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Transcript
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
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Continuum Solvation Models in
Chemical Physics
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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
Copyright © 2007
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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.
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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.
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D. Poulsen, Int J Quantum Chem., 100 (2004) 1136. and references cited therein.
[34] Polyrate 9.6, http://comp.chem.umn.edu/polyrate/
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[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.
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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
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[13] P. Laug and H. Borouchaki, Generation of finite element meshes on molecular surfaces, Int.
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[14] P. Laug and H. Borouchaki, BLSURF – Mesh Generator for Composite Parametric Surfaces –
User’s Manual, INRIA Rapport Technique 0235 (1999)
[15] J. L. Pascual-Ahuir, GEPOL: Un metodo para calculo de superficies moleculares, Tesis
Doctoral, Facultad de ciencias quimicas, Universitat de València 1988.
[16] J. L. Pascual-Ahuir, E. Silla and I. Tunon, GEPOL: An improved description of molecular
surfaces. III. A new algorithm for the computation of a solvent-excluding surface, J. Comput.
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[17] C. S. Pomelli, J. Tomasi and R. Cammi, A Symmetry adapted tessellation of the GEPOL
surface: applications to molecular properties in solution, J. Comput. Chem., 22 (2001)
1262–1272.
[18] G. Scalmani, V. Barone, K. N. Kudin, C. S. Pomelli, G. E. Scuseria, and M. J. Frisch,
Achieving linear-scaling computational cost for the polarizable continuum model of solvation,
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Modern Theories of Continuum Models
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[19] G. Scalmani, N. Rega, M. Cossi and V. Barone, Finite elements molecular surface in
continuum solvent models for large chemical systems, J. Comput. Meth. Science Eng., 2
(2001) 159–164.
[20] M. J. Wenninger, Polyhedron Models, Cambridge University Press, New York 1974.
[21] C. S. Pomelli and J. Tomasi, Variation of surface partition in GEPOL: effects on solvation
free energy and free-energy profiles, Theor. Chem. Acc., 99 (1998) 34–43.
[22] L. Frediani, R. Cammi, C. S. Pomelli, J. Tomasi and K. Ruud, New developments in the
symmetry-adapted algorithm of the polarizable continuum model, J. Comput. Chem., 25
(2004) 375–385.
[23] M. Cossi, B. Mennucci, R. Cammi, Analytical first derivatives of molecular surfaces with
respect to nuclear coordinates, J. Comput. Chem., 17 (1996) 57–73.
[24] J. W. Harris and H. Stocker, General spherical triangle, §4.9.1 in Handbook of Mathematics
and Computational Science, Springer-Verlag, New York 1998, pp 108–109.
[25] C. S. Pomelli, A tessellationless integration grid for the polarizable continuum model reaction
field, J. Comput. Chem., 25 (2004) 1532–1541.
[26] H. Li and J. H. 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,
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[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.
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[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.
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iterative procedure for energies and geometry optimizations, Chem. Phys. Lett., 293 (1998)
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(1995) 20–30.
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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.
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[28] T. HaDuong, S. Phan, M. Marchi and D. Borgis J. Chem. Phys., 117 (2002) 541.
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J. A. Montgomery, Jr., T. Vreven, G. Scalmani, K. N. Kudin, S. S. Iyengar, J. Tomasi,
V. Barone, B. Mennucci, M. Cossi, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara,
K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai,
X. Li, H. P. Hratchian, J. E. Peralta, A. F. Izmaylov, J. J. Heyd, E. Brothers, V. Staroverov,
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Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J.
Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P.
Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O.
Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui,
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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. A systematic coupling of CC theory to other continuum methods, like
the ASC based methods is still an open problem, and thus great advances are expected
in the near future.
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Continuum Solvation Models in Chemical Physics
1.5.6 Conclusion
Molecular solutes described within QM continuum solvation models are characterized
by an effective Hamiltonian which depends on the wavefunction of the solute itself.
This makes the determination of the wavefunction a nonlinear QM problem. We have
shown how the standard methods of modern quantum chemistry, developed for isolated
molecules, have been extended to these solvation models. The development of QM
continuum methods has reached a satisfactory stage for completely variational approaches
(HF/DFT/MCSFC/VB). More progress is expected for continuous solvation model based
on MPn or CC wavefunction approaches.
References
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R. Cammi and J. Tomasi, J. Chem. Phys., 109 (1998) 2798; (d) R. Cammi, L. Frediani,
B. Mennucci, J. Tomasi, K. Ruud and K. V. Mikkelsen, J. Chem. Phys., 117 (2002) 13.
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Valle, R. Bonaccorsi, R. Cammi and J. Tomasi, Theochem 76 (1991) 295; (d) F. J. Olivares
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1.6 Nonlocal Solvation Theories
Michail V. Basilevsky and Gennady N. 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.
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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. The future will reveal whether it can survive
as a computational tool competitive with these popular and more familiar computational
schemes.
Acknowledgement
MVB and GNCh thank the RFBR (grant 04-03-32445).
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1.7 Continuum Models for Excited States
Benedetta Mennucci
1.7.1 Introduction
For long time it has been well known that solvents strongly influence the electronic
spectral bands of individual species measured by various spectrometric techniques (UV
visible spectrophotometries, fluorescence spectroscopy, etc.). 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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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. However, it should be noted that even for the spin–
spin coupling constants the supermolecular approach and especially the supermolecular
approach combined with the continuum model are usually more successful than the
continuum model alone.
Although methods based on molecular dynamics seem very promising, and, with
increase in computer power, are likely to become more widespread, continuum models
will probably remain in use, especially in the calculation of NMR parameters. NMR
spectroscopy is inherently ‘slow’, that is, the time scale of interaction with incident radiation allows for multiple rearrangement of the solvent structure. This makes continuum
models more realistic for NMR than for optical spectroscopies with shorter time scales.
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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. A critical comparison between experimental and computational results
is thus always necessary.
Acknowledgements
The authors wish to thank the Italian Research Ministry (MIUR) and Gaussian Inc. for
financial support. The integration between quantum mechanical and stochastic approaches
is the result of an ongoing collaboration with the group of Prof. Antonino Polimeno
(Dipartimento di Chimica, Università di Padova).
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[16] V. Barone and A. Polimeno, PCCP 8 (2006) 4609.
[17] E. Meirovitch, D. Igner, G. Moro and J. H. Freed, J. Chem. Phys., 77 (1982) 3915.
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(c) M. Fixman and K. Rider, J. Chem. Phys., 51 (1969) 2429.
[19] R. McWeeny, Methods of Molecular Quantum Mechanics, Academic Press, London, 1992.
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and M. J. Frisch, J. Chem. Phys., 104 (1998) 5497.
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M. J. Kaupp J. Am. Chem. Soc., 122 (2000) 9206.
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[29] J. Tomasi, B. Mennucci and R. Cammi, Chem. Rev., (Washington, D.C.) 105 (2005) 2999.
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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.
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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.
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circular dichroism: trans-1(S), 2(S)-Dicyanocyclopropane, J. Am. Chem. Soc., 109 (1987)
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Lett., 142 (1987) 153–158; (f) K. J. Jalkanen, P. J. Stephens, R. D. Amos and N. C. Handy,
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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,
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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
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[18]
[19]
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[21]
[22]
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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
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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,
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rotational strengths using density functional theory, Mol. Phys., 89 (1996) 579–594.
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R. Ditchfield, Mol. Phys., 27 (1974) 789–807.
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(Theochem.), 357 (1995) 225–235.
A. D. Becke, J. Chem. Phys., 90 (1993) 1372, 5648.
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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. This means that the
solvent participates in the observed optical effect not only by a modification of the
geometric structure and electronic density of the solute, but that part of the observed
OR or circular dichroism arises from the chiral solvent shell rather than from the solute
molecule as such. This is not accounted for by the PCM, and can be rendered only by
an explicit quantum mechanical treatment of at least the first solvent shell, or preferably
by molecular dynamics simulations.
Acknowledgments
This work has received support from the Polish Ministry of Science and Informatics
through the 1TO9AO713OMNil KBN grant. KR has received support from the Norwegian
Research Council through a Strategic University Program in Quantum Chemistry (Grant
No 154011/420) and a YFF grant (Grant No 162746/V00).
Properties and Spectroscopies
217
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2.6 Raman Optical Activity
Werner Hug
2.6.1 Introduction
Raman optical activity (ROA) has many facets. It is a spectroscopic method in its own
right and a tool which provides unique insight into the vibrational and specific aspects of
the electronic structure of dissymmetric molecules. 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)
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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. For molecules measured in the condensed phase, further advances in the understanding of the structure of liquids and of the influence of intermolecular interactions on
vibrational spectra will have to be gained. Both aspects represent formidable theoretical
challenges.
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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. Luo and H. Ågren, J. Chem. Phys., 110 (1999) 7960; (c) P. Macak, P. Norman, Y. Luo
and H. Ågren, J. Chem. Phys., 112 (2000) 1868.
[4] (a) R. Cammi, B. Mennucci and J. Tomasi, J. Phys. Chem. A, 102 (1998) 870; (b) R. Cammi,
B. Mennucci and J. Tomasi, J. Phys. Chem. A, 104 (2000) 4690.
[5] (a) P.Th. van Duijnen, de A. H. Vries, M. Swart and F. Grozema, J. Chem. Phys., 117 (2002)
8442; (b) L. Jensen, M. Swart and P.Th. van Duijnen, J. Chem. Phys., 122 (2005) 034103.
[6] (a) J. Kongsted, A. Osted, K. V. Mikkelsen and O. Christiansen, J. Mol. Struct.: THEOCHEM
632 (2003) 207; (b) A. Osted, J. Kongsted, K. V. Mikkelsen, P. -O. Astrand and
O. Christiansen, J. Chem. Phys., 124 (2006) 124503.
[7] J. Tomasi, B. Mennucci and R. Cammi, Chem. Rev., 105 (2005) 2999.
[8] R. Cammi, B. Mennucci and J. Tomasi, in M. G. Papadopoulos (ed.), Nonlinear Optical
Responses of Molecules Solids and Liquids: Methods and Applications, Research Signpost,
Kerala, India, 2003, p. 113.
[9] R. W. Munn, Y. Luo, P. 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.
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(2002) 12331.
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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 s