Since reaction by tunneling may occur where the Boltzmann factor is much larger than that for overbarrier reaction, its contribution to the reaction rate may be large even when the tunneling probability is small. An analogous nonclassical effect is called nonclassical reflection. Tunneling and nonclassical reflection may both be understood in terms of a particle of energy E impinging on a one-dimensional barrier Vz with barrier height V ‡ . Classically the particle has zero transmission probability for E < V ‡ (this is classical reflection) and unit transmission probability for E > V ‡ (this is classical transmission). The fact that a quantum mechanical particle has nonzero probability of transmission for E < V ‡ is called tunneling (or nonclassical transmission), and the fact that it has nonunit probability of transmission for E > V ‡ is called nonclassical reflection (or diffraction by the barrier). The two phenomena have similar magnitudes; for example, for a purely parabolic barrier, the tunneling probability at energy E = V ‡ − is the same as the nonclassical reflection probability at E = V ‡ + ; however, tunneling usually has a much greater effect on reaction rates because it occurs at energies that have an exponentially larger Boltzmann factor [51]. One-dimensional treatments of tunneling are not reliable [52]. For gas-phase reactions, accurate multidimensional tunneling approximations have been developed [33,34,53–55] and are well validated against accurate quantum mechanical calculations [56, 57]. These tunneling approximations are nonseparable, and using them to calculate overcomes (at least partially) the separability assumption of transition state theory. In fact when tunneling dominates the reaction rate and is modeled by multidimensional tunneling approximations, the calculation is better viewed as a semiclassical multidimensional dynamics calculation than as transition state theory. To extend these methods to liquidphase reactions modeled by continuum solvation methods one needs to know the effective potential for tunneling. The PMF is already averaged over a canonical ensemble of solvent configurations and, like any free energy quantity, it includes entropy as well as potential energy; thus it is not a priori clear that it can be used to provide the effective barrier for tunneling. In principle one would calculate the tunneling from the potential energy and ensemble average [26, 27, 31] the tunneling probabilities. A more practical (but approximate) procedure is to calculate the tunneling from the ensemble-averaged potential energy. It can be shown [31] that the canonically averaged mean potential energy is given (within an additive constant, which is all that is required) by U = VR + GoS R − T GR T (3.25) Neglect of the last term yields Equation (3.22), which is called (in this context) the zero-order canonical mean-shape (CMS-0) approximation [31]. Separable Equilibrium Solvation (SES) In the SES approximation [32] we make some simplifications in the ESP formalism. First, the saddle point is optimized using V (R) rather W (R), and the MEP is also traced using 346 Continuum Solvation Models in Chemical Physics V (R). In calculating transition state partition functions along s and in calculating reactant partition functions, W (R) is used instead of V (R) at the minimum energy structure of each transition state or reactant, but vibrational frequencies are calculated using V (R). In tunneling calculations, as in ESP theory, U (R) is used instead of V (R). The CMS-0 approximation is usually made in computing U (R). SES theory can be used to illustrate a classic example of a solvent effect on a chemical reaction, namely the solvent effect on bimolecular nucleophilic substitution SN 2 reactions [58]. Figure 3.8 shows how an approximate potential of mean force changes with the solvent. We can see that in the gas phase, the barrier is very low. In aqueous solution, the anion is very well solvated, and the formation of the transition state leads to considerable charge delocalization, decreasing the favorable solvation effect. As a consequence, a very high effective barrier is generated. In dipolar aprotic solvents, such as dimethyl sulfoxide, because the ionic species are less solvated than in water, the solvent effect decreases, producing the well-known [59] rate acceleration of ionic SN 2 reactions on going from aqueous to dipolar aprotic solvents. H –δ Nu X C H δ– H Protic solvent Dipolar aprotic solvent Gas phase Nu– + CH3X Nu–CH3...X NuCH3 + X – NuCH3 ...X– Figure 3.8 Potential of mean force profile for a typical SN 2 reaction in different media. Nonequilibrium Solvation (NES) In the above treatments only the solute coordinates R appear explicitly and therefore the definition of the transition state does not depend on solvent coordinates. The NES approximation [60,61] provides a way to include solvent in the reaction coordinate while retaining a continuum description of the solvent by adding a coupling Hamiltonian for a collective solvent coordinate [60–70] (or more than one) to the Hamiltonian for the Chemical Reactivity in the Ground and the Excited State 347 R degrees of freedom. (Sometimes the collective solvent coordinate is assumed to be the reaction coordinate itself [70] rather than, as here, finding a reaction coordinate in the space obtained by augmenting the solute coordinates by the collective solvent coordinate.) As in ESP theory, in NES theory the equilibrium solvent effects are included by replacing V (R) by W (R) in nontunneling parts of the calculation and by U (R) in tunneling algorithms. Consider the case of a single collective solvent coordinate y. This coordinate is linearly coupled to the solute at the transition state by generalized Langevin theory [71–75]. (It is not necessary to couple the solvent to the solute for the calculation of reactant properties because we retain the equilibrium-reactant approximation.) The form of the coupling is [60, 61] Hcoupling = py2 2 + 2 1 F y − CT R − R‡ 2 (3.26) where py is the momentum conjugate to y is the reduced mass to which all coordinates are scaled (this will cancel out and have no effect on the dynamical results), F is a collective solvent force constant, C is a solute–solvent coupling vector with components Ci , and T denotes a transpose. The parameter F may be related to the solvent relaxation time, and the coupling constants may be related to some measure of the strength of the solute–solvent coupling, such as viscosity or diffusion coefficient from experiment or the force autocorrelation function from explicit-solvent molecular dynamics simulations [60]. Solvent relaxation times can be modeled either by explicit-solvent simulations [76–79] or by continuum models [62, 65, 74, 79–86]. The SES, ESP, and NES methods are particularly well suited for use with continuum solvation models, but NES is not the only way to include nonequilibrium solvation. A method that has been very useful for enzyme kinetics with explicit solvent representations is ensemble-averaged variational transition state theory [26, 27, 87] (EA–VTST). In this method one divides the system into a primary subsystem and a secondary one. For an ensemble of configurations of the secondary subsystem, one calculates the MEP of the primary subsystem. Thus the reaction coordinate determined by the MEP depends on the coordinates of the secondary subsystem, and in this way the secondary subsystem participates in the reaction coordinate. Other methods of including nonequilibrium solvation are reviewed elsewhere [86], and the reader is also referred to selected relevant and more recent original papers [66,88–100]. Particularly relevant to the present volume are methods that introduce extra degrees of freedom by using the solvent reaction field not only at the current value of R but also at nearby values [65, 66]. Many of the approaches introduce finite-time effects and additional degrees of solvent freedom by introducing different time scales for electronic and atomic polarization [88–97, 99, 100]. In the absence of discrete solvent molecules or a collective solvent coordinate, continuum solvation models do not allow the solvent to enter into the reaction coordinate, and in many cases that misses the primary role of the solvent. The solvent may enter the reaction coordinate only quantitatively, for example by having a slightly different strength of hydrogen bonding to the solute at the transition state than at the reactant, or it may enter qualitatively, for example by entering or leaving the first solvation shell, by 348 Continuum Solvation Models in Chemical Physics donating or accepting a proton (later being regenerated by another proton transfer so it remains a catalyst, not a reagent), and so forth. Some examples of solvent participation in the reaction coordinate that cannot be mimicked without explicit solvent molecules occur in the formamidine rearrangement [101, 102] and in the Beckmann rearrangement [103] of oximes. 3.3.3 Parameterization of Continuum Models for Dynamics Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104]. The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute–solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105]. 1º solvation shell: Specific interactions A Bulk solvent: Solute-Dipoles Interactions (dielectric constant) Figure 3.9 Interaction of the solute with the first solvation shell and with the bulk solvent. The solvation free energy calculated by considering only the bulk electrostatics is somewhat arbitrary because the boundary between the dielectric medium and the solute is not well defined, and in fact the treatment of the solvent as a homogeneous, isotropic, linear medium right up to a definite boundary is not valid. To obtain an accurate solvation Chemical Reactivity in the Ground and the Excited State 349 free energy, we can use various empirical procedures. These procedures may involve empirical adjustments of the location of the solute–solvent boundary, and/or they may involve introducing additional terms that depend on the solute–solvent boundary. An important point about the components of a solvation energy calculation is that they have no real meaning as thermodynamic variables; only the sum of all the components of the free energy of solvation is meaningful. This is well illustrated by a systematic comparison of three solvation models that was recently reported; the bulk electrostatic terms differed greatly, but after adding each model’s nonbulk electrostatic terms, the resultant free energies of solvation were in better agreement [106]. Adding additional terms to account for system-specific first-solvation shell interactions works well for neutral solutes, especially if the additional terms are well parameterized. This strategy has been used in the SMx models [69,107] x = 1–6, where the parameters are atomic surface tensions, where a surface tension is a free energy per unit area. In later versions of these models the area in question here is the solvent-exposed surface area of a given atom of the solute, and the atomic surface tensions depend on the local bonding geometry of the atom in question [107–115]. These dependences are built into parameterized, continuous functions of geometry in such a way that they are well defined at transition states, and furthermore the user is not required to assign molecular mechanics types to the atoms. One expects that the scheme works at least in part because the partial atomic charge, polarizability, and atomic size of each atom of the solute are functions of its local bonding geometry. This procedure is less satisfactory for charged solutes where these properties are not the same functions of geometry as for neutral solutes. Allowing the atomic surface tensions to depend explicitly on local charge would solve this problem, but would complicate the algorithm, requiring the first-solvation-shell effects to be self-consistently adjusted during the SCRF iterations. An alternative approach is to treat some or all of the solvent molecules in the first solvation shell, especially those near highly concentrated regions of partial charge in the solute, as parts of an extended solute, called the supermolecule. Pliego and Riveros [116] call this the cluster–continuum model, whereas other researchers call it a mixed discrete–continuum approach. Pliego and Riveros [116] have provided a protocol for how many explicit solvent molecules should be included, whereas Kelly et al. [117] have suggested that one solvent molecule is usually sufficient. Pratt and co-workers have proposed a quasichemical theory [118–122] in which the solvent is partitioned into inner-shell and outer-shell domains with the outer shell treated by a continuum electrostatic method. The cluster–continuum model, mixed discrete– continuum models, and the quasichemical theory are essentially three different names for the same approach to the problem [123]. The quasichemical theory, the cluster–continuum model, other mixed discrete–continuum approaches, and the use of geometry-dependent atomic surface tensions provide different ways to account for the fact that the solvent does not retain its bulk properties right up to the solute–solvent boundary. Experience has shown that deviations from bulk behavior are mainly localized in the first solvation shell. Although these first-solvation-shell effects are sometimes classified into cavitation energy, dispersion, hydrophobic effects, hydrogen bonding, repulsion, and so forth, they clearly must also include the fact that the local dielectric constant (to the extent that such a quantity may even be defined) of the solvent is different near the solute than in the bulk (or near a different kind of solute or near a different part of the same solute). Furthermore 350 Continuum Solvation Models in Chemical Physics since the atomic radii and the atomic surface tensions are usually determined empirically, they must also make up for systematic errors in the solute charge distributions and for the fact that actually the solute–solvent boundary is gradual and fluctuating, not sharp and fixed. Returning to the calculation of the bulk electrostatic contribution to the free energy of solvation, first one needs to define the size and shape of the cavity. Although some older continuum models used spherical or other idealized shapes (e.g. ellipsoids), most modern continuum models use realistic cavity shapes based on superposition of atomcentered spheres. There is, however, no consensus on the radii to be assigned to these atomic spheres. For the purpose of parameterizing the atomic radii, it is especially useful to consider the solvation of ionic species because the solvation free energy of these species has a high absolute value and is very sensitive to the atomic radii. Solvation data for organic ions are widely available for water [124–126], dimethyl sulfoxide (DMSO) solutions [126], and assorted other solvents [127], and these have been used to parameterize continuum models in many solvents [114, 115, 117, 127–129]. Comparing the performance of the parametrizations of continuum models for pKa calculations in water and DMSO illustrates the reliability that can be achieved. It was shown that polarized continuum models can predict pKa values in DMSO solution [130, 131] with an error of only two units. These results indicate that these models can be used for semiquantitative modeling of ionic reactions in dipolar aprotic solvents. On the other hand, in water or protic solvents, the performance of continuum models is worse because of strong hydrogen bonds between the ionic species and the water molecules or because of the unique cooperative hydrogen bonding structure of liquid water. No set of atomic radii is capable of producing very accurate solvation free energy values for all situations. Nevertheless, by including some explicit water molecules in the first solvation shell in order to account for critical solute–solvent hydrogen bonds and for strong electrostatic interactions, one can obtain more accurate results. In this approach, the solute–water cluster becomes the new solute (Figure 3.10). In the calculation of pKa values in water solution, Figure 3.10 Inclusion of the first-solvation-shell interaction for the solvation of the hydroxide ion in water solution. Chemical Reactivity in the Ground and the Excited State 351 it was shown that the cluster–continuum method works much better than pure continuum models [132]. One special difficulty of applying parameterized models to chemical reactions deserves a special mention, namely that transition states often have charge distributions quite different from those against which solvation models are parameterized. For example, the partial atomic charge on Cl in the Cl CH3 Cl−1 SN 2 transition state is about −07, midway between the values (−10 and about −04, respectively) found in Cl− monatomic anion and typical alky chlorides. Thus the atomic radii and atomic surface tensions optimized against equilibrium free energies needs to be re-validated for transition structures. In principle, we can distinguish two possible kinds of breakdown of continuum solvation models as they are usually applied to dynamics problems. One is breakdown of the linear response approximation, and the other is breakdown of equilibration solvation. Although these are different, they sometimes occur in tandem. It is worthwhile to sort these issues out in a little more detail. If one considers very small deviations from equilibrium, the solvent response is formally linear since one can always make the deviation from equilibrium so small that the first term dominates the expansion in powers of the deviation. However in practice, the computations may require more than this. For example to calculate an equilibrium free energy of solvation one may require linear response over the entire range of solute–solvent coupling from zero coupling (solute in gas phase) to full coupling (solute fully inserted in solvent). Even for this large range of coupling, it may be reasonable to assume linear response of the bulk electrostatic effect, but the response of the first solvation shell may show appreciable nonlinearity. The approaches mentioned above can account for this in various ways, e.g. by using a nonbulk dielectric constant in the first solvation shell, by treating all or part of the first solvation shell explicitly, or by including empirical atomic surface tensions on the solute. These same issues occur for dynamics, and they are compounded by the fact that the deviation from equilibrium is finite, so the formal justification for linear response is no longer applicable. In fact the assumption of linear response may even be qualitatively wrong for a nonequilibrium situation, such as the energy relaxation of a highly excited mode produced by photoexcitation or reaction [105]. A quantitative assessment of the effect of nonlinear response on calculated thermal reaction rates is not available, but the assumption that it is small or can be modeled by the methods mentioned above has worked well. 3.3.4 Absolute Free Energy Barriers, Reaction Mechanisms, and Kinetic Isotope Effects The standard-state free energy of activation for a liquid-phase reaction is ‡ ,o G‡ ,o = G‡o g + GS (3.27) where the first term on the right-hand side is the gas-phase (g) value, and the second term is the solvation contribution given by R,o ‡o G‡o S = GS − GS (3.28) 352 Continuum Solvation Models in Chemical Physics where the superscript R denotes the value for reactants. As discussed above, the solvation free energy includes a bulk electrostatic contribution and nonbulk electrostatic terms. The bulk electrostatic term can be calculated by the dielectric continuum model and is the largest contribution for ionic reactions. The other terms make a smaller contribution for ions, but for reactions involving neutral species, where the bulk electrostatic term is less important, nonbulk electrostatic solvation can be significant or even dominant. These terms can be calculated by empirical models. In addition to SMx and the cluster–continuum model, other continuum models have also been used to study reactions in liquids, including the polarized continuum model [133–135] (PCM), the conductor-like screening model (COSMO [136] and COSMO–RS [137, 138]), the generalized COSMO [139] (GCOSMO) model, conductorlike PCM [140] (CPCM), and isodensity PCM [141] (IPCM). Ionic reactions are especially interesting because they can have a large solvent effect. In the past 10 years, the important class of ionic SN 2 reactions have been studied through ab initio calculations coupled with continuum solvation models in different media [32, 142–150]. In the case of dipolar aprotic solvents, the performance of the continuum models is very good. Tondo and Pliego [147] have investigated the SN 2 reaction of CH3 COO− with ethyl halides: CH3 COO− + CH3 CH2 X → CH3 COOCH2 CH+X− X = Cl Br I. They have used MP4/CEP-31+G(d)//MP2/CEP31+G(d) electronic structure calculations and the Pliego–Riveros parametrization of the PCM model for DMSO solvent [128]. The standard-state free energies of activation were calculated to be 24.9, 20.0, and 185 kcal mol−1 , while the experimental values are 22.3, 20.0, and 166 kcal mol−1 , respectively. Thus, the theoretical calculations were able to predict the correct reactivity order. Another SN 2 reaction that was studied using this same PCM parametrization for DMSO solvent is the interaction of the cyanide ion with ethyl chloride [149]: CN− + CH3 CH2 Cl → CH3 CH2 CN + Cl− . The solution phase G‡o calculated at the CCSD(T)/6-311+G(2df,2p)//B3LYP/6-31G(d) electronic structure level is 241 kcal mol−1 and it is in excellent agreement with the experimental [151] value of 226 kcal mol−1 . On average, these theoretical calculations for SN 2 reactions overestimate the solution-phase barrier by only 2 kcal mol−1 , confirming the accuracy of the continuum model for ionic reactions in dipolar aprotic solvents, as anticipated from extensive pKa calculations [130]. In the case of protic solvents such as water, the continuum models are less accurate, especially for small ions or those with highly localized partial charges because of the importance of specific solute–solvent interactions in the first solvation shell [115, 132]. As an example, Pliego and Riveros [152] investigated the hydroxide ion addition to ethyl acetate in aqueous solution to form a tetrahedral intermediate. They used the PCM method with only electrostatic contributions. The liquid-phase free energy barrier for this step, calculated at MP2/6-311+G(2df,2p)//HF/6-31+G(d) level of electronic structure theory is 176 kcal mol−1 , while the experimental value is 188 kcal mol−1 . The small error is due to overestimation of the barrier by the MP2 calculations. In a similar system [158], MP4 calculations decrease the barrier by 2 kcal mol−1 in relation to MP2 energies. Thus, using more reliable gas-phase energies should lead to an underestimation of the barrier by about 3 kcal mol−1 . In addition, liquid-phase optimization could produce an even smaller barrier, increasing the deviation. The studies just reviewed were based on geometries optimized in the gas phase, and the nonbulk electrostatic contributions were not included. However these refinements Chemical Reactivity in the Ground and the Excited State 353 are included in some other work. As an example, Kormos and Cramer [144] have investigated the identity SN 2 reaction H2 C = CHCH2 Cl + Cl− in aqueous solution using DFT calculations and the SM5.42 method. Optimizations were done for both gas phase and solution in order to evaluate the solvent effect on the transition state structure and free energy. They found that liquid-phase optimization decreases the barrier by 15 kcal mol−1 in this case, and the gas-phase and liquid-phase geometries were reasonably close. In contrast, for the identity CH3 2 C = CHCH2 Cl + Cl− reaction, the barrier dropped by 6 kcal mol−1 , and a large difference was found between the geometry in aqueous solution and the gas-phase geometry. This behavior can be explained if we consider that in the latter case, the transition state resembles a stable tertiary carbocation species, allowing both of the chlorine atoms to be more distant from the carbon atom for the liquidphase transition state structure. Systems with solvent-dependent transition states include amide hydrolysis [153] and decarboxylations [154]. Such effects are sometimes studied by microhydration models [155–157]. It is expected that in DMSO and other dipolar aprotic solvents, liquid-phase optimization should be less important than in aqueous solution. The united atom for Hartree–Fock (UAHF) method [135] uses environment-dependent, charge-dependent atomic radii in order to try to improve the accuracy of continuum solvation calculations, but this has not always worked well. An example is the identity Cl− + CH3 Cl → ClCH3 + Cl− reaction in aqueous solution. Vayner et al. [145] have studied this symmetrical SN 2 reaction using the CPCM model with the UAHF parametrization. The gas-phase energies were determined at the CBS-QB3 level. The calculated free energy of activation was 353 kcal mol−1 , a very high value. They found that the solvent contribution to the barrier is 273 kcal mol−1 , which can be compared with an estimated experimental value of ∼ 23 kcal mol−1 . For the same system, Truong and Stefanovich [142] used the GCOSMO method to study these identity reactions in aqueous solution. In their calculations, the solvent contribution to the free energy of activation is in the range of 17–19 kcal mol−1 , depending on the ab initio method used. The lower reliability of the continuum models for modeling ionic reactions in situations where there are strong solute–solvent interactions can be partially overcome by introducing some explicit solvent molecules, as in the cluster–continuum model [116]. Two interesting systems that have been studied using this approach are the basic hydrolyses of methyl formate [158] and formamide [159]. An extensive analysis of the different reaction pathways of methyl formate was carried out. The free energy of activation profile for all the pathways was obtained at the MP4/6-311+G(2df,2p)/HF/6-31+G(d) level of electronic structure theory combined with the cluster–continuum model, where the IPCM method was used for the continuum electrostatics. Pliego and Riveros [158] calculated that the direct attack of the hydroxide ion on the carbonyl group, leading to a tetrahedral intermediate, has a free energy of activation of 152 kcal mol−1 , in good agreement with the experimental value of 153 kcal mol−1 . In addition, another reaction pathway was investigated, where the water molecule hydrating the hydroxide ion acts as the attacking species, and the hydroxide ion acts as a general base. Both the transition states for these pathways are presented in Figure 3.11. The free energy barrier for this general base catalysis mechanism was calculated to be 163 kcal mol−1 , indicating that this mechanism is less important. Therefore, the calculations resolved the experimental 354 Continuum Solvation Models in Chemical Physics O O O O O O O C H O O C Hydroxide ion attack O C H C O Coordination water attack Figure 3.11 Application of the cluster–continuum model for basic hydrolysis of the methyl formate. controversy [160] about the true reaction mechanism, indicating that the direct hydroxide ion attack is the most important pathway, although some products are generated from the coordination-water attack pathway. On the other hand, in the case of basic hydrolysis of formamide, Pliego [159] found that only the direct hydroxide ion attack can take place. This finding obtained using the cluster–continuum model, was recently confirmed by an explicit-solvation molecular dynamics study by Blumberger and co-workers [161, 162], where no coordination-water attack mechanism was found. A review [86] written in 1998 already gave 17 references for transition state geometries optimized in solution; this is particularly straightforward when stable analytic gradients [113, 163] are available. Liquid-phase optimization can be very important in some reactions. In Figure 3.8, although the relative energy of the stationary points changes due to solvation, gas-phase structures provide a good approximation. The situation is different − for neutral–neutral SN 2 reactions such as the example NH3 + CH3 Cl → NH3 CH+ 3 + Cl . The formation of two charged species has a large solvent effect, and the N − C and C − Cl distances in the transition state must be determined under liquid-phase conditions [32, 164–167]. However, SES and ESP models produce very similar free energy profiles as a function of Rb − Rf − Rb − Rf ‡ , where Rb is the breaking bond distance, and Rf is the forming bond distance [32]. Even more important than changing the geometries, in some cases the solvent can induce a different electronic structure than in the gas phase process, leading to a new reactive process. A very interesting example is the halogenation of alkenes [168–176] or alkynes [177, 178]. Figure 3.12 illustrates the products generated in an apolar solvent and in polar solvents. In low-polarity solvents, chlorination takes place through a radical mechanism [177], while in polar solvents, the stabilization of the charged species favors the ionic mechanism. In the case of bromination, the ionic mechanism also occurs in polar solvents [168, 171]. However, in apolar solvents, a second bromine molecule can participate of the process, forming the tribromide ion [168, 171]. Chemical Reactivity in the Ground and the Excited State ClH2C ent r solv X2 + H2C CH2 apola Br2 CH3 + Cl X = Cl Br3– X = Br Br CH2 H2 C 355 X polar solvent H2C CH2 + X– X = Cl, Br Figure 3.12 Halogenation in apolar and polar solvents. The halogenation reaction of ethylene has been modeled by many researchers [170, 172–176]. For chlorination in apolar solvents (or in the gas phase), the formation of two radical species requires the use of flexible CASSCF and MRCI electronic structure methods, and such calculations have been reported by Kurosaki [172]. In aqueous solution, Kurosaki has used a mixed discrete–continuum model to show that the reaction proceeds through an ionic mechanism [174]. The bromination reaction has also received attention [169, 170]. However, only very recently was a reliable theoretical study of the ionic transition state using PCM/MP2 liquid-phase optimization reported by Cammi et al. [176]. These authors calculated that the free energy of activation for the ionic bromination of the ethylene in aqueous solution is 82 kcal mol−1 , in good agreement with the experimental value of 10 kcal mol−1 . Xie et al. [179] calculated the free energies of reaction and activation in the gaseous and liquid phase for the following aryl ester hydrolysis reaction: His H2 O MeCOOC6 H4 CH3 → His HOC6 H4 CH3 + MeCOOH. In the gas phase they obtained a free energy of reaction Go = −45 kcal mol−1 and a free energy of activation G‡o of 443 kcal mol−1 . Solvation has only a small effect on Go ; PCM implicit solvent calculations yield −66 kcal mol−1 in HCCl3 and −80 kcal mol−1 in H2 O. However the transition state has considerable zwitterionic character and PCM calculations lower G‡o to 276 kcal mol−1 in HCCl3 and 161 kcal mol−1 in H2 O, which is consistent with experiment [180]. Rod et al. [181] compared continuum and discrete solvation models for the reaction by which SCH3 + 3 transfers a methyl group to an oxygen of a catecholate ligated to Mg2+ . The gas-phase reaction, which is calculated to be exoergic by 53 kcal mol−1 , was calculated to occur without a barrier. In solution, a simulation with 8800 T1P3P explicit solvent molecules yielded a reaction exoergicity of 21 kcal mol−1 and a free energy of activation of 131 kcal mol−1 . Four implicit solvation models yielded free energies of activation of 10.9, 11.2, 12.7, and 139 kcal mol−1 , in reasonable agreement with the explicit-solvent calculation. Although enzyme catalysis has usually been modeled with discrete solvent models [2, 182], Rod et al. [181] also performed an interesting comparison of discrete and continuum models for the methyl transfer from S-adenosylmethione to catecholate catalyzed by catechol O-methyltransferase. In this case the protein was explicit in both simulations, one with 7800 T1P3P water molecules and two others with implicit solvent, each having different atomic radii. The explicit calculation gave a free energy of activation of 163 kcal mol−1 , and the implicit ones both gave 155 kcal mol−1 . It is not surprising 356 Continuum Solvation Models in Chemical Physics that the implicit models are better for the enzyme reaction than the aqueous nonenzymatic reaction because the solvent is not interacting strongly with the reactants. A troubling result is that there are more protein hydrogen bonds and salt bridges in the implicit simulation than in the explicit one, but not all the hydrogen bonds and salt bridges of the explicit simulation are preserved in the implicit one. This kind of difference is highlighted in earlier studies [183–185] of protein dynamics where explicit and implicit solvent models were compared, and implicit models were found to be less accurate. Another comparison of discrete and continuum molecules was carried out for the decarboxylation of 4-pyridylacetic acid zwitterion in aqueous solution [157]. Transition states were optimized with the SM5.42 implicit solvation model with zero, one, or two discrete water molecules, yielding free energies of activation of 24.2, 21.2, and 183 kcal mol−1 , respectively, as compared to −47 kcal mol−1 in the gas phase. The fact that one can obtain quite different solvation free energies for ions without and with discrete solvent is not surprising in light of the discussion in Section 3.3.3. Including only two discrete waters without the continuum yields G‡ = +11 kcal mol−1 , very far from the results with implicit solvent, which again is not surprising since one expects that ionic solvation energies converge very slowly as solvent molecules are added. Kinetic isotope effects for this reaction were found to be less sensitive than G‡ to the inclusion of solvation effects. An important prototype reaction where both discrete and continuum solvation models have been applied is the Claisen rearrangement of allyl vinyl ether to 4-pentenal. This is an electrocyclic reaction proceeding by a [3,3] sigmatropic shift, and the interpretation of solvent effects is complicated by the difficulty of modeling the polarity of the transition state even in the gas phase, as reviewed elsewhere [186]. Severance and Jorgensen [187,188] provided the first correct account of the solvation effects on the basis of explicit-solvent calculations. Their calculated rate acceleration in aqueous solution, relative to the gas phase, corresponds to a lowering of G‡o by 38 kcal mol−1 , which may be compared to a lowering of 4.0 that they estimated from various experiments. The acceleration was attributed to enhanced hydrogen bonding at the transition state. A later explicit-solvent calculation by Gao [189] gave a lowering of 35 kcal mol−1 . Earlier predictions [190] by implicit-solvent models gave much smaller effect, 07 kcal mol−1 , but that is now attributed to inaccurate modeling of the charge distribution at the transition state in the early studies. Using a more accurate charge distribution based on MCSCF electronic structure calculations gave a lowering of G‡o of 43 kcal mol−1 [186]. These results indicate the high sensitivity of predicted solvent effects on rate constants to getting the charge distributions right in solution by self-consistent reaction fields. They also show that continuum models can sometimes account well for hydrogen bonding effects. Although ionic mechanisms are more common in aqueous solution than radical mechanisms, Nguyen and co-workers [191] presented an interesting application of continuum solvation models to a radical reaction, in particular CH3 + H2 O2 → CH4 + HO2 , in aqueous solution. They used the PCM/HF/6-31G∗∗ continuum solvation model, with UAHF radii and including electrostatics, cavitation, dispersion, and repulsion, to calculate the standard-state free energy of solvation of the transition state to be −54 kcal mol−1 and that of reactants to be −94 kcal mol−1 , with the difference being +41 kcal mol−1 . Chemical Reactivity in the Ground and the Excited State 357 A similar calculation with the COSMO–RS solvation model gave −47 kcal mol−1 for the transition state, −72 kcal mol−1 for reactants, and +25 kcal mol−1 for the difference. Experimentally, it is found that the reaction rate is 12 × 103 times slower in aqueous solution than in the gas phase [192], corresponding to an increase in the free energy of activation of 42 kcal mol−1 , in excellent agreement with PCM 41 kcal mol−1 but not with COSMO–RS 25 kcal mol−1 . The good agreement with PCM was attributed to successful parameterization. The solvation difference of the transition state from reactants was found to be very close 02–03 kcal mol−1 to the difference between that for products and that for reactants in this reaction. This is curious because the reaction is exothermic, and the transition state breaking and forming bond distances are (as expected from the Hammond postulate) more reactant-like than product-like, although the O − O − H bond angle is more product-like. Another application of continuum models to radical reactions is provided by the hydrogen abstracton H +CH3 OH → H2 +CH2 OH [61,193], for which experimental [194, 195] kinetic isotope effects are available. These studies [166,193] also include variational effects, multidimensional tunneling, nonequilibrium solvation, and kinetic isotope effects, as well as the coupling between these effects. Including an optimized multidimensional treatment of tunneling, the ESP rate constant for the perprotio reaction is 2.0 times larger than the SES one, and the NES rate constant is 52 % smaller than the ESP one for the ‘best’ nonequilibrium solvation parameters. The comparison of aqueousphase and gas-phase rate constants needs to be re-examined now because the gas-phase transition state is now better understood [196] than when the liquid-phase study was conducted. The 1,2-hydride shift in phenyl glyoxal hydrate to produce mandelate and the corresponding deuteride shift have been studied using continuum solvation models and VTST with multidimensional tunneling by Tresadern et al. [197]. They found that, starting from a reaction intermediate, the varational effect lowers the overbarrier rate constant k by 26 % and kinetic isotope effect (KIE) by 6 %. Tunneling, in contrast, raises k by a factor of 5.1 and the KIE by 71 %. Without corner cutting, the tunneling effect would be much smaller (factor of 3.6 and 51 %, respectively). A challenging test of all kinds of models of liquid-phase reaction rates is provided by the measurement of six different kinetic isotope effects (two secondary H/D at different positions, one 11 C/14 C, and 12 C/C13 at a different position, one 14 N/14 N, and one 35 Cl/37 Cl) for the SN 2 reaction between n − C4 H9 4 NCN and C2 H5 Cl in dimethylsulfoxide at 303 K [145]. The mean of the unsigned deviations from unity these six KIEs. The experimental values of these KIEs are 0.990, 1.014, 1.21, 1.001, l.000, and 1.007. All KIEs will then be calculated by transition state theory. In the same order, the best theoretical method, based on B3LYP/aug-cc-pVDZ gas-phase electronic structure calculations gave 0.994, 1.005, 1.17, 0.993, 1.000, and 1.007, which is very good for five of the six results. Adding solvation by the PCM/UAHF method yielded 0.973, 0.978, 1.17, 0.993, 1.000, and 1.007 (in the same order), which is less accurate in half the cases. Probably the conclusion to be drawn is that solvent effects are small, and the error is dominated more by the quality of the electronic structure theory than the quality of the solvation model. In some cases, the solvent effect on rate constants has been carefully investigated by continuum models, and it has been found to be small [198, 199]. 358 Continuum Solvation Models in Chemical Physics Continuum solvation models have also been used to rationalize the Hammett + parameters determined [200] from SN 1 solvolysis rate constants [201] of cumyl chlorides. In particular the SM5.42R/AM1 [112] model reproduces the experimental + within 24 %. The use of continuum models for placing the empirical correlations of physical organic chemistry on a firmer basis is in its infancy. 3.3.5 Competitive Reactions Many important chemical reactions have competitive parallel pathways. In some cases, this competition is very significant and can diminish the yield of desired product. The ability to predict branching ratios is helpful for planning synthetic routes, and as a consequence, an important goal of theoretical studies of chemical reactions is to be able to predict the product ratio. An interesting system where parallel pathways play an important role is the SN 2 nitration of alkyl halides (Figure 3.13). The nucleophilic attack of the nitrite ion can take place through the nitrogen, leading to nitroalkanes, or through the oxygen atom, producing the alkyl nitrite. Nitroalkanes are the desired products due to their importance as well as their utilization as chemical intermediates to further transformation. In the case of the reaction of the nitrite ion with n-hepthyl bromide, experimental studies show that only 67 % of the product is the nitroalkane, and a large amount of the alkyl nitrite (33 %) is formed [202]. C7H15NO2 + Br – (67%) – C7H15Br + NO2 C7H15ONO + Br – (33%) Figure 3.13 Experimental product ratio in the SN 2 reaction of n-hepthyl bromide with nitrite ion in dimethyl formamide (DMF) solution. Westphal and Pliego [203] have recently performed a high-level ab initio study of the prototypical system CH3 CH2 Br + NO− 2 , which is a good representation of the n-hepthyl bromide reaction. Gas-phase geometry optimizations were done at the MP2/6-31+G(d) level of electronic structure theory, followed by single-point energy calculations at the CCSD(T)/6-311+G(2df,2p) level (for bromine, Ahlrichs’ TZVPP basis set was used) and solvent contribution at PCM/B3LYP/6-31+G(d) level (DMSO solvent, Pliego and Riveros parametrization). Using transition state theory, they predicted that the nitroalkane and alkyl nitrite are formed in the proportion of 46:54, in good agreement with the experimental data for the hepthyl bromide, 67:33. This shows that continuum solvation models can be very useful for application in synthetic chemistry. Chemical Reactivity in the Ground and the Excited State 359 3.3.6 Catalyst Design and Control of Chemical Reactions Our present ability for modeling chemical processes using theoretical methods constitutes a very powerful tool in the design of new catalysts. Continuum solvation models have been used in the design and modeling of metallocene catalysts for olefin polymerization [204, 205] and Pd(II) ligand systems to perform aerobic oxidations of secondary alcohols [206]. Continuum solvation models have been used in the design of a novel supramolecular organocatalytic concept, aimed to selectively stabilize SN 2 transition states [148,150,207]. Such development can become a very useful tool for controlling the regioselectivity, chemoselectivity, and enantioselectivity of SN 2 reactions. It should also be mentioned that the solvent itself may be considered a catalyst [101–103, 153, 208–210]. 3.3.7 Conclusion Continuum solvation models can be used to predict the free energy of activation of chemical reactions and the effective potential for condensed-phase tunneling, and they can therefore be combined with transition state theory to predict chemical reaction rates. Acknowledgments D.G.T. is grateful to Christopher J. Cramer, Jiali Gao, Bruce C. Garrett, Gregory K. Schenter, and many students and postdoctoral research associates for collaboration on innumerable solvation projects. This work was supported in part by the National Science Foundation under grant no. CHE03-49122. J.R.P.Jr. thanks Professors Jose M. Riveros, Stella M. Resende and Wagner B. de Almeida, and students Gizelle I. Almerindo, Daniel W. 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