Download New Concepts in Inverse Quantum Chemistry - ETH E

DISS ETH NO. 21576
New Concepts in Inverse Quantum
Chemistry
A dissertation submitted to
ETH ZURICH
for the degree of
DOCTOR OF SCIENCES
presented by
THOMAS WEYMUTH
MSc Chemistry, ETH Zurich
born on November 26th, 1987
citizen of Winterthur (ZH)
accepted on the recommendation of
Prof. Markus Reiher, examiner
Prof. Hansjörg Grützmacher, co-examiner
2013
to my parents
Contents
1
Introduction
1.1 Rational Design of Chemical Compounds . . . . . . . . . . . . . .
1.2 Some Notational Conventions . . . . . . . . . . . . . . . . . . . . .
3
3
5
2
Theoretical Background of Quantum Chemistry
2.1 Introduction to Quantum Mechanics . . . . . . . . . . . . . . . . .
2.2 Quantum Field Theory of Electromagnetism: Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Relativistic Quantum Chemistry: The Dirac Equation . . . . . . . .
2.4 Nonrelativistic Quantum Chemistry: The Schrödinger Equation . .
2.4.1 Derivation of the Schrödinger Equation . . . . . . . . . . .
2.4.2 Calculation of Vibrational Frequencies . . . . . . . . . . . .
2.4.3 Electronic Structure Methods . . . . . . . . . . . . . . . . .
2.4.4 A Short Notice About Spin . . . . . . . . . . . . . . . . . .
2.5 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Classification of Exchange–Correlation Functionals . . . . .
2.5.2 Constructing Exchange–Correlation Functionals . . . . . . .
2.5.2.1 Investigating Model Systems: The Local Density
Approximation . . . . . . . . . . . . . . . . . . . .
2.5.2.2 Gradient Expansions of the Exchange–Correlation
Energy . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2.3 Constraint Satisfaction: The Generalized Gradient
Approximation . . . . . . . . . . . . . . . . . . . .
2.5.2.4 Empirical Fits . . . . . . . . . . . . . . . . . . . .
2.5.2.5 Modelling the Exchange–Correlation Hole . . . .
2.5.2.6 Admixture of Hartree–Fock Exchange . . . . . . .
2.5.3 Shortcomings of Density Functionals . . . . . . . . . . . . .
2.5.3.1 Delocalization Error/Self-Interaction Error . . . .
2.5.3.2 The Spin-Polarization/Static-Correlation Error . .
7
7
23
24
27
30
31
32
33
Concepts and Strategies for Rational Compound
3.1 Inverse Spectral Theory . . . . . . . . . . . .
3.2 Quantitative Structure–Activity Relationships .
3.3 Inverse Perturbation Analysis . . . . . . . . .
3.4 Model Equations . . . . . . . . . . . . . . . .
3.5 Optimized Wave Functions . . . . . . . . . .
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43
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48
3
i
Design
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3.6
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61
Assessment of DFT for Transition Metal Complexes
4.1 New Benchmark Set of Accurate Coordination Energies . . . . . .
4.1.1 WCCR10 Ligand Dissociation Energy Database of Large
Transition Metal Complexes . . . . . . . . . . . . . . . . . .
4.1.2 Computational Details . . . . . . . . . . . . . . . . . . . . .
4.1.3 Performance of Popular Density Functionals . . . . . . . . .
4.1.3.1 Structures . . . . . . . . . . . . . . . . . . . . . .
4.1.3.2 Zero-Point Vibrational Energies . . . . . . . . . .
4.1.3.3 Ligand Dissociation Energies . . . . . . . . . . . .
4.1.3.4 Comparison with Other Benchmark Studies . . .
4.1.4 Conclusions on the WCCR10 Benchmark Set . . . . . . . .
4.2 Investigating the Parameters of BP86 . . . . . . . . . . . . . . . . .
4.2.1 Computational Methodology . . . . . . . . . . . . . . . . .
4.2.1.1 Computational Details . . . . . . . . . . . . . . .
4.2.1.2 Investigation of Approximations . . . . . . . . . .
4.2.2 Dependence of WCCR10 Reaction Energies on the Parameters of BP86 . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Reparametrizing BP86 . . . . . . . . . . . . . . . . . . . . .
4.2.4 Conclusions on the Parameters of BP86 . . . . . . . . . . .
63
63
Gradient-Driven Molecule Construction
5.1 Concept of Gradient-Driven Molecule Construction . . . . . . . . .
5.2 Theory of Gradient-Driven Molecule Construction . . . . . . . . . .
5.3 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Model Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Direct Optimization by Positioning Nuclei and Adding Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1.1 Finding a Complex Binding N2 . . . . . . . . . .
5.4.1.2 Finding a Complex Activating N2 . . . . . . . . .
5.4.1.3 Effect of the Coordination Geometry . . . . . . .
5.4.2 Environment Potential Represented by Point Charges . . . .
5.4.3 Environment Potential Represented on the DFT Grid . . . .
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
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3.7
3.8
4
5
6
Advanced Sampling Techniques . . . . . . . . . . . . . .
3.6.1 Rational Design in Solid-State Chemistry . . . .
3.6.2 Inverse Band Structure Approach . . . . . . . . .
3.6.3 Linear Combination of Atom-Centered Potentials
3.6.4 Alchemical Potentials . . . . . . . . . . . . . . .
Mode- and Intensity-Tracking . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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Towards Inverse Design of Molecular Vibrational Properties
115
6.1 Finding Raman Optical Activity Signatures of Protein β-Sheets . . 115
6.2 Proposed Signatures of β-Sheets . . . . . . . . . . . . . . . . . . . 116
6.3 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . 118
6.4
6.5
7
6.3.1 Model Structures . . . . . . . . . . . . . . . . . . . .
6.3.2 Computational Details . . . . . . . . . . . . . . . . .
Results and Discussion . . . . . . . . . . . . . . . . . . . . .
6.4.1 Proposed Signatures . . . . . . . . . . . . . . . . . .
6.4.2 Additional Signatures . . . . . . . . . . . . . . . . .
6.4.3 Robustness of β-Sheet Signatures . . . . . . . . . .
6.4.4 Influence of Side Chain . . . . . . . . . . . . . . . .
6.4.5 Twisted β-Sheets . . . . . . . . . . . . . . . . . . . .
6.4.6 Microsolvation . . . . . . . . . . . . . . . . . . . . .
6.4.7 Comparison to Other Secondary Structure Elements
6.4.8 Differentiating Parallel and Antiparallel β-Sheets . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Conclusions and Outlook
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A List of Important Abbreviations
141
B List of Important Symbols
143
C List of Publications
145
D Standard and Inverse Vibrational Calculations with M OV I PAC
D.1 Introduction into M OV I PAC . . . . . . . . . . . . . . . . . . .
D.2 The M OV I PAC Philosophy . . . . . . . . . . . . . . . . . . . .
D.3 Performance of Numerical and Analytical Derivatives . . . . .
D.4 Technical Aspects of M OV I PAC . . . . . . . . . . . . . . . . .
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E Description of Program Implementations
155
E.1 Program for the Optimization of the Jacket Potential . . . . . . . . 155
E.2 Interfacing A DF with M ATHEMATICA . . . . . . . . . . . . . . . . 156
Abstract
Of special interest in chemical research is the design of new molecular compounds
and materials with favorable properties. This is a great challenge for quantum
chemistry, since standard computational methods rely on the a priori definition of
a molecular structure (in terms of a fixed framework of atomic nuclei and a given
overall charge and spin state). However, exactly such a structure is not known in the
design process. This problem is addressed in the field of inverse quantum chemistry,
in which one tries to formally invert the Schrödinger (or similar) equations in order
to be able to find a molecular structure compatible with a predefined property. In
a first part of this work, we review existing inverse quantum chemical approaches
and analyze their potential for future applications.
Then, we propose a new concept for the rational design of molecular compounds,
namely gradient-driven molecule construction. Starting from a predefined structural
fragment, one systematically creates a ligand sphere around this fragment such that
the geometric gradients on all atoms vanish and, thus, the entire structure represents
a stable molecule. In applying this concept, we present first steps towards the design
of a new transition metal catalyst for the homogeneous fixation of dinitrogen.
In this design procedure, we rely on density functional theory (DFT), which
is the only current quantum chemical theory capable of treating relatively large
systems at acceptable computational cost. The fact that the accuracy of a given
density functional for a particular molecular system cannot be known a priori,
creates a need for datasets of reliable reference values such that benchmark studies
can be carried out. However, data for large transtition metal compounds are hardly
available. Therefore, we present a new set of ten ligand dissociation energies of
large transition metal complexes for which accurate experimental gas-phase data are
available, and assess the performance of nine popular density functionals. Moreover,
we systematically investigate the specific empirical parameters in the popular BP86
functional.
Finally, with respect to potential future developments in the inverse design of
molecular vibrational properties, we present reference calculations on large model
systems of protein β-sheets. An in-depth analysis of these spectra provides characteristic Raman optical activity signatures for β-sheets and allows also to discrimintate
parallel from antiparallel sheet structure.
v
Zusammenfassung
Die Synthese von neuen Verbindungen mit wünschenswerten Eigenschaften war und
ist in der Chemie seit jeher ein wichtiges Forschungsgebiet. Für die Quantenchemie stellt dies eine besondere Herausforderung dar, denn die molekulare Struktur
(definiert in der Quantenchemie als räumliche Anordnung von Atomkernen sowie
Gesamtladungs- und -spinzustand) ist zu Beginn des Entwicklungsprozesses zumindest nicht vollständig bekannt. Genau diese wird aber für quantenchemische
Rechnungen benötigt, da der Hamiltonoperator vom Molekül abhängt. Die inverse
Quantenchemie versucht, Verbindungen mit gewissen, vordefinierten Eigenschaften
mit Hilfe computergestützter quantenchemischer Berechnungen zu finden. In einem
ersten Teil der vorliegenden Arbeit werden bestehende inverse quantenchemische
Vorgehensweisen vorgestellt und beurteilt.
Sodann stellen wir ein neues Konzept für das Design neuer Verbindungen vor,
die sogenannte “gradient-driven molecule construction”. Dabei geht man von einem
vordefinierten Molekülfragment aus, das dann durch eine geeignete Ligandensphäre
stabilisiert wird. Wir stellen verschiedene Ansätze für das Konstruieren einer solchen Ligandensphäre vor und zeigen erste Schritte in der Entwicklung eines neuen
Übergangsmetallkomplexes für die homogen-katalytische Distickstofffixierung.
Solche (grossen) Übergangsmetallverbindungen können gegenwärtig nur mit der
Dichtefunktionaltheorie beschrieben werden, da andere quantenchemische Methoden zu rechenintensiv sind. Allerdings ist die Genauigkeit eines gewissen Dichtefunktionals für ein gegebenes Molekül nicht a priori klar. Zur Bewertung eines
Dichtefunktionals braucht man deshalb genaue Vergleichswerte. Solche sind aber
für grössere Übergangsmetallkomplexe kaum vorhanden. Wir haben deshalb einen
eigenen Datensatz (genannt WCCR10) geschaffen, der die experimentellen Energien der Ligandendissoziationsreaktionen zehn grosser Übergangsmetallkomplexe
enthält. Diese experimentellen Referenzwerte sind sehr genau und wurden in der
Gasphase bestimmt, was für quantenchemische Berechnungen ideal ist. Zum Abschluss dieses Teils der Arbeit werden die Reaktionsenergien des WCCR10-Satzes
mit neun verschiedenen, bekannten Dichtefunktionalen berechnet und mit den experimentellen Referenzwerten verglichen. Hierbei wird auch auf die im Funktional
BP86 verwendeten empirischen Parameter eingegangen.
Im dritten Teil der vorliegenden Arbeit schliesslich werden theoretische Spektren
der Raman optischen Aktivität von β-Faltblattmodellen präsentiert und analysiert.
Die Analyse identifiziert charakteristische Signaturen von parallelen und antiparallelen β-Faltblättern und vergleicht diese Ergebnisse mit experimentellen Befunden.
Diese Rechnungen können für spätere Entwicklungen im Bereich des inversen Designs von Schwingungseigenschaften als Referenz verwendet werden.
Chapter 1
Introduction
“The purpose of computing is insight,
not numbers.”
R. W. Hamming
1.1
Rational Design of Chemical Compounds
The last decades have witnessed the fast-paced development of a wide range of
quantum chemical methods [1], such as highly accurate but also computationally
demanding wave-function-based approaches like Coupled Cluster and Configuration
Interaction [2] and density functional theory [3], which allows for the description
of large molecular systems consisting of hundreds of atoms — usually at reduced
accuracy, however. Today, these computational approaches are firmly established
in chemistry, and they are essential to all areas of chemical research.
Underlying all these methods is the time-independent Schrödinger equation1 ,
ĤΨ = EΨ,
(1.1)
where Ψ is the sought-for wave function characterizing the system under investigation, E the energy associated with this wave function, and Ĥ is the Hamiltonian
operator. In nonrelativistic quantum chemistry, the Hamiltonian of an assembly of
(point-like) atomic nuclei and electrons is given by
X ZI X 1
X ZI ZJ
1X 1
1X
Ĥ = −
∆I −
∆i −
+
+
.
(1.2)
2 I mI
2 i
r
r
r
iI
ij
IJ
i,j>i
i,I
I,J>I
The indices I, J and i, j run over all nuclei and electrons, respectively. mI denotes
the rest mass of nucleus I, ZI its charge number, and rij is the spatial distance
between particles i and j. The Laplacian is defined as
∆i =
1
∂2
∂2
∂2
+
+
.
∂x2i
∂yi2 ∂zi2
(1.3)
A short review of the most important theoretical concepts in quantum chemistry is given in
chapter 2.
4
Introduction
The first two terms in Eq. (1.2) are the kinetic energy operators of the nuclei and
electrons, while the last three terms describe the electron–nucleus attraction, the
electron–electron repulsion, and the nucleus–nucleus repulsion, respectively. The
rather complicated Hamiltonian of Eq. (1.2) is usually simplified by invoking the
well-known Born–Oppenheimer approximation, in which the nuclear coordinates are
treated as (fixed) parameters. With that, we can define the electronic Hamiltonian
X ZI X 1
X ZI ZJ
1X
∆i −
+
+
,
(1.4)
Ĥel = −
2 i
r
r
r
iI
ij
IJ
i,j>i
i,I
I,J>I
and the corresponding electronic Schrödinger equation
Ĥel Ψel = Eel Ψel ,
(1.5)
where the individual variables have a similar meaning as above, but referring to the
electronic structure (in a fixed nuclear framework) only. The molecular structure,
defined as an assembly of atomic nuclei fixed in space, is a direct consequence of
this approximation.
For a given assembly of atomic nuclei and electrons, the nonrelativistic electronic Hamiltonian is unequivocally defined. The methods mentioned above aim at
an approximate solution of the electronic Schrödinger equation. In principle, all
observables and molecular properties of interest can then be calculated from the
wave function, which contains all information that can possibly be known about
the system [4]. Therefore, quantum chemical methods provide us with a wealth
of information about a certain system, but only after this very system has been
specified in terms of the nuclear framework — the structure — for which a given
property shall be calculated. For the design of new materials one encounters the
reverse situation, where a desired property is known and a molecule having this
very property is searched for. Given the “forward” direction from structure to
property, extensive screening of structures is necessary to find a molecule with a
predefined property (one example for this approach is drug design). It is thus highly
desirable to develop computational approaches which would render such time- and
cost-intensive trial-and-error procedures obsolete. In order to achieve this, the direction needs to be reversed, i.e., from properties to structures. Such approaches
are called “inverse approaches”.
Inverse problems do not only occur in the field of chemistry, but play an important
role in diverse areas such as geophysics (where one tries to infer the composition of
the Earth’s mantle from gravimetric measurements [5]), medical imaging (where one,
for example, attempts to clarify the internal structure of tissues by ultrasonic pressure
waves [6]) and computer vision (where one tries to construct three-dimensional data
from lower-dimensional images [7]). Accordingly, the theory of inverse problems
constitutes a whole branch of mathematics with important contributions made in the
first half of the 20th century by Russian researchers (see Ref. [8] for an extensive
bibliography). Clearly, since inverse problems represent an important aspect in
many fields of research, the study of inverse problems in quantum chemistry may
well benefit from developments made in other fields [9, 10].
As we will see in chapter 3, where we will review existing inverse quantum
chemical approaches, a direct inversion of the Schrödinger equation is not possible.
1.2 Some Notational Conventions
5
Therefore, contemporary inverse approaches are not truly inverse from a mathematical point of view, but rather rely on sophisticated sampling and optimization
algorithms. The greatest challenge that such an application faces is the huge size
of chemical space, i.e., the space of molecules accessible by contemporary synthetic protocols. It has been estimated that this size is between 1020 and 1024
molecules [11]. However, an estimation of the total size of chemical space highly
depends on the assumptions made; for example, the number of molecules having
up to 30 carbon, nitrogen, oxygen, and sulfur atoms has been estimated to be larger
than 1060 [12], while the number of proteins that could theoretically exist is roughly
10390 [13] (for an average size of 300 amino acid residues per protein). This might
be the reason for the fact that inverse quantum chemical approaches have only
emerged during the past 20 years, although the basic idea has been present much
earlier [14].
In this work, we propose a new concept for the design of stable compounds
starting from a predefined molecular fragment. As a proof of principle, we present
first steps towards the rational design of a transition metal catalyst suitable for
homogeneous nitrogen fixation in chapter 5. In doing so, we make heavy use of
density functional calculations. Since the electronic structure of transition metal
complexes is known to pose a challenge for many quantum chemical methods,
a reliable assessment of the performance of density functionals for such systems
is mandatory. In section 4.1, we therefore present the WCCR10 database, which
contains ligand dissociation reactions for ten large transition metal complexes, for
which accurate experimental data is available, and investigate nine different density
functionals. We also take a closer look at the exchange–correlation functional BP86,
which we will utilize in most applications presented in this work in section 4.2.
Finally, with regards to an inverse design of molecular vibrational properties, we
present in chapter 6 an extensive vibrational analysis of protein β-sheets, which
can be used as reference in future inverse applications.
1.2
Some Notational Conventions
In general, we follow the recommendation of IUPAC throughout this thesis [15],
but we may deviate somewhat from these recommendations in order to improve
clarity and readability. Below, we summarize some of the more important formal
conventions adopted in this work.
Any vectors and matrices are printed in boldface; we do not further distinguish the
dimensionality or tensorial nature of these. Scalar and (yet) unspecified quantities
are printed in normal type. For classical quantities and the corresponding operators
in a quantized framework, we will generally employ the same symbol. Particularly,
we will usually not specifically denote an operator by the typical hat which one
often encounters. Instead, the actual meaning of a given symbol will become
obvious from its context. We refer the reader to appendix B for a list of important
symbols used in this thesis.
In general, we use Hartree atomic units, i.e., we set the dielectric constant of
the vacuum multiplied by 4π, 4π0 , the reduced Planck constant, ~, the elementary
6
Introduction
charge, e, and the electron rest mass, me , all equal to one. However, we might
occasionally switch to a different unit system, in which case this will be stated
explicitly.
Chapter 2
Theoretical Background of
Quantum Chemistry
“Nothing great can be achieved
without the elementary curiosity of the
philosopher.”
M. Born
Driven by human curiosity, mankind was always striving to explain their world on
a rational basis. As a result, we have nowadays at our disposal a huge amount
of knowledge in the field of physics and natural sciences in general. Needless
to say that this thesis does not provide the space for even the shortest outline of
the impressive history of physical science; in any case, we do neither want to
provide the reader with a thorough introduction into physics nor with a complete
recapitulation of the scientific theories established today, as this cannot be the
task of a dissertation. (For more detailed treatises, we refer the reader to the
extensive literature available, e.g., Refs. [16, 17]. Moreover, we also would like
to point the reader’s attention to a very interesting historical perspective of the
early days of quantum chemistry [14].) Instead, the intention of this chapter is
rather to briefly review the concepts most important to this thesis — namely, those
of quantum mechanics and the application of these to chemical problems — and to
bring them into relation with each other, while the reader is supposed to already
have a general familiarity with them.
2.1
Introduction to Quantum Mechanics
Quantum mechanics allows us to describe matter at microscopic scales, as opposed
to classical Newtonian and Einsteinian mechanics. In quantum mechanics, one
introduces the concept of a state of a given system. This state is associated with a
state (or wave) function Ψ, from which any information that can possibly be known
about the system can be obtained by acting upon it with the hermitean operators
associated with the respective observables [4]. Every hermitean operator O features
8
Theoretical Background of Quantum Chemistry
a set of orthonormal eigenfunctions {Φi } for which we have
OΦi = oi Φi ,
(2.1)
i.e., to every eigenfunction corresponds exactly one eigenvalue (the index i is
often referred to as the so-called quantum number). In any measurement of the
observable represented by O one will obtain exactly one of the eigenvalues {oi }.
As the eigenfunctions {Φi } can always be chosen to form a complete orthonormal
basis set, we can expand any state Ψ into these eigenfunctions,
X
Ψ=
ci Φi ,
(2.2)
i
where ci are the expansion coefficients. The probability of measuring oi is then
simply given by ci 2 . Furthermore, we can define the expectation value
R∞
ΨOΨdτ
hΨ| O |Ψi
=
,
(2.3)
hOi = R−∞
∞
hΨ|Ψi
ΨΨdτ
−∞
where τ denotes all dynamic variables of the system, and where we introduced
Dirac’s well-known bracket notation in the last step. It should be noted here
that although we can extract physical information from it, the state function itself
does not have any direct physical interpretation. However, according to Born, the
absolute square of it represents a probability density distribution; for a system of
N spinless particles, the expression |Ψ(r 1 , r 2 , . . . , r N )|2 dr 1 dr 2 . . . dr N , where r i
denotes the spatial variable of particle i, gives the probability of finding particle
1 in the volume element dr 1 while at the same time finding the other particles in
their respective volume elements. From the trivial fact that all particles have to be
somewhere, we can define the normalization condition
!
hΨ|Ψi = 1.
(2.4)
We further postulate the time evolution of a quantum mechanical state Ψ to be
given by
i~
∂
Ψ = HΨ,
∂t
(2.5)
where t is the time variable and H the so-called Hamiltonian operator, the mathematical form of which remains yet to be determined (see subsequent sections1 ).
However, from a dimensional analysis we understand already at this point that H
must represent some energy, which for now obvious reasons has to be the energy of
the system represented by Ψ. If the Hamiltonian does itself not explicitly depend
1
We should note here that the mathematical expression for the operator representing a given
quantity is not always obvious. Often, one makes use of the so-called correspondence principle
in order to deduce the mathematical form for an operator from its classical analog, by promoting
the variables for space and momentum to their respective operators. The correspondence principle
thus represents the fact that quantum mechanics must essentially reduce to classical mechanics
in macroscopic dimensions.
2.2 Quantum Electrodynamics
9
on time, one can trivially separate time and spatial variables in Eq. (2.5) to get the
two equations
i~
∂
Ψ = EΨ,
∂t
(2.6)
and
HΨ = EΨ,
(2.7)
where E now denotes the energy of the system, which is the central quantity
in quantum chemistry. Especially in the context of nonrelativistic quantum mechanics, the last two equations are often referred to as the time-dependent and
time-independent Schrödinger equations, respectively.
2.2
Quantum Field Theory of Electromagnetism:
Quantum Electrodynamics
To the extent of our current knowledge, all natural phenomena are governed by the
concerted action of four fundamental forces: the gravitational interaction, leading to
a mutual attraction of any two or more massive bodies; the electromagnetic interaction, present in any system of electrically charged particles; the strong interaction,
holding together quarks and possibly antiquarks in hadronic particles (e.g., atomic
cores); and finally the weak interaction, which affects all fermions (i.e., particles
with half-integer spin) and which is, among other things, vital for the explanation
of the radioactive decay. A complete physical theory, i.e., one which would be
capable of explaining satisfactorily all natural phenomena, would thus have to take
into account all these four forces. However, despite all efforts undertaken so far,
such a theory has not been found to the present day. Furthermore, although a
thorough discussion of such issues would be far beyond the scope of this work, we
should mention here for the sake of completeness that there are advanced physical
theories speculating about an additional fifth force (see, e.g., Ref. [18]).
Among the four fundamental forces, both gravitation and the electromagnetic
interaction can be experienced by human beings in their everyday life. The electromagnetic interaction is particularly important for explaining essentially all chemical
phenomena, ranging from macroscopic properties such as color and odor of a particular substance to microscopic observations at a molecular level, e.g., structural
parameters such as bond lengths, and spectroscopic data of all kinds. Therefore, it
would be desirable in chemistry to have a complete theory of the electromagnetic
interaction at hand, as such a theory would enable one to precisely describe mathematically any chemical system. Indeed, by quantizing observable quantities (e.g.,
energy) and the associated interaction particle fields, Feynmann, Schwinger, Tomonaga and others developed the theory of quantum electrodynamics (QED) [19, 20]
in the middle of the 20th century, which yields impressively accurate results in
all cases encountered so far (for example, the agreement between calculated and
experimentally determined values for the dissociation energies of all isotopomers
10
Theoretical Background of Quantum Chemistry
of the dihydrogen molecule is excellent [21–25]). However, while it is certainly a
very sophisticated theory, QED is also highly complicated, leading to mathematical expressions the evaluation of which is computationally extremely demanding.
Therefore, QED is not applicable to molecules containing more than only two or
three atoms. In order to be able to make theoretical predictions for larger systems,
we thus need to introduce further approximations not present in QED2 , making the
resulting formulas less costly from a computational point of view.
2.3
Relativistic Quantum Chemistry:
The Dirac Equation
One approximation which proves particularly beneficial is to renounce from a
complete quantum field theoretical description of both radiation and matter, and
instead only treat the former quantum mechanically, while the latter is described
classically, i.e., by Maxwell’s equations. In this case, the Hamiltonian in Eq. (2.5)
for a single particle in an external electromagnetic field can be cast into the form [26]
q
H (D) = cα · p − Aext + βmc2 + qφext ,
c
where c denotes the speed of light in the vacuum,
0 σi
(1)
(2)
(3)
(i)
α = (α , α , α )
α =
σi 0
(2.8)
(2.9)
and
12 0
β=
0 −12
are the Dirac matrices in standard representation with
1 0
0 1
0 −i
σ1 =
σ2 =
σ3 =
1 0
i 0
0 −1
(2.10)
(2.11)
being the Pauli spin matrices, the momentum operator is usually chosen to be
p = −i∇
(in the case of which the position operator is simply r̂ = r), q
resent charge and mass of the particle, respectively, and Aext and
(external) vector and scalar potentials, which completely determine
electromagnetic field. We denote this special form of the Hamiltonian
Hamiltonian and indicate it by the subscript “(D)”. Since the Dirac
2
(2.12)
and m repφext are the
the external
as the Dirac
matrices are
Also QED is not entirely free of approximations; most notably, actual calculations take
usually advantage of perturbative expansions truncated at a finite order of terms which of course
leads to some truncation error.
2.3 Relativistic Quantum Chemistry
11
four dimensional, it is easy to see that the state function corresponding to the Dirac
Hamiltonian is a four dimensional vector, a so-called 4-spinor. In many situations,
it is convenient to resort to a two-dimensional notation by collecting the first two
elements of the 4-spinor into what is called the large component, and likewise the
last two components into the small component, i.e., we write the 4-spinor as
 
Ψ1
(L) Ψ2 
Ψ


Ψ= =
.
(2.13)
Ψ3
Ψ(S)
Ψ4
With that, we can separate the Dirac equation with the Hamiltonian from Eq. (2.8)
into the two equations
q
(2.14)
cσ · p − Aext Ψ(S) + mc2 Ψ(L) + qφext Ψ(L) = EΨ(L) ,
c
and
q
cσ · p − Aext Ψ(L) − mc2 Ψ(S) + qφext Ψ(S) = EΨ(S) .
c
(2.15)
Starting from the Eq. (2.8) for a single particle, one can define what is usually
referred to as a quasi-relativistic many-particle Hamiltonian suitable to describe
molecular systems by employing a single (absolute) time frame for all particles
and invoking the well-known Born–Oppenheimer approximation (see also below).
We then obtain
X (D) X
X ZI ZJ
X p2
I
+
hi +
gi,j +
H (qr) =
.
(2.16)
2m
|r
−
r
|
I
I
J
i
i<j
I<J
I
The first term represents the kinetic energy of the nuclei, mI being the mass of
nucleus I and the sum running over all nuclei, which is treated entirely nonrelativistic. The second term, in which the sum goes over all electrons, contains the
kinetic energy of these together with the attractive nuclear potential, and is largely
inspired by Eq. (2.8),
1
(D)
(2.17)
hi = cαi · pi + Aext + (β i − 14 ) c2 − φext + Vnuc
c
(the charge of the electron qe equals −1 in Hartree atomic units). Note that we
subtracted the rest energy in the above equation, as opposed to Eq. (2.8). The
nuclear potential energy is given by
Vnuc = −
X
I
ZI
.
|r i − r I |
(2.18)
The third term in Eq. (2.16) denotes the mutual repulsion of the electrons; in its
simplest form it is given by
gi,j =
1
,
|r i − r j |
(2.19)
12
Theoretical Background of Quantum Chemistry
in the case of which one calls H (qr) usually the Dirac–Coulomb Hamiltonian. It
does not take into account any retardation effects arising from the finite speed of
light. Such effects can, however, be incorporated by employing modified forms of
gi,j as is the case, e.g., in the Dirac–Coulomb–Breit Hamiltonian. The last term
in Eq. (2.16) finally captures the nucleus–nucleus interaction, which is constant in
the Born–Oppenheimer approximation. Nowadays, such quasi-relativistic Hamiltonians are implemented in computer programs such as D IRAC [27] and are heavily
used in four-component calculations. Still, we want to point out here that they
are unsatisfactory from a conceptual point of view; most notably, they are not
invariant under Lorentz transformations nor do they feature a bounded spectrum
(which, however, is already not satisfied in the original Dirac equation). Nevertheless, the numerical data obtained employing, e.g., the Dirac-Coulomb Hamiltonian
are in good agreement with experimental values. In fact, especially for molecules
containing heavy elements such as transition metal complexes, so-called relativistic
effects (i.e., the difference between expectation values arising from a relativistic
and a fully nonrelativistic treating) must not be neglected [26, 28, 29]. However,
four-component relativistic computations are hardly affordable for large molecules
such as the transition metal complexes dealt with in this thesis. Therefore, a number
of approximations have been developed which reduce the original four-component
structure of the spinors to only two components and generally are computationally
much less expensive. In this context, one can distinguish between two main directions of research. While transformation techniques such as the Douglas–Kroll–Hess
procedure [30–32] focus on a unitary transformation of the Dirac equation to arrive
at a block diagonal structure of the respective operators, elimination techniques try
to express two components by the other two components by rearranging the Dirac
equation (which, in its original form, represents four coupled integro-differential
equations). One of the best known of these elimination techniques is called Zeroorder regular approximation (ZORA) [33–37], and is implemented in A DF [38]
(note that this method has also been proposed independently by different authors in
an earlier paper [39]). ZORA is a very efficient means of incorporating relativistic
effects in a standard nonrelativistic calculations; it is also used for many of the
calculations done in this work.
Another very successful approximation with which relativistic effects can be easily
incorporated into standard nonrelativistic quantum chemical procedures and at the
same time greatly reduce computational cost are effective core potentials (ECPs)
[40, 41]. ECPs are basically model potentials which mimick the core electrons of
heavy nuclei. These electrons then have not to be treated explicitly in the rest
of the Hamiltonian, which is the reason for the speedup just mentioned. If these
model potentials are now chosen such that they properly reflect the (true) relativistic
reference, they can successfully account for relativistic effects (we should mention
here that the core regions are most important for relativistic effects) [42]. Recently,
von Lilienfeld et al. demonstrated the optimization of ECP parameters such that
not only atomic, but also molecular properties are correctly reproduced [43].
2.4 Nonrelativistic Quantum Chemistry
2.4
13
Nonrelativistic Quantum Chemistry:
The Schrödinger Equation
2.4.1
Derivation of the Schrödinger Equation
The nonrelativistic limit of Eq. (2.16) can be obtained by letting the speed of light
approach infinity, c → ∞. Obviously, this only affects the one-electron operator
(D)
hi , as the speed of light does not occur in the other three terms of the Dirac–
Coulomb Hamiltonian. When analyzing the one-electron operator in Eq. (2.17) we
immediately see that any magnetic effects mediated by the external vector potential
Aext vanish. In order to deal with the divergent terms of first and second order in c,
we investigate the one-electron operator for a single particle in split notation which
allows us to represent the small component in terms of only the large component,
Ψ(S) =
2c2
cσ · p
Ψ(L)
+ Ẽ + φext − Vnuc
(2.20)
where Ẽ is the energy eigenvalue of the one-electron operator only. With this,
we can now eliminate the small component in the Dirac equation of the large
component; we obtain
(σ · p)(σ · p)
2+
Ẽ−Vnuc +φext
c2
Ψ(L) + Vnuc Ψ(L) − φext Ψ(L) = ẼΨ(L) .
(2.21)
By taking the limit for c → ∞ and realizing that
(σ · p)(σ · p) = p2
(2.22)
(which is actually a special form of the so-called Dirac relation [26]) we find that
(D)
lim hi Ψ(L) =
c→∞
p2 (L)
Ψ + Vnuc Ψ(L) − φext Ψ(L) = hi Ψ(L) ,
2
(2.23)
and, thus, the Dirac equation of the large component reduces to the well known
Schrödinger equation with the nonrelativistic Hamiltonian being given by
H (nr) =
X
X
X ZI ZJ
X p2
I
+
hi +
gi,j +
.
2mI
|r I − r J |
i
i<j
I<J
I
(2.24)
Therefore, in the nonrelativistic Hamiltonian, we account for the kinetic energy
of nuclei and electrons, respectively, for the mutual repulsion of electrons and
nuclei among themselves, as well as for the attractive interactions between nuclei
and electrons (and possibly for any effects stemming from external electromagnetic
potentials). Note that one has only to take care of pairwise interactions, since in
nonrelativistic quantum chemistry, one treats all nuclei and all electrons as being
point-like, and hence, no polarization effects can be present (this approximation
has been found to be excellent in the nonrelativistic context [44] while they may
be nonnegligible in a fully relativistic treatment [45]).
14
Theoretical Background of Quantum Chemistry
2.4.2
Calculation of Vibrational Frequencies
The Hamiltonian given in Eq. (2.24) is the fundamental operator for most quantum
chemical applications, and is also underlying almost all computations done in this
work. However, it is easy to see that this operator is highly complex leading to a very
complicated Schrödinger equation the solution of which is very involved. Therefore,
a range of approximations has been developed which considerably simplify the
Schrödinger equation but introduce only minor errors. One recognizes that the
nuclear masses are typically several orders of magnitude larger than the electron
mass; in fact, this size difference permits a separation of nuclear and electronic
motions, an approximation which has been proposed by Born and Oppenheimer
[46] and is therefore known as the Born–Oppenheimer approximation. Pictorially
speaking, the nuclei move much slower than the electrons due to their larger
masses. The electrons, in turn, can almost instantaneously adapt to any change in
the nuclear configuration. Therefore, one can set the nuclear coordinates to fixed
values, and treat them as parameters instead of as true dynamic variables. This
essentially removes all nuclear degrees of freedom from the calculation and therefore
simplifies matter considerably. The resulting simplified Schrödinger equation can
then be solved for given nuclear configurations, and in general, the energy eigenvalue
will be different for every single such configuration. The collection of all these
energies for different nuclear configurations (of the same molecule) is referred to
as the potential energy surface (PES).3 In a rigorous treatment, one first defines
the electronic Hamiltonian,
X p2
I
,
(2.25)
Hel = H (nr) −
2m
I
I
which is the full nonrelativistic Hamiltonian without the kinetic energy of the nuclei.
Note that here and in the following, we omit the superscript “(nr)” for the sake of
clarity; if not stated otherwise, we will always deal with nonrelativistic operators.
The solution of the Schrödinger equation with the electronic Hamiltonian yields the
electronic state functions Ψel,n and electronic energies, Eel,n , respectively, where
n shall denote a general (set of) quantum numbers. Note that Hel still contains
the nuclear coordinates as variables! We can now write the total state function by
expanding it into the basis of electronic state functions, i.e.,
X
Ψk ({r i }, {r I }) =
χk,n ({r I })Ψel,n ({r i }, {r I }).
(2.26)
n
With this ansatz, one can show that the full Schrödinger equation finally takes on
the form
X
X p2
I
χk,m +
Cm,n χk,m + Eel,m χk,m = Ek χk,m ,
(2.27)
2mI
n
I
3
As an interesting side remark, we note that the notion of a molecular framework, where
the positions of the atomic nuclei are precisely known, is a direct consequence of the Born–
Oppenheimer approximation. If the nuclear coordinates were treated on the same ground as the
electronic ones, the positions of the nuclei would not be known due to the well-known Heisenberg
uncertainty relation [47] (see also Refs. [48, 49]).
2.4 Nonrelativistic Quantum Chemistry
15
where Cm,n is given by
Cm,n =
X
I
1
2 hΨel,m | ∇I |Ψel,n i −
∇I
2mI
+
X
I
1
hΨel,m | −
∆I |Ψel,n i .
2mI
(2.28)
These double sums are called nonadiabatic couplings, and represent couplings between near-lying vibrational levels of different electronic states. In the original
Born–Oppenheimer approximation, we set all these couplings to zero, which implies that all nuclear coordinates are not treated as variables anymore, but set to
some fixed values. We then find
X p2
I
χk,m + Eel,m χk,m ≈ Ek χk,m .
(2.29)
2mI
I
While the first term represents the kinetic energy of all nuclei, the second term represents the potential energy. Interestingly, this is exactly the electronic energy. One
can interpret this as the nuclei moving in the potential generated by the electrons.
Due to its simplicity, the above equation is most often employed when information
about the nuclear motion is to be obtained. However, the potential energy term
is not straightforward to compute. In the Born–Oppenheimer approximation, the
electronic energy can only be obtained point-wise (namely, for individual nuclear
configurations). While it would, in principle, be possible to compute this energy
for a set of points, and then use some interpolation in order to come up with an
analytical expression for Eel , this approach is practically not feasible for molecules
containing more than just a few atoms. In these cases, one typically resorts to
the so-called harmonic approximation: let us expand the electronic energy into a
Taylor series around the molecular equilibrium structure, and truncate it after the
term of second order. Furthermore, we are free to subtract the constant zero-order
term from the total energy (leading to an energy term that we shall denote in the
(vib)
following by Eel,m ), and the linear term is, of course, zero for a molecule in its
equilibrium structure, as stable molecules correspond to minima on the PES. We
then find
2
∂ Eel,m
1 XX
(vib)
rI,α
rJ,β
α, β ∈ {x, y, z}.
(2.30)
Eel,m ≈
2 I,J α,β
∂rI,α ∂rJ,β
These second derivatives can also be written in matrix form; the resulting matrix
is usually called the Hessian matrix. It is convenient for the further mathematical
treatment to include the individual atomic masses into the corresponding coordinates.
We thus obtain the so-called mass-weighted Hessian matrix H (m) , the elements of
which are given by
2
1
∂ Eel,m
(m)
HIα,Jβ = √
.
(2.31)
mI mJ ∂rI,α ∂rJ,β
It is straightforward to see that the coordinates of the individual nuclei are coupled
via the Hessian matrix. However, it is possible to decouple the resulting differential
16
Theoretical Background of Quantum Chemistry
equations by means of a suitable coordinate transformation. In practice, this transformation is achieved by subjecting the mass-weighted Hessian matrix to a unitary
transformation which diagonalizes the matrix. These new coordinates obviously no
longer couple among each other; they are usually called normal modes. Following
such a procedure, one obtains a set of 3Nnuc independent differential equations
(Nnuc denotes the number of nuclei present in a given molecule) which all take the
form of the well-known harmonic oscillator and are therefore straightforward to
solve [50]. It turns out that the eigenvalues of the mass-weighted Hessian matrix
are equal to the squares of the harmonic angular frequencies. The eigenvectors
of the Hessian are the so-called normal modes; they correspond to the collective
motion of the nuclei in a given vibration.
2.4.3
Electronic Structure Methods
The electronic energy Eel is the key property for computing the harmonic frequencies. Furthermore, many other properties (e.g., dipole moments) are directly
obtained from the electronic energy or derivatives thereof. Therefore, Eel is the
central quantity in quantum chemistry, and its computation has received a great
amount of attention during the last decades. Today, we have a range of methods at
our disposal to obtain this energy. These can largely be divided into two groups.
The first group makes use of the state function in order to calculate molecular
properties, and thus, these methods are often referred to as wave function-based
approaches. The most basic approach of these is the well-known Hartree–Fock
(HF) method. Here, the state function is approximated by a product of auxiliary
one-electron functions, the so-called molecular orbitals. This product has to be
antisymmetrized in order to account for the Pauli principle, which can easily be
done by creating a so-called Slater determinant from the molecular orbitals. This
determinantal function is then varied such that the resulting electronic energy is
as small as possible (in practical implementations, every molecular orbital is expanded into a set of basis functions, such that the optimization reduces to finding
the optimal expansion coefficients of these basis functions — which essentially is a
generalized eigenvalue problem). In the Hartree–Fock approach, every electron is
treated as moving in an averaged charge cloud, generated by all other electrons.
By definition, the deviation from the HF energy and the true energy is the socalled correlation energy. Typically, the HF energy amounts to roughly 97% of
the true energy. However, the remaining few percent are of utmost importance
for the correct description of chemical processes and properties. Therefore, more
advanced schemes have been developed in order to capture as much correlation
energy as possible. These methods are typically very accurate, but also extremely
time-consuming. We have not the space here for a detailed description of all these
methods; rather, we refer the reader to the wealth of excellent text books on this
topic [2, 51, 52].
2.5 Density Functional Theory
2.4.4
17
A Short Notice About Spin
Before proceeding further we shall note in passing here the intricate fact that the
electron spin does not arise naturally within the context of nonrelativistic quantum
chemistry, but instead has to be introduced artificially. This is usually done by
augmenting the spatial variables of each electron, r i , by another variable representing
the spin. Throughout this thesis, we will often drop this additional variable for
the sake of both brevity and clarity. Nevertheless, it is to be understood that the
electron spin has always to be properly accounted for.
2.5
Density Functional Theory
The second main electronic structure method discards the state function in favor
of the electron density, which is computationally more feasible. This approach is
therefore called density functional theory (DFT). The aim of DFT is to find an
expression for the electronic energy in terms of the electron density
Z
Z
ρ(r) = Nel · · · |Ψ0 (r, r 2 , . . . , r Nel )|2 dr 2 . . . dr Nel ,
(2.32)
where Nel denotes the total number of electrons present. Note that we are only interested in the ground-state electron density, indicated by the subscript ”0“. However,
for ease of notation we will subsequently drop this index.
DFT traces its origin back to the 1920s when Fermi, based on earlier work by
Thomas [53], presented a model which related the energy of a homogeneous electron
gas to its density [54]. However, this as well as related models proved to be of little
help in chemistry as they cannot properly account for bonding in molecules [55].
The fundamental theoretical basis of DFT has been developed several years after
these first trials, namely in the 1960s by Hohenberg and Kohn with their famous
theorems [56], stating that the electron density completely specifies (within a trivial
additive constant) the external potential and by that the Hamiltonian and the wave
function; and that any trial density yields an upper estimate of the true ground-state
energy.
We can thus express the ground-state energy as a functional (indicated by square
brackets) of the ground-state electron density:
Z
Eel [ρ] = hΨ[ρ]| Hel |Ψ[ρ]i = F [ρ] + Vext [ρ] = F [ρ] + ρ(r)νext (r)dr. (2.33)
Here, F [ρ] is the so-called Hohenberg–Kohn functional which is universal in the
sense that it does not depend on the system. The so-called external potential νext (r),
however, depends on the system under investigation; it models the mutual attraction
of electrons and nuclei,
X −ZI
.
(2.34)
νext (r) =
|r
−
r
I|
I
The basic idea of DFT is to minimize the energy functional given in Eq. (2.33)
subject to the boundary condition that the electron density integrates to the correct
18
Theoretical Background of Quantum Chemistry
number of electrons,
Z
Nel = ρ(r)dr,
(2.35)
which can be achieved with Lagrange’s method of undetermined multipliers. Unfortunately, the analytical form of the Hohenberg–Kohn functional is not known.
Therefore, it is essential to develop approximations to it, for which it is advantageous
to split it into different terms,
(c)
F [ρ] = T [ρ] + J[ρ] + Exc
[ρ],
(2.36)
where T stands for the kinetic energy, J for the Coulomb repulsion energy, and
(c)
Exc for the conventional exchange–correlation energy (conventional in the sense
that it occurs in this form in wave function-based methods). While the second term
of Eq. (2.36) is known exactly,
ZZ
ρ(r 1 )ρ(r 2 )
1
dr 1 dr 2 ,
(2.37)
J[ρ] =
2
|r1 − r2 |
the remaining two terms are unknown. Although the conventional exchange–
correlation energy can be assumed to yield a rather small contribution to the total
energy, this is not at all true for the kinetic energy. For a long time, all approximations to this term which have been developed, for example the Thomas–Fermi
functional mentioned above,
Z
2/3
3
3π 2
,
(2.38)
TTF [ρ] = CF ρ5/3 (r)dr,
CF =
10
proved to be too inaccurate for practical applications. In an effort of improving
on the Thomas–Fermi and related models, Kohn and Sham had the idea of approximating the kinetic energy of the actual system under consideration by that
of a non-interacting one, thus effectively reducing the problem to the solution of
one-particle equations, the so-called Kohn–Sham equations [57]. For such a system,
the exact wave function is known to be a Slater determinant, constituted from the
so-called Kohn–Sham orbitals ϕi fulfilling the one-particle equations
1
(2.39)
T̂s + V̂ext ϕi (r) = − ∆ + νext (r) ϕi (r) = i ϕi (r),
2
and yielding the density
ρ(r) =
Nel
X
|ϕi (r)|2 .
(2.40)
i=1
With this, a large part of the kinetic energy of the actual (i.e., interacting) system
can be captured, and we rewrite Eq. (2.36) to
F [ρ] = Ts [ρ] + J[ρ] + Exc [ρ]
(2.41)
2.5 Density Functional Theory
19
where Exc [ρ] is what is called the DFT exchange–correlation energy, given by
(c)
Exc [ρ] = T [ρ] − Ts [ρ] + Exc
[ρ],
(2.42)
i.e., the remaining difference between the actual and the approximated kinetic energy,
as well as other unknown contributions to the total energy are then collected in
the exchange–correlation functional. One can now insert Eq. (2.41) into Eq. (2.33)
and carry out the minimization of the resulting energy functional. Then one finds
the so-called Kohn–Sham equations,
1
− ∆ + νext (r) + νcoul (r) + νxc (r) ϕi (r) = i ϕi (r)
(2.43)
2
which are used to determine the Kohn–Sham orbitals in a self-consistent manner.
Here,
Z
ρ(r 0 )
νcoul (r) =
dr 0 ,
(2.44)
|r − r 0 |
stems from the Coulomb repulsion energy, and
νxc (r) =
δExc [ρ]
δρ
(2.45)
is the functional derivative of the exchange–correlation functional, which is the only
unknown part in the Kohn–Sham equations. The development of approximations to
this functional is at the heart of all efforts of improving DFT. In the next sections,
we will give a detailed account on how to construct such exchange–correlation
functionals (note that in the remaining part of this text, we will use the term
”exchange–correlation functional“ where we actually mean ”approximation to the
(universal) exchange–correlation functional“).
2.5.1
Classification of Exchange–Correlation Functionals
An exchange–correlation functional is usually attributed to one of five major classes
depending on the quantities used in its construction. The more information is used
during this construction procedure, the better is in general the approximation to
the true exchange–correlation functional. Thus, these five classes are furthermore
ordered according to how well they generally approximate the true exchange–
correlation functional. Perdew refers to this ordering as the ”Jacob’s ladder“, raising
from the ”Hartree world“ to the ”heaven of chemical accuracy“ [58].
The first rung of this ladder is represented by the local density approximation
(LDA) exchange–correlation functionals. They are derived by solely using the electron density ρ. These functionals are often obtained by investigating a homogeneous
electron gas. As in a real molecule the electron distribution is far from being homogeneous, one expects that the inclusion of the gradient of the density, ∇ρ, leads
to better exchange–correlation functionals. These are termed generalized gradient
approximation (GGA) functionals and build the second rung of ”Jacob’s ladder“.
20
Theoretical Background of Quantum Chemistry
It is also possible to include higher orders of the gradient of the electron density,
thereby generating what is generally called meta-GGA functionals, standing on the
third rung of ”Jacob’s ladder”. These can additionally (or instead of higher powers
of the gradient of the density) include the kinetic energy density τ , which is given
by
N
τ (r) =
el
1X
|∇ϕi (r)|2 .
2 i=1
(2.46)
Advancing to the fourth rung, one furthermore includes information about the
occupied orbitals and/or incorporates the exact exchange energy, which is given for
spin-compensated, i.e., closed-shell, systems by
ZZ
|ρ(r 1 , r 2 )|
1
(exact)
dr 1 dr 2
(2.47)
Ex
=−
4
|r1 − r2 |
where ρ(r 1 , r 2 ) is the so-called Kohn–Sham one-electron density matrix,
ρ(r 1 , r 2 ) =
Nel
X
ϕi (r 1 )ϕ∗i (r 2 ).
(2.48)
i=1
Here, the asterisk denotes the complex conjugate of a given quantity. Hybridfunctionals, which contain a certain admixture of Hartree–Fock exchange, belong to
this class of exchange–correlation functionals. However, note that the exact exchange
energy within the framework of DFT is in general different from the Hartree–Fock
exchange energy although the two terms are often used as synonyms [59]. Finally,
the utilization of information also of unoccupied Kohn–Sham orbitals leads us to the
so called generalized random phase approximation (RPA) functionals, representing
the fifth and highest rung of all exchange–correlation functionals. Also the socalled double hybrid functionals belong to this class. In these functionals, a certain
amount of MP2 correlation energy is included besides Hartree–Fock exchange.
2.5.2
Constructing Exchange–Correlation Functionals
As has already been stated, the development of exchange–correlation functionals
is one of the most important tasks in DFT. There are different approaches used to
construct new exchange–correlation functionals, which we can divide into several
major classes [59, 60]:
• investigating model systems (e.g., the homogeneous electron gas),
• gradient expansions of the exchange–correlation energy,
• constraint satisfaction,
• empirical fits,
• modelling the exchange–correlation hole, and
• admixture of Hartree–Fock exchange.
In the next subsections, we will give a short account of these strategies, where we
will also highlight some of the more popular density functionals available today.
2.5 Density Functional Theory
2.5.2.1
21
Investigating Model Systems: The Local Density Approximation
In a first approach, the basic idea is to investigate an analytically tractable model
system, and then subsequently to use the results obtained in this way for practical
applications. For example, Loos and Gill recently reported on the correlation energy
in the high-density limit (see below) of several two-electron model systems such as
helium (where the electrons move in a Coulombic external potential) or hookium
(where the external potential is harmonic) [61, 62]. However, the prime model
system is certainly the homogeneous electron gas having a constant density
ρ=
Nel
.
V
(2.49)
The state functions of such a system are simply given by the eigenfunctions of
noninteracting particles in a three-dimensional box; from these state functions the
corresponding electron densities are built straightforward. Usually, one splits the
exchange–correlation energy up into an exchange and a correlation part and defines
a mean exchange–correlation energy per particle,
xc = x + c =
Ex
Ec
+
.
Nel Nel
(2.50)
Thus, given the mean exchange–correlation energy per particle, the total exchange–
correlation energy is easily constructed as
Z
Exc = xc Nel = xc ρV = xc ρdr.
(2.51)
A straightforward extension of Eq. (2.51) to inhomogeneous systems yields
Z
Exc ≈ xc (ρ(r))ρ(r)dr,
(2.52)
where we also indicated the dependence of the exchange–correlation energy on the
density.
A formula for the exchange energy was presented by Bloch in 1929 [63], and
by Dirac in 1930 [64]. By recalling that the exact Hartree–Fock exchange energy
can be written as in Eq. (2.47), one finds, using
ϕkx ,ky ,kz =
1
V
1/2
eikx x+ky y+kz z =
1
V 1/2
eik·r ,
(2.53)
what is called the local density approximation expression for the exchange energy,
Z
Ex(LDA)
= −Cx
3
Cx =
4
1/3
3
.
π
ρ(r)4/3 dr,
(2.54)
where
(2.55)
22
Theoretical Background of Quantum Chemistry
This formula is known to underestimate the exchange energy in inhomogeneous
systems [59]. The correlation energy of a homogenoeus electron gas is known
analytically in the limiting cases of very high and very low electron densities. For
very high electron densities, the quantity
1/3
3
,
(2.56)
rs =
4πρ
which is called the Wigner–Seitz radius, is much smaller than 1, i.e, r s 1. In
this case, the correlation energy can be approximated by
c (r s ) ≈ aln(r s ) + b + r s (cln(r s ) + d),
r s 1,
(2.57)
where a = 0.031091, b = −0.046644, c = 0.00664, and d = −0.01043 [65] (note,
however, that values for the coefficients a, b, and c have been given earlier by
other authors with slightly different values [66, 67]). In the low-density limit, i.e.,
if r s 1, one can write [68, 69]
U1
U2
1 U0
+ 3/2 + 2 ,
r s 1,
(2.58)
c (r s ) ≈
2 rs
rs
rs
where Uj are constants. For intermediate values of r s , exact numerical solutions
are known from Monte-Carlo simulations carried out by Ceperley and Alder [70]
(note that such numerical simulations are still an active field of research [71]).
It appears thus natural to develop a correlation functional by parametrizing these
numerical data by a suitably chosen function. The best known of these functionals
is the one suggested by Vosko, Wilk, and Nusair [72], given by
2b
Q
x2
(VWN)
+ tan−1
c
(x) =A ln
X(x) Q
2x + b
(2.59)
2
bx0
(x − x0 )
2(2x0 + b)
Q
−1
−
ln
+
tan
,
X(x0 )
X(x)
Q
2x + b
1/2
where x = r s , X(x) = x2 + bx + c, Q = (4c − b2 )1/2 , and A = 0.0621814,
b = 13.0720, c = 42.7198, and x0 = −0.409286. Note that these values are only
valid for spin-compensated systems; for open-shell cases, Vosko, Wilk and Nusair
examined five additional interpolation formulas [72].
2.5.2.2
Gradient Expansions of the Exchange–Correlation Energy
The formulas presented in the preceding section are exact for a uniform electron
density. However, in atoms and molecules the electron density is not constant but
rather spatially varying. Therefore, one tries to include this spatial dependence into
exchange–correlation functionals. One way of doing this is to expand the exchange
and the correlation energy in terms of powers of the gradient of the electron density,
i.e.,
Z
Z
(0)
Exc [ρ] ≈ ρ(r)xc (ρ(r))dr + ∇ρ(r)(1)
xc (ρ(r))dr
Z
(2.60)
2 (2)
+ |∇ρ(r)| xc (ρ(r))dr
2.5 Density Functional Theory
23
(note that there are two terms of quadratic order which are, however, not linearly
independent such that either of the two can be eliminated by partial integration).
By realizing that the energy is a scalar quantity we understand that the second term
of Eq. (2.60) must vanish, such that
Z
Z
(0)
Exc [ρ] ≈ ρ(r)xc (ρ(r))dr + |∇ρ(r)|2 (2)
(2.61)
xc (ρ(r))dr.
It is advantageous to expand the exchange energy, Ex , in terms of the so-called
reduced density gradient
s(r) =
|∇ρ(r)|
,
2ρ(r)4/3 (3π 2 )1/3
so that we get
Z
2
Ex [ρ] ≈ ρ(r)(0)
x (ρ(r))(1 + µ2 s(r) )dr.
(2.62)
(2.63)
It is very difficult to determine the value of the coefficient µ2 (and, of course, the
values of coefficients belonging to higher orders of the gradient of the density). In
fact, different authors presented quite different numerical values for µ2 : according
to Antoniewicz and Kleinman, µ2 = 10/81, while Sham found a value of 7/81 for
the same coefficient [59].
The expansion of the correlation energy proceeds as outlined above for the exchange energy; not very surprisingly, one encounters the same mathematical problems. Experience shows that the improvements of such density gradient expanded
exchange–correlation functionals over LDA functionals (i.e., exchange–correlation
functionals only depending on the electron density, see Section 2.5.1) are only
modest, demonstrating that the expansion presented in Eq. (2.60) is not valid for
the rapidly varying electron densities encountered in molecules. However, today
one regards these functionals as producing exact results in what is called the limit
of a slowly varying density [59].
2.5.2.3
Constraint Satisfaction: The Generalized Gradient Approximation
Today, many analytic properties of the exact exchange–correlation functional are
known [58, 73, 74]. For example, the exchange energy is always negative,
Ex < 0,
(2.64)
while the correlation energy is smaller or equal to zero,
Ec ≤ 0.
(2.65)
Another very well known constraint which the exact exchange–correlation functional
satisfies is the so-called Lieb–Oxford bound,
Z
4
(2.66)
Ex [ρ] ≥ CLO ρ(r) 3 dr,
24
Theoretical Background of Quantum Chemistry
where −1.44 ≥ CLO ≥ −1.68 [75]. Moreover, we shall take notice here of the socalled spin-scaling relation [76], which relates spin-compensated and spin-polarized
systems to each other,
Ex [ρα , ρβ ] =
1
(Ex [2ρα ] + Ex [2ρβ ]) .
2
(2.67)
One particularly appealing strategy of developing new functionals consists of
starting from a suitable functional form containing some variable parameters, and
determining all these parameters by imposing the known analytical properties as
constraints. This leads to so-called nonempirical density functionals, as they need no
empirical data for their parametrization. There are only a few nonempirical density
functionals. One of the better known of these is the PBE functional, developed by
Perdew, Burke, and Ernzerhof in 1996 [77, 78]. It uses the ansatz
Z
Ex [ρ] ≈ ρ(r)x (ρ(r))(1 + Fx (s(r)))dr,
(2.68)
which is called generalized gradient approximation (GGA, see Section 2.5.1). Note
that this formulation is reminiscent of the gradient expansion of the exchange–
correlation energy (cf., Eq. (2.63)). In the case of PBE, the so-called enhancement
factor Fx is chosen to be
Fx(PBE) (s(r)) = 1 +
µs(r)2
1+
µs(r)2
κ
.
(2.69)
The parameter κ = 0.804 is chosen such that the Lieb–Oxford bound (2.66) is
satisfied, while µ ≈ 0.21951 is determined from the fact that in the limit of a
homogeneous electron gas, the gradient contribution to the exchange energy must
cancel that for the correlation energy [77].
In continuation of this work, Tao, Perdew, Staroverov, and Scuseria developed
also a nonempirical meta-GGA functional in 2003, dubbed TPSS [79, 80].
2.5.2.4
Empirical Fits
In a mathematically less strict approach, one determines the parameters in a density
functional by fitting it to some reference values. This reference can be based on
high-level ab initio data or experimental measurements. One very well known
example of such a functional is the BP86 exchange–correlation functional [81, 82],
which we shall describe here in some detail as it is also the most used functional
of this very thesis.
The BP86 exchange–correlation functional consists of the B88 exchange functional
which has been proposed by Becke in 1988 [81] and the P86 correlation functional
which has been developed by Perdew in 1986 [82]. We can write the BP86
exchange–correlation energy as
(BP86)
Exc
= Ex(B88) + Ec(P86) ,
(2.70)
2.5 Density Functional Theory
25
i.e., exchange and correlation are treated independently from each other and simply
summed up. The B88 functional has the following functional form (note that we
explicitly take into account the electron spin)
XZ
4
(B88)
(LSDA)
Ex
= Ex
−β
(2.71)
ρσ (r) 3 Fx(B88) dr,
σ
(LSDA)
where Ex
is the exchange energy in the local spin density approximation
(LSDA), i.e., the exchange energy of a homogeneous gas of non-interacting electrons. This energy is a functional of only the spin density ρσ (r),
1
Z
4
3 3 3X
(LSDA)
(LSDA)
Ex
= Ex
[ρ] = −
ρσ (r) 3 dr,
(2.72)
2 4π
σ
where the sum runs over “up” and “down” electron spins (σ ∈ {α, β}, where α
and β denote spin up and spin down, respectively). The above formula is obtained
by applying the spin-scaling relation (2.67) to the LDA expression (2.54) for the
exchange energy as obtained by Bloch [63] and Dirac [64]. The central part of
B88 is the enhancement factor
Fx(B88) = Fx(B88) (xσ (r)) =
xσ (r)2
,
1 + 6βxσ (r)arcsinh(xσ (r))
(2.73)
where β is an empirical parameter, and
xσ (r) =
|∇ρσ (r)|
4
ρσ (r) 3
(2.74)
is the reduced (dimensionless) gradient of the density; the operator ∇ denotes the
first partial spatial derivative. The parameter β has originally been determined by
a fit to the exact exchange energies of the six noble gas atoms helium through
radon, which gave a value of 0.0042 for β.
The P86 correlation functional can be written as
Z
|∇ρtot (r)|2
1 −Φ
(P86)
(LSDA)
e C(ρtot (r))
dr.
(2.75)
Ec
= Ec
+
4
d
ρtot (r) 3
As already stated, no closed analytical expression is known for the correlation
(LSDA)
energy in the local spin density approximation, Ec
. Nowadays, one mostly
relies on interpolations to numerical data obtained by Ceperley and Alder [70]; the
best known of these interpolations is the one recommended by Vosko, Wilk, and
Nusair [72], which we will also employ. To continue with Eq. (2.75), d is given
by
s
5 5
1
1 + ξ(r) 3
1 − ξ(r) 3
d = 23
+
,
(2.76)
2
2
where
ξ(r) =
ρα (r) − ρβ (r)
ρtot (r)
(2.77)
26
Theoretical Background of Quantum Chemistry
is the so-called spin polarization, and ρtot (r) = ρα (r) + ρβ (r). Furthermore, the
exponent Φ is given by
C(∞) ∇ρtot (r)
Φ = 1.745f˜
.
C(ρtot (r)) ρtot (r) 76
(2.78)
In this equation, f˜ is an empirical parameter, and
0.002568 + 0.023266rs (r) + 7.389. 10−6 rs (r)2
, (2.79)
1 + 8.723rs (r) + 0.472rs (r)2 + 7.389. 10−2 rs (r)3
(according to Perdew, the numerical parameters in this formula have been obtained by Rasolt and Geldart, but unfortunately their exact procedure is no longer
reproducible [82]) where
13
3
.
(2.80)
rs (r) =
4πρtot
In the original work by Perdew, a value of 0.11 for the parameter f˜ was determined
from the exact correlation energy of the neon atom.
While in the BP86 functional, the empirical fit is only made in the end to
determine the actual numerical values of two parameters, one can of course also
use an approach where the mathematical form of a given density functional is
not so much motivated by physical reasons, but simply designed such that it can
be fitted well to some trusted reference data. Such functionals usually involve
several empirical parameters, and are fitted to quite large datasets. One of the first
examples of this kind of functionals is the B97 exchange–correlation functional
proposed by Becke [83]. This author proposed both an exchange and correlation
functional following the general GGA ansatz (2.68). He then proposed to write the
enhancement factor in terms of a polynomial series
X
Fx(B97) (s(r)) =
ci u(s(r))i ,
(2.81)
C(ρtot (r)) = 0.001667+
i
where the coefficients ci are determined by a least-squares fitting procedure, and
the function u is chosen suitably [83]. In this first application, Becke used the
well-known G2 test set [84,85] as “training set” (i.e., the data fitted to). In a similar
fashion, many other density functionals have been developed, such as the HCTH
functional [86], proposed by Hamprecht, Cohen, Tozer and Handy, as well as their
closely related reparametrizations HCTH/120 [87], HCTH/147 [87], HCTH/407 [88],
and τ -HCTH [89] (actually, the number after the slash in the HCTH family of
functionals corresponds directly to the number of molecules in the respective training
set). Another very well known family of fitted density functionals is the Minnesota
collection of functionals, which includes the functionals M05 [90], M05-2X [91],
M06 [92], M06-L [93], M06-2X [92], M06-HF [94], M08-HX [95], M08-SO [95],
M11 [96], and M11-L [97] (note that Truhlar and co-workers have also developed
several other functionals which are often also regarded as being members of the
Minnesota collection, e.g., the functionals SOGGA11-X [98] and N12 [99]). The
Minnesota functionals mainly differ in their amount of exact exchange (note that
a reduction of the amount of exact exchange was found earlier to be most crucial
for the spin splitting energetics [100–103]).
2.5 Density Functional Theory
2.5.2.5
27
Modelling the Exchange–Correlation Hole
The electron–electron repulsion, Vee , can be written using the so-called pair density,
Nel (Nel − 1)
P2 (r 1 , r 2 ) =
2
Z
Z
···
|Ψ(r 1 , r 2 , . . . , r N )|2 dr 3 . . . dr N ,
(2.82)
as
ZZ
Vee =
P2 (r 1 , r 2 )
dr 1 dr 2 .
|r1 − r2 |
(2.83)
We can now split this expression up into a (classical) Coulomb part and a (quantum
mechanical) exchange–correlation part by writing the pair density as
1
(2.84)
P2 (r 1 , r 2 ) = ρ(r 1 ) (ρ(r 2 ) + hxc (r 1 , r 2 )) ,
2
where we introduced the so-called exchange–correlation hole hxc (r 1 , r 2 ). With this,
we can write the exchange–correlation energy as
ZZ
1
ρ(r 1 )hxc (r 1 , r 2 )
(c)
Exc =
dr 1 dr 2
(2.85)
2
|r1 − r2 |
(note that this is again the conventional exchange–correlation energy, as it does
not include the difference between the approximated and the actual kinetic energy
which is arising in the Kohn–Sham scheme, see above).
Eq. (2.85) suggests that a reasonable model of the exchange–correlation hole
should provide accurate exchange–correlation energies. There are many different
approaches to model this hole. For example, one can simply expand it into powers
of the gradient of the electron density or one can start out from a correlated
wave function. Furthermore, it is also possible to study the properties of an
analytically solvable model system. This last approach has the advantage of revealing
many interesting properties of the exchange–correlation hole as well as boundary
conditions the exact exchange–correlation hole satisfies. For example, it has been
shown that the exact exchange–correlation hole represents the deficit of exactly one
electron, i.e., its value integrated over all space amounts to −1 [74].
For conveniency one usually splits up the exchange–correlation hole into an
exchange hole hx (r 1 , r 2 ), and a correlation hole, hc (r 1 , r 2 ), by writing [74]
hxc (r 1 , r 2 ) = hx (r 1 , r 2 ) + hc (r 1 , r 2 ),
(2.86)
allowing us to handle the two parts separately. In 1983, Becke presented the Taylor
expansion of the spherically averaged exact exchange σ-spin hole [104]:
1 (∇ρσ (r 1 ))2
1
2
(σ)
∇ ρσ (r 1 ) − 4τσ (r 1 ) +
s2 , (2.87)
hx (r 1 , s) ≈ −ρσ (r 1 )−
6
2
ρσ (r 1 )
where s = r 2 − r 1 . Taking this equation as starting point, Becke and Roussel
presented an expansion of the exchange–correlation hole for an arbitrary manyelectron system,
a
(a|b − s| + 1)e−a|b−s| − (a|b + s| + 1)e−a|b+s| , (2.88)
h(BR)
(a, b, s) = −
x
16πbs
28
Theoretical Background of Quantum Chemistry
where a and b are assumed to have positive values only and have no physical
significance. Using Eq. (2.88) one finds for the exchange energy
Z
1 X ρσ (r)
1
(BR)
−ab
=−
1 + ab dr.
Ex
1−e
(2.89)
2 σ
b
2
Becke presented also an expression of the correlation energy [105], where he also
used the Taylor expansion shown in Eq. (2.86). After some tedious mathematical
manipulations, he found
Ec(B88c) = Ec(αα) + Ec(ββ) + Ec(αβ)
(2.90)
where
Ec(σσ)
Z
= Cσσ
1
ρσ (r) τσ (r) −
8
(∇ρσ (r))2
ρσ (r)
4
zσσ
2
zσσ 1−
ln 1 +
dr,
zσσ
2
(2.91)
and
Ec(αβ)
Z
= Cαβ
2
ρα (r)ρβ (r)zαβ
1
1−
ln (1 + zαβ ) dr.
zαβ
(2.92)
Here, Cσσ and Cαβ are adjustable parameters while zσσ and zαβ are the so-called
correlation lengths for which Becke proposed [105]
(σ)
(σ 0 )
(2.93)
zσσ0 = cσσ0 RF + RF
(σ)
where cσσ0 is a simple proportionality constant and RF is what is called the Fermi
hole radius, which is given by
(σ)
RF
1
=
3
3
4π
−1/3
ρ−1/3
.
σ
(2.94)
Another strategy is to find the correlation energy with the aid of a suitable wave
function, and then convert it into an expression suitable for DFT. As the correlation
functional of Lee, Yang, and Parr (LYP) [106] has been constructed using such an
approach we will use it as an example to illustrate the basic concept. According
to Colle and Salvetti [107, 108], the correlated wave function Ψ of a given system
can be written as
Y
Ψ(r 1 , r 2 , . . . , r N ) = Ψ(HF) (r 1 , r 2 , . . . , r N )
(1 − f (r i , r j ))
(2.95)
i<j
where Ψ(HF) is the corresponding single-determinantal Hartree–Fock wave function,
and the function f (r i , r j ) is defined as
s0
−γ 2 s02
f (r i , r j ) = e
1 − g(R) 1 +
.
(2.96)
2
2.5 Density Functional Theory
29
Here, s0 = |r i − r j |, R = (r i + r j )/2, and the parameter γ is related to the electron
density via
γ = 2.29ρ(R)1/3
with which Colle and Salvetti [107] approximated the function g(R) as
√
πβ
√ .
g(R) ≈
1 + πβ
(2.97)
(2.98)
Colle and Salvetti could subsequently relate the correlation energy to the electron
density and found [107]
0
0
−1/3
(HF)
Z
1 + bρ(R)−8/3 ∇2s0 P2 (R − s2 , R + s2 )
e−cρ(R)
0
s =0
Ec(CS) = −a ρ(R)
dR,
1 + dρ(R)−1/3
(2.99)
(HF)
in which P2
is the pair density employing the Hartree–Fock orbitals and a, b, c,
and d are adjustable parameters. Lee, Yang, and Parr took Eq. (2.99) as a starting
point and restated it to take the following form [106]:
Z
−1/3
1 + bρ(r)−5/3 τ (HF) (r) − 2τ̃ (W) (r) e−cρ(r)
(CS)
dr, (2.100)
Ec
= −a ρ(r)
1 + dρ(r)−1/3
where τ (HF) is the Hartree–Fock kinetic energy density and
τ̃ (W) (r) =
1 |∇ρ(r)|2 1 2
− ∇ ρ(r)
8 ρ(r)
8
(2.101)
is reminiscent of the Weizsäcker kinetic energy density. As can be seen, Eq. (2.100)
depends through the term τ (HF) on the orbitals. Lee, Yang, and Parr replaced this
term by its second order gradient expansion
1
1
τ (HF) (r) ≈ CF ρ5/3 (r) + τ (W) (r) + ∇2 ρ(r),
9
18
(2.102)
(note that the first term is simply the Thomas–Fermi kinetic energy density given in
Eq. (2.38)) thereby arriving at the final expression for the correlation energy [106],
Ec(LYP)
−1/3
ρ(r) + bρ(r)−2/3 CF ρ(r)5/3 − 2τ̃ (W) (r) e−cρ(r)
=−a
dr
1 + dρ(r)−1/3
(2.103)
Z
−1/3
1
bρ(r)−2/3 91 τ̃ (W) (r) + 18
∇2 ρ(r) e−cρ(r)
−a
dr.
1 + dρ(r)−1/3
Z
Today, there are several of these Colle-Salvetti-type correlation functionals available.
For example, in 2003 Imamura and Scuseria constructed a correlation functional
based on this method, which yields correlation energies comparable to that of the
LYP correlation functional [109].
30
Theoretical Background of Quantum Chemistry
2.5.2.6
Admixture of Hartree–Fock Exchange
Modifying the two-electron operator (2.18) by multiplication with a parameter λ,
(λ)
gi,j → gi,j = λ
1
,
|r i − r j |
0 ≤ λ ≤ 1,
(2.104)
allows us to tune the magnitude of the electron–electron interactions. Then, by
recalling the Hellmann–Feynmann theorem, we can write the derivative of the
Hohenberg–Kohn functional (2.41) with respect to this parameter λ as
(λ)
(λ) 1
∂F (λ) [ρ] (λ) ∂Hel (λ) (λ) Ψ
= Ψ
,
= Ψ
Ψ
∂λ
∂λ
|r i − r j |
and obtain
Z 1
∂F (λ) [ρ]
dλ = F (1) [ρ] − F (0) [ρ] = Exc [ρ] + J[ρ].
∂λ
0
Reverting this equation yields
Z 1
Z 1
(λ) (λ) 1
(λ)
Ψ
Exc [ρ] =
Ψ
dλ − J[ρ] =
Exc
[ρ]dλ,
|r
−
r
|
i
j
0
0
(2.105)
(2.106)
(2.107)
which is known as the adiabatic connection formula (cf., Refs. [110–113]). Based
on this formula, Becke argued that the true exchange–correlation energy can be
approximated as [114]
Exc =
1
(LDA)
Ex(exact) + Exc
2
(2.108)
In order to implement this formula in an actual DFT code, the true exchange energy
is replaced by the corresponding Hartree–Fock expression. In a generalization of
this approach, the amount of Hartree–Fock exchange is varied freely. In principle,
any functional (constructed using one of the above strategies) can be combined with
Hartree–Fock exchange. The resulting functionals are denoted as hybrid functionals.
Probably the best known hybrid functional is the B3LYP functional, for which,
however, no proper citation exists [59]. According to Ref. [59], B3LYP is given
by
(B3LYP)
Exc
= a0 Ex(HF) +(1−a0 −ax )Ex(LDA) +ax Ex(B88) +(1−ac )Ec(LDA) +ac Ec(LYP) ,
(2.109)
where the parameters a0 = 0.20, ax = 0.72, and ac = 0.81 have been determined
by a fit of a different density functional (namely, B3PW91 [115]) to a dataset
containing atomization energies, ionization potentials, and proton affinities. Even
more confusion exists about the actual implementation of this functional in different
programs. While the LDA correlation energy is usually taken from a given interpolation form (namely, the so-called “form V”) given by Vosko, Wilk, and Nusair [72]
to numerical data from Ceperley and Alder, the developers of the widely known
2.5 Density Functional Theory
31
G AUSSIAN program package [116] apparently [59] used a different interpolation
form, applied to different numerical data. To make things even more complicated,
the interpolation form actually used is apparently [59] “form I”, while it is stated
in the G AUSSIAN manual that “form III” is used. It should also be noted here that
meanwhile, the G AUSSIAN developers added the correct “form V” interpolation
to their code (keyword VWN5), while the original Gaussian implementation has
also been added to other programs (e.g., T URBOMOLE [117], with the keyword
b3-lyp_Gaussian).
B3LYP is called a global hybrid functional, as the amount of Hartree–Fock
exchange is the same everywhere in space. In contrast to this, some researchers
have constructed so-called local hybrid functionals, where the amount of Hartree–
Fock exchange depends on a spatial variable [118]. This approach can help to
reduce the self-interaction error, one of the shortcomings of almost any density
functionals available today (see below). So-called long-range corrected functionals
constitute a closely related class of density functionals. In these functionals, the
expression for exchange–correlation energy is solely used for short distances, while
at longer distances, only the exact Hartree–Fock exchange is employed (see, e.g.,
Refs. [119–122]). This separation further reduces the so-called self-interaction
error [123].
The admixture of Hartree–Fock exchange leads to several functionals which have
a good overall performance [124]. It is thus natural to further improve density
functionals by combining them with a (scaled) wave function-based expression
for the correlation energy. The inclusion of some correlation energy as obtained
by Møller–Plesset perturbation theory of second order (MP2) leads to so-called
double hybrid density functionals. This new family was proposed in 2006 by
Grimme [125], even though a similar approach has been developed by Truhlar
already in 2004 [126]. Grimme was inspired by the perturbation theory of Görling
and Levy (GLPT) [127,128], which is comparable to the well-known Møller–Plesset
perturbation theory in wave function theory. In GLPT, correction terms are derived
that resemble the MP2 expressions. Grimme therefore proposed an ansatz of the
general form
Exc = (1 − ax )Ex(GGA) + ax Ex(HF) + bEc(GGA) + cEc(MP2) ,
(2.110)
(MP2)
to include a certain amount of MP2 correlation energy, Ec
. Grimme further
set b = 1 − c, and found ax = 0.53 and c = 0.27 by a fit to the G2/97 test set [85].
The resulting functional is dubbed B2-PLYP, and it performs superior to many
traditional functionals such as B23LYP [125]. In 2011, Sharkas et al. presented a
rigorous derivation of double hybrid density functionals [129], and today, many of
these functionals exist and are routinely applied, often with good results.
2.5.3
Shortcomings of Density Functionals
Despite the great efforts undertaken so far to develop ever better density functionals,
even the most recent developments suffer from a range of insufficiencies. In order
to overcome these, it is mandatory to systematically analyze the shortcomings of
present-day functionals [122, 130–141]. In recent work, these insufficiencies have
32
Theoretical Background of Quantum Chemistry
been attributed to two sources of error [130,138]: the delocalization error — referred
to as self-interaction error in the earlier literature — and the static-correlation or
spin-polarization error. Apart from these two errors, it was long recognized that
standard DFT lacks a proper description of weak noncovalent interactions because
traditional density functionals cannot account for the long-range correlation effects
responsible for dispersion forces [142–145]. However, empirical correction terms
can very effectively account for this error in DFT calculations [146–151] (see also
below).
2.5.3.1
Delocalization Error/Self-Interaction Error
One can very nicely illustrate the concept of delocalization error, introduced by Yang
and co-workers in 2008, using the example of the bond cleavage of H+
2 [130]. Upon
dissociation (i.e., when the two nuclei are infinitely separated from each other), the
single electron present in this system must reside at either of the two nuclei as it
must be somewhere and fractional electrons do not exist in reality (note that the
third possible situation, namely when the three particles are all infinitely separated
from each other, clearly features a higher energy than the aforementioned case and
is therefore omitted in the following discussion). Now, it is a priori impossible
to say at which of the two nuclei (in the following, we denote them by 1 and
2, respectively) the electron resides. Thus, we can formulate the electronic wave
function Ψ describing the electronic structure of dissociated H+
2 as
1
1
Ψ = √ Ψ1 + √ Ψ2 ,
2
2
(2.111)
where Ψi describes a state where the electron is located at nucleus i, and the
coefficients must have the same values due to the symmetry of the system at hand.
We note in passing that from the very same symmetry considerations it follows
that Ψ1 and Ψ2 describe degenerate states. The electron density ρ resulting from
the wave function given in Eq. (2.111) is
1
1
ρ = ρ1 + ρ2 ,
2
2
(2.112)
where ρi is the electron density built from Ψi . In calculating this electron density,
we took advantage of the fact that hΨ1|Ψ2 i = 0 as the two nuclei are infinitely
separated from each other (and thus, Ψ1 has to be zero everywhere where Ψ2 is
different from zero, and vice versa). The electron density of Eq. (2.112) can be
interpreted as representing a system featuring fractional, i.e., noninteger, electrons:
half an electron is located at atom 1, whereas another half an electron resides at
atom 2.
In principle, the energy resulting from an electron density built from a superposition of degenerate states has to be equal to the energy obtained from the electron
density built from one of these states, i.e.,
E[ρ] = E[ρ1 ] = E[ρ2 ].
(2.113)
2.5 Density Functional Theory
33
However, experience shows that the condition formulated in Eq. (2.113) is violated
by current exchange–correlation functionals. In fact, one usually finds that E[ρ] <
E[ρi ]. Hence, one says that DFT favors systems with fractional charges.
This wrong behavior of contemporary density functionals can be associated with
the so-called self-interaction error, which describes the artificial interaction of an
electron with itself. The self-interaction error has been well known in DFT [152–
163] and recently, it has been generalized to the many-electron self-interaction
error [133, 153]. It is ultimatively due to the fact that with any approximate
functional, the Coulomb part and the exchange–correlation part do not exactly
cancel each other in the case of a one-electron system as opposed to Hartree–Fock
and other wave-function theories. In fact, the dominating Coulomb part which
pushes electrons apart [130, 164] leads to the result that an electron density like
ρ given in Eq. (2.112) features a lower energy than a density like, e.g, ρ1 , where
the electronic charge is spatially more confined. Very generally speaking, this
dominance of the Coulomb part leads to electron densities which are artifically
spread out too much.
The erroneous behavior of approximate density functionals has of course implications for quantum chemical calculations. For example, the binding energy of systems
with fractional charges is overestimated, which results in an incorrect prediction of
reaction energies and barrier heights. Barriers are usually underestimated because of
strongly delocalized electron distributions in the transition states [135,165,166]. An
example for which this effect is particularly dominant is the Diels–Alder reaction
where a substituted alkene is added to a conjugated diene in a cycloaddition. Typical
products are compact structures with highly localized electron density. Therefore,
these species are underbound because the non-bonded repulsion is overestimated
by the currently available density functionals [135]. Thus, large errors in thermochemistry and reaction barrier heights are observed [135, 167, 168].
2.5.3.2
The Spin-Polarization/Static-Correlation Error
The concept of spin-polarization error, also introduced by Yang ang co-workers, can
be explained using the example of dissociating H2 [130]. The line of reasoning is
very similar as in the case of the delocalization error. Let us imagine a homolytic
dissociation of H2 . In the dissociation limit, we have two hydrogen atoms, in each
of which the electron can have either α-spin or β-spin. As it is again a priori not
clear in which spin state the electron residing at one of the two fragments is, we
can write the electronic wavefunction Ψ describing this fragment as
1
1
Ψ = √ Ψα + √ Ψβ ,
2
2
(2.114)
where Ψσ describes the electron in spin state σ and equal coefficients appear to be
the most natural choice. From this wave function, we get the so-called spherically
averaged density [169], representing a net spin of zero,
1
1
ρ = ρα + ρβ ,
2
2
(2.115)
34
Theoretical Background of Quantum Chemistry
with ρσ being the density constructed from Ψσ (note that hΨα|Ψβ i = 0 as the two
spin states are orthogonal to each other). The density ρ can now be viewed as
featuring fractional spins — half an electron having α-spin while another half an
electron has β-spin. The so-called constancy condition requires now that, in the
absence of external electromagnetic fields,
E[ρ] = E[ρα ] = E[ρβ ].
(2.116)
For any degenerate system — including open-shell transition-metal clusters — the
energy for the spherically averaged density should be equal to the ground state
energy. However, this is not fulfilled for any available approximate density functional
[132, 169], as present-day functionals have problems describing systems featuring
degenerate ground states. Both DFT and Hartree–Fock theory predict too high
energies for such systems with the largest discrepancy in the case of a net spin
of zero and fractional spins corresponding to 0.5 α-electrons and 0.5 β-electrons.
Hence, the spin-polarization error stems from the biased description of spin-polarized
and non-spin-polarized densities, which should be equal in energy [138].
Of course, fractional charges as well as fractional spins are both associated with
fractional numbers of electrons, as they are properties connected to the electron rather
than being capable of existing on their own. Thus, we may conclude that the two
errors, being both connected with fractional electrons, have an intimate relation,
possibly even sharing the same roots. However, the nature of this relationship
remains yet unclear.
The systematic failure of density functionals creates a need for new improved
functionals that reduce or are free of the (many-electron) self-interaction error
and the spin-polarization/static-correlation error. Different approaches for modifying existing functionals have been investigated, ranging from a simple scaling of
the exchange part [170] to more elaborate schemes like the Perdew–Zunger self–
interaction correction [156,157,171], the real-space correlation models proposed by
Becke [172,173], or the so-called Coloumb-attenuated functionals which usually rely
on a separation of the Coulomb part into a short-range and a long-range part which
are subsequently treated differently (see, e.g., Refs. [164,174]). Additionally, several
new functionals have been proposed over the last years that should account for the
shortcomings mentioned in the preceeding section (see for example Refs. [122,175])
and their performance has been evaluated [135,138,153]. Although all density functionals still deviate from the correct description, self-interaction corrected functionals
such as rCAM-B3LYP and MCY3 perform better for the prediction of Diels–Alder
reaction energies or dimerization energies of aluminum complexes than traditional
functionals as, for example, B3LYP and BLYP [135]. However, these new functionals still show qualitatively the same trends as older density functionals, i.e.,
even though the errors have been reduced to a great extent, they have not been
eliminated in a rigorous fashion. This raises the yet unanswered question whether
such a rigorous elimination of, e.g., the self-interaction error is possible at all.
As has already been stated, standard density functional theory cannot account for
a correct description of dispersion interactions. Many approaches were proposed
for the incorporation of dispersion interactions (see, e.g., Refs. [176–186]). As
already mentioned, a widely used empirical dispersion correction was developed by
2.5 Density Functional Theory
35
Grimme [146–151]. He suggested to add an energy term of the form 1/ (r I − r J )6
to any standard functional. Furthermore, he developed new functionals that include
these correction terms and are especially parametrized to accurately describe systems
with non-covalent interactions (one of the best known of these functionals is the B97D functional [146]). These empirical dispersion-corrected functionals apparently
work well for many (organic) molecules, but we will show some problematic cases
in this work.
Chapter 3
Concepts and Strategies for
Rational Compound Design
“When intuition joins exact research,
the progress of understanding will be
accelerated astoundingly.”
P. Klee
In this chapter, we shall review existing approaches in the field of inverse quantum
chemistry and analyze their potential for future applications. We will first review
in section 3.1 the mathematical concepts developed for inverse Sturm–Liouville
problems, where important results emerged as early as 1929 and strongly influenced
research carried out between 1950 and 1970 on inverse quantum scattering. Then, we
review inverse techniques which have either already found widespread use in rational
compound design or exhibit a great potential for doing so. We discuss quantitative
structure–activity relationships in section 3.2, inverse perturbation analysis in section
3.3, and the use of model equations in section 3.4. Then, we move on to more recent
approaches, which rely on sophisticated sampling and optimization techniques, such
as the optimization of wave functions described in section 3.5, and the optimization
of solid state compounds in sections 3.6.1 and 3.6.2. We review two very recent
approaches, namely the linear combination of atomic potentials in section 3.6.3
and alchemical potentials in section 3.6.4. Finally, we also highlight developments
made by our research group in section 3.7 and then give a conclusion in section
3.8.
3.1
Inverse Spectral Theory
Inverse spectral theory is the branch of mathematics which studies what can be
deduced about the structure of (differential) operators given some of their properties
are known [187]. In quantum chemistry, this translates to the question whether it
is possible to reconstruct the Hamiltonian from a predefined set of eigenvalues.
The Soviet–Armenian physicist Viktor Ambarzumian1 investigated to which extent
1
Note that one often encounters the English transliterations “Ambartsumian” and “Ham-
38
Concepts and Strategies for Rational Compound Design
a differential equation is defined by its eigenvalue spectrum [188]. Basic inverse
questions — although not explicitly stated as such — have been studied earlier in a
purely mathematical context (see, for example, Ref. [189]). In fact, Lord Rayleigh
briefly discussed an inverse problem already in 1877 in his work about acoustic
waves [190]. Moreover, Ambarzumian was well aware of the fact that the ability
of reconstructing the Hamiltonian from its eigenvalue spectrum would yield great
insight into the structure and properties of matter [188]. He studied the simple
case of an oscillating string, mathematically modeled by a differential equation of
second order,
−
d2 φ
+ q(x)φ(x) = αφ(x)
dx2
0≤x≤1
(3.1)
with the boundary conditions
dφ(0)
dφ(1)
=
= 0,
dx
dx
(3.2)
where φ(x) describes the oscillation as a function of position x, q(x) is a continuous
function, and α is a constant, and showed that the series of eigenvalues
αn = n2 π 2
n∈N
(3.3)
is only consistent with the differential equation, Eq. (3.1), if q(x) ≡ 0. It is easily
seen that Eq. (3.1) resembles the Schrödinger equation for a particle in a onedimensional box if we take α as proportional to the energy eigenvalue and q(x) as a
position-dependent potential. Note, however, that the boundary conditions employed
by Ambarzumian are different from the ones of the particle in a one-dimensional
box. Ambarzumian studied so-called natural boundary conditions, where the first
derivative of the function must vanish at the box boundaries, whereas for the particle
in the box the function itself must be zero at the boundaries.
From the results of Ambarzumian, it follows that we can reconstruct the potential
q(x) — and concomitantly the full Hamiltonian — if we know the (complete) set of
energy eigenvalues (see also Ref. [187]). For example, we can choose a potential q
such that the resulting Hamiltonian features an eigenvalue spectrum of µ2 π 2 , n2 π 2
with n ∈ N and 1 < µ < 2, i.e., we can selectively shift only the lowest eigenvalue
while leaving all others unchanged. In this case, the (unnormalized) wave function
corresponding to the lowest eigenvalue µ2 π 2 can be shown [187] to be given by
ψµ (x) = sin(πx)
,
sin(µ(1 − x)) − sin(µx)
, sin(πx)
sin(µ)
(3.4)
where the denominator
[f (x), g(x)] = f (x)
dg(x) df (x)
−
g(x)
dx
dx
(3.5)
bardzumyan” for the last name. In this work, however, we adopt the transliteration as reported
in Ambarzumian’s paper [188].
3.1 Inverse Spectral Theory
39
is the so-called Wronskian of f and g. The function given in Eq. (3.4) is depicted
in Fig. 3.1 a) for µ2 = 3, and compared to the “standard” ground state wave
function of an electron in a one-dimensional box, i.e., the wave function obtained
if q 0 (x) ≡ 0.
Figure 3.1: a): Normalized wave functions for an electron in a one-dimensional box of
length 1 a.u. Red: ground state wave function when no potential energy term is present in
the Hamiltonian. Blue: ground state wave function if an additional potential is introduced
such that the lowest eigenvalue is shifted from π 2 to 3π 2 while all others stay unchanged.
b): Potential q(x) producing the wave function of Eq. (3.4) with µ2 = 3.
40
Concepts and Strategies for Rational Compound Design
The corresponding potential can easily be constructed numerically by inserting
Eq. (3.4) into the Schrödinger equation for a particle in a one-dimensional box and
solving for q. It is plotted in Fig. 3.1 b). When analyzing the expectation values of
the potential and the kinetic energy associated with ψ√3 , one finds that the kinetic
energy accounts for the major part (namely, approximately 80%) of the total energy,
while the potential energy constitutes the remaining 20%. We can thus conclude
that the potential q causes the build-up of kinetic energy.
During the last decades, several different methods to construct a Hamiltonian
from a given energy spectrum have been developed. However, in 1946, the Swedish
mathematician Göran Borg generalized Ambarzumian’s work [191]. In his seminal
paper, Borg showed that in the general case, one cannot determine the function q(x)
from only one eigenvalue spectrum. Instead, one also needs the eigenvalue spectrum
of a “complementary” eigenvalue problem (e.g., the same differential equation
with different boundary conditions) in order to fully determine the corresponding
differential operator. Thus, in the general case we cannot deduce the form of the
Hamiltonian by predefining a suitable set of energy eigenvalues, as is done in
the above example. It was pointed out that such an inversion is in general not
possible, since different Hamiltonians (molecules) can actually feature comparable
properties [192,193]. Even if it would be possible to strictly invert the Schrödinger
equation in the most general way — meaning that every molecule features a different
value for a certain property — it can well be imagined that among the infinitely many
different molecules (and even the number of molecules accessible by contemporary
synthetic procedures is almost infinitely large [11, 12]), there would always be
molecules that feature almost identical values for that property. One can thus
intuitively understand that for a given application, several molecules can exhibit
the correct property. Therefore, different approaches have to be devised to tackle
this difficult problem. In the following, we will review the most important of these
approaches.
3.2
Quantitative Structure–Activity Relationships
Quantitative Structure–Activity Relationships (QSAR) and the closely related Quantitative Structure–Property Relationships (QSPR) are central for design attempts in
chemistry, biology, and materials research [194–200]. The origins of QSAR trace
back to as early as 1863, when Cros described a relation between the toxicity of
primary aliphatic alcohols and their solubility in water [201]. However, the actual
mechanistic basis of contemporary QSAR approaches was laid in 1964 by Hansch and Fujita with the development of the linear Hansch equation [202], which
was based on the famous work of Hammett, who proposed an equation to relate equilibrium constants of a particular reaction involving benzene derivatives to
some structural parameter, which depends on the particular type of benzene substituents, and a reaction parameter, which depends on the type of reaction under
study [203, 204].
The basic assumption of QSPR is that every physical, chemical, and biological
property depends in a systematic way on the underlying molecular structure. The
3.2 Quantitative Structure–Activity Relationships
41
functional form of this dependence is then searched for, by trying to establish a
model relating a given property to one or more descriptive parameters of the structure
(so called descriptors). In a typical QSPR study, a training set of molecular structures
and corresponding properties is taken as input. The descriptors are then usually
calculated directly from the molecular structure. As descriptors, diverse quantities
such as the molecular weight, the number of ring systems, dipole moment, or
the solvent-accessible molecular surface can be employed [197]. Even though one
could also use descriptors which are obtained experimentally, it is advantageous
to employ descriptors which can be obtained directly from the (three-dimensional)
molecular structure without the need of time-consuming measurements. Once all
desired descriptors have been obtained, a statistical analysis of these data is done
in order to find a relation between the descriptors and the target property in form
of a mathematical expression. In some cases, a simple linear function can already
be used to relate one descriptor to a target property, while in some cases, more
complex, nonlinear, multivariate expressions might be necessary. In contemporary
QSPR studies, sophisticated methods such as artificial neural networks and support
vector machines are employed in order to establish a functional dependence between
properties and descriptors [199]. Once such a dependence is established, one can
easily predict the described properties for arbitrary molecules (although one has to
be careful when predicting a property for a molecule which is very different from
any molecule of the original training set [200]).
Let us illustrate this with a very simple example. We have the set of nine trigonalbipyramidal molybdenum complexes shown in the top right corner of Fig. 3.2, and
would like to find an expression giving the Mo–N binding energy as function of
some descriptor(s). As the Mo–N bond length can be expected to depend on the
binding energy, it appears natural to choose this bond length as a descriptor.
Fig. 3.2 shows the binding energies as a function of the Mo–N bond length2 . In
a first approximation, this dependence can be described by a linear function. With
a least-squares fit, the binding energy E can be obtained from the Mo–N bond
length d as
E(d) = 8.31 kJ mol−1 pm−1 d + 1818.73 kJ mol−1 .
(3.6)
As one can see from Fig. 3.2, this relation is overly simplified. Even though the
Mo–N bond length can be used as a first, rough criterion to estimate the bond
strength, it is certainly not the only factor influencing this quantity. This can be
seen from two complexes in Fig. 3.2, which have almost the same bond length
(namely, about 198.5 pm), but largely differ in their binding energy. Our calculated
expression above would predict the same binding energy for both complexes, thus
overestimating the value for one complex and underestimating it for the other. A
descriptor which could be better suited to judge the Mo–N binding energy would
be the stretching frequency of the Mo–N bond. With such an approach, Schenk
and Reiher established a relationship between the N–N stretching frequency and
2
These data are calculated with the program A DF, version 2010.02b [38], utilizing the BP86
exchange–correlation functional [81, 82] and the TZP basis set without frozen cores [205] at all
atoms. Scalar-relativistic effects were taken into account by means of the zeroth order regular
approximation [37].
42
Concepts and Strategies for Rational Compound Design
the degree of activation of dinitrogen for a series of Schrock-type nitrogen fixating
catalysts [206].
Figure 3.2: Dependence of the Mo–N binding energy E on the Mo–N bond length d of
the trigonal-bipyramidal molybdenum complexes with the general structure shown in the
top right corner. Calculated data for the molecules are shown as black squares (labeled
with the individual residues R). A linear least-squares regression to these data is given
by the black line.
Of course, the development of a robust QSPR for a given quantity is not trivial,
and cannot be simply automatized. In particular, it is often not a priori clear which
descriptors should be chosen. Furthermore, a given QSPR is usually only valid for a
comparingly small subset of chemical compound space, namely for molecules which
are very similar to each other in terms of their molecular structures. Therefore,
these techniques are only of limited applicability for a truly inverse approach, to
which the entire chemical compound space should be accessible. Moreover, even
though it is easy to identify the descriptor values necessary for a given desired
property once a mathematical relation between this property and its descriptors are
found, the construction of a molecule corresponding to a given set of descriptors is
not straightforward at all. An approach to tackle this problem is denoted as inverse
QSAR [207, 208]. In this approach, one aims at constructing a library of chemical
compounds similar to a given lead compound (which is known to exhibit some
favorable property). In this context, chemical similarity is defined as the Euclidean
distance of two molecules in the space spanned by all descriptors used. Therefore,
the more similar the descriptor values are for two compounds, the more chemically
similar they are expected to be. The generation of new molecules is done by
connecting different predefined building blocks with each other in order to obtain
3.3 Inverse Perturbation Analysis
43
a chemically reasonable structure. Then, the descriptors of this new structure are
computed. They can then be used in an optimization algorithm such as simulated
annealing in order to find structures which exhibit a predefined similarity to the lead
compound. One can thus identify further compounds worth to be closer investigated
without the need of screening huge molecular libraries.
3.3
Inverse Perturbation Analysis
In molecular spectroscopy, one is often interested in determining the potential
leading to a certain energy spectrum as accurately as possible. Nowadays, there
exist many different approaches to accomplish this task [209–212], and there are
also specialized computer programs available (see, e.g., Ref. [213]). For the sake of
brevity, we shall not extensively dwell here on the details of all these approaches.
However, we would like to highlight one method, which is very general in its scope
of use and can also serve as motivation in the development of more advanced inverse
approaches, as we shall see shortly.
In 1975, Kosman and Hinze developed a method they called Inverse Perturbation
Analysis (IPA) which allows to calculate accurate potential energy curves [214].
They started from the one-dimensional Schrödinger equation for a diatomic molecule
(note, however, that the method itself is not limited to diatomics),
1 d2
+ U (r) R(r) = ER(r),
(3.7)
−
2µ dr2
where µ is the reduced mass of the molecule under consideration, r the internuclear
distance, R(r) the nuclear wave function, and E the corresponding energy eigenvalue. U (r) is the potential energy function, which is usually not known exactly.
However, many approximate potentials are known, which are often very close to
the unknown exact potential, e.g., the well-known Rydberg–Klein–Rees potential.
One of the central ideas of IPA is now to write the exact potential as a sum of a
known approximate potential U (0) and a (small) correction term ∆U ,
U (r) = U (0) (r) + ∆U (r).
(3.8)
If the approximate potential is a good approximation to the exact curve, the correction term is expected to be small, and can consequently be searched for with an
iterative perturbative approach, where ∆U is the perturbation: First, equation (3.7)
is solved (numerically) for the wave functions R(0) and energy eigenvalues E (0)
only employing the approximate potential U (0) . Since the true energy eigenvalues
E can be obtained experimentally by spectroscopic techniques, one can calculate
an energy correction
∆E = E − E (0) ,
which can be approximated as the first-order energy correction
∆E ≈ R(0) (r) |∆U (r)| R(0) (r) .
(3.9)
(3.10)
44
Concepts and Strategies for Rational Compound Design
If one expands the potential correction into a set of predefined suitable basis
functions, fi (r),
X
∆U (r) =
ci fi (r),
(3.11)
i
then the energy correction becomes
X ∆E ≈
ci R(0) (r) |fi (r)| R(0) (r) ,
(3.12)
i
which can be solved by standard methods to obtain the best set of coefficients ci .
Once this set is obtained, one can use these coefficients to come up with a better
representation of the actual potential energy curve by employing Eq. (3.8). With
this improved potential, one can go again through the procedure outlined above
to obtain an even better potential function. Thus, one can stepwise improve the
description of the potential energy in an iterative fashion until the energy correction
term ∆E is below a given threshold.
In their original paper, Kosman and Hinze successfully applied their method to
HgH. However, they found that their results heavily depend on the particular choice
of basis functions fi (r), which is in addition by far not trivial. While Kosman
and Hinze employed global polynomials (i.e., polynomials defined over the entire
definition range of r, −∞ ≤ r ≤ ∞), Hamilton and coworkers several years
later relied on local Gaussian functions and found accurate and rapidly convergent
results [215].
Still, they experienced some numerical instabilities (the results were very sensitive
to changes in the parameters of the basis functions). These instabilities are due
to a characteristic of inverse problems: they are typically ill-conditioned. In our
case, the behavior of the potential energy function U (r) in regions where the wave
function R(r) is zero is not determined, as can be seen by inspection of Eq. (3.10),
and, therefore, the solution of this equation is not unique. Here, we note again
that it can be mathematically proven that a single energy eigenvalue spectrum is
generally not enough to determine the underlying potential uniquely (c.f., section
3.1). To remedy this problem of numerical instability, Wu and Zhang proposed to
solve Eq. (3.10) by means of a singular value decomposition, and found that their
approach is fast, accurate and numerically stable [216].
We thus have a working method to produce accurate molecular potential energy
functions, given the exact energy spectrum is known and a good approximative
potential can be found as starting point. This is certainly very interesting from an
inverse point of view. However, we are still confronted with two major problems:
first, we are usually not so much interested in a molecule featuring a given energy
eigenvalue spectrum, but rather a molecule with a given other property (such as, e.g.,
dipole moment). These properties can, however, usually not be directly related to
a specific energy spectrum. Furthermore, even if we could somehow overcome this
problem, and come up with a certain molecular potential, it would not automatically
be clear how the underlying molecular structure would be composed of atomic nuclei
and electrons. However, this problem could be solved in quite an elegant fashion by
employing some sort of atom-based basis functions, such that each basis function
3.4 Model Equations
45
would represent one (abstract) atom — the actual type of the atom would then be
determined by the final expansion coefficient ci . In the above optimization procedure,
one would of course not only have to search for the expansion coefficients, but also
the position at which the basis functions are centered (which enter as a parameter
in these local functions, and denote the position of the nuclei in space), unless
one uses a comparingly large number of such basis functions, such that they can
simply be distributed evenly in space — at a position where no nuclei should be,
we can expect that the respective expansion coefficient is close to zero. In fact,
such approaches have been emerging during the past few years, see sections 3.6.3
and 3.6.4.
Even though there appears to be no general solution of the first problem mentioned
above, we shall see in the sections to come that we might find other ways of
directly relating the molecular potential to properties other than the energy eigenvalue
spectrum, thereby making this approach highly interesting.
3.4
Model Equations
A completely different, but nevertheless very interesting approach is the development
of (simple) model equations. This technique has been successfully employed by
Marder et al. in their famous paper from 1991, in which they describe possibilities
to increase the first electronic hyperpolarizability of conjugated organic molecules
[217], and we shall use this specific publication to illustrate the underlying design
approach, but of course many other workers have studied the design of molecules
with optimal optoelectronic properties (see also Refs. [218–220]).
If a molecule is placed in an external electric field, its charge density is distorted,
an effect which is known as polarization. When the external electric field has only
a comparingly small influence on the electronic structure of a given molecule, one
can write the polarization P as a power series
P = αE + βE 2 + γE 3 + O(E 4 ),
(3.13)
where E is the external electric field, and the constants α, β, and γ are denoted as
polarizability, first hyperpolarizability, and second hyperpolarizability, respectively.
We note in passing that in the general case, the (hyper-)polarizabilities are no
scalar quantities but instead described by tensors. However, for the following
discussion, our approximation does not constitute any loss of generality. Since the
electric field constitutes only a minor perturbation of the molecule, perturbation
theory is a suitable method to calculate α, β, and higher order corrections (in
cases where the wavelength is much larger than the size of the molecule, the
perturbation Hamiltonian describes the interaction of the molecular dipole moment
with the external electric field). β can thus be obtained by correcting the state
functions to second order in the electric field [221]. The resulting expression is
rather complicated, involving sums over all occupied orbitals. However, one often
observes that the first hyperpolarizability is dominated by contributions from only
the first few excited states. This motivates a simplification by taking only the
ground state and the first excited state into account. In the resulting two-state
46
Concepts and Strategies for Rational Compound Design
approximation [221],
β∝
2
Xge
(Xgg − Xee )
(Eg − Ee )2
,
(3.14)
where the subscripts “e” and “g” denote the excited and ground state, respectively.
The matrix elements Xij are given by
Xij = hΨi | r |Ψj i
i, j ∈ {e, g}.
(3.15)
2
, Xgg − Xee , and 1/ (Eg − Ee )2 on the first
The influence of the three quantities Xge
hyperpolarizability can be studied using a simple two-orbital system as example
[221]. We can thus write the wave function as linear combination of these two
orbitals,
Ψ = c1 φ1 + c2 φ2 .
(3.16)
We furthermore define the energy expectation values
hφ1 | H |φ1 i = ∆,
(3.17)
hφ2 | H |φ2 i = −∆,
(3.18)
and
such that the energy difference between the two orbitals φ1 and φ2 is 2∆. The
coupling strength between the two orbitals is defined as
hφ1 | H |φ2 i = t.
(3.19)
With these definitions, we can write the time-independent Schrödinger equation in
matrix form as
∆−E
t
c1
0
=
.
(3.20)
t
−∆ − E
c2
0
For this system of homogeneous equations to have a nontrivial solution, i.e., c1 ,
c2 6= 0, the determinant of the 2×2 matrix needs to be zero, which gives us an
equation to determine the energy E. We find the two solutions
√
(3.21)
E± = ± ∆2 + t2 .
Taking the first equality of Eq. (3.20), namely
(±)
(∆ − E± ) c1
(±)
+ tc2
= 0,
we can establish the relation
s
2
(±)
c1
E± − ∆
∆
∆
=
=± 1+
− .
(±)
t
t
t
c2
(3.22)
(3.23)
3.4 Model Equations
47
For each of the two energy eigenvalues, we get a separate set of coefficients ci ,
such that we can actually write two wave functions,
(−)
(−)
(3.24)
(+)
(+)
(3.25)
Ψg = Ψ− = c1 φ1 + c2 φ2 ,
and
Ψe = Ψ+ = c1 φ1 + c2 φ2 ,
where the subscripts “g” and “e” again denote the ground and excited state, respectively. Expressions for the individual coefficients can be obtained by making
use of the normalization condition, i.e.,
hΨ± |Ψ± i = 1.
(3.26)
2
, Xgg − Xee , and 1/ (Eg − Ee )2 as
We can thus now calculate expressions for Xge
well as for β as a function of ∆. These functions are given in Fig. 3.4. We see
that the combination of the individual expressions leads to a maximum of β at
about ∆ = 0.25 when the overlap between φ1 and φ2 and the coupling strength t
are set to 0.5. The task is now to find a molecule corresponding to these values,
which is by far not trivial.
Figure 3.3: The first hyperpolarizability β of a simple two-orbital system and its components (cf., Eq. (3.14)) as a function of ∆, where 2∆ is the difference between the
two orbital energies. In this figure, the overlap between the two orbitals as well as the
coupling strength t were set to 0.5.
In their publication, Marder et al. also made use of the mathematical expression
for the first hyperpolarizability β given in Eq. (3.14), but they relied on a somewhat
48
Concepts and Strategies for Rational Compound Design
more sophisticated model system consisting of four orbitals (corresponding to donor,
acceptor, and bridge orbitals) [217]. Marder et al. could relate the above expressions
to actual molecular building blocks, such that they could come up with very specific
design strategies to obtain promising molecules [217].
Probably one of the biggest advantages of this approach is the fact that it does not
necessitate time-consuming computations, instead relying on rather simple yet very
elegant equations. However, the development of such a simplified model expression
is not always trivial. Moreover, the deduction of an actual molecular (sub-)structure
is not straightforward at all, but requires often a considerable amount of chemical
knowledge and intuition. Unfortunately, especially this last fact is a big drawback,
limiting the applicability of this method to a rather narrow subspace of chemical
compound space. Therefore, this approach suffers from the same fundamental
problem as QSPR techniques (see above).
3.5
Optimized Wave Functions
This severe limitation has long been recognized (also by the authors of the study
taken as an example in Section 3.4) [222]. In 1996, Kuhn and Beratan proposed a
new strategy in order to overcome this problem, which can be regarded as a first
truly inverse approach in rational compound design. They suggested to search for an
optimal molecular structure by optimizing the corresponding mathematical objects
describing it, i.e., the Hamiltonian and its wave function (in a quantum mechanical
framework). The authors also point out that this inverse problem is ill-conditioned
as not enough information is present in order to carry out the optimization (see also
Section 3.3). This problem can be lifted by introducing additional constraints. For
example, bonding in molecules is local in nature, leading to characteristic patterns
in the Hamiltonian matrix. This can be conveniently illustrated with the very simple
Hückel molecular orbital method [223,224], where a linear conjugated hydrocarbon
chain corresponds to a tridiagonal Hamiltonian matrix. This is because the overlap
of orbitals centered on atoms not directly bound to each other is assumed to be
zero. While this assumption is of course not strictly true, the overlap of basis
functions located at distant atoms is in fact usually negligibly small, and also
in more advanced electronic structure theories the Hamiltonian matrix is usually
sparse. With the Hückel method, for a linear chain of three orbitals centered at
x = {0, 1, 2}, we can write the Hamiltonian in matrix form as


α1 β1 0
H =  β1 α2 β2  ,
0 β2 α3
(3.27)
where the αi represent the orbital energies and the βi the coupling between adjacent
orbitals. Since we are free in our choice of energy origin, we can arbitrarily set
one of the αi to zero. Furthermore, we can also choose any suitable unit for the
3.5 Optimized Wave Functions
49
energy, which allows us to set one of the βi to −1. With this, we can write


0 −1 0
(3.28)
H → H 0 = −1 α20 β20  .
0
0
0 β2 α3
The task is now to find values for the remaining parameters such that a given
property is optimized. As an example, Kuhn and Beratan took the electric dipole
operator, the matrix elements of which can be written as
µkm = −
3
X
(k) (m)
x i ci ci ,
(3.29)
i=1
(k)
when overlap contributions are neglected (ci is the expansion coefficient of the
i-th orbital in wave function k as in section 3.4) and optimized the transition dipole
(2) (3)
(2) (3)
moment µ23 = c2 c2 + 2c3 c3 . Within our simple approach, µ23 is a function of
only three variables, namely α20 , α30 , and β20 . If we arbitrarily set β20 = β1 = −1,
we can plot µ23 as a function of only α20 and α30 , as shown in Fig. 3.5.
Figure 3.4: Dependence of the transition dipole moment O23 on the two parameters α20
and α30 , respectively. All data are given in arbitrary units.
α20
It is straightforward to optimize µ23 with respect to these two parameters. For
≥ −5, one finds that µ23 is minimized for α20 = −5.00 and α30 = 0.00, which
50
Concepts and Strategies for Rational Compound Design
yields µ23 = −0.97 [222]. The global minimum, however, appears to be situated
at α20 = −4848.59 and α30 = 0.00. With these values, one finds µ23 = −1.00.
In their pioneering study, Kuhn and Beratan proposed also to optimize not only the
elements in the Hamiltonian matrix, but also the coefficients of the wave function
(while the energy eigenvalues are held fixed). They applied their method to a
few very simple “toy” examples as the one given above in order to demonstrate
the general concept. However, for many years after its publication, the method
could not be applied to find a “real” optimum structure, as was hoped by the
authors. The problem was not so much the sheer size of chemical compound
space, which makes advanced sampling schemes necessary (see also below), but
rather the fact that — as is also the case for many methods reviewed above — it is
extremely difficult to construct a molecule from its Hamiltonian matrix or wave
function [225]. However, in 2006, Beratan and coworkers established a general
framework which should surround this problem (see Section 3.6.3).
3.6
Advanced Sampling Techniques
Today, powerful computer systems and clever optimization algorithms (e.g., simulated annealing [226], genetic algorithms [227], and Monte Carlo sampling [228])
allow us to sample a comparingly huge part of chemical space. The application
of such algorithms is nowadays the most important approach in rational compound
design. Therefore, we shall also review these approaches, although they are not
strictly inverse, since they do not directly aim at predicting a structure featuring a
predefiend set of properties, but rather follow the more traditional approach where
the properties of a predefined molecular structure are calculated.
3.6.1
Rational Design in Solid-State Chemistry
During the early 1990s, Schön and Jansen developed a sophisticated technique for
synthesis planning [229, 230]. They proposed a modular computer program, which
searches chemical space by means of a global optimization technique and identifies promising structure candidates, that can subsequently be refined and analyzed.
After some external boundary conditions (e.g., initial number of atoms, external
pressure, etc.) have been specified, the algorithm sets up an elementary cell. The
unit cell vectors (length and orientation) are chosen randomly, as are the positions
and composition of atoms. Then, all these parameters are globally optimized employing a fast but not very accurate electronic-structure method. The final result is
then subjected to a more accurate refinement. The global optimization is carried
out multiple times, starting with different random simulation cells. The resulting
collection of promising compounds can then be analyzed in terms of their composition and physical properties. Jansen and coworkers have applied their methodology
to a range of example systems, for instance noble gas mixtures [231] and binary
ionic compounds [232].
3.6 Advanced Sampling Techniques
3.6.2
51
Inverse Band Structure Approach
A similar approach has been mentioned in 1994 by Werner et al. who proposed to
tailor a band structure reflecting the desired properties and to subsequently search
for a solid exhibiting this very band structure [233]. They termed this problem of
finding a solid with a suitable band structure the “inverse band structure problem”.
Unfortunately, they could not come up with a working solution to this problem. At
this point, we should illustrate how complex this optimization problem is. Consider
a comparingly simple system, such as the pseudo-binary alloy A0.25 B0.75 C with a
unit cell of 128 lattice sites (64 cation sites, occupied by either A or B, and 64
anion sites, all of which are occupied by C), then the total number of possible
configurations is given by the binomial coefficient of 64 and 16, i.e., it is on
the order of 1014 ! Needless to say that exhaustive enumeration of such a big
search space is simply not possible. However, only five years later, Franceschetti
and Zunger published a computational procedure which successfully addresses the
inverse band structure problem [234]. Similarly to the approach by Schön and
Jansen, they start with an elementary cell of a given configuration. They define an
atomic configuration in terms of a vector
σ = (S1 , S2 , . . . SN ),
(3.30)
where Si is the atom type at lattice site i. A given property P (such as the bandgap)
is a function of the atomic configuration. The goal of the optimization is to find
a configuration with one or several predefined target properties P (target) , i.e., one
attempts to minimize the function
O(σ) =
X
ωα Pα − Pα(target) ,
(3.31)
α
where ωα is the weight assigned to the property Pα . In the original approach
by Franceschetti and Zunger, this function is minimized by a simulated annealing
algorithm. Atomic configurations are generated by Monte Carlos moves (such as,
e.g., changing the type of a given atom). Even though this algorithm is able to
find the global minimum significantly faster than would be possible with exhaustive
enumeration, it still requires sampling of several thousand atomic configurations. It
is therefore very important to have a fast method at hand to evaluate the properties
of a given atomic configuration. Franceschetti and Zunger relied on a valence-forcefield method to optimize the positions of the atoms of a given atomic configuration
[235], and utilized a semiempirical Hamiltonian, where each atom is replaced by a
pseudopotential [236], in order to evaluate the properties. Furthermore, they made
use of a special technique for the diagonalization of the Hamiltonian, which focuses
on the few eigenvalues around the bandgap [237].
Utilizing this approach, Franceschetti and Zunger predicted in their original publication [234] the configuration an Al0.25 Ga0.75 As alloy should have in order to
exhibit a maximum bandgap, and very recently, Zunger and coworkers studied a
quaternary (In,Ga)(As,Sb) semiconductor [238]. This original approach has continuously been further developed and improved. For example, it is also possible
52
Concepts and Strategies for Rational Compound Design
to employ a genetic algorithm instead of simulated annealing [239, 240], which
generally leads to a faster convergence.
We should note here that the general method of finding the optimal species with
regard to a given property in a subset of chemical space is also employed and
further developed by many other researchers, although they do not specifically aim
at an inverse approach. For example, Nørskov and coworkers have identified the
20 most stable four-component alloys of 32 transition, noble and simple metals in
both face-centered cubic and body-centered cubic structures [241]. In doing so,
they searched a subspace of no less than 192’016 compounds by means of an
evolutionary algorithm [241]. Due to the limited space of this work, we refer the
reader to the literature [242–246] for further examples of computational materials
design.
The approach by Franceschetti and Zunger is specifically tailored for the design
of solid-state materials, and its extension to the design of isolated molecules is
not necessarily straightforward. In the above procedure, the total number of atoms
is defined from the outset, as a certain unit cell with a given number of lattice
sites is defined. However, for an isolated molecule, this approach is not valid.
One would therefore have to abandon this constraint and introduce different ones.
For example, by means of simple valence rules, one could define the total number
of atoms coordinating to a given atom (i.e., the number of nearest neighbors).
Furthermore, in the design of solid-state materials one can apply periodic boundary
conditions, and therefore in principle study a crystal of infinite size. For an isolated
molecule, however, this is not possible, and one therefore encounters the problem of
unsaturated valences. In a first approach, one could simply saturate all valencies with
hydrogen atoms. In fact, such rules have been found to be useful in the gradientdriven construction of molecules as proposed by Weymuth and Reiher [247].
3.6.3
Linear Combination of Atom-Centered Potentials
One of the more daunting problems in the field of rational design is the fact that
individual molecules represent discrete points in chemical space, whereas most
optimization methods are developed to work with continuous functions. In 2006,
Beratan, Yang and coworkers addressed this problem [248]. They started from the
well-known fact that within the framework of density functional theory any molecular system is characterized by the external potential ν(r) [249], which represents
the attraction between electrons and nuclei, and the total number of electrons. They
then treated this external potential as a continuous variable to be optimized. Yang
et al. even presented a functional of external potentials together with a corresponding variational principle [250]. One major issue with this approach is that not all
external potentials correspond to a molecule, even though all molecules map unequivocally to a given ν(r). An external potential which corresponds to a chemical
structure is called C-representable. Beratan, Yang and coworkers very elegantly
solved this problem of C-representability by expanding the potential ν(r) into a
set of atomic potentials, i.e.,
X (r ) (r )
ν(r) =
bA A νA A (r),
(3.32)
A,r A
3.6 Advanced Sampling Techniques
(r )
53
(r )
where bA A is the expansion coefficient of the potential term νA A (r), which
denotes the potential of atom A at position r A . This approach has been called
“linear combination of atomic potentials” (LCAP), and it constrains the external
potential to be necessarily C-representable. The optimization problem then reduces
to finding the best set of expansion coefficients for these atomic potentials. It is
important to note that not only individual atoms can be represented with such atomic
potentials; rather, a collection of such potentials can be contracted to represent e.g.,
a functional group,
X (r ) (r )
(r )
νA A (r) =
bB A νB A (r),
(3.33)
B
and can be optimized as a whole. In order to correspond to an actual molecule,
(r )
the coefficients bA A must be either 0 or 1 denoting either the absence or the
(r )
presence of the atom or functional group, respectively, represented by νA A (r).
However, since the expansion coefficients are treated as continuous variables during
the optimization, one can find non-integer values for them. In this case, rounding to
the nearest integer is necessary. While this approach of continuous optimization is
rather new in quantum chemistry, Zunger and coworker pointed out that Bendsøe and
Kikuchi presented a very similar approach in the context of materials design [251].
In their original publication [248], Beratan, Yang and coworkers investigated
as a proof of concept a rather simple problem, constructed from two sites (fixed
in space), for which they searched for the best potential such that the resulting
hyperpolarizability is maximized. In doing so, they started from a small library of
six different chemical substituents modeled by different potentials, and investigated
how the optimal potential is composed of these “atomic” potentials. They finally
found that H2 S2 has the largest hyperpolarizability of all molecules present in the
small chemical space considered (six substituents at two sites give a total of 26 = 64
possible molecules) which was in agreement with the results found by exhaustive
enumeration. With the same methodology, Keinan et al. successfully optimized
the first hyperpolarizability of porphyrin-based nonlinear optical materials [252].
d’Avezac and Zunger recently published results showing that the approach by
Yang and coworkers is very efficient for optimizing the atomic configurations of
alloys [253].
In recent years, the group of Yang continuously developed and improved their
method further. In 2007, they implemented it into an AM1 semiempirical framework [254]. In 2008, a gradient-directed Monte Carlo approach, which is able to
“jump” between chemical structures, and thus to overcome barriers between local
minima, was presented [255], and the approach was also implemented within a tightbinding framework [256]. Furthermore, these authors studied different optimization
procedures and found that a combination of their LCAP approach with a best-first
search algorithm (BFS, an algorithm searching a graph by stepwise expanding the
most promising node [257]) improves the performance of the optimization [258].
The LCAP and related approaches have reached a state of maturity which allows
them to be applied to rather complex problems [252]. However, we should note
that one disadvantage is that the library of potentials to be investigated has a finite
size and, thus, not the entire chemical space is sampled. Furthermore, the position
54
Concepts and Strategies for Rational Compound Design
of the potentials is fixed in space (and only optimized in the course of a standard
structure optimization), which further restricts the space of chemical compounds
studied. This bears the danger of overlooking a good candidate, simply because
it is not in the sampling space. This problem could, of course, be remedied by
employing an extremely large library and allowing potentials at many different sites.
However, this also dramatically increases the computational cost of the resulting
optimization problem. Nevertheless, as computer systems become more and more
powerful, the LCAP approach will certainly be applied to increasingly complex
problems.
3.6.4
Alchemical Potentials
Another interesting approach to convert chemical space into a continuous space has
been developed by von Lilienfeld et al. Starting again from the Hohenberg–Kohn
theorem [249], von Lilienfeld et al. noted that any molecular property O can be
written as a functional of the nuclear charge distribution Z(r) and a function of
the total electron number Nel [193], i.e., O ≡ O[Z(r)](Nel ). Pictorially speaking,
molecules are superpositions of electron and nuclear charge distributions. The
electron density, however, is (up to a constant factor) fully determined by the
total electron number and the external potential, which itself is a functional of the
nuclear charge distribution. Rational compound design can thus be described as
an optimization problem within the space spanned by nuclei and electrons, which
leads us to the minimization problem
min |O[Z(r)](Nel ) − O0 |2 ,
Nel ,Z(r)
(3.34)
where O0 is the desired target property. A continuous optimization in the space of
electrons and nuclei implies, of course, fractional electronic and nuclear charges.
At this point, we should note that some controversy concerning fractional electrons
emerged some years ago and is still not fully resolved (see, e.g., Ref. [130]). In
practice, in von Lilienfeld’s approach fractional atomic and electronic charges are
generated by means of a suitably parametrized effective core potential (ECP) [259].
Therefore, there are no explicit fractional electrons occurring in the computation,
but rather implicitly dealt with in the ECP.
In order to efficiently search the space spanned by electrons and nuclei, von
Lilienfeld et al. worked out a mathematical expression for the derivative of the
total electronic energies with respect to the nuclear charge distribution [193]. This
has subsequently become known as the nuclear chemical potential µn (r) (in analogy
to the well known electronic chemical potential), which, for a discrete nuclear charge
distribution and perturbation theory to the first order, is found to be the electrostatic
field E (1) (r),
Z
X ZI
ρ(r 0 )
(1)
dr 0 +
,
(3.35)
µn (r) ≈ E (r) = −
|r − r|
|r − r I |
I
where the integration is over the entire space. The nuclear chemical potential is a
function of space and is related to the proton affinity. At the position of a nucleus,
3.7 Mode- and Intensity-Tracking
55
it measures the tendency of this nucleus to “mutate” into a different type. At
these specific points, the nuclear chemical potential is usually called the alchemical
potential.
During their studies, von Lilienfeld et al. found it necessary to come up with a
rigorous description of chemical space [260]. They found that such a description
is tightly connected to grand-canonical ensemble theory and, hence, could benefit
a lot from earlier developments. It was soon recognized that molecular properties
such as the potential energy represent a state function within the space of Z(r)
and Nel , and hence, any two points in this space can be connected by an order
parameter, which is usually denoted as λ. One can then use this parameter to
interpolate between any two molecules (similar to well-known techniques such as
thermodynamic integration) and study the variation of a given property as one
continuously changes one molecule into the other [260–263]. In such studies, von
Lilienfeld and coworkers found that the path connecting any two molecules is
generally not linear [262]. This makes the prediction of properties of unknown
molecules based on results of known compounds very difficult. A general method
of linearizing these connection paths has yet to be found, and von Lilienfeld even
offers a prize (the equivalent of an ounce of gold) to the person finding a solution
to this complicated problem [259].
So far, this approach has not been applied to many problems. As a proof
of principle von Lilienfeld et al. [193] designed a nonpeptidic anticancer drug
candidate by optimizing the atoms constituting the peptide unit in a peptidic starting
structure (even though this peptidic structure already showed strong antitumoral
activity, it would be cleaved in vivo by proteases, such that it can essentially not
be employed as drug). In 2007, Marcon et al. showed that the energy of the
highest occupied molecular orbital in benzene and its BN-doped derivatives can be
precisely tuned [264].
3.7
Mode- and Intensity-Tracking
Both Mode- and Intensity-Tracking are rather different from all the above mentioned
approaches in the sense that they do not aim for the rational design of a molecular compound or material. However, they can be counted as inverse methods
since they turn the usual procedure in quantum chemical computations upsidedown [265]. More specifically, Mode-Tracking allows for the direct determination
of selected molecular vibrations without the need of first calculating the entire
Hessian matrix (as would be the case in the standard approach) [266]. Similarly,
Intensity-Tracking enables one to obtain the most intense vibrations without the
full Hessian matrix [267–270]. Thus, these two methods directly target molecular
vibrations, and allow one to obtain these at lower computational cost than with the
traditional methodologies.
In Mode-Tracking, one starts from a guess mode, which is then iteratively refined
until a normal mode is found that is similar to the guess vibration (measured by an
overlap criterion), but is also an eigenvector of the Hessian (within a given numerical
precision). Mathematically speaking, a Krylov subspace iteration is carried out based
56
Concepts and Strategies for Rational Compound Design
on Davidson’s method [271, 272]. It sets out to calculate the Davidson matrix for
a user-defined guess, from which a residual vector is obtained after diagonalization
which is a measure for the deviation from an exact eigenvector and thus provides
information on how to improve the guess vector. From this residual and the
preconditioner, which is an approximation to the inverse of the Hessian (minus the
sought-for eigenvalue), a correction vector is generated, which is added as a new
basis vector in the iterative subspace diagonalization. The theory has been explained
in great detail in the original articles [266,273] and in recent reviews [265,267,274],
so we will only shortly review it here.
The standard approach to calculate the k-th vibrational frequency of a molecule
is to solve the well-known eigenvalue problem
H (m) q k = λk q k ,
(3.36)
where the eigenvector q k is called the k-th normal mode, and the eigenvalue λk is
proportional to the square of the k-th vibrational angular frequency. H (m) is the
mass-weighted Hessian matrix (in Cartesian coordinates), the entries of which are
given by [275]
2
∂ Eel
1
(m)
,
(3.37)
HIα,Jβ = √
mI mJ ∂rIα ∂rJβ
where mI and r I denote the mass and spatial position of nucleus I (α ∈ {x, y, z};
in the following, we will abandon these two-component subscripts in favor of
single-component subscripts ranging from 1 to 3N written in lower case). The
idea of Mode-Tracking is to selectively calculate certain normal modes without
first computing the entire Hessian matrix (which is the most time-consuming step
of a standard vibrational analysis) [266]. The motivation behind Mode-Tracking is
that it is often possible to guess a vibration that is characteristic for a particular
spectroscopic phenomenon or a molecular structure. In these cases, it is straightforward to create an approximate normal mode for a desired vibration. One can
therefore start with a guess normal mode
b=
3N
X
(m)
bi e i ,
(3.38)
i=1
where the sum runs over all Cartesian coordinates of the N nuclei, bi is the
(m)
displacement magnitude of the mode in the i-th component, and ei are the massweighted nuclear Cartesian basis vectors. Taking b as a first approximation to the
desired normal mode q k , one can calculate the k-th element of the left-hand side
of Eq. (3.36) as
X
∂ 2 Eel
1 ∂ 2 Eel
1
bj = √
.
σk = H (m) q =
√
mj mk ∂rj ∂rk
mk ∂rk ∂b
k
j
(3.39)
The last equality means that σk can be calculated by computing the directional
derivative of the nuclear gradient along the vector b, which can conveniently be
3.7 Mode- and Intensity-Tracking
57
done in a semi-numerical fashion [266]. In Mode-Tracking, the target mode is
iteratively improved by a Davidson-type subspace iteration method [266]. In every
iteration, a new vector b(l) is generated (l denoting the iteration), and concomitantly,
also a new vector σ (l) is obtained. The i vectors generated up to iteration i can
be assembled into the matrices B (i) and Σ(i) , respectively. With them, one can
calculate the matrix
H̃
(m,i)
= B (i)T H (m) B (i) = B (i)T Σ(i) ,
(3.40)
which can be obtained from B (i) and Σ(i) alone, i.e., it is not necessary to compute
the full Hessian matrix H (m) . We then solve the small eigenvalue problem
H̃
(m,i)
u(i) = ρ(i) u(i) ,
(3.41)
where the matrices ρ(i) and u(i) contain the i-th approximations to the target
eigenvalue λk and target normal mode q k , respectively. The best approximations,
(i)
(i)
which we will denote as ρs and us , are selected from ρ(i) and u(i) , and the
residuum vector
r (i)
s =
i
X
(i)
(l)
b
us,l σ (l) − ρ(i)
s
(3.42)
l=1
is calculated. From this residuum vector, a new basis vector b(i+1) is generated,
b(i+1) = X (i) r (i)
s ,
(3.43)
where X (i) is a preconditioner. The convergence characteristics of the ModeTracking algorithm strongly depend not only on the initial guess mode, but also
on the particular choice of X (i) . However, experience shows that also very simple
choices for the preconditioner (e.g., a unit matrix) facilitate fast convergence [273].
The Mode-Tracking approach has successfully been applied in studies of a large
[(Ph3 PAu)6 C]2+ cluster [276], molecular wires (i.e., long carbon chains) [277], adsorbate vibrations [278], and tetrameric methyl lactate clusters [279]. Furthermore,
in 2008, Herrmann et al. integrated the Mode-Tracking methodology into a QM/MM
framework [280]. Very recently, Kovyrshin and Neugebauer [281, 282] presented
a similar tracking methodology for the selective calculation of electronic excitations in the framework of time-dependent DFT. Subspace iteration methods for
vibrational problems have been used before in the context of molecular mechanics
applications [283–286], where huge Hessians must be diagonalized and where subspace iteration techniques are a natural means to solve the diagonalization problem.
In quantum chemistry, however, the Mode-Tracking idea established new principles [266, 276, 277, 287–291]: (1) isolated, structure-characteristic vibrations can be
directly targeted, (2) vibrations that involve only a subset of atoms (e.g., in the
case of adsorbates on surfaces [278] or in QM/MM partitionings [280]) can be
directly optimized, (3) low-dimensional Hessians also benefit if their entries are
time-consuming to calculate.
In our former work [273], it was shown that the proper construction of a guess
vector has an important influence on the convergence characteristics of the algorithm.
58
Concepts and Strategies for Rational Compound Design
Often, such guess vectors can be constructed based on intuition (e.g., for localized
stretch vibrations [273]) or on the basis of model calculations [277]. In the following,
we will demonstrate this feature for the example of the N−H stretch vibrations
in the adenine–thymine (AT) base pair shown in Figure 3.5. This is an important
example from the class of supermolecular assemblies, which appear in an increasing
number of quantum chemical applications, e.g., in explicit solvation studies or in
studies on models for protein binding pockets.
Structure optimizations and single-point calculations needed for the mode-tracking
calculations as reported in the following have been carried out with the program
package T URBOMOLE 6.0 [117] employing the BP86 density functional [81, 82]
and Ahlrichs’ def-TZVP basis set [292] for all atoms.
O
N
N
N
N
N
N
N
O
Figure 3.5: Structure of the adenine–thymine base pair used as an example for modetracking calculations.
Table 3.1: Vibrational wavenumbers (in units of cm−1 ) for the AT base pair. The values
reported in the column “complex“ refer to the reference, i.e., full vibrational calculation;
”M.-T.” stands for Mode-Tracking results in the first (1) and final iteration.
isolated
complex
M.-T. (1)
3500.1
3518.8
3544.8
3552.6
3652.1
2673.6
3209.9
3541.4
3551.8
3588.7
2668.8
3318.6
3541.7
3551.9
3483.0
M.-T. (final)
assignment
2673.8 T: NH· · · N
3210.1 A: NH2· · · O sym
3541.7 T: NH isolated
3552.0 A: NH isolated
3588.6 A: NH2· · · O asym
A straightforward way to construct guess vibrations in supermolecular systems
is to start from the vibrations calculated for isolated subsystems. The AT base
pair represents a fairly simple example in this context, but the procedure outlined
here can easily be extended to several subsystems. In the present case, we first
carried out frequency analyses for the two building blocks, i.e., for adenine and
for thymine. Both molecules were fully optimized, and the resulting frequencies
3.7 Mode- and Intensity-Tracking
59
for the N−H stretch vibrations are listed in the first column of Table 3.1. For the
isolated bases, all N−H stretch vibrations lie between 3500 and 3660 cm−1 .
The Hessians of the two constituent molecules were then employed as blocks for
an approximate Hessian of the base pair, and the normal modes were adopted as
guess normal modes in a mode-tracking calculation on the complex. In the present
case, each vibration has been tracked separately; results are given in Table 3.1 for
the first and the final iteration of the mode-tracking calculation (we report the data
with an accuracy of 0.1 cm−1 to illustrate the small numerical differences between
mode tracking and full vibrational calculations, even though typical discrepancies
between measured fundamental frequencies and calculated harmonic ones are on
the order of 10 – 20 cm−1 ). The results of the first iteration just reflect the change
in wavenumber when assuming that a given monomer normal mode is unchanged
upon complex formation. The guessed vibrations are shown in Figure 3.6, and the
normal modes of the complex are displayed in Figure 3.7.
Figure 3.6: Normal modes obtained for isolated thymine or adenine, respectively, used
as guess modes in the mode-tracking calculation. The frequency associated with each
normal mode is given in brackets (all values in cm−1 ).
From Table 3.1, we can distinguish three different types of vibrations. The
first type comprises vibrations which are hardly affected upon base pairing, neither
normal mode nor the vibrational frequency are affected. Hence, already the results
for the isolated molecules are very close to the results obtained for the complex.
The two “isolated” N−H vibrations, i.e., the N−H stretch vibrations not involved
in hydrogen bonding, belong to this class. They are found at 3545 and 3553 cm−1
in the isolated case, and at 3542 and 3552 cm−1 in the base pair. The changes in
wavenumber are thus of the order of 1 – 3 cm−1 , and already the first mode-tracking
iteration essentially leads to the converged result.
The second type involves modes which significantly change in wavenumber,
but the normal modes are hardly changed upon base pairing. In our example, the
thymine NH· · · N mode belongs to this class. It decreases by more than 820 cm−1 in
60
Concepts and Strategies for Rational Compound Design
wavenumber, from 3500 to 2674 cm−1 . Interestingly, the largest part of this change
is already captured in the first mode-tracking iteration, which is an indication that
the exact normal mode is already well approximated by the guess vibration taken
from the set of modes of the isolated molecule. This guess deviates by less than
5 cm−1 from the converged wavenumber, and thus overestimates the wavenumber
change by only 0.6 %. By comparing Figures 3.6 and 3.7, it can be seen that these
normal modes are essentially unchanged.
Figure 3.7: Normal modes for the AT dimer. The frequency associated with each normal
mode is given in brackets (all values in cm−1 ).
This situation changes for the third type of vibrations, for which both the
wavenumber and the normal modes change significantly. In the present example, this involves the two N−H stretch modes of the NH2 group, which can be
classified as “symmetric” and “antisymmetric” in the free adenine molecule. But
this symmetry is completely broken by the hydrogen-bonding interaction of one
of the two hydrogen atoms with the thymine oxygen atom, as can be seen in
Figure 3.7. Also here, already the results obtained with the guess mode in the
first iteration of the mode-tracking calculation indicate significant changes in the
wavenumbers, but they still deviate from the final results by about ±100 cm−1 . Most
of this deviation is corrected in the second iteration, in which an approximation for
the other NH2· · · O mode is obtained as a by-product. We note in passing that all
converged wavenumbers from the mode-tracking calculation agree within 0.4 cm−1
with the results from a conventional frequency analysis.
The present example indicates that the construction of guess vibrations based
on isolated molecule calculations is a powerful tool when calculating vibrational
spectra for molecules in explicit environments. The weaker the interaction with
the environment is, the faster the convergence will be. In the present example, all
vibrations were converged within four iterations or less, based on a preconditioner
constructed from the subsystem Hessians. A more challenging study in a similar
spirit has recently addressed the effects of a protein binding pocket on the resonance
3.8 Summary
61
Raman spectrum of the carotenoid spheroidene in the photosynthetic reaction center
of the purple bacterium Rhodobacter sphaeroides [293]. Using a similar approach
as employed here, guess vibrations were constructed for the isolated spheroidene
(102 atoms), while subsequently the vibrations have been refined in the binding
pocket (507 atoms), thus demonstrating the power of Mode-Tracking for challenging
vibrational problems.
Intensity-Tracking generally works with the same principle as explained above
for Mode-Tracking. However, the approach does not aim to refine a given userdefined vibration, but to yield those normal modes which are associated with highest
intensities (for a given vibrational spectroscopy). Therefore, the most important
difference to Mode-Tracking consists in the choice of the starting guess mode.
This guess depends strongly on the kind of vibrational spectroscopy for which the
highest-intensity modes are to be obtained, but can generally be obtained from
an eigenvalue problem. Our group has developed and successfully applied ideal
starting guesses for conventional infrared spectroscopy [269], Raman and Raman
optical activity spectroscopy [270], as well as resonance Raman spectroscopy [268].
Even though neither Mode- nor Intensity-Tracking are intended for the rational
inverse design of molecular compounds, they are very interesting from an algorithmic point of view. They allow for a very time-efficient evaluation of molecular
vibrations. Similar techniques, aimed at the direct determination of different molecular properties would be very beneficial especially in the inverse design of new
molecules and materials. We should remember at this point that the most promising
contemporary approaches (see section 3.6) all rely on sophisticated optimization
algorithms. Even though these can find the optimal value of a function without
the need of exhaustive enumeration, they still need a significant amount of quantum mechanical computations, which in fact is the most time-consuming step of
the whole optimization process. It is thus highly desirable to develop methods
which enable one to evaluate the target molecular property (i.e., the property to
be optimized) as efficiently as possible. Mode- and Intensity-Tracking demonstrate
that subspace iteration methods are a feasible way to iteratively solve an inverse
problem — here, to optimize normal modes from pre-defined molecular distortions.
3.8
Summary
The rational design of molecules and materials remains a challenging topic, despite
the many improvements made in this field over the past decades. The most advanced
contemporary methods all rely on sophisticated optimization algorithms and powerful
computer hardware. Therefore, future advancements in these areas will automatically
allow one to tackle increasingly complicated problems. Still, even the most advanced
optimization algorithm will eventually encounter limits, especially when considering
time limitations (we note in passing that the so-called mean-time between failure
tends to decrease with increasing size and complexity of a computer system).
Therefore, further progress in inverse quantum chemistry heavily depends not only
on the development of very efficient optimizing routines, possibly tailored to the
specific problem at hand, but also on possibilities of evaluating desired molecular
62
Concepts and Strategies for Rational Compound Design
properties as efficiently as possible. In this respect, Mode- and Intensity-Tracking are
examples how one can obtain selectively interesting molecular properties (namely,
specific normal modes describing molecular vibrations with associated harmonic
frequencies) without first having to calculate — in a time-consuming way — the
entire quantum chemical information (in our case, the full Hessian matrix).
Chapter 4
Assessment of DFT for Transition
Metal Complexes
“It is very difficult to make an accurate
prediction, especially about the future.”
N. H. D. Bohr
Density functional theory (DFT) is an important tool in computational quantum
chemical research [1, 124, 294]. This is due to the fact that DFT offers a good
compromise between accuracy and computing time, which allows to investigate
large molecules with hundreds of atoms. Also in this thesis, DFT will be the main
workhorse. However, a drawback of DFT is the fact that the essential ingredient
of DFT, the exact exchange–correlation functional, is not known. During the last
decades, efforts were made to develop better and better approximations to the
exact exchange–correlation functional (by increasingly advanced analytic ansätze
and improved reference data sets for their parametrization) [295]. As a result, a
plethora of (approximate) density functionals is now at our disposal. Unfortunately,
there is no truly rigorous way to systematically develop such approximations, and
their reliability is assessed in statistical analyses by comparison to reference data
sets. As a consequence, a given density functional might provide reasonable or even
excellent results in some application, but completely fail in another one. A thorough
assessment of the performance of DFT for large transition metal complexes —
presented in this chapter — is mandatory when considering the fact that we will
heavily rely on DFT in the inverse approaches presented in this work.
4.1
New Benchmark Set of Accurate Coordination
Energies
Comparing the values of one or several chemical properties calculated with different
density functionals with accurate reference data, obtained either experimentally or
by quantum chemical methods of controllable accuracy (e.g., coupled cluster [2]),
provides a means to measure the quality of these density functionals, at least with
64
Assessment of DFT for Transition Metal Complexes
respect to the properties studied. Pople and coworkers were among the first to
realize the importance of benchmark studies, which they applied to assess the
quality of the so-called G1 and G2 composite approaches [84, 296]. In 1981, they
proposed a comparatively small database containing the atomization energies of
31 small molecules such as LiH, CH4 , CO2 , and H2 NNH2 [296]. This database
has been continuously extended; in 1997, G2/97 was presented which comprised
already 148 molecules [84, 85], followed by the G3/99 test set with 222 molecules
in 1999 [297]. In 2005, the latest revision of this Pople benchmarking database
was presented — G3/05 includes a total of 454 reference data points [298].
More recently, Truhlar and coworkers assembled a large collection of databases
including atomization energies, ionization potentials, electron and proton affinities,
barrier heights, anharmonic vibrational zero point energies, saddle point geometries, and a broad range of dissociation energies (covering systems with hydrogen
bonds, charge-transfer complexes, dipole interactions, and weakly interacting systems) [92, 299–308]. This broad range of different properties shall assure a rather
complete picture of the performance of a given density functional. In a similar spirit, Grimme and Korth proposed “mindless” benchmarking relying on a
large library of automatically generated “artificial” molecules, which should remove any chemical biases [309]. Goerigk and Grimme incorporated this test set
into the GMTKN24 database covering main group thermochemistry, kinetics, and
noncovalent interactions [310], which was recently extended to form the so-called
GMTKN30 database [311].
A drawback of all of these databases is the fact that the coverage of large
transition metal complexes is rather weak. This deficiency can be significant for
practical purposes as coordination chemistry in the condensed phase usually involves
complex, spatially extended ligand environments. By contrast, a compact small
diatomic like Cr2 with unsaturated valencies can be a true challenge for electronic
structure methods, but it is no prototypical system for coordination chemistry.
While the Grimme databases are specifically set up for main group chemistry
and do therefore not cover any transition metals, the Truhlar databases include
some (very) small transition metal compounds like Ag2 , CuOH2 + , and Fe(CO)5 .
Jiang et al. developed a database with the heats of formation of 225 molecules
containing 3d-transition metals [312]. However, most of these molecules are also
rather small. In addition, many of the experimental reference data have a rather
large uncertainty, which might result in wrong conclusions as to the performance
of a given density functional. Very recently, Zhang et al. therefore selected the 70
molecules which are estimated to have experimental uncertainties in their enthalpies
of formation equal or less than 2 kcal mol−1 [313] and tested no less than 42 density
functionals on that database. However, this reduced dataset contains again only very
small compounds; furthermore, only a minority of them contains organic ligands.
Hughes and Friesner assembled a rather large database of spin-splittings of 57
octahedral first-row transition metal complexes [314]. Furche and Perdew also
presented a database of 3d transition metals, including reaction energies. Again,
also this database includes only very small compounds (and reactions such as
the dissociation of the vanadium dimer) [315]. By contrast, large organometallic
compounds are key players in important fields such as homogeneous catalysis [316].
4.1 New Benchmark Set of Accurate Coordination Energies
65
It is often not possible to obtain conclusions by quantum chemical computations on
small complexes which are also valid for larger compounds, as large transition metal
complexes often have a very distinct electronic structure. Furthermore, dispersion
effects have a much larger influence in bigger complexes. Therefore, a database of
reliable values for important chemical properties like ligand dissociation energies
for a variety of large transition metal complexes is highly desirable.
For such large complexes, there are no ab initio computational data available.
Therefore, we have to resort to experimental data. Ideally, these data should not
only be accurate but also obtained in the gas phase, such that strong (environmental)
intermolecular interactions like solvent effects can be excluded. Chen and coworkers
accurately measured ligand dissociation energies of a broad range of large transition
metal complexes in the gas phase by means of tandem mass spectrometry. The
experimental ligand dissociation energies were obtained by injecting the full complex
into a mass spectrometer and letting it subsequently collide with an inert noble gas
such as argon or xenon. This collision led to a dissociation of the ligand, and the
dissociation cross sections were measured as a function of the nominal collisional
energy. The actual value of the ligand dissociation energy was then determined by
Monte Carlo simulations based on a rather involved model of the entire dissociation
process involving the ligand dissociation energy as a fit parameter [317]. This
procedure assures a sufficient and determinable accuracy of the ligand dissociation
energies. Here, we present a database of ten selected ligand dissociation energies
from Refs. [318–325] and investigate the performance of nine widely used density
functionals.
In section 4.1.1, we introduce our new database (which we will abbreviate in the
following as WCCR10), and in section 4.1.2 the density functionals investigated
and the computational benchmarking procedure. The results are discussed in section
4.1.3 before providing a conclusion together with an outlook in section 4.1.4.
4.1.1
WCCR10 Ligand Dissociation Energy Database of
Large Transition Metal Complexes
Our database consists of ten dissociation reactions depicted in Fig. 4.1. These
reactions were selected out of all such reactions that have been published so far [318–
324] with the experimental methodology mentioned in the introduction. Specific care
was taken to achieve a good balance of different transition metals and different ligand
types. Hence, duplication of similar decoordination reactions has been avoided. All
molecules in this database feature a closed-shell electronic structure. All transition
metal complexes are charged, which is an implication of the fact that the ligand
dissociation energies have been determined by mass spectrometry.
Table 4.1 lists the experimental ligand dissociation energies for the ten reactions
depicted in Fig. 4.1. The error bounds were estimated in the original literature
from approximately 10 to 20 regressions with the L-CID program [317] for three
independent data sets. Furthermore, a lab-frame energy uncertainty of 0.15 eV was
assumed [318–324]. These experimental ligand dissociation energies are based on reaction rates derived within approximate Rice–Ramsperger–Kassel–Marcus (RRKM)
theory and can therefore be considered to be free energies at 0 Kelvin. Therefore,
66
Assessment of DFT for Transition Metal Complexes
these energies have to be compared to computed total electronic energies which
are corrected for their zero-point vibrational energies.
Reaction 1
Reaction 5
Reaction 2
M = Au: Reaction 7a
M = Cu: Reaction 7b
M = Ag: Reaction 7c
Reaction 3
Reaction 8
Reaction 9
Reaction 4
Figure 4.1: Overview of the ten reactions of the WCCR10 database. The aromatic
substituent abbreviated as “Ar” is 2,6-C6 H3 Cl2 .
4.1 New Benchmark Set of Accurate Coordination Energies
67
Table 4.1: Experimental reference values for the ten ligand dissociation energies in the
WCCR10 database.
reaction
1
2
3
4
5
7a
7b
7c
8
9
Eref / (kJ mol−1 )
Ref.
a
[318]
[319]
[319]
[326]
[326]
[322, 323]
[322, 323]
[322, 323]
[324]
[325]
120.2 ± 2.7
204.4 ± 5.0
207.3 ± 2.9
136.7 ± 3.8b
192.5 ± 5.0c
211.9 ± 6.7
218.2 ± 5.4
194.4 ± 8.4
162.7 ± 4.2
102.6 ± 2.5
a
Obtained by taking the arithmetic mean of two independent values which are obtained
with different collisional gases. With argon, a value of 121.6 ± 3.8 kJ mol−1 is found, while
with xenon, the ligand dissociation energy is found to be 118.7 ± 3.8 kJ mol−1 . Moreover,
these two values were obtained with the program C RUNCH [327]. In this case, the error
bounds are estimated from the reproducibility across data sets and fitting assumptions (also
here, a lab-frame energy uncertainty of 0.15 eV is assumed).
b The value of 139.6 kJ mol−1 published in Ref. [320] neglects the reaction path degeneracy.
For the WCCR10 set, the reaction energy has been redetermined with the correct reaction
path degeneracy.
c The value of 189.8 kJ mol−1 published in Ref. [321] neglects the reaction path degeneracy.
For the WCCR10 set, the reaction energy has been redetermined with the correct reaction
path degeneracy.
4.1.2
Computational Details
In this study, seven different density functionals are investigated, namely BP86
[81, 82], B3LYP (including 20% of Hartree–Fock exchange) [106, 328], B97-DD2 [146], TPSS [79], TPSSh (10% Hartree–Fock exchange) [329,330], PBE [77,78],
and PBE0 (25% Hartree–Fock exchange) [331]. Note that we adopt the special
designation “B97-D-D2” for the functional which is usually just referred to as “B97D” in order to make clear that this functional is a reparametrized version of the
original “B97” functional, and that Grimme’s second generation dispersion correction
(“D2”) [146] should be used in conjunction with this functional. Furthermore, for
BP86 and B3LYP we investigate Grimme’s third generation dispersion correction
[150], denoted as BP86-D3 and B3LYP-D3, respectively. This collection of nine
functionals includes generalized gradient approximation (GGA, BP86, PBE0), metaGGA (TPSS), and hybrid functionals (B3LYP, TPSSh, PBE0).
For the benchmark study, the program package T URBOMOLE version 6.3.1 in
its shared-memory parallelized implementation was employed [117]. We made an
attempt to fully optimize all structures in C1 symmetry (i.e., without any structural
constraints) with each of the above-mentioned density functionals. Hence, the ligand
68
Assessment of DFT for Transition Metal Complexes
dissociation energies were in general not obtained by simple single-point calculations
(apart from a few exceptions — to be discussed below). This methodology allows us
to eliminate any errors stemming from an inconsistent description of the molecular
structure.
In order to minimize the basis set superposition error, Ahlrichs’ very large
quadruple-ζ basis with two sets of polarization functions (def2-QZVPP) was employed at all atoms [332] (note that for transition metal complexes, the basis set
size may have a much larger effect on the accuracy than previously assumed [333]).
For all non-hybrid functionals, advantage was taken of the resolution-of-the-identity
(RI) approximation with the corresponding auxiliary basis sets [334]. Scalar relativistic effects were taken into account for all second- and third-row transition metal
elements by means of Stuttgart effective core potentials [335] as implemented in
T URBOMOLE. In all structure optimizations, the maximum norm of the Cartesian
geometry gradient was converged to 10−4 hartree/bohr. If not stated otherwise,
all total electronic energies were converged to within 10−7 hartree. For all other
program settings, the default values were found to be suitable for our purpose.
The reaction energies have to be corrected for the zero-point vibrational energies
(ZPE). These have been computed with BP86 within the harmonic approximation.
For all other density functionals, the ZPE resulting from BP86 was taken since
our tests showed (see also below) that the different density functionals yield very
similar ZPEs. For all molecules except for the modified Grubbs catalyst of reaction
4 (see Fig. 4.1), the vibrational analysis was carried out with the methodology
just described, but using the fine grid “m4” and electronic energies converged to
10−8 hartree. The total number of basis functions for the modified Grubbs catalyst
(having 174 atoms), however, was too large for T URBOMOLE ’ S AOFORCE module.
Therefore, in this case the vibrational analysis was carried out with S NF 5.0.0 as
provided with M OV I PAC 1.0.0 [336]. This program implements a semi-numerical
differentiation scheme which allows to construct the Hessian matrix as numerical
first derivatives of analytical geometry gradients of the total electronic energy (see
appendix D) [275]. The geometry gradients have been computed with the serial
version of T URBOMOLE 6.3.1. For the numerical differentiation, a 3-point central
difference formula with a step size of 0.01 bohr was utilized.
The entire benchmark is computationally very expensive, as very accurate calculations with tight self-consistent field convergence criteria and huge basis sets on
large molecules had to be conducted. A typical single-point computation required
a total computing time of about 100 days (featuring only a small number of about
20 SCF iterations), corresponding to a wall time of a few days when employing
a parallel version. Consequently, the full structure optimizations required several
months to complete. Unfortunately, it turned out that for some combinations of
reactions and density functionals (namely, reactions 2, 3, and 4 for B3LYP, B3LYPD3, TPSSh, and PBE0, and reaction 9 for B3LYP-D3) the structure optimizations
were not feasible for some of the molecules involved. However, we found for all
non-dispersion-corrected functionals that the individual structures agree very well
with each other (see section 4.1.3.1). Therefore, we decided to include data for
the above-mentioned reactions and functionals obtained by single-point calculations
(taking the necessary precautions in our analysis). The structures resulting from
4.1 New Benchmark Set of Accurate Coordination Energies
69
optimizations with dispersion-corrected functionals, however, are rather different
from each other. Therefore, we cannot present any data for reactions 2, 3, 4,
and 9 obtained with the B3LYP-D3 functional. However, these restrictions do not
compromise the complete and coherent picture that emerges for the functionals
investigated.
4.1.3
Performance of Popular Density Functionals
4.1.3.1
Structures
The first crucial step is to analyze the effect that the individual functionals have
on the molecular structures optimized. To this end, we compare the structures
resulting from different density functionals to each other in terms of the root mean
square deviation (RMSD) between the atomic positions of a given structure and
the corresponding BP86-optimized structure (the BP86 functional is known to yield
reasonable structures [124]). The resulting RMSD values are shown in Fig. 4.2.
Figure 4.2: Distribution of RMSD values for all optimized structures, taking the structures
resulting from BP86 as reference and using a bin size of 10 pm. Note that we plot a
cumulative count on the abscissa (i.e., the total number of molecules having a RMSD
between 0 and 10 pm is 145).
We see that about 91% of all structures have a RMSD below 30 pm. Visual
inspection of these structures shows that they are almost identical to the BP86
reference. Especially the PBE optimized structures agree almost perfectly with the
corresponding BP86 structures (we note that the exchange part of PBE is similar to
B88, while the correlation part of PBE differs from P86 [77, 81, 82]). This similarity of structures suggests that a full structure optimization might not be necessary
70
Assessment of DFT for Transition Metal Complexes
for each individual density functional. Instead, single-point calculations may be
sufficient. To investigate this aspect further, we calculated the ligand dissociation
energy of reaction 8 for selected density functionals not only from a full structure
optimization, but also by means of single points on BP86 structures. These data
are given in Table 4.2 (last two columns). We see that the individual coordination energies agree very well with each other, with deviations hardly exceeding
0.5 kJ mol−1 . This finding allows us to present ligand dissociation energies also
for the few cases where a full structure optimization was not possible (cf., section
4.1.2).
Figure 4.3: Overlay of the structures of the reactant complex of reactions 2 and 3 resulting
from optimization with BP86 (solid structure) and BP86-D3 (translucent structure). It is
clearly visible that the phenyl substituents align in an almost coplanar fashion with the
five-membered rings when dispersion corrections are considered.
However, there are also a few structures with large RMSD values (i.e., larger
than 30 pm); interestingly, all except one are obtained with a dispersion-corrected
functional. Typically, the structures with the largest RMSD values are the transition
metal complexes of reactions 2 and 3 (see Fig. 4.1). When analyzing the structures
of these complexes we found that the overall molecule “contracts” when dispersion
corrections are taken into account (as one would expect), and the terminal phenyl
substituents adopt a stacked, almost coplanar, conformation with respect to the
4.1 New Benchmark Set of Accurate Coordination Energies
71
five-membered rings of these complexes (see Fig. 4.3). Thus, large intramolecular
dispersion effects are responsible for the large differences observed here.
It is hardly surprising that the structures resulting from dispersion-corrected functionals are rather different from their BP86 reference, since the latter does not take
any long-range dispersion effects into account. However, we found that the structures optimized with B3LYP-D3 and B97-D-D2 deviate also rather strongly from
the ones obtained with BP86-D3 (see Fig. 4.4). Therefore, in the case of dispersioncorrected density functionals, a full optimization is necessary in order to avoid any
artificial errors stemming from an unoptimized and thus for a given functional not
well-defined structure. For example, the ligand dissociation energy of reaction 3
is calculated to be 297 kJ mol−1 when the structures are optimized with BP86-D3.
When only single points on the BP86 structures are considered instead, the reaction energy is found to be 256 kJ mol−1 , i.e., there is a significant difference of
41 kJ mol−1 . At this point, we should emphasize that it is not possible to determine how reliable either structure is as there are no experimental data with which
one may compare. However, the fact that the structures obtained with different
dispersion-corrected functionals differ significantly from each other requests caution
when applying dispersion corrections in structure optimizations.
Figure 4.4: Distribution of RMSD values for the structures optimized with the dispersioncorrected functionals B3LYP-D3 and B97-D-D2, taking the structures resulting from BP86D3 as reference and using a bin size of 10 pm.
Many of the complexes studied in this work feature moieties (often the methyl
substituents) that have very low rotational barriers (for example, the rotation around
the P–Au bond of reactant complex 5 has a barrier of roughly 3.4 kJ mol−1 ). The
optimization algorithm may not necessarily find the exact minimum with respect
72
Assessment of DFT for Transition Metal Complexes
to these rotations, even though tight convergence criteria were employed. However,
since these rotational barriers are so low, this should have a negligible effect on
the overall ligand dissociation energies. Furthermore, in most cases these ligands
with low rotational barriers are present both in the reactant as well as the product complexes, such that one can expect a cancellation of errors to occur when
calculating the reaction energies.
4.1.3.2
Zero-Point Vibrational Energies
For a sensible comparison with the experimental reference energies of the WCCR10
reference set, we have to correct the total electronic energies by their respective
zero-point energies (ZPE), which requires a full vibrational analysis. However,
for the large transition metal complexes investigated in this work, this is hardly
feasible for every density functional. Therefore, we decided to calculate the ZPE
only with BP86, and to subsequently use this ZPE for all other density functionals
as well. The unscaled harmonic frequencies obtained with BP86 are known to be
in good agreement with experimental fundamental frequencies, which must be due
to a fortunate cancellation of errors [102, 337]. Still, we tested the validity of this
approximation at the example of the ligand dissociation energy of reaction 8 (cf.,
Fig. 4.1). These data are given in Table 4.2.
Table 4.2: Zero-point vibrational energies (ZPE) and ligand dissociation energies (∆Ei ) of
reaction 8 for selected density functionals. ∆E1 is the ligand dissociation energy calculated
from a full structure optimization to a local minimum on the respective potential energy
surface, followed by vibrational analysis to compute the ZPE. ∆E2 represents the ligand
dissociation energy calculated from a structure optimization after which, however, no
vibrational analysis has been carried out — instead, the ZPE is always calculated with
BP86. ∆E3 finally is the ligand dissociation energy calculated from single points, using
both the structures and the ZPE from BP86. All values are given in kJ mol−1 .
functional
ZPE
∆E1
∆E2
∆E3
BP86
BP86-D3
PBE
TPSS
B3LYP
0.68
0.66
0.82
0.48
1.46
137.16
163.70
146.09
143.39
107.93
137.16
164.89
146.29
143.07
109.23
137.16
164.35
146.91
143.37
109.37
For this selected reaction, we calculated the ligand dissociation energies from
a structure optimization followed by a full vibrational analysis. These data are
denoted as ∆E1 in Table 4.2. This computational reference has to be compared
with ∆E2 , which represents the ligand dissociation energies obtained from a structure
optimization, but taking the ZPE from the BP86 calculation.
We learn from Table 4.2 that the ZPE corrections vary between 0.5 kJ mol−1 in
the case of TPSS and 1.5 kJ mol−1 for B3LYP. Clearly, these variations are tiny
compared to those found in the corresponding ligand dissociation energies, which
4.1 New Benchmark Set of Accurate Coordination Energies
73
vary over a range of almost 60 kJ mol−1 . In fact, we find that the values of ∆E2
agree well with the corresponding values of ∆E1 . Therefore, we conclude that the
∆E2 methodology adopted for our benchmark is sufficiently accurate.
4.1.3.3
Ligand Dissociation Energies
We are now in a position to analyze the ligand dissociation energies of the individual
reactions as calculated with the nine different density functionals. These energies are
plotted together with their experimental reference values in Fig. 4.5. While for some
reactions all density functionals yield very similar energies (e.g., reaction 7c), there
is an enormous spread for other reactions, most notable for reactions 2 and 3. The
energies calculated with the dispersion-corrected functionals BP86-D3 and B97-DD2 (recall that no data are available for B3LYP-D3) deviate strongly not only from
the values of all other density functionals, but also from the experimental reference
(the same is true for reaction 4 but not for all other reactions). Obviously, in these
cases, the dispersion corrections do not lead to an improved performance, but,
on the contrary, make the result worse compared to the non-dispersion-corrected
reference BP86. In some cases, the ligand dissociation energy is overestimated
by more than 100 kJ mol−1 . We therefore conclude that dispersion corrections in
the empirical form proposed by Grimme do not necessarily lead to an improved
performance for coordination energies at least with regard to absolute accuracy. Very
recently, Kobylianskii et al. reported similar findings for the gas-phase Co–C bond
energies of adenosylcobinamide and methylcobinamide [338]. We should mention
here that in a previous study, Jacobsen and Cavallo found that the inclusion of
dispersion interactions led to worse phosphane substitution energies of a range of
transition metal complexes [339]. In that study, this effect was, however, attributed
to an incomplete treatment of solvent effects [339], which was also confirmed
by Grimme [340, 341]. As solvation effects do not play a role in our study,
this cannot be a reason for the failure of dispersion corrections observed here.
As we could only speculate on possible reasons for this failure we should recall
the basic assumptions that underly the construction of dispersion corrections by
Grimme. Grimme’s dispersion corrections were parametrized for pairs of atoms in
specific valence states within a reference molecule. As the dispersion coefficients
are calibrated against accurate results on neutral systems, this might lead to too large
dispersion coefficients for cationic systems. Since there are no anionic counterions
present in our study (the dispersion coefficients of which might then be estimated to
be too small), there cannot be any cancellation of errors as discussed in Ref. [150].
Another possible cause for the failure of empirical dispersion corrections observed
here might be rooted in the large size of the complexes studied (note that the
dispersion-corrected functionals perform particularly badly for reactions 2, 3, and 4,
which involve the largest compounds of the WCCR10 test set). It has been found by
Tkatchenko and von Lilienfeld that for large, bulky molecules (and even more so for
condensed-phase systems), three-body interactions must not be neglected [342,343].
This has also been confirmed by Risthaus and Grimme [344]. While it is in
principle possible to account for these three-body dispersion effects by means of an
Axilrod–Teller–Muto term, this correction is not used by default in Grimme’s set
74
Assessment of DFT for Transition Metal Complexes
of dispersion corrections [344]. In order to assess their magnitude, we calculated
the three-body correction term for the structures obtained with the standard D3
dispersion correction. The resulting values for this three-body correction are very
similar for the functionals BP86-D3, B3LYP-D3 and B97-D-D2, so that we may
focus on the results obtained with BP86-D3. We find that the three-body interaction
leads in all cases to a repulsive term, thus slightly destabilizing the molecule. This
destabilization is more pronounced the larger the molecules are. For example, in the
case of the modified Grubbs catalyst (174 atoms; reaction 4), the three-body term
amounts to a contribution of +14.0 kJ mol−1 , while it is +8.9 kJ mol−1 for the azabox
copper(I) complex of reaction 3 (90 atoms, cf., Fig. 4.3), and only +1.6 kJ mol−1 for
the aquo platinum(II) complex of reaction 1 (42 atoms). Including the three-body
correction thus leads to slightly smaller ligand dissociation energies. In the case of
reaction 3, the ligand dissociation energy is 7.5 kJ mol−1 smaller, respectively, while
the ligand dissociation energy of reaction 4 is lowered by 5.3 kJ mol−1 . Clearly,
these corrections do not significantly improve the agreement with the experimental
values and can therefore not explain the discrepancies.
Figure 4.5: Experimental and calculated ligand dissociation energies of the ten benchmark
reactions studied. For the B3LYP-D3 functional, not all data are available (see also section
4.1.2). For the experimental data, the black bars denote the error bounds.
4.1 New Benchmark Set of Accurate Coordination Energies
75
Ligand dissociation energies could be sensitive to the spatial cut-off function, since
the reactions studied involve the formation of a short metal–ligand bond (dispersion
is difficult to model at short distances [345]). In order to investigate this issue, we
calculated the ligand dissociation energy of reaction 3 with BP86-D3 in combination
with the Becke–Johnson (BJ) damping function (the standard damping function
of the D3 correction is often called zero-damping, as the dispersion interaction
between two atoms approaches zero if the internuclear distance approaches zero.
With BP86-D3-BJ, the reaction energy is calculated to be 298 kJ mol−1 , while it is
found to be 297 kJ mol−1 with BP86-D3. Therefore, the damping function has only
a minor effect on the overall ligand dissociation energy of reaction 3. It is thus not
likely that it can account for the discrepancies observed in this work. This result
is in line with the work by Grimme and coworkers who found that the damping
function in general has a negligible effect on the results (such as ligand dissociation
energies) [346].
Figure 4.6: Distribution of the absolute deviations (AD) of the ligand dissociation energies
as calculated with the individual density functionals from their experimental reference
values.
Apart from the very strong deviations the dispersion-corrected functionals produce
for some reactions, most of the calculated ligand dissociation energies deviate
between 15 and 55 kJ mol−1 from their experimental reference. However, in some
cases, an almost perfect agreement is found (for example, reaction 7c calculated
with BP86 and reaction 7b calculated with B3LYP). The distribution of the absolute
76
Assessment of DFT for Transition Metal Complexes
deviations from the experimental reference values is shown in Fig. 4.6. We can
hardly establish any trends. Based on these results no functional may be clearly
preferred over the others.
By investigating the mean absolute deviations (MADs) of the individual functionals (see Table 4.3), we see that most functionals have MADs of about 30 kJ mol−1 .
The MAD of the dispersion-corrected functionals BP86-D3 and B97-D-D2 is clearly
larger, which is due to the above-mentioned strong errors in the case of reactions
2, 3 and 4 (note again that there is no data available for these reactions for
B3LYP-D3, which is why the MAD of this functional is seemingly low; we can,
however, expect it to be significantly larger if the data on reactions 2, 3 and 4
could be included). The largest absolute deviation (LAD) from the experimental
values is about 50 kJ mol−1 for most functionals. For BP86, the largest deviation
is 67 kJ mol−1 , and for B3LYP, it is even as large as 83 kJ mol−1 .
Table 4.3: Mean absolute deviations (MAD) as well as largest absolute deviations (LAD)
of the ligand dissociation energies as calculated with the individual functionals (first two
columns) as well as MAD values for selected functionals obtained in three previously
published studies. All values are given in kJ mol−1 .
functional
BP86
BP86-D3
B3LYP
B3LYP-D3
PBE
B97-D-D2
TPSS
TPSSh
PBE0
MAD
LAD
(this work)
(this work)
39.58
44.60
39.08
20.48a
31.83
36.09
32.93
31.99
26.88
66.57
91.62
83.42
39.15a
53.14
68.95
52.06
55.93
54.81
MAD
MAD
MAD
(study 1) (study 2) (study 3)
[308]
[313]
[315]
n/a
n/a
43.05
n/a
n/a
n/a
46.15
20.90
50.16
n/a
n/a
n/a
16.89
25.50
45.14
n/a
n/a
n/a
17.97
n/a
42.64
15.97
17.56
40.55
12.58
19.23
n/a
a
Note that for B3LYP-D3, no data are available for reactions 2, 3, 4, and 9. For reactions
2, 3, and 4, the other dispersion-corrected density functionals (BP86-D3 and B97-D-D2)
have particularly large errors. We might therefore expect that the values presented in
this Table would be significantly worse for B3LYP-D3 if these data could be taken into
account.
Neglecting the dispersion-corrected functionals, B3LYP performs worst considering both the MAD as well as the largest absolute deviation from experiment. BP86
has a comparable MAD, but its LAD is significantly better than in the case of
B3LYP. A considerable improvement can be obtained by using the non-empirical
PBE functional. With 32 kJ mol−1 its mean absolute deviation is about 7 kJ mol−1
lower than that of BP86. The LAD is improved by roughly 14 kJ mol−1 , being about
53 kJ mol−1 . Moving from this GGA functional to the non-empirical meta-GGA
functional TPSS, no further improvement is obtained; both the MAD as well as the
4.1 New Benchmark Set of Accurate Coordination Energies
77
LAD are almost identical to those of PBE. In the case of these two functionals,
however, admixture of Hartree–Fock exchange can further decrease the mean absolute deviations. In particular for PBE0, the MAD decreases to 27 kJ mol−1 , which
is the lowest value of all density functionals investigated in this work. However,
as exemplified by B3LYP, adding Hartree–Fock exchange does not generally lead
to an improved description of the ligand dissociation energies. Note that in an
earlier benchmark study by Fuche and Perdew [315], the TPSS and TPSSh functionals were found to be best functionals for dipole moments, reaction energies, and
harmonic frequencies of 3d transition metal complexes. Still, the best-performing
functionals in our study have errors of up to about 50 kJ mol−1 for certain reactions
and thus no functional performs well for our complete WCCR10 set.
4.1.3.4
Comparison with Other Benchmark Studies
It is instructive to compare our results presented above with the findings obtained in
previous studies. From the plethora of benchmark studies published so far we shall
select here a few which are comparable to our work in the sense that they also focus
on transition metal compounds. The data obtained in these studies is included in
Table 4.3. In the first study (which we dub “study 1” here), Zhao and Truhlar [308]
studied a model system for the Grubbs II catalyst. They compared the energies of
important stationary points in the catalytic cycle of the metathesis reaction obtained
with different density functionals to accurate reference energies obtained with a
CCSD(T)-based composite approach. As the system studied in this work is very
similar to reaction 4 of our work, one would expect a similar performance for
the different density functionals. In fact, we observe from the data in Table 4.3
the same qualitative ordering, namely, PBE0 performs best, while PBE, TPSS, and
TPSSh are only slightly worse. B3LYP, however, is significantly worse compared
to the other functionals. Nevertheless, we also observe some interesting differences
compared to our study. B3LYP performs 7 kJ mol−1 worse in the work of Zhao and
Truhlar, while the MADs of other functionals are more than 10 kJ mol−1 better than
in our benchmark. This is in agreement with our study as the deviation between
the experimental ligand dissociation energy of reaction 4 and the one calculated
with B3LYP is 48.7 kJ mol−1 , while PBE0 reproduces this energy rather accurately,
having an error of only 4.5 kJ mol−1 (i.e., this error is even smaller than the one
found in study 1). However, the functionals TPSS and PBE have an absolute error
of roughly 30 kJ mol−1 for reaction 4, and are thus found to be significantly worse
in our study than in study 1.
In a second, very recent study (hereafter called “study 2”), Zhang el al. investigated 70 small 3d transition metal compounds and compared DFT results to accurate
experimental data [313]. These compounds are different compared to those in our
WCCR10 set. Nevertheless, the results of this study are in rather good agreement
with our results, which might be an indication of a certain transferability of both
results. The overall errors of PBE0 and PBE are approximately 7 kJ mol−1 to our
findings, while TPSSh performs roughly 14 kJ mol−1 better in study 2. However,
it might come as a surprise that B3LYP has a MAD of only 21 kJ mol−1 in study
2, which is in sharp contrast to our findings.
78
Assessment of DFT for Transition Metal Complexes
Finally, Furche and Perdew [315] also assessed the performance of different
density functionals on a data set of small 3d transition metal compounds. Here,
we focus on their results from the analysis of 18 different reaction energies (such
as the dissociation of V2 , CoH, and CrF), for which experimental data are also
available. We denote this third study as “study 3” in Table 4.3. Overall, Furche
and Perdew found larger errors for the individual functionals than in our study,
which might be due to the unsaturated valencies in the small molecules leading to
peculiar electron-correlation effects. The qualitative ordering is, however, roughly
comparable to our study.
In summary, we have seen that the actual numerical values for the absolute
error of a given density functional can vary significantly, depending on the data
set employed for the tests. The qualitative ordering of different density functionals
appears to be more robust in this respect, although also here, different test sets can
lead to different conclusions. Therefore, when considering the performance of a
given density functional one has to pay special attention to how this performance
was assessed. While very large, diverse databases might give a good indication of
the average performance of a density functional, they might under- or overestimate
the errors of that density functional in a special application. Likewise, a very
specialized database covering only a certain class of compounds can hardly be
considered representative for a broad area of chemistry.
4.1.4
Conclusions on the WCCR10 Benchmark Set
We have presented the database WCCR10, a set of ten selected ligand dissociation energies for large transition metal complexes for which accurate, consistently
measured experimental gas-phase data are available in the literature. We then investigated the performance of nine different density functionals with respect to these
ligand dissociation energies. The density functionals selected encompass a broad
range of different functional classes, namely GGA, meta-GGA, and hybrid functionals; in addition, for some functionals also Grimme’s third generation dispersion
correction was employed. The calculations were set up such that any deviation
from the experimental reference has to be attributed to the density functional, and
not to other factors such as an incomplete basis set or numerical artifacts due to
loose thresholds.
In our benchmark study, we found that Grimme’s dispersion correction significantly overshoots the energies for some reactions, thereby impacting agreement
with experiment negatively as compared to the results obtained without dispersion
corrections. Furthermore, we found that different dispersion-corrected functionals
lead to significantly different structures, whereas the structures resulting from the
uncorrected parent functionals were in much better mutual agreement. Unfortunately, a definitive assessment of the quality of the structures is not possible since
there are no reliable reference data to compare with, but our study gives evidence
that special care should be taken when applying empirical dispersion corrections
in structure optimizations.
Among all functionals not corrected for dispersion, B3LYP performs worst with a
mean absolute deviation of about 39 kJ mol−1 , while BP86 is a little bit better, at least
4.2 Investigating the Parameters of BP86
79
in terms of its MAD. The non-empirical density functionals PBE and TPSS have a
clearly smaller error both in terms of mean absolute deviation and largest absolute
deviation. This result is particularly appealing from a theoretical point of view, since
these density functionals are constructed by choosing a mathematical expression that
satisfies constraints which would also be fulfilled by the true exchange–correlation
functional, instead of simply fitting it to a reference set of chemical data. Adding a
portion of Hartree–Fock exchange to these functionals improves their performance
further. We have compared our findings to selected benchmark studies from the
literature and found that the actual numerical values for the errors of the individual
density functionals depend heavily on the test set chosen. The relative performance
of different density functionals is — at least from a qualitative point of view —
somewhat more robust with respect to the choice of the test set, but also here one
finds astonishing differences. Therefore, care must be taken when utilizing such
benchmark results to assess the performance of density functionals.
Even the best functional identified for the WCCR10 set, PBE0, has a rather large
mean absolute deviation. On average, the ligand dissociation energies are wrong by
more than 20 kJ mol−1 , while the maximum deviation is even as large as 55 kJ mol−1 .
As these values lie well within the range of chemically relevant energies (which
are on the order of a few kJ mol−1 to a few hundred kJ mol−1 ), the accuracy of
contemporary density functionals often cannot be expected to be sufficient for a
given application. It is therefore of utmost importance to further analyze the density
functionals available today in order to find possible cures to their weaknesses. A
viable alternative to the development of new functionals might be the study of
composite approaches [347] as exemplified by the so-called correlation consistent
composite approach (ccCA method) of Wilson and coworkers [312, 348, 349], the
Gaussian 4 method of Curtiss and coworkers [350], the method of Peterson and
coworkers [351] and a very recent approach developed by Bross et al. [352]
4.2
Investigating the Parameters of BP86
As stated repeatedly in this thesis, the most severe drawback of DFT is the fact
that the exact expression for the exchange–correlation energy functional, its central
ingredient, is not known, and, thus, reliable approximations to it have to be found.
Hundreds of these exchange–correlation functionals have been developed so far,
and the development of new functionals is a vivid field of research (see, e.g.,
Refs. [59, 353]). One of the very early generalized-gradient-approximation (GGA)
functionals is BP86 [81, 82]. It is among those functionals for which the good
balance of accuracy vs. efficiency was demonstrated in chemistry [124,354]. Many
others of these early GGAs are also still in use. This fact is remarkable because
of the much more elaborate, balanced, and extended data sets which have become
available in recent years and to which an increasing number of parameters in the
functionals is fitted. One may even view the recent developments of range separated
functionals [355, 356] as an attempt to introduce a continuous, position-dependent
parametrization.
In this section, we reconsider the BP86 parameters for the most challenging
80
Assessment of DFT for Transition Metal Complexes
systems which are transition metal complexes [357]. In this context it is important
to note that BP86 has recently been found to provide very accurate results for the
Co–C bond strength of adenosylcobinamide and methylcobinamide when compared
to gas phase data [338], while its performance for the WCCR10 set was rather
weak (see section 4.1; we should emphasize, however, that the performance of a
given density functional significantly depends on the test set employed).
The BP86 exchange–correlation functional is explained in detail in section 2.5.2.4.
Considering the limitations under which the empirical parameters β and f˜ have been
obtained, it is interesting to ask how strong the influence of a variation of these two
parameters is on results of actual computations. This question has apparently never
been addressed. We should note, however, that Fabiano et al. carried out a study
on the variability of the parameters present in the PBE functional and its many
empirical revisions [358]. Here, we study the dependence of the ligand binding
energies of the ten transition metal compounds in the WCCR10 reference set on both
β and f˜ in BP86 (see Fig. 4.1 for a detailed list of reactions). We should explicitly
state that the coordination energies in the WCCR10 set have been determined for
0 K and, thus, we do not need to take any temperature effects into account (only
the zero-point vibrational energy has to be taken into account). Furthermore, it
is important to note that we directly address investigating a chemically relevant
property, namely ligand binding energies, rather than fundamental properties such
as atomization energies.
4.2.1
Computational Methodology
4.2.1.1
Computational Details
In order to be able to vary the empirical parameters of BP86 [81, 82], we reimplemented this functional in a local version of T URBOMOLE 5.7.1 [117]. In a
first step, the B88 and P86 energy expressions have been programmed with the
computer algebra system M APLE [359]. Note that in T URBOMOLE, the summation
over the two spins is done implicitly by calling the respective subroutine twice,
first with ρα , and then with ρβ . However, the (numerical) integration routine as
well as the LSDA parts of the energy do not have to be changed. Therefore, it
suffices to focus on the integrands of Eqs. (2.71) and (2.75). Obtaining the functional derivatives necessary for the self-consistent field (SCF) equations was carried
out with M APLE ’ S differentiation algorithms. The resulting expressions were then
automatically be converted to Fortran code and optimized; this final code was then
implemented into T URBOMOLE.
For this study, we take the BP86 optimized structures from the WCCR10 set
(see section 4.1.1). They were fully optimized with T URBOMOLE [117], employing
Ahlrichs’ def2-QZVPP basis set at all atoms [332]. Advantage was taken of the
resolution-of-the-identity (RI) approximation with the corresponding auxiliary basis
sets [334]. For all second and third row transition metal atoms, scalar relativistic
effects were considered by suitable effective core potentials [335]. The zero-point
vibrational energies (ZPE) of all structures have been calculated within the harmonic
approximation.
4.2 Investigating the Parameters of BP86
81
For all ten reactions, the parameter β was scanned in single-point calculations
between 0 and 1.0, while for f˜, values between 0.05 and 0.25 were considered.
An equidistant stepsize of 0.05 was employed except for the interval 0 < β <
0.05, where the stepsize was reduced to 0.001. For these single-point calculations, Ahlrichs’ valence triple-zeta basis set with one set of polarization functions
(def-TZVP) [292] plus the corresponding auxiliary basis sets for the RI approximation were used [360, 361]. The resulting data were then further analyzed with
M ATHEMATICA 9 [362]. In order to optimize the empirical parameters β and f˜
with respect to the individual reaction energies, the numerical grid data was interpolated by a Delaunay piecewise linear triangulation method employing zero as
extrapolation value [363]. For comparisons with experimental reference energies,
these have been corrected for their ZPE (taken from the WCCR10 set). Note that
the harmonic frequencies of BP86 are known to be in good agreement with experimentally measured fundamental ones, which is due to some error cancellation
effect [102, 337].
4.2.1.2
Investigation of Approximations
For feasibility reasons, we introduce a range of approximations into our calculations,
and the influence of them has to be carefully evaluated. The most important
approximations are i) the limited basis set size (we should note, however, that
triple-zeta basis sets are generally considered as being reliable enough), ii) effective
core potentials (ECP) to treat the core electrons of transition metal atoms, iii) the
RI approximation, and iv) general numerical thresholds such as the grid used to
numerically integrate the exchange–correlation potential. In order to investigate the
error introduced by these approximations, we carried out a series of calculations
on reaction 1 (see Fig. 4.7) where we eliminate these approximations.
The largest effect on the final ligand binding energies is owed to the basis set.
In fact, the energy of reaction 1 is 97.5 kJ/mol when calculated with a def-TZVP
basis set, but it changes by 21.8 kJ/mol to 75.8 kJ/mol when a def2-QZVPP basis
set is employed. The fact that the smaller def-TZVP basis set produces a result
which is closer to the true experimental binding energy of 120.2 ± 2.7 kJ/mol (this
value changes to 132.1 ± 2.7 kJ/mol if corrected for the ZPE) can only be taken
as an indication of an error compensation. Of course, the result obtained with the
def2-QZVPP basis set has to be regarded as being more accurate, since a larger
basis set naturally allows for a more accurate description of the total wave function.
In particular, the basis set superposition error can be assumed to be considerably
smaller when employing the def2-QZVPP basis set. In fact, for reaction 1 the basis
set superposition error is approximately 8.9 kJ/mol with the def-TZVP basis set,
but only 2.2 kJ/mol with the quadruple zeta basis set. Although we will use the
smaller def-TZVP basis set for our study here for computational reasons, we do
not anticipate a drawback of our parameter analysis as we are mostly interested on
relative energy changes. Furthermore, we will take into account a basis set error
of 25 kJ mol−1 in our analysis.
The RI approximation leads to a change of only 0.18 kJ/mol of the overall
reaction energy. Likewise, choosing the finer grid “m4” along with a very tight SCF
82
Assessment of DFT for Transition Metal Complexes
convergence criterion (the total electronic energy is converged to within 10−8 hartree)
lowers the ligand binding energy by only 0.07 kJ/mol. Therefore, the standard
grid “m3” along with the SCF convergence criterion chosen (10−7 hartree) for our
calculations are adequate.
Finally, we investigate the effect of the effective core potentials employed for all
second and third row transition metal atoms. The basis set employed in this study
(def-TZVP) is specifically designed to be used with Stuttgart ECPs for the heavier
elements. Therefore, we have to resort to a different (but comparable) basis set for
an all-electron calculation. Recently, Peng et al. constructed a basis set which they
called rSV(P)alls1 and which is specifically tailored for one-component all-electron
calculations with silver [364]. As reaction 7c is the only one of the WCCR10 set
involving a silver atom, we calculated its reaction energy with the rSV(P)alls1 allelectron basis set on silver, the standard def-TZVP basis set on all other atoms, and
including scalar-relativistic effects with the third-order Douglas–Kroll–Hess method
(in our standard calculations, the effective core potential accounts for this effect).
The resulting reaction energy of 200.71 kJ mol−1 is in excellent agreement with the
reference calculation where an ECP in conjunction with the def-TZVP basis set
on the silver atom was used, in which case the reaction energy is found to be
201.01 kJ mol−1 . We may therefore conclude that the ECP does in fact introduce
a negligible error.
4.2.2
Dependence of WCCR10 Reaction Energies on the Parameters of BP86
The dependence of the ten reaction energies on the two parameters β and f˜ is
shown in Figs. 4.7 to 4.8. Even though we analyzed β in the range of zero to 1.0,
we will focus on the values below 0.05, since we noticed that the reaction energies
do not show any significance β-dependence for larger values of β. Within the
parameter range considered, the dissociation energies vary within several hundred
kJ/mol for a given reaction; most notably, the energy of reaction 9 becomes even
slightly exothermic for very small values of β and large values for f˜, suggesting that
the ligand is not bound for these parameter values. For all reactions, the qualitative
dependence is the same; the dissociation energy becomes larger for smaller values of
f˜. This similarity is both interesting and reassuring considering the fact that BP86
was not parametrized against reaction energies of such large systems as investigated
in this work. Still, the exact quantitative behavior of the individual dissociation
energies is quite different in each case. In fact, the parameter f˜ generally affects
the dissociation energy more than β. In order to clarify this issue, we split the
coordination energy defined in reaction 1 into exchange, correlation, and remaining
contributions. These contributions are summarized in Table 4.4. However, both
have roughly the same absolute value and largely cancel each other in the resulting
ligand dissociation energy. In fact, we found that the exchange contribution to
the total electronic energies is much larger than the correlation contribution (total
energies not shown in Table 4.4). For the standard values of β and f˜, we see from
Table 4.4 that (differential) exchange as well as (differential) correlation contribute
approximately 20 kJ/mol to the overall reaction energy, but with opposite sign. This
4.2 Investigating the Parameters of BP86
83
situation remains unchanged when we vary the parameters β and f˜; both energies
have approximately the same absolute values, but different signs. However, the
absolute values vary significantly with β and f˜, dropping to ∼5 kJ/mol when f˜
is changed from 0.11 to 0.20. While for β = 0.0042, the exchange contribution
is a little larger than the correlation part, this situation is reversed for β=0.50.
In this case, the correlation energy contributes about 3 kJ/mol more to the ligand
dissociation energy. However, this small difference cannot fully explain the fact
that for β = 0.50, the reaction energy depends almost only on f˜.
Figure 4.7: Dependence of the reaction energies of reactions 1 – 4 on the empirical
parameters β and f˜ of BP86. The black dots indicate points for which actual computations
have been carried out; between these points the energies were interpolated. The red dot
indicates the original values of β and f˜ as proposed by Becke and Perdew, respectively.
The green circle represents the values of β and f˜ for a given reaction for which the best
approximation to the experimental values is obtained. The dashed green lines show the
range of values for β and f˜ which leads to a reaction energy which is within 25 kJ mol−1
of the experimental value.
84
Assessment of DFT for Transition Metal Complexes
Figure 4.8: Dependence of the reaction energies of reactions 5 – 9 on the empirical
parameters β and f˜ of BP86. The black dots indicate points for which actual computations
have been carried out; between these points the energies were interpolated. The red dot
indicates the original values of β and f˜ as proposed by Becke and Perdew, respectively.
The green circle represents the values of β and f˜ for a given reaction for which the best
approximation to the experimental values is obtained. The dashed green lines show the
range of values for β and f˜ which leads to a reaction energy which is within 25 kJ mol−1
of the experimental value.
4.2 Investigating the Parameters of BP86
85
Table 4.4: Decomposition of the total (“tot”) ligand coordination energy of reaction 1 into
exchange (“x”) and correlation (“c”) contributions for different values of the parameters
β and f˜.
β, f˜
EBP86
react, tot
EBP86
react, x
EBP86
react, c
EBP86
react, rest
0.0042, 0.11
0.5, 0.11
0.0042, 0.2
97.53
104.57
81.67
−19.58
−12.54
−5.92
19.66
15.48
3.77
97.48
101.63
83.82
The observed changes in the ten reaction energies (cf., Figs. 4.7 and 4.8) could
stem from either the reactant complex, the two product molecules, or both. In
order to shed light onto this aspect, we show the energies for the three molecules
of reaction 1 in Fig. 4.9. For the sake of clarity, we divided the total electronic
energies obtained for the individual values of β and f˜ by a reference energy, which
was calculated for the original values β = 0.0042 and f˜ = 0.11. We see that the
qualitative dependence of the total electronic energy on the two parameters β and
f˜ is the same for all three molecules. From this, we conclude that the dramatic
changes of the reaction energies stem from both the description of the reactants as
well as the products.
One may determine the optimal values for β and f˜ reproducing exactly the
experimental dissociation energies for the individual reactions. From all value
pairs yielding the experimental binding energies, we chose the one which is most
closes to the original values. These optimal parameters are indicated by a green
circle in Figs. 4.7 and 4.8. We should note here that in the case of reaction 7b,
the reoptimized values do not lead to a perfect agreement with the experimental
reaction energy, but deviate 0.2 kJ mol−1 from it. This is, however, acceptable for
the purpose of this work. For all reactions the optimal value for β varies rather
strongly. However, regarding f˜, the original value of 0.11 appears to be very good
for all reactions except reaction 7b. The fact that β varies strongly among the
individual reactions demonstrates that there exists no universal set of values for
this parameter, which can be taken as an indication of the inappropriateness of
BP86 as a universal exchange–correlation functional. We also see that for many
reactions, both β as well as f˜ may change significantly while still leading to a
reaction energy which is within 25 kJ mol−1 of the experimental reference energy.
From this we may conclude that with different basis sets, one would obtain rather
different optimal values for these parameters.
86
Assessment of DFT for Transition Metal Complexes
Figure 4.9: Ratio of the total electronic energies to their total electronic energy references
(for which β = 0.0042 and f˜ = 0.11) for the three molecules of reaction 1. The plot in
the lower right corner shows the the ligand binding energy of this reaction.
4.2.3
Reparametrizing BP86
For the entire set of the WCCR10 dissociation energies considered in this work,
the original BP86 functional (i.e., with β = 0.0042 and f˜ = 0.11), gives a mean
absolute deviation of 31.5 kJ/mol (the mean absolute deviation is the arithmetic
average of all absolute deviations of the computed values from their respective experimental reference values; when employing the large def2-QZVPP basis set, the
mean absolute deviation is 39.6 kJ mol−1 ; see 4.3). It is possible to reoptimize the
values for β and f˜ such that the mean absolute deviation is minimized. Carrying
out this minimization yields β = 0.00759734 and f˜ = 0.0758972. It is particularly noteworthy that the value for f˜ changes significantly, even though for every
individual reaction, the original value of f˜ was found to be almost perfect. However, this new parametrization lowers the mean absolute deviation by only about
9.5 kJ/mol to 22.0 kJ/mol. This is mostly due to the fact that the optimal value of
4.2 Investigating the Parameters of BP86
87
β is quite different for the individual reactions. For example, for reactions 7a, 7b,
and 7c the optimal value for β is much larger than for the other reactions. If we
exclude these reactions from the minimization, we obtain a lower mean absolute
deviation of 10.9 kJ mol−1 . However, also in this case the overall improvement
is clearly smaller than the error stemming from the basis set for which a value
roughly 25 kJ mol−1 may be assumed.
Therefore, depending on the subset for which the empirical BP86 parameters
are optimized, we obtain different results. This suggests that the optimal values
for both parameters β and f˜ are not system-independent. For a calibration against
the full set of ten reactions, there is hardly any improvement over the original
set of parameters. Therefore, we understand that a reparametrization of BP86 at
the WCCR10 test set is not advisable. Clearly, this can be taken as a general
conclusion.
4.2.4
Conclusions on the Parameters of BP86
We have analyzed the dependence of ligand dissociation energies of transition
metal complexes on the empirical parameters of a density functional, taking the
well-established BP86 exchange–correlation functional as an example. We studied
ligand coordination to ten large transition metal complexes, for which very accurate
experimental gas-phase data is available in the WCCR10 test set. We found that the
dependence of the reaction energies on the two parameters β and f˜ is qualitatively
the same for all reactions, even though the individual complexes are chemically
rather different. Within the parameter intervals analyzed, the reaction energies
span a wide range and these dramatic changes were found to depend on both
the reactants as well as the products of the reactions. Taken separately, exchange
and correlation energies contribute roughly the same to the overall reaction energy.
However, while the correlation contribution is positive, the exchange energy has
a negative sign, leading to an almost full cancellation of the two contributions in
the reaction energies. This beneficial behavior is long known and becomes evident
if an exact local exchange expression like Görling’s local Hartree–Fock [365] is
combined with a standard correlation functional.
Using the experimental data as reference, it is possible to re-calibrate the two
parameters β and f˜. However, such an optimization with the full set of ten ligand
dissociation energies decreases the overall error by only 9.5 kJ/mol. We found that
this is due to the fact that the optimal parameters for every individual reaction
are quite different from each other, suggesting that the two parameters in BP86
(especially β) are not system-independent. This motivates the development of
exchange-correlation potentials without such system-dependent parameters. In this
respect, the explicit reconstruction and analysis of the Kohn–Sham potentials of
chemically different systems may be helpful [366, 367].
Chapter 5
Gradient-Driven Molecule
Construction
“Chemistry is a creative science. . . and
the first chapter of its Book of
Genesis is not yet written.”
A. D. Little
The design of new molecules and materials exhibiting favourable properties is
an ever ongoing quest in a wide range of research fields. With the advent of
powerful computer systems, theoretical methods and tools are becoming increasingly
important in this design process (see also chapter 3) [234]. These methods allow
one to pre-screen properties of a set of molecules or materials without the need of
time- and resource-intensive synthesis. However, as explained in chapter 1, standard
quantum chemical methods rely on the definition of a molecular structure, which
is often not known, but searched for. In this chapter, we develop an approach for
the design of molecules and materials with specific stability in this work. We shall
first define the general principle of our inverse approach and then illustrate different
implementations at a specific example, namely, the design of a new transition metal
catalyst for the homogeneous fixation of dinitrogen.
5.1
Concept of Gradient-Driven Molecule
Construction
Rational design and thus also inverse approaches should rely on the free energy as
the ultimate thermodynamic criterion to decide on the stability of a computationally
designed molecule or material. However, for reactions which involve the formation
of strong bonds, the reaction free energy is determined mostly by the difference in
electronic energy of the reactants. In such cases, electronic structure theory at zero
Kelvin is usually sufficient for the design process and temperature corrections may
be neglected. Then, the minimum of the electronic energy of a product molecule
or material can serve as a stability criterion. This minimum is determined by
its vanishing geometry gradients (under the assumption that all eigenvalues of the
90
Gradient-Driven Molecule Construction
geometric Hessian are positive). Hence, following our previous initial investigation
[247] we may formulate the design principle as:
Gradient-driven Molecule Construction (GdMC): Given a target structural
element or property, but an unknown overall structural composition of the target
molecule or material, then the unknown parts of the compound can be designed
such that the geometry gradient of the electronic energy on all atoms (of known
substructures and of the constructed subsystems) approaches zero componentwise.
An example for the application of this design principle is the search for smallmolecule activating metal complexes. Important small molecules, for which activation procedures are sought, are H2 , N2 , O2 , CO2 , and CH4 . Upon fixation of such a
small molecule by a transition metal ion embedded in a surface or in a sophisticated
chelate ligand, a strong bond can be formed, which weakens the bonds within the
small molecule and may induce other structural changes (like bending due to an
evolving lone pair).
A specific example is the task of finding a transition metal catalyst that reduces
molecular dinitrogen to ammonia under ambient conditions at high turnover numbers
[368–371]. In industry, the process is accomplished by a heterogeneous iron catalyst
within the Haber–Bosch process, but under harsh conditions [372]. Only three
different families of synthetic homogeneous catalysts, two relying on molybdenum
and one on iron as central transition metal atom, have achieved this task so far
[373–377]. However, the molybdenum complexes are suffering from very low turnover numbers (no information about the turn-over number of the iron-based catalyst
could be found). Intense research efforts [206, 378–380] focused on increasing the
stability of one of them, the Schrock catalyst depicted in Fig. 5.1.
iPr
HIPT
N2
HIPT
HIPT
N
N
iPr
Mo
N
HIPT =
iPr
N
iPr
iPr
Figure 5.1: Lewis structure of the original Schrock catalyst, [MoN(NC2 H2 HIPT)3 )]N2
(HIPT: hexa-iso-propyl terphenyl), the first homogeneous catalyst capable of activating
nitrogen [373].
5.2 Theory of Gradient-Driven Molecule Construction
91
With quantum chemical methods, we were able to identify [206, 368, 381–391]
the most crucial steps in catalytic processes that facilitate N2 fixation: 1), feasibility
of dinitrogen binding, 2), transfer of the first hydrogen atom (e.g., as some kind
of proton plus electron transfer) onto the bound dinitrogen ligand, 3), exchange of
the (second) ammonia molecule produced by the next incoming dinitrogen ligand,
and 4), prevention of side reactions.
For a nitrogen-fixating catalyst, the first step is thus to bind dinitrogen to one,
two or even several transition metal centers in some ligand environment. This task
may be very difficult to achieve, as is highlighted, for instance, by the case of iron
complexes in a sulfur-rich ligand environment [389, 391–394] (note that the active
site of nitrogenase is an iron-sulfur cluster [395]). These transition metal complexes
bind dinitrogen comparatively weakly and feature hardly any bond activation as
monitored by the N–N bond length that remains almost unchanged. However, an
N2 -fixating system may directly activate N2 upon binding such that the triple bond is
broken and a diazenoid or hydrazinoid structure with corresponding elongated N–N
bond length emerges [396]. If the N–N bond is not activated upon coordination, a
subsequent reduction by electron transfer onto the ligand is necessary in order to
start the chemical reduction process [385, 386].
In a first application of our GdMC design principle to the task of identifying
a dinitrogen-binding or dinitrogen-activating complex, we started [247] from a
predefined central fragment, which consists of a transition metal atom and N2 at
a certain distance with the N–N bond length fixed to some reasonable value. The
task then is to find a ligand environment that reduces the geometry gradients on
all atoms in the compound. While we have studied already some options and
possibilities for Schrock-type dinitrogen fixation within this framework [247], a
rigorous investigation is mandatory and shall be provided in this work. Therefore,
we first consider the basic formalism of GdMC in the subsequent section and
discuss possible realizations of gradient-reducing potentials to be substituted by a
fragment scaffold in a second step. The whole analysis in the subsequent results
section is guided by the knowledge that we have gained in the past decade about the
Schrock dinitrogen activating molybdenum complex. The idea is to understand what
measures need to be taken in order to reconstruct a complex where functionality
has already been confirmed and investigated.
5.2
Theory of Gradient-Driven Molecule Construction
The definition of a molecular fragment structure that shall be stabilized by chemical
embedding into a larger molecular structure is central to GdMC. Without the chemical embedding, the fragment structure will in general not represent a stationary
point on the potential energy surface. In other words, without the chemical embedding the fragment will feature a non-vanishing geometry gradient on the nuclei
of the fragment (i.e., the derivative of the total electronic energy with respect to
the nuclear coordinates will be significantly different from zero). Moreover, the
GdMC concept requires to construct a molecular environment such that the overall
92
Gradient-Driven Molecule Construction
gradient will vanish and the resulting compound will represent a stable structure in
toto on the electronic energy hypersurface. For an algorithmic realization of this
concept, atomic nuclei and electrons must be placed to form a suitable scaffold that
can host and stabilize the fragment. Accordingly, the number of electrons as well
as the number, position, and charge of atomic nuclei are optimization parameters.
This optimization problem is very complex (even if we neglect for the moment
the spin degrees of freedom). Not only the lengths of all Cartesian gradient components must vanish at each nucleus of the fragment, they must also vanish at all
nuclei which are added in the chemical embedding process. However, this second
requirement can be separated from the first one if a two-step optimization procedure
is adopted. For this, the gradient-reducing environment may be represented by a
jacket potential νjac that mediates all interactions with the fragment [247]. In a
second step, a molecular realization of the optimized jacket potential needs to be
found. It is the purpose of this paper to study the design strategies in such oneand two-step approaches.
We choose the jacket potential νjac to enter the electronic fragment Hamiltonian
Hel (in Hartree atomic units) as a one-electron operator,
Hel = −
X ZI
X 1
X ZI ZJ X
1 X
∆i −
νjac (i), (5.1)
+
+
+
2 i∈frag
r
r
r
iI
ij
IJ
i,I∈frag
i,j>i∈frag
I,J>I∈frag
i∈frag
where the indices I, J and i, j run over all nuclei and electrons, respectively, of
the fragment. ZI denotes the charge of nucleus I and rij is the spatial distance
between particles i and j. The first term describes the kinetic energy of the fragment
electrons with the Laplacian ∆i . Although the jacket potential is written as a oneelectron operator, it may contain contributions from nuclear repulsions, electronic
kinetic energy operators and so forth when it is replaced by a viable molecular
structure in the second optimization step. In fact, the particular choice of the
jacket potential will determine the approximation adopted in the second step. If a
purely classical (electrostatic) embedding is considered to be adequate for the design
problem (i.e., if exchange and quantum correlation effects can be neglected), then
the jacket potential will exactly be a one-electron operator. If, however, quantum
mechanical superposition effects are non-negligible — as this would be the case
for fragments that require cuts through chemical bonds to separate them from the
environment — then two-electron effects or nonadditive kinetic energy terms could
become important. The total electronic energy can be obtained as the expectation
value of the above Hamiltonian,
Eel = hΨel |Hel |Ψel i.
(5.2)
If we approximate the many-electron wave function Ψel of the subsystem by a
determinant expansion, the orbitals can be obtained from a self-consistent-field-type
equation, which we may write for our purposes in case of the fragment (denoted
in the following as “the fragment” (frag)) as
1
γii ∆frag + νfrag (r) + νjac (r) ϕi (r) = εi ϕi (r),
(5.3)
2
5.2 Theory of Gradient-Driven Molecule Construction
93
where ϕi (r) and εi represent the i-th orbital and the associated orbital energy. γii
is a generalized occupation number, which is equal to one for the spin orbitals
of a single-determinant theory like Hartree–Fock or Kohn–Sham density functional
theory (DFT) and which is a real number in multi-determinant theories. The first
term of the operator on the left-hand side represents the kinetic energy of the central
fragment, and the second term collects all potential energy terms important for the
fragment,
(ne)
(coul)
(xc)
(nn)
νfrag (r) = νfrag (r) + νfrag (r) + νfrag (r) + νfrag (r),
(5.4)
in which the individual terms denote the attraction between all nuclei (n) and all
electrons (e), the repulsion between all electrons, a non-classical term representing
exchange–correlation effects and finally the (constant) nucleus–nucleus repulsion
within the fragment.
The third term on the left-hand side of Eq. (5.3), i.e., the jacket potential,
represents both the ligand sphere (to which we will refer as “the environment”) as
well as its interaction with the fragment. We may divide it into a potential energy
operator for the environment and one for the interaction between the fragment and
the ligand environment,
νjac (r) = νenv (r) + νint (r).
(5.5)
Formally, we may write
(ne)
(coul)
(xc)
(nn)
(kin)
νenv (r) = νenv
(r) + νenv
(r) + νenv
(r) + νenv
(r) + νenv
(r),
(5.6)
where the first three terms have a similar meaning as in Eq. (5.4). Note that we
need to introduce potential energy operators modeling the nucleus–nucleus repulsion
as well as the electronic kinetic energy terms (cf., the fourth and fifth terms on
the right hand side). Analogously, we may define for the interaction part
(ne)
(coul)
νint (r) = νint (r) + νint
(xc)
(nn)
(kin)
(r) + νint (r) + νint (r) + νint (r).
(5.7)
In this equation, the first term represents the interaction between the nuclei of the
fragment and the electrons of the environment, and the interaction between the
nuclei of the environment and the electrons of the fragment. The second term
stands for the mutual Coulomb repulsion of the electrons of the fragment and the
environment, respectively, while the third and the last terms represent the exchange–
correlation term between the fragment and the ligand sphere and the non-additive
kinetic-energy contribution for the joined fragment and environment, respectively.
The fourth term in Eq. (5.7) denotes the mutual repulsion between the nuclei of
the fragment and the ones of the environment.
We can now define the absolute value of the gradient of the total electronic
energy with respect to the Cartesian coordinates of one nucleus I as
s
2 2 2
∂Eel
∂Eel
∂Eel
+
+
,
(5.8)
|∇I Eel | =
∂rI,x
∂rI,y
∂rI,z
94
Gradient-Driven Molecule Construction
For the analysis to come, we define an overall absolute gradient |∇tot Eel | as the
sum of all individual absolute gradients of a given fragment, i.e.,
X
|∇tot Eel | =
|∇B Eel |.
(5.9)
B∈frag
The individual terms in the square root expression of Eq. (5.8) are given by
X ZI ZB (rI,α − rB,α ) X ∂ϕi (r) ∂Eel
∆ ϕi (r)
=
−
γii
∂rI,α B∈frag
|r I − r B |3
∂r
I,α
i
Z
Z
∂νfrag (r)
∂ρ(r)
(5.10)
ρ(r)dr + νfrag (r)
dr
+
∂rI,α
∂rI,α
Z
Z
∂νjac (r)
∂ρ(r)
+
ρ(r)dr + νjac (r)
dr,
∂rI,α
∂rI,α
where α ∈ {x, y, z}. In Eq. (5.10), the first two terms are the derivatives of the
nucleus–nucleus repulsion energy and the electronic kinetic energy of the fragment.
The remaining terms are the derivatives of the potentials occurring in Eq. (5.3); ρ(r)
is the total electron density of fragment and environment and r a spatial electron
coordinate. Of course, the potential terms can be broken down into derivatives of
their individual contributions as laid out in Eqs. (5.4) – (5.7). In particular, we have
(ne)
∂νfrag (r)
−ZI (rI,α − rα )
=
,
∂rI,α
|r I − r|3
(5.11)
and
(coul)
∂νfrag (r)
=
∂rI,α
Z Z
∂ρ(r) ρ(r 0 )
drdr 0 .
∂rI,α |r − r 0 |
(5.12)
All other derivatives are not straightforward to evaluate, in particular because their
explicit form is not known in this general formalism. It depends on the approximations made in a specific exchange–correlation functional as well as on the environment and interaction potentials. However, from Eqs. (5.10) – (5.12) we can see
that an important quantity in the gradient of the electronic energy is the derivative
of the electron density ρ(r) with respect to the nuclear coordinates.
For the sake of simplicity, we carry out the derivations for the simple case of a
one-determinant approximation to Ψel (e.g., Hartree–Fock or Kohn–Sham DFT). In
this case, the density is written as a sum over the absolute squares of all occupied
orbitals,
X
ρ(r) =
|ϕi (r)|2 ,
(5.13)
i
from which we obtain
X
∂ρ(r)
∂ϕi (r)
=2
ϕi (r)
.
∂rI,α
∂rI,α
i
(5.14)
5.3 Computational Details
95
In molecular calculations, the orbitals are usually expanded in terms of atom-centered
basis functions,
X
ϕi (r) =
cµi χµ (r − r I ),
(5.15)
µ
so that we find
X ∂χµ (r − r I )
∂ϕi (r) X ∂cµi
=
χµ (r − r I ) +
cµi
.
∂rI,α
∂rI,α
∂rI,α
µ
µ
(5.16)
If we employ Cartesian Gaussian-type orbitals (GTOs) as basis functions, i.e.,
χµ (r − r I ) = χµ (rx − rI,x , ry − rI,y , rz − rI,z )
2
= Nµ (rx − rI,x )u (ry − rI,y )v (rz − rI,z )w e−ζµ |r−rI | ,
(5.17)
the derivative of χµ (r − r I ) in Eq. (5.16) is straightforward to calculate. Nµ is a
normalization constant, ζµ a known parameter, and the sum u + v + w denotes the
“angular momentum” of the orbital.
Our GdMC concept requires that all geometry gradients vanish, i.e.,
!
|∇I Eel | = 0
∀I,
(5.18)
which is only possible if the individual Cartesian components vanish, as can be
seen by Eq. (5.8). Utilizing Eq. (5.10), we immediately find the requirement that
X ZI ZB (rI,α − rB,α ) X ∂ϕi (r) −
+
γii
∆ ϕi (r)
3
|r
∂r
I − rB |
I,α
i
Z
ZB
∂ρ(r)
∂νfrag (r)
(5.19)
ρ(r)dr − νfrag (r)
dr
−
∂rI,α
∂rI,α
Z
Z
∂νjac (r)
∂ρ(r)
!
=
ρ(r)dr + νjac (r)
dr
∀I and α.
∂rI,α
∂rI,α
This equation can be used as a working equation to determine νjac (r). Its solution
is not immediately obvious, but it can certainly be approximated with iterative
numeric methods.
5.3
Computational Details
All calculations presented in this work are within the density functional theory
framework. For the calculations in Section 5.4.1, we employed the program package
A DF, version 2010.02b [38], with the BP86 exchange–correlation functional [81,82]
and the Slater-type TZP basis set without frozen cores [205] at all atoms. In these
calculations advantage was taken of the resolution-of-the-identity technique for the
evaluation of the two-electron Coulomb integrals. Scalar-relativistic effects were
taken into account for all atoms by means of the zeroth order regular approximation
(ZORA) [37].
96
Gradient-Driven Molecule Construction
The calculations in Section 5.4.3 employed a local version of T URBOMOLE
5.7.1, which was modified such that the jacket potential can be represented on the
DFT grid. In all T URBOMOLE calculations, we also applied the BP86 exchange–
correlation functional [81, 82], but with Ahlrichs’ def-TZVP GTO basis set at all
atoms [292]. Note that the resolution-of-the-identity approximation was not invoked.
We should explicitly state here that the actual numerical values for nuclear
gradients depend on the basis set used. Therefore, it is important that such gradients
are calculated using exactly the same procedure in order to be able to compare
them to each other.
5.4
Model Hierarchies
Being a central quantity in GdMC, the potentials representing the environment and its
interaction with the fragment can be modeled with a range of different approaches.
As already stated above, the exact mathematical form of the environment and
interaction potentials depends strongly on the ansatz chosen, and we will therefore
obtain different expressions for the gradients. In a formally exact treatment, we
take the ligand sphere as being constituted by nuclei and electrons (although, of
course, we do not yet know their exact nature, number, and spatial distribution).
However, the additional potential terms need to be approximated in order to arrive
at a computationally feasible approach. In the most simplistic approach, one may
attempt to represent the jacket potential by a collection of point charges (see Section
5.4.2). In the sections to follow, we will investigate three different approaches.
5.4.1
Direct Optimization by Positioning Nuclei and Adding
Electrons
As a first approach, we attempt to decrease the gradient on the central fragment
by adding nuclei and electrons to its environment. A highly nontrivial optimization
problem emerges as it is not known how many electrons and how many nuclei
of which type and at which position will be required to model the environment.
Note that this means that the overall charge and spin state remain unknown, and
eventually have to be optimized, too.
Therefore, finding a way to reduce the number of optimization parameters would
be very beneficial. One option is the stepwise build-up of a ligand sphere exploiting
chemical principles for the scaffold construction. Clearly, the full beauty of inverse
design will only flourish if this can be done in a first-principles way, but for this
first attempt we are advised to exploit chemical knowledge in order to establish a
feasible algorithm.
In a first step, one could start with only a few atoms coordinating directly to
the central fragment, building up a first ligand shell. Due to their close spatial
proximity to this central fragment, one can expect that the atoms of this first ligand
shell will have the largest influence on it. Moreover, a reasonable starting guess for
the number and position of these initial atoms can be made by exploiting van der
Waals or covalent radii and the chemical knowledge about atomic valence. In the
5.4 Model Hierarchies
97
present case, we can find good starting configurations by studying the coordination
numbers and geometries (bond lengths, etc.) of molybdenum complexes [397]. In a
next step, the position and the type of atoms in the first ligand sphere are optimized
such that the gradient is minimized considering suitable capping atoms (see below)
to saturate all valencies. We then proceed by iteratively adding atoms to the ones
already present, thereby creating an increasingly complex ligand shell, in order to
minimize the gradient further.
In order to investigate this approach in terms of its practical usability, we may
start studying a reasonably well-defined complex. We thus deduced a range of
model complexes from the original Schrock complex as depicted in Fig. 5.2. All
models contain the central Mo–N2 fragment; the molecular environment is modelled
by two ligand shells with substituents R1 and R2 , respectively. Accordingly, we
denote these models [Mo]R1 , R2 –N2 , where the subscripts indicate which substituents
are present in the two ligand shells. For the sake of simplicity, we restrict our
model systems to a trigonal-bipyramidal coordination geometry as found in the
original Schrock catalyst and varied only the type of the three equatorial atoms
(their position is optimized in the course of a standard structure optimization). Any
remaining unsaturated valencies are saturated with hydrogen capping atoms.
N
b
optimization of first
ligand shell: R2 = H:
optimization of second
ligand shell: R1 = NH:
optimization of second
ligand shell: R1 = O:
R1 = NH
R1 = PH
R1 = O
R1 = CH2
R2 = OH
R2 = NH2
R2 = CH3
R2 = CH3
R2 = NH2
N
R1
a
Mo
R2R1
NH3
R1
R2
R2
Figure 5.2: Structural hierarchy of the model complexes.
The choice of the fixed Mo–N and N–N bond lengths (denoted as a and b in
Fig. 5.2) is important. There are two obvious choices for a. Either one takes the
bond length from the original Schrock catalyst (198.4 pm [206]), or one can let this
bond length relax to its equilibrium covalent bond length. If one sets b to the value
found in the original Schrock complex (i.e., 114.2 pm [206]), the N–N distance
represents an activated molecular dinitrogen ligand (the N–N distance for isolated
N2 is calculated to be 110.4 pm [206]). If one would like to find a complex which
directly transforms dinitrogen upon binding to a diazenoid or even hydrazinoid
species, one could set b to a value of 125.2 pm and 144.9 pm as calculated for
isolated diazene or hydrazine. First, we set both a and b to the values they have
in the Schrock complex; later in this study, we will investigate different values
for these two bond lengths. Then, the doublet states of the model complexes have
been optimized under the constraint that the distances b (and a) of Fig. 5.2 are
kept fixed.
98
Gradient-Driven Molecule Construction
5.4.1.1
Finding a Complex Binding N2
If we aim to construct a complex which binds molecular dinitrogen, then we can set
the distances a and b to the values found in the original Schrock N2 complex with
the HIPT ligands (for the initial placement of the other atoms, idealized bonding
angles and bond lengths deduced from covalent radii may be used). For this case,
Table 5.1 lists the absolute values of the gradients for the model systems mentioned
above.
Table 5.1: N2 binding: Absolute values of the Cartesian gradients on the atoms of the
central fragment (N(1) is the nitrogen atom bound to the molybdenum atom) for the
different model systems shown in Fig. 5.2. In all these model systems, the distances a
and b (cf., Fig. 5.2) have been fixed to the values found in the full Schrock complex
depicted in Fig. 5.1. All data are given in atomic units.
model
Mo–N2
full Schrock
[Mo]O, H –N2
[Mo]NH, H –N2
[Mo]CH2 , H –N2
[Mo]PH, H –N2
[Mo]O, CH3 –N2
[Mo]O, NH2 –N2
[Mo]NH, OH –N2
[Mo]NH, NH2 –N2
[Mo]NH, CH3 –N2
|∇Mo Eel |
|∇N(1) Eel |
|∇N(2) Eel |
|∇tot Eel |
|∇env Eel |
−2
−2
−3
−2
0.00
8.74·10−2
2.42·10−3
9.28·10−3
4.19·10−3
3.31·10−3
1.05·10−2
6.18·10−3
6.46·10−3
5.84·10−3
5.06·10−3
3.76·10
3.56·10−3
1.40·10−3
8.47·10−3
5.37·10−3
1.10·10−2
1.86·10−3
2.16·10−2
1.39·10−2
1.21·10−3
1.25·10−2
3.48·10
2.32·10−3
1.68·10−2
2.11·10−2
2.04·10−2
2.81·10−2
9.40·10−3
3.45·10−2
1.35·10−2
1.56·10−2
3.47·10−2
2.73·10
7.06·10−3
1.58·10−2
1.26·10−2
1.50·10−2
1.73·10−2
1.03·10−2
1.31·10−2
9.23·10−4
1.72·10−2
2.22·10−2
7.51·10
1.17·10−2
3.40·10−2
3.45·10−2
4.18·10−2
5.64·10−2
2.16·10−2
6.92·10−2
2.83·10−2
3.40·10−2
6.94·10−2
According to these data, the overall gradient |∇tot Eel | is decreased significantly in
all model systems compared to its value in the original central fragment. However,
while the gradients on the molybdenum atom and on the nitrogen atom attached
to it are decreased, the gradient on the second nitrogen atom is larger in all
model systems compared to the isolated central fragment. The model system with
phosphorus in the first ligand shell, [Mo]PH, H –N2 , features larger gradients than
the other complexes. With a value of 5.64·10−2 hartree/bohr, it clearly has the
largest overall gradient of this series of model systems. Apparently, phosphorus
does not stabilize the central fragment as well as carbon, nitrogen, or oxygen. From
these three elements, oxygen is obviously best suited, as [Mo]O, H –N2 features the
smallest overall gradient (namely, 3.40·10−2 hartree/bohr) on the central fragment,
although the overall gradient is not much larger if nitrogen (as in the original
Schrock catalyst) is present instead. The overall gradient on the nuclei of the
ligand sphere (denoted as |∇env Eel |) is very small, since the positions of its atoms
are fully optimized (also note that as |∇env Eel | includes all atoms of the ligand
sphere, it tends to be larger for larger ligand spheres). If we would not optimize
these positions, and simply use idealized angles and covalent radii to construct a
5.4 Model Hierarchies
99
ligand sphere, then both, the atoms of the ligand sphere and the central fragment,
feature large overall gradients between 10−2 and 10−1 hartree/bohr for all model
systems.
We carry out the second-shell construction step with model systems [Mo]O, CH3 –
N2 and [Mo]O, NH2 –N2 (see Fig. 5.2) for which we found lowest absolute gradients.
While the overall gradient of [Mo]O, NH2 –N2 (6.92·10−2 hartree/bohr) is larger than
in its parent model [Mo]O, H –N2 , we can decrease it to 2.16·10−2 hartree/bohr when
binding methyl groups to the oxygen atoms. We also carried out some preliminary
calculations to investigate the third optimization cycle, and found that a model
system employing ethanolate ligands, [Mo]O, CH2 CH3 –N2 , further decreases the overall
gradient to 1.57·10−2 hartree/bohr.
Table 5.2: N2 binding: Absolute values of the Cartesian gradients on the atoms of the
central fragment (N(1) is the nitrogen atom bound to the molybdenum atom) and Mo–N
distance a for the different model systems shown in Fig. 5.2. In all these model systems,
the distance b (cf., Fig. 5.2) has been fixed to the value found in the full Schrock complex
depicted in Fig. 5.1, while a was allowed to relax. All data are given in atomic units,
if not stated otherwise.
model
Mo–N2
full Schrock
[Mo]O, H –N2
[Mo]NH, H –N2
[Mo]CH2 , H –N2
[Mo]PH, H –N2
[Mo]O, CH3 –N2
[Mo]O, NH2 –N2
[Mo]NH, OH –N2
[Mo]NH, NH2 –N2
[Mo]NH, CH3 –N2
|∇Mo Eel |
|∇N(1) Eel |
|∇N(2) Eel |
|∇tot Eel |
2.5·10−5
1.04·10−4
5.70·10−4
8.43·10−4
9.69·10−4
9.19·10−4
9.91·10−4
1.25·10−3
1.15·10−3
1.29·10−3
1.07·10−3
2.21·10−2
9.47·10−4
1.64·10−2
1.68·10−2
1.79·10−2
2.39·10−2
1.01·10−2
2.36·10−2
6.58·10−3
1.61·10−2
2.84·10−2
2.22·10−2
2.06·10−4
1.63·10−2
1.67·10−2
1.77·10−2
2.38·10−2
9.00·10−3
2.41·10−2
6.51·10−3
1.70·10−2
2.84·10−2
4.44·10−2
1.26·10−3
3.33·10−2
3.44·10−2
3.66·10−2
4.86·10−2
2.01·10−2
4.90·10−2
1.42·10−2
3.44·10−2
5.79·10−2
|∇env Eel | a / pm
0.00
3.80·10−2
1.51·10−3
4.39·10−3
2.93·10−3
8.24·10−3
1.05·10−2
8.85·10−3
7.02·10−3
9.07·10−3
5.69·10−3
206.9
199.0
198.6
196.9
197.4
200.4
198.0
202.9
201.1
198.5
196.1
An open question is whether an even smaller final overall gradient could possibly
be obtained with a complex which does not feature the smallest gradient in the first
construction steps. In order to address this question, we investigate model complexes
[Mo]NH, OH –N2 , [Mo]NH, NH2 –N2 , and [Mo]NH, CH3 –N2 (cf., Fig. 5.2). The parent complex of these systems is [Mo]NH,H –N2 , which features nitrogen atoms coordinating
to the molybdenum atom. With a value of approximately 2.83·10−3 hartree/bohr,
[Mo]NH, OH –N2 features the smallest overall gradient within these three systems (Table 5.1). However, this value is larger than the overall gradient of [Mo]O, CH3 –N2 .
Thus, in this case it is not advantageous to start from a non-optimal parent complex
in the second construction cycle. However, we will present in the next paragraph
a different example where in fact the opposite is the case.
100
Gradient-Driven Molecule Construction
In order to investigate the flexibility and capabilities of the shell-wise construction
approach, we let the Mo–N distance a (see Fig. 5.2) relax during the optimization
procedure instead of keeping it fixed. With this additional degree of freedom, we
can expect to find smaller overall gradients compared to the case studied above
where a was kept fixed. The resulting data are given for all model systems
in Table 5.2. As expected, we obtain smaller overall gradients for all model
systems. However, we observe the same general trend as in the previous case.
Oxygen atoms in the first ligand shell lead to the smallest overall gradient, although
nitrogen atoms are also about equally good, while phosphorus is again not well
suited to decrease the gradient. In the second ligand shell, the methyl groups
decrease the overall gradient further. Nevertheless, we note that now [Mo]NH, OH –
N2 features with 1.42·10−2 a.u. clearly a smaller overall gradient than [Mo]O, CH3 –N2
with 2.01·10−2 a.u. This shows that it is indeed possible to obtain a smaller overall
gradient by starting from a parent complex which does not feature the smallest
gradient.
Table 5.3: N2 binding: N2 and ligand (i.e., the ligand in the equatorial position) binding
energies for the different model systems shown in Fig. 5.2. For the intrinsic binding
(fixed)
energies E(fixed)
bind, N2 and Ebind, ligand , the (dinitrogen) ligand as well as the complex fragment
were kept fixed, while they were fully relaxed in the calculation of E(rel)
bind, N2 . Finally, in
(N
rel)
2
the case of Ebind, N2 only dinitrogen was optimized, while the model system fragment was
kept fixed. In all these model systems, the distance b (cf., Fig. 5.2) has been fixed to
the value found in the full Schrock complex depicted in Fig. 5.1, while a was allowed
to relax (with the exception of E’(fixed)
bind, ligand , where also a was kept fixed). All data are
given in kJ mol−1 .
model
E(fixed)
bind, N2
E(rel)
bind, N2
2 rel)
E(N
bind, N2
E(fixed)
bind, ligand
E’(fixed)
bind, ligand
full Schrock
[Mo]O, H –N2
[Mo]NH, H –N2
[Mo]CH2 , H –N2
[Mo]PH, H –N2
[Mo]O, CH3 –N2
[Mo]O, NH2 –N2
[Mo]NH, OH –N2
[Mo]NH, NH2 –N2
[Mo]NH, CH3 –N2
n/a −151.8a
−155.9 −119.5
−187.3 −137.7
−183.2 −146.6
−150.4 −117.0
−158.0 −126.1
−134.6
237.0b
−154.4
71.0b
−177.5 −129.9
−194.7 −136.1
n/a
−147.5
−178.9
−174.5
−141.9
−149.5
−126.1
−146.0
−169.1
−186.3
n/a
−941.6
−914.2
−912.6
−789.8
−859.8
−919.5
−941.6
−919.5
−863.8
n/a
−941.7
−914.0
−911.2
−791.3
−860.1
−921.5
−942.3
−919.5
−862.1
a
This value was obtained from Ref. [206] and is calculated with a slightly different
methodology (see Ref. [206]). However, since in both methodologies a high degree of
accuracy was aimed at, the differences of the actual numerical results are negligible (for
example, the distance a was calculated to be 198.4 pm in Ref. [206], while it is found
to be 199.0 pm in this work).
b In these two cases, the model system fragment decomposes during relaxation.
However, we also note that for the [Mo]NH, OH –N2 model system, the Mo–N
5.4 Model Hierarchies
101
distance a is quite long with 201.1 pm. The [Mo]OH, CH3 –N2 model system, on the
contrary, features a Mo–N bond length which is very similar to the one found
in the full Schrock complex. Taking this bond length as a convenient means to
estimate the N2 binding energy (note that this value can be directly obtained during
the optimization at no additional computational cost), one would thus conclude that
the binding energy in the case of [Mo]OH, CH3 –N2 model system is comparable with
the full Schrock complex, while it is much smaller for the [Mo]NH, OH –N2 complex.
One might even conclude that the binding energy might be too small in this case,
such that the N2 fragment is not properly activated. However, while there is a
rough correlation between the bond length and the corresponding binding energy,
this relation is not very strict, as we can see by explicitly calculating the intrinsic
binding energies of the N2 fragment for the individual complexes (see Table 5.3).
We calculated the so-called intrinsic binding energy, where both the dinitrogen
ligand as well as the model system fragment (i.e., the model complexes without
the dinitrogen ligand) were kept fixed (E(fixed)
bind, N2 ), as well as the binding energy
when both fragments were fully relaxed (E(rel)
bind, N2 ). In addition, we calculated the
2 rel)
binding energy for the case where only N2 was allowed o relax (E(N
bind, N2 ). In
the calculation of E(rel)
bind, N2 , many of the fragments experienced significant structural
changes during relaxation. Two model system fragments ([Mo]O, NH2 and [Mo]NH, OH )
even decomposed completely. Therefore, we will omit these binding energies in the
following dicussion and rather focus on the intrinsic binding energies. We see that
(N2 rel)
the values for E(fixed)
bind, N2 are always larger than the corresponding values for Ebind, N2 ,
but the qualitative trends are literally the same in both cases.
When comparing the binding energies E(fixed)
bind, N2 of the two systems [Mo]NH, OH –N2
and [Mo]OH, CH3 –N2 (see Table 5.2), we see that both values are slightly larger than
the binding energy observed in the full Schrock complex. The binding energy of
the [Mo]NH, OH –N2 model complex (154.4 kJ mol−1 ) is even more similar to the one
of the Schrock catalyst (at least regarding the true intrinsic binding energies —
2 rel)
this picture changes when considering E(N
bind, N2 ). We therefore conclude that also
the [Mo]NH, OH –N2 complex activates the N2 fragment to a sufficient amount even
though the Mo–N distance a might suggest otherwise. Therefore, we see that it is
indeed possible to find a smaller overall gradient by starting from a complex which
was not optimal in the previous optimization cycle. This has to be regarded as a
drawback of the entire methodology of a stepwise build-up of the ligand sphere
in a sense that no primary parts of a chelate-ligand scaffold may be disregarded
in early optimization steps. Hence, the computational effort should not be reduced
by eliminating intermediary scaffolds too early.
One can raise the question why oxygen is the best element to decrease the
overall gradient in the first optimization cycle. We calculated also the intrinsic
binding energies of the ligands in the equatorial position; these data are also
given in Table 5.3. The reason for the large values of these binding energies is
the fact that these ligands are negatively charged, which leads to a separation of
positive and negative charges during the dissociation. For the model systems of
the first optimization cycle, the ligand binding energies correspond with the ability
of a given ligand to decrease the overall gradient. The hydroxo ligand of the
[Mo]O, H –N2 complex, which leads to the smallest overall gradient, is most strongly
102
Gradient-Driven Molecule Construction
bound to molybdenum (941.6 kJ mol−1 ), while it is much smaller (789.8 kJ mol−1 )
for [Mo]PH, H –N2 . However, in the second optimization cycle, this relation is not
strictly valid anymore. The methanolate ligand in [Mo]O, CH3 –N2 clearly decreases
the gradient compared to the parent system [Mo]O, H –N2 , but it is less strongly bound
(859.8 kJ mol−1 ). The ONH2 ligand, which leads to a very large overall gradient,
on the other hand is rather strongly bound with 919.5 kJ mol−1 . Therefore, one
cannot rely on the ligand binding energies in order to assess the ability of a given
ligand to decrease the overall gradient. A thorough decomposition analysis of the
overall absolute gradient might be necessary in order to clarify this aspect.
In order to further investigate how many optimization cycles might be necessary to decrease the gradient on the central fragment to an acceptable threshold,
we constructed a series of model systems gradually approaching the full Schrock
complex as depicted in Fig. 5.3. We then optimized the positions of all atoms
except the ones of the central fragment so that both distances a and b were kept
fixed. The resulting gradients on this central fragment are given in Table 5.4. One
can see that the overall gradient does not monotonically decrease when enlarging
the model systems. On the contrary, it even reaches a value which is higher than
the one obtained in the original Mo–N2 fragment. A similar behavior is observed
for the gradients on the individual atoms. The largest two model systems feature
overall gradients which are only slightly larger than the one of the full Schrock
complex. We also see that the overall gradient of the full Schrock complex drops
by one order of magnitude if the Mo-N and N–N distances are allowed to relax.
Upon relaxation, the Mo–N distance is elongated from 198.4 to 199.0 pm, while the
N–N distance remains almost unchanged. This shows that the overall gradient is
highly sensitive to the specific computational methodology employed. The value of
198.4 pm for the Mo–N distance was taken from Ref. [206], where a different basis
set (namely, Ahlrichs’ GTO-type TZVP basis [292] at all atoms except for carbon
and hydrogen, for which the SVP basis [398] was used) was employed. Since
in our methodology, the Slater-type TZP basis set is employed, the equilibrium
distance of the Mo–N distance is slightly different (0.6 pm). Using a Mo–N bond
length of 198.4 pm therefore leads to overall gradients which are rather large at
first sight.
5.4 Model Hierarchies
103
H3C
N2
H3C
N2
CH3
N
N
Mo
N
Mo
N
H
N2
H
Mo
Ph
N
Ph
N
N
H
NH3
H
Mo
N
N
H
tph
N2
H
N
H3C N
Mo
N2
tph
N
CH3
H
tph
N
N
NH3
H3C
Mo
N
N
HIPT
N2
H
N2
N
H N
H
Ph
H
HIPT
N
N
N2
H
Mo
N
N
HIPT
N
N
Mo
N
N
Figure 5.3: Overview of the model system series gradually approximating the full Schrock
complex depicted in the bottom right corner. The abbreviation “tph” stands for terphenyl
(i.e., the HIPT ligand without the isopropyl substituents).
104
Gradient-Driven Molecule Construction
Table 5.4: N2 binding: Absolute values of the Cartesian gradients on the atoms of the
central fragment (last three columns; N(1) is the nitrogen atom bound to the molybdenum
atom) for the model systems gradually approaching the full Schrock complex (see Fig. 5.3).
In all model systems, the Mo–N distance a is constrained to 198.4 pm, while the N–N
distance b is kept fixed at 114.2 pm. For the full Schrock catalyst, data for both constrained
as well as unconstrained distances a and b is given. All data are given in atomic units.
model
Mo–N2
[Mo]NH, H –N2
[Mo]NH, CH3 –N2
[Mo]NH, CH2 CH2 –N2
[Mo]NCH3 , CH2 CH2 –N2
[Mo]NPh, CH2 CH2 –N2
[Mo]Ntph, CH2 CH2 –N2
full Schrock
full Schrocka
a
|∇Mo Eel |
|∇N(1) Eel |
|∇N(2) Eel |
|∇tot Eel |
|∇env Eel |
−2
−2
−3
−2
0.00
9.28·10−3
5.06·10−3
4.31·10−3
9.04·10−3
7.29·10−3
2.44·10−2
8.74·10−2
3.80·10−2
3.76·10
8.47·10−3
1.30·10−2
2.74·10−3
7.65·10−4
7.37·10−3
7.46·10−3
3.56·10−3
1.04·10−4
3.48·10
2.11·10−2
3.39·10−2
3.25·10−2
3.84·10−2
3.17·10−3
1.67·10−3
2.32·10−3
9.47·10−4
2.73·10
1.26·10−2
2.14·10−2
2.97·10−2
3.79·10−2
4.14·10−3
5.98·10−3
7.06·10−3
2.06·10−4
7.51·10
3.45·10−2
6.82·10−2
6.49·10−2
7.70·10−2
1.47·10−2
1.51·10−2
1.17·10−2
1.26·10−3
(a, b relaxed)
5.4.1.2
Finding a Complex Activating N2
The N–N bond length b (c.g., Fig. 5.2) is a convenient descriptor to (pre)define the
degree of activation of the dinitrogen molecule. In order to find a complex which
does not only bind, but also activate molecular nitrogen, one can carry out a similar
optimization procedure as laid out above, setting b to a value as found in diazene
or hydrazine. For isolated diazene b is found to be 125.2 pm while it has a value
of 144.9 pm for isolated hydrazine. Here, we study the gradients of the models
[Mo]NH, H –N2 , [Mo]PH, H –N2 , [Mo]O, H –N2 , and [Mo]CH3 ,H –N2 , setting b arbitrarily
to a value of 130 pm. This corresponds to a significant activation; very similar
values are found for the N–N bond length in the original Schrock complex and
derivatives thereof when dinitrogen has been doubly protonated and reduced [206].
The resulting gradient data are given in Table 5.5.
As can be seen, our optimization procedure does not have any success: only
the absolute gradient on the nitrogen atom bound to the molybdenum center can
be decreased somewhat, but the resulting overall gradients are all slightly larger
(0.1·10−1 – 0.5·10−1 hartree/bohr) than in the free Mo–N2 fragment. This result was
expected as our model systems are inspired by the original Schrock complex, which
itself does not activate dinitrogen directly. The activation of nitrogen is done in
this case by a sequence of protonation and reduction steps. In fact, we find that
the overall gradient does already slightly decrease if we reduce the model complex
[Mo]NH, H –N2 by one elementary charge, thus forming [Mo]NH, H –N2 − . However, if
we protonate the complex at the terminal nitrogen atom of the N2 fragment insetad
of reducing it, the overall gradient is significantly reduced. The gradient can be
further optimized by employing a combined reduction and protonation; the overall
absolute gradient of [Mo]NH, H –N2 –H is only about half as large as it is in the
5.4 Model Hierarchies
105
original fragment. Therefore, these preliminary calculations show that also in this
more challenging case, our method can be applied to decrease the gradients on all
nuclei.
Table 5.5: N2 activation: Absolute values of the Cartesian gradients on the atoms of the
central fragment (last three columns; N(1) is the nitrogen atom bound to the molybdenum
atom) for the different model systems studied. In all these model systems, the distance
a (cf., Fig. 5.2 has been fixed to the value found in the Schrock complex, while b was
set to 130 pm. All data are gvien in atomic units.
model
Mo–N2
[Mo]NH, H –N2
[Mo]PH, H –N2
[Mo]O, H –N2
[Mo]CH2 , H –N2
[Mo]NH, H –N2 −
[Mo]NH, H –N2 –H+
[Mo]NH, H –N2 –H
|∇Mo Eel |
|∇N(1) Eel |
|∇N(2) Eel |
|∇tot Eel |
|∇env Eel |
8.43·10−3
5.11·10−2
3.34·10−2
4.66·10−2
4.91·10−2
6.74·10−2
1.02·10−1
1.51·10−1
3.11·10−1
2.77·10−1
3.19·10−1
3.06·10−1
2.75·10−1
2.08·10−1
8.58·10−2
4.28·10−2
3.19·10−1
3.28·10−1
3.52·10−1
3.52·10−1
3.24·10−1
2.76·10−1
1.79·10−1
1.19·10−1
6.39·10−1
6.56·10−1
7.05·10−1
7.04·10−1
6.49·10−1
5.51·10−1
3.67·10−1
3.13·10−1
0.00
4.56·10−3
2.92·10−3
4.31·10−3
7.74·10−3
2.73·10−3
6.41·10−3
3.38·10−3
In the original Schrock catalyst, one finds that the N2 fragment is subject to
a significant bending motion upon activation, i.e., the diazenoid and hydrazenoid
structures have Mo–N–N angles of about 120◦ . We can thus modify our model
systems accordingly in order to decrease the overall gradient. However, we found
that the [Mo]NH, H –N2 complex does not support an Mo–N–N angle of 120◦ . During
the structure optimization, the angle relaxes to 180◦ . One knows that the activation
of the dinitrogen fragment (and concomittantly, its sidewards bending) requires
molybdenum d-electrons. However, also in the [Mo]NH, H –N2 − and in the [Mo]NH, H –
N2 2− complex the Mo–N–N angle is stable only at 180◦ . Therefore, in this specific
case, this structural modification does not lead to a lower overall gradient.
5.4.1.3
Effect of the Coordination Geometry
Almost all data reported above has been obtained on model complexes featuring
a trigonal-bipyramidal coordination environment as observed in the full Schrock
complex. However, also in the full Schrock complex, a stable six-coordinate intermediate has been found in a computational investigation of the reaction mechanism [381]. We have modified the test system [Mo]O, H –N2 such that it includes
another hydroxo ligand and adopts an octahedral coordination geometry. In this
case, we can see rather large effects (see Table 5.6). In an octahedral coordination
environment, the model complex features absolute gradients which are almost one
order of magnitude larger than in the trigonal-bipyramidal case. This suggests that
this latter coordination geometry might be better suited for a molybdenum-based
N2 -activating complex. This might be due to the increased steric crowding in the
case of a six-coordinate complex. We have also investigated a square-pyramidal
106
Gradient-Driven Molecule Construction
coordination environment by displacing the NH3 ligand such that it does no longer
form a N–Mo–N angle of 180◦ . When adopting an initial N–Mo–N angle of 135◦
or larger, the ammonia ligand returns to its axial position during the optimization,
and we essentially find again the original model system [Mo]O, H –N2 as schematically depicted in Fig. 5.2. However, when the initial N–Mo–N angle is set to
132◦ (or a smaller value), then the NH3 ligand does not return to an axial position.
Still, we do not obtain a square-pyramidal coordination environment, but observe
a conformational change which is reminiscent of the Berry pseudo-rotation, such
that we end up with an almost perfect trigonal-bipyramidal coordination geometry,
where the N2 fragment lies in an equatorial position now, and one of the OH groups
has taken the axial position. This conformation features absolute gradients which
are much smaller than in the octahedral case, but they are still larger than in the
original model complex.
Table 5.6: Effect of the coordination geometry on the absolute gradients of model system
[Mo]O, H –N2 . “Distorted” refers to a trigonal-bipyramidal-like coordination where the N2
ligand lies in an equatorial position, while the axial positions are occupied by NH3 and
OH (see text). All data are given in atomic units.
coordination
trigonal-bipyramidal
octahedral
distorted
5.4.2
|∇Mo Eel |
|∇N(1) Eel |
|∇N(2) Eel |
|∇tot Eel |
|∇env Eel |
1.40·10−3
3.04·10−2
1.62·10−2
1.68·10−2
5.83·10−2
3.21·10−2
1.58·10−2
2.80·10−2
1.61·10−2
3.40·10−2
1.17·10−1
6.44·10−2
2.42·10−3
4.06·10−3
5.38·10−3
Environment Potential Represented by Point Charges
The approach discussed in section 5.4.1 is feasible, but requires significant computational resources to combinatorially construct and evaluate ligand scaffolds. Moreover, its disadvantage from a fundamental theoretical point of view is clearly the
fact that it heavily relies on chemical intuition in order to find good starting configurations (i.e., where to put which nucleus, charge and spin state). As a consequence,
fundamentally new coordination environments — especially those that produce unusual bonding situations in the chelate ligand — are likely to be overlooked.
We are therefore advised to establish a more abstract optimization scenario, in
which the ligand sphere is represented by a more abstract interaction potential,
which in a first approximation may simply be represented by a collection of point
charges (these may adopt fractional charges and may be placed anywhere in space —
not necessarily located at the position of an atomic nucleus). With enough such
charges, we should be able to represent a charge density that produces a potential
via Poisson’s equation (neglecting quantum superposition (entanglement) effects).
From this charge density, the corresponding ligand sphere could be deduced — the
positions of the nuclei could be found by finding the maxima of the distribution,
while the kind of nucleus could be determined by inspecting the Kato cusp of this
charge distribution at the respective maximum, provided that charge distributions
5.4 Model Hierarchies
107
which do not resemble those of molecular scaffolds can be avoided. A further
advantage of this approach is that it is straightforward to implement as most quantum
chemical programs can deal with arbitrary fields of point charges although present
SCF convergence accelerators may not be optimal to converge orbitals in such fields.
A solution could be imposing a strict overall charge neutralization condition. Still,
the resulting optimization problem is extremely complex. We have to optimize the
number of point charges, as well as their spatial distribution and the values of the
individual charges.
In a first attempt, we started with few point charges in order to keep the variational
problem feasible and the simplex algorithm was employed in order to minimize
the overall gradient. This algorithm has the advantage that only the function to
be minimized has to be evaluated and not its derivative. Furthermore, the simplex
algorithm is generally known to be robust, although it converges rather slowly.
Since the simplex algorithm is not suited to find the global minimum of a given
function, but usually converges to the closest local minimum, the outcome of our
minimization attempts heavily depends on the starting guess, i.e., number, position,
and magnitude of point charges. Moreover, in many cases the quantum mechanical
evaluation of the gradients did not converge. In order to circumvent this problem,
we modified the optimization procedure such that a very large gradient was returned
to the optimizer routine in case the gradient evaluation did not converge.
The fact that the point charge arrangements resulting from such an optimization
procedure are dependent on the initial conditions is undesirable. We therefore implemented a simple interface between M ATHEMATICA [362] and A DF [38] in order
to exploit the sophisticated optimization routines implemented in M ATHEMATICA
(see Appendix E). This new setup allows us to carry out global optimizations of
the overall gradient with methods such as simulated annealing, random walks, and
differential evolution [399, 400]. In our studies, we employed the latter algorithm
which belongs to the class of evolutionary algorithms. At the start of the optimization, a set (called the population) of several different, randomly chosen solution
vectors xi (containing the parameters to be optimized) is created. Experience shows
that the population should be five to ten times larger than the number of parameters
to be optimized [399]. In each iteration of the optimization, the population is newly
created as follows. For each of the solution vectors, three other vectors xu , xv ,
and xw are chosen randomly. From these three vectors, a single new vector xs is
calculated,
xs = xw + s (xu − xv ) ,
(5.20)
where s is a real scaling factor. This step is called mutation, as it introduces
random changes into the set of solution vectors. Usually s = 0.5 is a good
choice [399]. Note that due to the fact that the mutations are depending on
the solution vectors themselves, large mutations are created in components of the
solution vectors where they span a broad range, which means that the optimization
is still rather far away from a solution. In contrast to this, the mutations tend to be
small for converged components of the solution vectors. In a second step, called
recombination, a new solution vector is created from xs and xi by iterating over
all vector components, taking a component from xs with a given probability (the
108
Gradient-Driven Molecule Construction
cross probability), and from xi otherwise. Then, the function to be minimized (i.e.,
in our case the overall gradient) is evaluated with the newly generated solution
vector. If the new vector leads to a lower overall gradient than its parent vector,
xi , it replaces this vector in the population. Otherwise, xi is not replaced in the
population. It is found that a large cross probability often accelerates convergence,
such that a value of, say, 0.9 for the cross probability is often employed in first
trials [399]. The solution is assumed to be converged if the difference between the
best function values of the newest and the second newest population are below a
given threshold, and the difference between the two best solution vectors is below
a second threshold. M ATHEMATICA allows to combine this advanced optimization
routine with constraints such as equalities and hard and soft inequalities, which is
very convenient for our purposes (see below).
Figure 5.4: Positioning of one (a) and two (b; shown from two different perspectives)
point charges (green), respectively, with respect to the Mo–N2 fragment. Nitrogen atoms
are drawn in blue, while the molybdenum atom is marked in rose. In c), the numbering
of the ten point charges (cf., Table 5.7) is given.
Following the recommendations given in Ref. [399], we employed a scaling factor
s of 0.5, a cross probability of 0.9, and a population size ten times the number
of optimization parameters. In all optimizations, a random seed of 0 was used.
The solution was assumed to be converged when two subsequent best function
values differed by less than 10−5 hartree/bohr and the two best solution vectors
differed by less than 10−3 hartree/bohr (i.e., the parameters “AccuracyGoal” and
“PrecisionGoal” were set to 5 and 3, respectively). In a first study, we optimized
the position and magnitude of a single point charge, i.e., the number of optimization
5.4 Model Hierarchies
109
parameters was only four. This point charge was confined to be between −8 and
8 bohr in each of the three coordinates (which encompasses the central fragment),
while its magnitude was confined to lie between −2 and 2. The overall optimization
time was several days (the parallel version of A DF was employed on four Intel
Xeon E3-1240 CPU cores). It was found that a point charge of −0.609325 e can
minimize the overall gradient of the Mo–N2 fragment from 7.51·10−2 hartree/bohr to
5.18·10−3 hartree/bohr. The placement of this point charge with respect to the central
fragment is shown in Fig. 5.4 a).
It is very interesting to note that the point charge is rather far away from the
central fragment, namely 488.5 pm from the terminal nitrogen atom. One could
now try to further decrease the gradient by adding a second point charge while
keeping the first point charge fixed (i.e., an approach similar to the one practiced
in section 5.4.1). However, the overall gradient cannot be decreased any further,
but instead slightly increases to 5.21·10−3 hartree/bohr. The second point charge is
even further away from the central fragment than the first one (namely 618.8 pm;
see Fig. 5.4 b)). Utilizing a cross probability of 0.1 (but keeping all other settings
identical) does not improve this situation. Instead, the overall gradient is now found
to be 0.126374 hartree/bohr with a charge of 1.57547 e (the position of which is
rather similar to the previous case described above). The large differences between
the two optimization runs can be explained by the rather loose convergence criteria
employed. However, tighter convergence criteria can be expected to lead to even
longer run times, which significantly limits the practicality of this approach.
Table 5.7: Point charges (in elementary charges) obtained after an optimization with
M ATHEMATICA (second column) compared to Hirshfeld and MDC-q charges calculated
for complex [Mo]O, H –N2 (third and fourth column, respectively). The individual positions
are shown in Fig. 5.4 c). For each set of point charges, the resulting overall gradient is
given in hartree/bohr in the last line.
position
optimized
Hirshfeld
MDC-q
1
2
3
4
5
6
7
8
9
10
gradient
−0.421
0.133
0.256
−0.542
0.125
0.261
0.030
0.117
0.592
0.698
−0.169
−0.738
−0.029
0.117
0.603
−0.660
0.133
0.261
0.339
0.095
0.514
0.155
−0.333
−0.883
0.052
−0.333
−0.894
0.378
−0.334
−0.818
−2
−2
2.44·10
8.70·10
2.00·10−1
An important issue is the reconstruction of a ligand sphere from a given point
charge arrangement. Therefore, in a second exploratory investigation study, a total
110
Gradient-Driven Molecule Construction
of ten point charges were placed at the positions of the ligand atoms of the
model system [Mo]O, H –N2 and kept fixed at these positions. However, during the
optimization, a significant number of gradient evaluations did not converge, such
that no satisfying solution could be obtained. We therefore added the constraint that
the total sum of all point charge must be zero. With this constraint, we were able to
converge a solution within a few days with the standard settings of M ATHEMATICA,
while with the settings described above, the optimization did not finish after several
weeks. The optimal point charges, together with the corresponding overall gradient,
is given in Table 5.7. Also given are Hirshfeld charges [401] and multipole-derived
charges [402] obtained from a computation on the model complex [Mo]O, H –N2 .
For the multipole-derived charges, the atomic multipoles are obtained from the
electron density up to a given order (in our case quadrupole moments — the charges
are therefore referred to as “MDC-q”) which are then reconstructed exactly by
distributing charges at the atom positions.
We see that the optimized point charges decrease the overall gradient significantly
better compared to the Hirshfeld and MDC-q charges. Most notably, the overall
gradient is even smaller than in the full [Mo]O, H –N2 model. However, the reconstruction of a ligand sphere is not obvious at all. Different charge analysis schemes
give significantly different numerical values for the same atoms, since they rely
on different formalisms. Therefore, one cannot expect that it is possible to assign
a given atom type to a given charge value. However, the comparison with the
Hirshfeld and MDC-q charges allows us nevertheless to make some interesting observations. First of all, we see that the optimized charges deviate strongly from both
the Hirshfeld as well as the MDC-q charges. Moreover, the symmetry present in
the Hirshfeld and MDC-q point charge arrangements, originating from the threefold
symmetry axis present in the [Mo]O, H –N2 model complex, can no longer be found
in the optimized charges. This would suggest that in each of the three equatorial
positions, a different ligand would be present. The Hirshfeld and MDC-q charges
agree qualitatively with each other in the sense that the more electronegative atoms
oxygen and nitrogen all have a negative partial charge, with larger negative charges
on the oxygen atoms (which in fact has a larger electronegativity than nitrogen). All
hydrogen atoms have a positive partial charge. The partial charges of the hydrogen
atoms bound to the oxygen atoms are larger than the charges of the H-atoms of
the ammonia ligand, which one would also expect in view of the electronegativities
of oxygen and nitrogen, respectively. Applying this qualitative reasoning now to
the optimized charges, one would replace the terminal positions 1, 2, and 6 (the
positions of the hydrogen atoms of the NH3 ligand) with a strongly electronegative
element (regarding the valences, fluorine could be a good choice as terminal atom),
while at position 5, an atom with a medium electronegativity would be placed. It is
important to realize that one should not only rely on the concept of electronegativity
for assigning atom types to point charges. Instead, the valence a given atom type in
a given position needs to have in order to allow for a reasonable molecule provides
important additional information. For example, examining position four, it becomes
clear that an atom from the fifth main group (nitrogen, phosphorus, etc.) would
perfectly fit into the local coordination environment. From the optimized charge
of this position, which is 0.698 e, we understand that the corresponding atom type
5.4 Model Hierarchies
111
should have a rather low electronegativity, as only this would be compatible with
this charge. One would therefore rather employ phosphorus in position four instead
of nitrogen.
We have seen that the overall gradient on the Mo–N2 fragment can be signficantly decreased already with only a single point charge. Furthermore, for more
complicated point charge arrangements, the concept of electronegativity can be used
to deduce atom types from a given charge magnitude. Even though assignment
of atom types clearly needs to be further elaborated on, the greater challenge is
most likely to be the optimization itself. A local optimization depends highly on
the starting conditions, which means that the quality of the inital guess is decisive
for the outcome of such an optimization. However, it is questionable whether such
a good inital guess can possibly be obtained. A global optimization is extremely
time-consuming already for very small search spaces. The global optimization of a
large set of point charges can thus be expected not to be possible in a straightforward
fashion.
5.4.3
Environment Potential Represented on the DFT Grid
Within the framework of DFT a grid is usually employed for the numerical integration of the exchange–correlation energy (and its functional derivative). It is
therefore very easy to represent a jacket potential on this grid and to introduce it
into the Kohn–Sham equations. With established optimization algorithms (simplex,
simulated annealing, genetic algorithms, etc.) it should be possible to optimize an
additional potential such that the geometry gradient is as small as possible. In
practical cases, the grid is tailored to represent the potential in the close vicinity
to the central fragment. This determines the number of grid points (and thus the
number of potential values to be optimized) unequivocally. But, if a large ligand
sphere is required to stabilize a comparatively small central fragment, the grid centered around this central fragment will most likely not extend far enough into space
to allow for the accurate representation of the entire ligand sphere. However, one
could again aim for a stepwise build-up of the ligand sphere by first reconstructing
the ligand atoms directly connected to the central fragment, and then performing
additional optimization runs until the gradient is sufficiently small.
Of course, also this approach has a number of drawbacks. First of all, from
the observations made so far, it can be anticipated that the optimization strongly
depends on the starting guess. Since we have already found a few molecules which
reduce the gradient on the central atom (cf., model system 5 above), we could
employ the Kohn–Sham potential of this system as a starting point. However,
such a procedure would reintroduce the dependence of chemical intuition, since the
model system designed by the approach of directly adding atoms makes heavy use
of chemical knowledge. A global optimization, on the other hand, is very likely
not to be feasible, since also very coarse DFT grids for small model system have
a rather large number of points. For example, the coarsest Turbomole grid (grid
”1“) has almost 7000 points for the Mo–N2 fragment and, thus, 7000 parameters
would have to be optimized. Moreover, the reconstruction of an actual molecular
ligand sphere from such an abstract potential is not obvious.
112
Gradient-Driven Molecule Construction
Nevertheless, the approach is appealing and we shall therefore report here about
preliminary investigations. We modified Turbomole 5.7.1 such that an arbitrary
potential can be read in, which will then be added to the Kohn–Sham potential.
Then, we used the simplex algorithm in order to minimize the overall gradient on the
central fragment. As a starting guess for the additional potential, we utilized a zero
vector, as this appears to be the most natural choice without relying on any intuition.
In order to keep the optimization problem as simple as possible, we refrained from
taking the Mo–N2 fragment in a first study, as this would already include almost
7000 parameters to be optimized. In fact, trial optimizations with this fragment
did not finish after several weeks of runtime. Instead, we chose the optimized
structure of a water molecule and removed the oxygen atom, thereby creating a
dihydrogen molecule with a signficantly enlarged bond. The goal was then to find
a potential stabilizing this H· · · H fragment. In this case, the variational problem
involved 1706 parameters. It was indeed possible to find a potential minimizing
the gradient, namely from 1.50·10−1 hartree/bohr to 8.66·10−3 hartree/bohr. The
stabilizing potential is depicted in Fig. 5.5.
Figure 5.5: Contourplot of the jacket potential in a.u. in the xz-plane, in which the
H· · · H fragment is oriented. All data are given in atomic units.
One can clearly see that the potential features a maximum at the position of the
two hydrogen nuclei. However, there are some interesting features, most notably the
5.5 Summary
113
local minima in the vincinity of the nuclei. It appears that a ligand sphere cannot be
easily constructed from this potential. In this respect, especially the maxima at the
positions of the hydrogen nuclei are an obstacle, since the placement of additional
nuclei so close to existing ones is certainly not preferable. To change the type of
nuclei within the central fragment, as might be suggested by a potential such as the
one of Fig. 5.5, is not an option either. However, Wang et al. showed how to solve
this problem of C-representability in a very elegant fashion, namely by expanding the
external potential into a set of atomic potentials [248]. Still, the implementation of
this method does not solve the problem that even for comparatively small central
fragments and very coarse DFT grids, the number of grid points and therefore
parameters to be optimized is very large. Also efficient optimization algorithms,
combined with a parallel evaluation of the geometry gradient on fast contemporary
computer hardware, need several days to weeks (possibly even months) to find a
solution.
5.5
Summary
In this chapter, we have proposed a new idea for rational compound design, namely
Gradient-driven Molecule Construction (GdMC). This approach has been applied
to the case of a small-molecule activating catalyst. We started with a predefined
central fragment and searched for a ligand sphere stabilizing this fragment, i.e.,
nullifying the nuclear gradient on the fragment such that this conformation is stable.
We have investigated different approaches for finding such a ligand sphere. In a
first approach, we have directly added nuclei and electrons to the central fragment
and optimized their numbers and (in the case of the nuclei) their type and spatial
location. The resulting optimization problem is highly nontrivial. First tests turned
out to be promising. The most important next step would be to automatize this
method such that it can be combined with an established optimizing method (such as
the differential evolution algorithm employed in this work for the other approaches),
which would allow to explore the limits of this approach. In this respect, the
automatic generation of reasonable starting structures appears imperative. Such a
structure generation could rely on standard covalent radii and idealized bond and
dihedral angles in order to generate starting structures. We should also note here that
significant work has already been done in this area. One can therefore benefit from
these developments (for reviews, see Refs. [403–406]). For example, computational
frameworks such as O PEN BABEL [407,408] provide functions which can generate
three-dimensional molecular structures from simple S MILES (Simplified Molecular
Input Line Entry System) [409–411] representations.
In a second approach, we minimized the gradient by means of point charges.
Already a single point charge can significantly decrease the overall gradient, but
this does not allow to reconstruct an actual ligand sphere. When several point
charges are utilized, one can make use of the concept of electronegativity in order
to assign a given nucleus to a given point charge. One could in principle also
imagine a brute-force approach which employs a large number of point charges,
such that the ligand sphere can be represented as a true charge distribution. From
114
Gradient-Driven Molecule Construction
this charge distribution, one could easily reconstruct a ligand sphere by analyzing
the cusps, but this approach is likely not to be practical due to the huge number
of point charges (and therefore, optimization parameters).
As a third possible method, we attempted to represent the ligand sphere as
an additional potential in the Kohn–Sham equations. Also here, the variational
problem is extremely complicated. In order to guarantee the representability of
such a potential in terms of an actual molecular structure, one could try to expand
the additional potential in terms of atom-based potentials as introduced by Beratan
and coworkers [248].
Chapter 6
Towards Inverse Design of
Molecular Vibrational Properties
“In physics, you don’t have to go
around making trouble for
yourself — nature does it for you.”
F. Wilczek
Besides the concept of gradient-driven molecule construction, which has been explained in the last chapter, there are also other promising approaches in inverse
quantum chemistry. As laid out in chapter 3, subspace iteration methods like Modeand Intensity-Tracking provide an efficient algorithmic framework for the iterative
solution of inverse problems (in the case of Mode- and Intensity-Tracking, this
is the design of molecular vibrations). The further development of these methods
deserves special attention — eventually allowing them to be utilized for the design
of new molecular compounds. However, for such investigations, accurate reference
calculations are needed. In this chapter, we shall present possible reference calculations, namely Raman optical acitivity spectra of large protein β-sheet models.
We will also present a detailed analysis of the calculated spectra, which allows
us to gain considerable insight into the Raman optical activity spectroscopy of
such β-sheets. This chapter serves also to illustrate the capabilities of the software
package M OV I PAC, which is developed by our research group and documented in
appendix D. M OV I PAC unites several programs for the calculation of vibrational
spectra, focussing especially on large molecules.
6.1
Finding Raman Optical Activity Signatures of
Protein β-Sheets
Together with α-helices, β-sheets belong to the most important and well-known
elements of protein secondary structure. While X-ray crystallography and nuclear
magnetic resonance (NMR) allow one to elucidate the overall structure of a protein
at an atomic level, electronic circular dichroism (CD) spectra provide a simple way
116
Towards Inverse Design of Molecular Vibrational Properties
to determine the amount of different secondary structure elements present in a given
protein [412]. Another successful chiroptical technique in this respect is Raman
optical activity (ROA) [413], which measures the difference of Raman scattering
intensity of right and left circularly polarized light (i.e., it is the chiral variant of
Raman spectroscopy).
Over the past years, both experimental and theoretical studies have established
characteristic ROA signatures for many types of secondary structure elements
[414–416]. In experimental studies, the identification of such signatures is usually based on the comparison of spectra of proteins featuring the same as well as
different secondary structure elements. However, most proteins have a rich secondary structure, such that it is difficult to unequivocally establish signatures for
the individual secondary structure elements. As an additional challenge, the fact
that the normal modes are not directly accessible in experimental studies makes it
often very difficult (if not impossible) to assign a given spectral band to a certain
vibration. Furthermore, the different factors contributing to the overall spectrum
(e.g., molecular structure, conformational dynamics and averaging, solvent effects,
etc.) are very difficult to separate experimentally. Here, a theoretical approach can
provide additional insight, as one is basically not restricted in the choice of model
systems. It is thus possible to study a polypeptide featuring only the secondary
structure element of interest, which makes the identification of characteristic signatures considerably simpler, although it does not guarantee that such a characteristic
spectral signature can be detected in a crowded spectrum.
Our group has carried out theoretical studies on the signatures of α-helices [417],
310 -helices [417], and β-turns [418, 419], but no theoretical study has addressed
the ROA signatures of β-sheets yet. It is the aim of this work to fill this gap. To
this end, we have calculated theoretical ROA spectra for various large models of
parallel as well as antiparallel β-sheets. A detailed analysis of these, together with
a range of experimental spectra published in the literature, should provide us with
characteristic signatures of protein β-sheets.
We first review the signatures which have already been proposed to be indicative
of β-sheets, based on experimental spectra, in section 6.2. Then, in section 6.3 we
explain our computational procedure as well as the different β-sheet structures we
have chosen to investigate. After this, we present the theoretical ROA spectra of
these model systems and analyze them in section 6.4. Finally, we give a conclusion
together with a short outlook in section 6.5.
6.2
Proposed Signatures of β-Sheets
The first experimental study on ROA signatures of β-sheets was conducted in
1994 by Barron and coworkers [420]. They proposed that positive ROA intensity
between ∼1000 – 1060 cm−1 originates from β-sheet structures. This assignment
is still assumed to hold true [414]. However, the experimental spectrum of αhelical poly(L-alanine) in dichloroacetic acid clearly also shows a positive peak at
∼1047 cm−1 (if the solvent is changed to only 30% dichloroacetic acid and 70%
chloroform, this peak is found at ∼1044 cm−1 ) [421]. We should also note here that
6.2 Proposed Signatures of β-Sheets
117
Barron et al. explicitly state that this spectral feature of β-sheets can greatly vary
in intensity and wavenumber range [414], which might reflect structural variations
within the strands.
Furthermore, Barron and coworkers also observed a sharp positive peak at
∼1313 cm−1 , which they assigned to an antiparallel β-sheet structure. Five years
later, the same group observed similar signals in ROA spectra of bovine βlactoglobulin, a protein which is rich in antiparallel β-sheet [422]. However, adjacent
strands of antiparallel β-sheets are connected by β-turns, and it is thus not a priori clear whether this signal arises from the strands or from the turn structures.
In this context it is important to note that our group recently found a similar
band in small β-turn model systems [419]. Moreover, in experimental spectra of
disordered poly(L-lysine) and poly(L-glutamic acid), a positive signal is observed
at ∼1320 cm−1 [414]. It is therefore very likely that a positive signal around
∼1310 – 1320 cm−1 is characteristic for turn structures rather than β-strands.
Finally, Barron and coworkers found a characteristic couplet, negative at lower
and positive at higher wavenumbers and centered around ∼1650 – 1670 cm−1 to be
indicative of β-sheets [420]. This assignment has been confirmed multiple times
since then [414,422–424]. Note that the α-helix shows a similar couplet, which is,
however, shifted to lower wavenumbers by ∼5 – 20 cm−1 [414, 417].
As another signature of β-sheets, Blanch et al. proposed a negative band at
∼1248 cm−1 [422]. In fact, a similar negative peak was observed between ∼1244
and ∼1253 cm−1 in a range of other proteins incorporating β-sheets [414,423,424],
such that this signature may also be regarded as being firmly established.
We should also note here that intermittently, a negative band around ∼1220 cm−1
was thought to be a signature for β-sheets as well [414,422,423], even though it was
originally assigned to β-turns [420]. A detailed analysis of the experimental spectra
published so far shows that there is no justification for this reassignment [419],
which is further confirmed by our theoretical study on model β-turns, which all
show negative ROA intensity at ∼1190 cm−1 [419].
Table 6.1: Summary of ROA signatures proposed in the literature to be characteristic of
β-sheet.
signature
wavenumber range / cm−1
ROA intensity
1
2
3
∼1000 – 1060
∼1240 – 1255
∼1650 – 1670
positive
negative
−/+ couplet
In summary, based on an analysis of the experimental spectra published so far,
there are two signatures which can be regarded as being reliable characteristics
of β-sheets. First, negative intensity between roughly 1240 and 1255 cm−1 , and
a −/+ couplet centered around ∼1650 – 1670 cm−1 . Furthermore, positive ROA
intensity in the range of ∼1000 – 1060 cm−1 has also been found in many spectra
of proteins featuring β-sheets, although we should recall that this signature is
118
Towards Inverse Design of Molecular Vibrational Properties
quite variable [414], and does also occur in experimental spectra of α-helical
polypeptides [421]. All relevant signals are summarized in table 6.1.
6.3
6.3.1
Computational Methodology
Model Structures
In β-sheets, several almost fully extended strands of the protein polypeptide chain are
held together by hydrogen bonds. As in the case of the α-helix, all possible backbone
hydrogen bonds are formed. One distinguishes two main forms, in which β-sheets
occur in proteins; these are denoted as parallel and antiparallel β-sheets [425]. The
two forms feature distinct hydrogen bonding patterns, as shown schematically in
Fig. 6.1.
O
a)
R
O
H
N
R
H
N
N
O
R
NH
N
H
O
O
H
R
R
H
O
N
R
R
N
N
H
N
H
R
O
b)
O
H
O
R
R
O
O
R
H
N
H
N
N
R
N
H
O
O
R
R
H
H
O
O
R
H
N
N
N
H
R
N
H
O
R
O
Figure 6.1: Schematical Lewis structures of a section of two adjacent strands of antiparallel
(a) and parallel (b) β-sheets. The two strands of the antiparallel β-sheet are connected
by a β-turn, which is drawn in red. The dashed lines at the end of the polypeptide chain
denote the continuation of the protein backbone.
While the hydrogen bonds are spaced evenly in the parallel β-sheet, there is an
alternation of narrower and wider spaced hydrogen bond pairs in the antiparallel β-
6.3 Computational Methodology
119
sheet. Furthermore, the hydrogen bonds between adjacent strands of an antiparallel
β-sheet are almost parallel to each other, while they form significant angles in the
case of parallel β-sheets.
Figure 6.2: Model structures investigated in this study: a), antiparallel β-sheet, the two
strands of which are connected by a β-turn; b), the same antiparallel sheet with the turn
removed; c), parallel β-sheet.
For our study, we constructed idealized models of both, parallel and antiparallel
β-sheets. All models consist of two strands. This will allow us to reproduce generic
signatures of β-sheets while still maintaining a reasonable size of the models (note
also that antiparallel β-sheets are often observed as a twisted ribbon of just two
strands [425]). In analogy to a previous study [417], we employed (S)-alanine as
the only residue type, which is the simplest chiral amino acid. In the case of the
antiparallel β-sheet, we investigate two models: in the first one, the two strands are
connected by a β-turn of type I, whereas we removed the four amino acid residues
participating in this turn in a second model. This allows us to study the influence
of the turn structure on the overall ROA spectrum. For the parallel β-sheet model,
the individual strands cannot be connected by short turns, but only by longer loops.
These three model systems are depicted in Fig. 6.2. The antiparallel β-sheet with
120
Towards Inverse Design of Molecular Vibrational Properties
turn consists of a total of 20 (S)-alanine residues, while the other two models feature
only 16 amino acids. At their N-terminus, the polypeptides are terminated with
an additional hydrogen atom, leading to an NH2 group, while at the C-terminus,
the carboxy group was replaced by an acetyl moiety. These terminal groups will
be associated with vibrations that are not found in β-sheets found in proteins. In
particular, we will see some artifacts in the amide I region, which, however, have
no adverse effect on our investigations.
In order to further analyze ROA signatures of β-sheets, we also set up a second
series of model systems, depicted in Fig. 6.3. We introduced perturbations to the
idealized β-sheet models of Fig. 6.2; in one case, we change either one or all
amino acid residues from (S)-alanine to glycine (Fig. 6.3 a, b), while in the second
case, we study microsolvation by adding one and two water molecules to one and
two peptide units, respectively (Fig. 6.3 c). Finally, we also consider the typical
twist found in most β-sheets in order to study the robustness of the signatures
with respect to large-scale conformational changes. To this end, we extracted two
strands of antiparallel β-sheet from concanavalin A (residues 60 – 66 and 73 – 79).
In order to guarantee consistency with the rest of our study, we changed all amino
acids to (S)-alanine in this model system (see Fig. 6.3 d) and optimized the whole
model in an unconstrained optimization.
6.3.2
Computational Details
All structures were fully optimized (i.e., without any constraints) with the BP86
exchange–correlation functional [81, 82] and Ahlrichs’ valence triple-zeta basis set
with one set of polarization functions [292] (dubbed def-TZVP) on all atoms as
implemented in the T URBOMOLE program package (version 5.10) [117]. Furthermore, in all calculations, advantage was taken of the resolution-of-the-identity (RI)
approximation with the auxiliary basis sets corresponding to def-TZVP. During
the structure optimization, electronic energies were converged to 10−6 a.u., while
the maximum norm of the Cartesian electronic energy gradient was converged to
10−4 a.u. It was made sure by vibrational analysis that all structures correspond to
true minima on the potential energy surface. The harmonic wavenumbers, normal
modes and the derivatives of the polarizability tensors necessary for the evaluation
of the ROA backscattering intensity were computed in a semi-numerical fashion
using the program S NF [275] of the package M OV I PAC [336], employing a stepsize of 0.01 Bohr. For the individual distorted structures, analytical energy gradients
and polarizability tensors were computed with our local version of T URBOMOLE’s
ESCF module [426], utilizing the same density functional and basis sets as described
above. To avert too large errors resulting from the numerical differentiation procedure, electronic energies were converged to 10−8 a.u. for the distorted structures,
using the tight m4 grid of T URBOMOLE. The vibrational frequencies obtained
were not scaled since it is known that BP86 due to some error cancellation yields
harmonic frequencies which are in good agreement with measured fundamental
ones [102, 337]. The velocity representation of the electric dipole operator was
applied in order to ensure gauge invariance.
6.3 Computational Methodology
121
Figure 6.3: Second set of model systems studied: a), parallel β-sheet with one residue
changed to glycine; b), antiparallel β-sheet with all residues changed to glycine; c),
parallel β-sheet with one and two amino-acid residues microsolvated by one and two
water molecules, respectively; d) twisted antiparallel β-sheet, extracted from concanavalin
A.
122
Towards Inverse Design of Molecular Vibrational Properties
The ROA backscattering intensities were obtained for a wavelength of 799 nm and
it was ensured in all cases that the energy corresponding to this wavelength is far
below any electronic excitations. The line spectra resulting from this methodology
were finally broadened by means of a convolution with a Lorentzian line shape
featuring a full width at half maximum (FWHM) of 15 cm−1 . This line broadening
as well as the analysis and graphical representation of all spectra were produced
with the program M ATHEMATICA (version 7.0) [427].
6.4
6.4.1
Results and Discussion
Proposed Signatures
We first investigate the ROA spectra of the three idealized model systems of
antiparallel and parallel β-sheets (Fig. 6.2) for the existence of the signatures
proposed in the literature. The three ROA spectra are shown in Fig. 6.4 with the
three regions where signatures are expected are highlighted in blue. Concerning
the first signature (positive intensity between ∼1000 and 1060 cm−1 ), we identify
a weak positive peak in the ROA spectrum of the antiparallel β-sheet containing a
turn while the other two spectra do not show positive ROA intensity in this range.
However, in these latter cases, there are always several normal modes associated
with positive intensity in the relevant spectral range, but they are always annihilated
by near-lying normal modes which are associated with negative ROA intensity. This
signature thus appears not to be a sufficiently reliable general signature for β-sheets,
which is in line with the experimental observation that it varies in both intensity
and wavenumber range [414].
The second signature, namely a positive peak between ∼1240 and 1255 cm−1 ,
cannot be identified in any of the three spectra in Fig. 6.4. However, this signature
is found in all experimental spectra published so far, except in the spectra of
poly(L-lysine) in water at 50◦ C, which also features a β-sheet conformation [423].
In this spectrum, one can only see a broad negative peak at 1218 cm−1 , which rises
to a strong positive peak at 1260 cm−1 . It might thus be that also in this spectrum,
there could actually be a somewhat weaker negative band at ∼1240 cm−1 , which
is simply annihilated by the negative peak nearby. Hence, we are left with a clear
contradiction of experimental and theoretical results. For this to resolve, we need
to investigate the calculated results in more detail. In fact, further investigations
(see Section 6.4.4) give some evidence that this signature could be dependent on
the amino acid side chains.
Finally, a −/+ couplet in the amide I region is proposed to be characteristic for
β-turns. Indeed, we find such a couplet only somewhat above the proposed spectral
region, namely, centered around roughly 1700 cm−1 . Note that the strong positive
peak at ∼1650 cm−1 is partially an artifact of the model systems, as NH2 distortion
vibrations also occur in this wavenumber range. However, in a protein-embedded
β-sheet such an amino group would not be present. If we would discard the normal
modes associated with these distortion vibrations, this peak would be somewhat
less prominent, but still clearly visible (most of the normal modes in this region are
6.4 Results and Discussion
123
associated with vibrations delocalized over the entire turn structure). The normal
modes building up the characteristic couplet represent the typical amide I vibrations,
i.e., C–O stretching vibrations, with some admixture of N–H in-plane bending. All
vibrations are usually delocalized over the entire sheet structure, as can be seen
in the top panel of Fig. 6.5, which depicts a typical normal mode found in this
spectral range.
Figure 6.4: ROA spectra of the first set of (unperturbed) model structures shown in
Fig. 6.2. Highlighted in blue are the spectral regions in which characteristic signatures of
β-sheets have been proposed in the literature, whereas the red region marks a signature
identified in this work.
124
Towards Inverse Design of Molecular Vibrational Properties
Figure 6.5: Typical normal modes of characteristic signatures of β-sheets, taking the
parallel β-sheet model as an example. a), amide I vibration at 1676 cm−1 , which is
responsible for the negative part of the couplet in Fig. 6.4, bottom; b), mode leading to
the sharp negative peak at 1347 cm−1 in Fig. 6.4, bottom; c), CH2 deformation vibration
of the glycine residue responsible for the band at 1246 cm−1 in Fig. 6.6, middle.
6.4.2
Additional Signatures
When comparing the three spectra in Fig. 6.4, we note that they are very similar,
despite the fact that the three model systems locally feature quite different structural
elements. For example, one antiparallel sheet involves a β-turn, while the other
does not. However, the presence (or absence) of this turn cannot be unequivocally
be established simply based on the two ROA spectra. Therefore, we conclude that
the ROA signals stemming from the turn [419] are simply covered by the signals
arising from the rest of the β-sheet models.
Due to the similarity of the three spectra, we recognize many spectral regions
where all spectra show either positive or negative ROA intensity. However, not
6.4 Results and Discussion
125
all of these regions should be regarded as true signatures. To be considered as a
reliable signature, we should identify a significantly strong peak, the normal modes
of which are associated with the peptide backbone of the sheet structure. One peak
which is particularly strong and almost identical in all three spectra is the negative
band at ∼1350 cm−1 . It is the strongest peak in the spectra of the two antiparallel
β-sheet models, and, although somewhat weaker, also pronounced in the case of
the parallel β-sheet. Furthermore, the shape of this peak is the same in all spectra,
being constituted of only a few very close-lying normal modes. These normal modes
are associated with Cα –H bending and CH3 deformation vibrations (see Fig. 6.5,
middle). The spectral range of this characteristic signature is highlighted in red in
Fig. 6.4. It can also be found in all experimental spectra of β-sheet models and
proteins incorporating a significant amount of β-sheets [414,420,422,423], and can
thus be considered as being a reliable signature of β-sheets.
6.4.3
Robustness of β-Sheet Signatures
Of course, one important issue to clarify is how robust these signatures are. I.e.,
how strongly do they depend on the exact conformation of the β-sheet model,
the particular type of side chain employed and the environment such as solvent
molecules? To investigate robustness against perturbations, we study the second set
of model systems of Fig. 6.3.
6.4.4
Influence of Side Chain
In order to investigate to which extent the characteristic β-sheet signatures depend
on the actual amino acid side chain, we changed one single residue to glycine.
The resulting ROA spectrum is shown in the middle panel of Fig. 6.6.
When comparing this spectrum to the one of the original, all-(S)-alanine parallel
β-sheet (top panel of Fig. 6.6), we find that the two spectra are very similar, and all
signatures proposed above can be found in the spectrum of the modified β-sheet.
This is not very surprising, since the original structure is only slightly changed.
Nevertheless, there is one particular band which allows to unequivocally distinguish
the two spectra from each other, namely a negative peak at 1246 cm−1 (highlighted
in red in Fig. 6.6). This band is caused by a single normal mode, which represents
a bending vibration of the two hydrogen atoms of the glycine residue, coupled
with N–H in-plane bending motions of the two neighboring peptide units (bottom
panel of Fig. 6.5). Interestingly, this negative band lies exactly in the region where
Barron and coworkers proposed a signature of β-sheets, which, however, could not
have been found in our all-(S)-alanine model systems (see above).
To replace only a single amino-acid residue is certainly a small perturbation.
Clearly, we need to study a larger modification of the model-sheet primary structure.
Therefore, we changed all residues in the antiparallel β-sheet model from (S)-alanine
to glycine (Fig. 6.3 b). Since glycine is achiral, all ROA signals stem from the
chirality of the β-sheet structure and can thus be taken as an ultimate fingerprint
of a sheet structure. The resulting ROA spectrum is shown in the lower panel
of Fig. 6.6. As can be seen, the overall intensity of all signals is diminished,
126
Towards Inverse Design of Molecular Vibrational Properties
which is due to the fact that all elements of local chirality are eliminated for the
achiral amino acid. From Fig. 6.6 it can be concluded that resulting ROA spectrum
specifically shows the intrinsic features of the β-sheet structure. However, we can
expect that the overall chirality of the β-sheet structure dominates also the spectra
of the all-(S)-alanine model systems. In fact, for large helical systems it has been
found that the helical chirality dominates over the local chirality of the individual
amino acids [291], even though for smaller systems, the configuration of individual
residues plays an important role [428].
Figure 6.6: ROA spectra of the idealized parallel β-sheet consisting of (S)-alanine only
(top panel) and with one residue changed to glycine (middle panel); the bottom panel
shows the ROA spectrum of an antiparallel all-glycine β-sheet. The blue marks highlight
the characteristic signatures established in section 6.4.1, while the red mark shows the
spectral region with which the all-(S)-alanine β-sheet can be distinguished from both other
structures.
Concerning the β-sheet signatures, we see that both, the couplet in the amide I
region as well as the negative peak at ∼1350 cm−1 can clearly be recognized. In
6.4 Results and Discussion
127
addition, we also find negative intensity in the range between 1240 and 1255 cm−1 .
These observations can be considered as evidence that this latter signature depends
on the side chains within the sheet structure, being not present when only (S)alanine builds up the strands, but being already clearly visible when only a single
residue is changed to glycine. We should emphasize again that a negative peak
at ∼1250 cm−1 is not visible in the spectrum of a model β-sheet consisting of
(S)-lysine only (see above).
6.4.5
Twisted β-Sheets
The idealized structures of the β-sheet models investigated above is often not found
as such in protein structures. A characteristic structural distortion of protein-building
β-sheets is that their strands are usually twisted. In order to examine the effect
of such large-scale conformational changes, we extracted a twisted β-sheet from
the structure of the protein concanavalin A, and calculated its ROA spectrum (all
residues were changed to (S)-alanine and the structure was then fully optimized),
which is shown in the lower panel of Fig. 6.7.
Figure 6.7: ROA spectra of the idealized antiparallel β-sheet (top panel), and the same
structure in the twisted conformation depicted in Fig. 6.3 at the bottom.
We see that the two spectra in Fig. 6.7 are again quite similar from a global
perspective, but there are some specific spectral regions where the two spectra of
Fig. 6.7 differ significantly. For example, the characteristic −/+ couplet in the amide
I region is no longer visible in the spectrum of the twisted structure, but only the
positive part of it is. Nevertheless, also in this spectrum there are some normal
128
Towards Inverse Design of Molecular Vibrational Properties
modes which are associated with negative ROA intensity just below the positive
peak, but these modes cannot build up a negative band since they are canceled
by close-lying normal modes associated with positive backscattering intensity. The
strong negative peak at ∼1350 cm−1 , however, is clearly present in the spectrum
of the twisted β-sheet.
6.4.6
Microsolvation
Another important issue of concern is the influence of the environment as, e.g.,
exerted by solvent molecules. Unfortunately, a full-fledged solvation study — (as
carried out for a sugar molecule recently by Cheeseman et al. [429] — of the βsheet models is beyond the scope of this work and computationally very demanding.
However, we can selectively add only a few water molecules to the sheet structure
and study the resulting microsolvation effects. Surely, this will not yield a spectrum
resembling a fully solvated case, but it will allow us to understand solvation in
a stepwise fashion. In this work, we only consider adding one and two water
molecules to one and two peptide units, respectively (Fig. 6.3 c). The spectrum of
such microsolvated parallel β-sheet models are shown in Fig. 6.8 and compared
to the unsolvated counterpart (Fig. 6.8, top). Clearly, the addition of one and two
water molecules constitutes a very small perturbation of the overall structure. Still,
the local effect and its spread among the normal modes can provide insights into
the change of the ROA spectrum upon hydration.
For this, the analysis of the spectra in terms of localized modes is a suitable way
to dissect the microsolvation effects. The concept of localized modes [430, 431]
allows us to spatially localize selected vibrations, which are usually delocalized over
the entire polypeptide chain in the basis of normal modes. Mathematically, this is
achieved by a suitably chosen unitary transformation of the Hessian matrix (see
Ref. [430] for details on the approach and Refs. [336, 432] for publicly available
implementations). In our case, we localized all 16 amide I vibrations, i.e., all
normal modes between 1630 and 1720 cm−1 . The resulting coupling matrix, i.e.,
the Hessian matrix in the basis of localized modes, is shown graphically in Fig. 6.9.
In this representation, we color-code the value of the coupling constants (i.e., the
off-diagonal elements of the coupling matrix). When first investigating the coupling
matrix of the unsolvated β-sheet (see the left-hand side of Fig. 6.9), we see that all
adjacent residues within one strand (i.e., residues one to eight for the first strand,
and residues nine to 16 for the second strand), couple strongly to each other with
coupling constants of approximately 8 cm−1 . Interactions of a given residue with
its second-nearest neighbor are much less pronounced and only slightly positive.
All other couplings are almost zero, indicating that such long-range interactions can
be neglected. However, the two strands are held together by hydrogen bonds, and
accordingly there is a strong negative coupling constant between residues which
are bound by hydrogen bonds (cf., the off-diagonal blue squares in Fig. 6.9).
6.4 Results and Discussion
129
Figure 6.8: ROA spectra of the idealized parallel β-sheet (top panel), and the same
structure microsolvated with one and two water molecules, respectively (cf. Fig. 6.3 c).
We found that neglecting any coupling constants smaller than 2.5 cm−1 yields
vibrational frequencies in close agreement with the reference values from the full
coupling matrix, namely with a mean absolute deviation of only 1.2 cm−1 (see Table
6.2). As only the coupling between adjacent residues and residues which are bound
by hydrogen bonds is larger than 2.5 cm−1 , this shows that even the interactions
between second-nearest neighbor residues can be neglected without introducing a
large error.
Comparing the coupling pattern of the unsolvated β-sheet to the one of the
microsolvated β-sheet (top right-hand side of Fig. 6.9), we see that it does not
change much. The largest differences can be seen for the coupling of residue five
(the one which has been microsolvated) to its neighbors. Therefore, we can conclude
that the microsolvation has only a minor effect on the coupling pattern. Still, small
changes can have unexpected strong effects on the normal mode spectrum.
130
Towards Inverse Design of Molecular Vibrational Properties
Figure 6.9: Coupling matrices of the localized amide I vibrations of the unsolvated
parallel β-sheet (top left), and its microsolvated counterparts with one (top right) and
two water molecules (bottom). The colored squares represent the coupling constants,
ranging from −7 to 9 cm−1 , while the grayscale squares on the main diagonal of the
right coupling matrices represent the difference between the localized frequencies of the
solvated structures with the one of the unsolvated β-sheet (also given in wavenumbers).
Apart from the coupling constants, the localized frequencies (i.e., the main diagonal elements of the coupling matrix) are important for the normal mode spectrum.
In the coupling matrix of the microsolvated β-sheet (right-hand side of Fig. 6.9),
we show the difference between the localized frequencies of the solvated structure
and the one of the unsolvated β-sheet are shown in a grayscale. We see that the
localized frequencies do not change much between the two coupling matrices, with
the exception of the localized frequency of residue five, which is shifted by almost
30 cm−1 to lower wavenumbers in the solvated case.
An interesting question is whether this general pattern changes in a fundamental
fashion if more than one residue is microsolvated. In order to study this question,
we microsolvated also a second peptide unit. Note that the microsolvation of a
unit directly adjacent to the one already microsolvated is impossible, as the C–O
and N–H bonds of this peptide unit are oriented towards the inside of the β-sheet.
Therefore, we added a second water molecule to the peptide unit of residue three in
the same fashion as we treated residue five. The resulting coupling matrix is shown
at the bottom of Fig. 6.9. Again, the coupling pattern between the unsolvated and the
6.4 Results and Discussion
131
microsolvated case does not change dramatically. Also for the localized frequencies,
we make the same observation as above. All localized frequencies remain essentially
unchanged, except the one of the two residues which are microsolvated. These
frequencies are shifted by almost 30 cm−1 to lower wavenumbers.
Table 6.2: Comparison of frequencies obtained by neglecting any coupling between nonadjacent residues as well as between resdiues not bound by hydrogen bonds (so-called
strong-coupling frequencies) with their reference values. All values are given in cm−1 .
6.4.7
strong-coupling frequencies
reference frequencies
error
1638.5
1641.6
1645.6
1648.1
1648.8
1650.2
1654.8
1669.9
1671.8
1674.1
1674.1
1677.5
1680.1
1683.3
1708.7
1710.6
1641.8
1642.6
1644.8
1646.3
1647.3
1648.7
1653.4
1669.7
1672.3
1673.2
1675.8
1676.9
1680.3
1684.8
1707.3
1712.2
3.35
1.07
0.77
1.75
1.42
1.52
1.38
0.15
0.45
0.91
1.64
0.54
0.18
1.45
1.39
1.67
Comparison to Other Secondary Structure Elements
After having established the −/+ couplet in the amide I region as well as the
sharp negative peak at about 1350 cm−1 to be truly characteristic for β-sheets, we
shall now investigate the ROA spectra of other secondary structure elements for
an interference of these signatures with vibrations characteristic for, e.g., helices.
We have already carried out theoretical studies on α- and 310 -helices [417] (see
also Refs. [433, 434]), as well as on β-turns [419]. The resulting ROA spectra
are reproduced in Fig. 6.10, together with the spectrum of the antiparallel β-sheet
model. Again, the two β-sheet signatures mentioned above are highlighted in blue.
We can see that these signatures can well serve to distinguish the β-sheet from the
other secondary structure elements. The sharp negative peak at ∼1350 cm−1 does
not exist in the spectra of the helical polypeptides. Also in the case of the small
β-turn, such a peak is not visible, even though there is a single normal mode which
is associated with negative ROA intensity in this spectral region. Furthermore, all
four secondary structure elements can be distinguished from each other by relying
on the band shape of the amide I region. In the case of the β-turn, we find
132
Towards Inverse Design of Molecular Vibrational Properties
only positive ROA intensity in this spectral range, while the 310 -helix produces
a characteristic +/− couplet. Both α-helix as well as (parallel and antiparallel)
β-sheets feature a −/+ couplet in the amide I region, but also these two secondary
structure elements can be distinguished from each other, as the couplet is shifted
approximately 20 cm−1 to lower wavenumbers in the case of the α-helical model
system (upon solvation, it is shifted to even lower wavenumbers [434]). Note that
this very finding is also experimentally confirmed [414].
6.4.8
Differentiating Parallel and Antiparallel β-Sheets
The discrimination of parallel and antiparallel β-sheet is a very important topic
which has already been addressed in infrared and vibrational circular difference
spectroscopy [435–437]. It is thus highly interesting to analyze the data presented
in this work in order to investigate whether one can distinguish these two types of
secondary structure element with ROA spectroscopy.
It is known that the extended amide III region, i.e., the spectral region between
roughly 1200 and 1400 cm−1 is especially sensitive to secondary structure [438].
Therefore, it is not very surprising that we find a difference between parallel and
antiparallel β-sheets exactly in this region. In fact, when closely analyzing the
spectra depicted in Figs. 6.4, 6.6, 6.7, and 6.8, we see that all parallel β-sheets
exhibit only positive ROA intensity around ∼1275 cm−1 , while we find weakly
negative peaks for all antiparallel β-sheet structures. All the small perturbations
studied above (different residue(s), twisted conformation, solvent molecule) have
no influence on this spectral feature. Therefore, it appears to be quite robust and
would thus be very well suited to differentiate parallel from antiparallel β-sheet
structure. When analyzing the experimental ROA spectra of jack bean concanavalin
A, human immunoglobulin G, and bovine β-lactoglobulin (which all incorporate a
significant amount of antiparallel β-sheet), we identify a characteristic ∼1250 cm−1 ,
which might correspond to the feature we proposed above [414].
However, the spectrum of rabbit aldolase apparently shows a negative peak at
∼1253 cm−1 [414], and it has been suggested that this band originates in the
parallel β-sheet structure found in this protein. Therefore, further investigations
are necessary to clarify whether the negative peak at ∼1275 cm−1 can be used as
reliable indicator of antiparallel β-sheets.
Apart from this feature in the amide III region, we find one additional difference
between the spectra of parallel and antiparallel sheets, namely a peak at ∼1525 cm−1 .
This peak is negative in the case of parallel β-sheets, and positive for antiparallel
sheets. However, this feature appears not to be very reliable. It vanishes when we
add a single water molecule to the parallel β-sheet (cf., Fig. 6.8). Moreover, the
spectrum of the antiparallel all-glycine β-sheet shows only (weak) negative ROA
intensity in this range, and not a positive peak as expected. Therefore, this feature
is not consistent across our entire data, and one should not regard it as a reliable
means to differentiate generic parallel β-sheet from antiparallel β-sheet structures.
6.4 Results and Discussion
133
Figure 6.10: Comparison of the ROA spectra of a model α-helix (top panel), the
antiparallel β-sheet model investigated in this work, a type I β-turn, and a 310 -helix
(bottom panel; all structures composed from (S)-alanine residues). The data for the helical
systems have been reproduced according to Ref. [417] and those for the β-turn according
to Ref. [419].
134
6.5
Towards Inverse Design of Molecular Vibrational Properties
Summary
We have calculated ROA spectra of a range of antiparallel and parallel β-sheet
model polypeptides in search for characteristic signatures. The stability of these
fingerprints against structural and electronic perturbations has been investigated for
(1) a replacement of a single (S)-alanine residue of one of the original all-(S)alanine sheets by a glycine residue, (2) a replacement of all (S)-alanine residues
by glycine, which allowed us to study the β-sheet specific ROA features, (3) the
addition of a single water molecule, and (4) the twisting of the sheet structure.
From the three signatures which have been proposed so far in the literature to be
indicative for sheet structures (see Table 6.1), the first signature — positive intensity
between ∼1000 and 1060 cm−1 — could not be found in our theoretical spectra.
This is in accord with experimental findings [414]. A second signature, namely a
negative peak between ∼1240 and 1255 cm−1 , cannot be found in the all-(S)-alanine
model systems, but is clearly visible when glycine is present in the sheet structure.
Therefore, our work gives some evidence that this signature is rather dependent
on the amino acid side chains. The third signature, a −/+ couplet in the amide I
region, could be fully confirmed in this study. This couplet is also particularly well
suited to distinguish the β-sheet structure from other secondary structure elements,
such as α- and 310 -helices.
In addition, we were able to propose another signature characteristic for parallel
as well as antiparallel β-sheets, namely a sharp negative peak at approximately
1350 cm−1 . This band is very robust with respect to large-scale conformational
changes of the sheet structure and can also be identified in all experimental spectra
of β-sheet structures (or proteins incorporating a significant amount of this secondary
structure element).
In order to obtain transferable knowledge on microsolvation effects, we investigated the addition of a single water molecule. While the resulting ROA spectrum is
almost identical to the unsolvated reference, the concept of localized modes allowed
us to gain additional insight. For the amide I region, we found that the coupling
pattern does not change significantly upon microsolvation. Adjacent amino acids
couple quite strongly, while interactions between second-nearest neighbors are already much weaker, and any further interactions can be neglected (except residues
which are bound by hydrogen bonds). Hydrogen bonds lead also to a significant
coupling of amino acids, which are, however, negative (as opposed to the coupling
within one strand). The only significant change introduced by the microsolvation
is the localized frequency of the solvated residue, which is shifted almost 30 cm−1
to lower wavenumbers. We should explicitly state here that the addition of a single
water molecule, although providing valuable insight, cannot account for all effects
of a full solvation. Therefore, further analysis of extended microsolvation patterns
is required for a complete understanding of the changes in the coupling matrix,
such that the complete ROA spectrum of a solvated polypeptide chain could eventually be estimated from its unsolvated reference case. This would provide access
to solvation effects at much less computational cost than required for a full-fledged
QM/MM approach [429]. Such work is currently in progress in our laboratory.
Finally, we analyzed our spectra in order to investigate whether parallel and
6.5 Summary
135
antiparallel β-sheets can be distinguished from each other by means of ROA features.
We only found one spectral feature which allows for a constant discrimination of all
our parallel β-sheet models from their antiparallel counterparts. This is a small peak
at ∼1275 cm−1 , i.e., in the amide III region, which is negative for antiparallel βsheets, but positive for parallel β-sheet structures. However, we need to emphasize
that the validity of this spectral feature needs to be established by further theoretical
and experimental studies.
Chapter 7
Conclusions and Outlook
“Pessimism is not a healthy thing in
science, but neither is unrealistic
optimism.”
N. Rashevsky
Inverse quantum chemical approaches aim at deducing a molecular structure compatible with a given predefined target property. During the last decades, different
inverse approaches have been developed, such as the inverse perturbation analysis,
the usage of model equations, and the optimization of wave functions. The most
successful contemporary inverse approaches (the inverse band structure approach,
the linear combination of atomic potentials, and the usage of alchemical potentials)
are essentially very sophisticated sampling and optimization schemes, as they do
not truly invert the Schrödinger equation. With the further development of ever
better optimization algorithms and computer hardware, one can expect that such
methods are successfully applied to increasingly complex problems.
In this work, we proposed a new concept for the rational design of molecular compounds, namely gradient-driven molecule construction. Starting from a
predefined central fragment, one searches for a ligand sphere stabilizing this fragment, i.e., leading to a vanishing geometric gradient on all atoms. Mathematically,
this (unknown) ligand sphere can be represented as a jacket potential entering the
Hamiltonian of the central fragment. We presented a working equation for the determination of this jacket potential, but the actual solution of it is not immediately
obvious. However, one should be able to solve it with iterative numeric methods,
and further research efforts should certainly focus on this aspect.
Apart from an exact determination of the jacket potential, one can try to approximate it by a range of different methodologies. In a first approach, one can directly
add atomic nuclei and electrons to the central fragment and optimize the spatial
position and the type of the nuclei as well as the overall charge and spin state of the
entire molecule. This extremely complex optimization problem can be simplified
by building the ligand sphere in a stepwise fashion. Following this approach, we
presented first steps towards the rational design of a new transition metal catalyst
for the homogeneous fixation of molecular dinitrogen. The automatization of this
approach is important to further explore its limitations.
138
Conclusions and Outlook
The jacket potential may also be represented as a collection of point charges. If
enough point charges are employed, one can expect to find a charge distribution from
which the ligand sphere could eventually be reconstructed. However, this would
require thousands of point charges, and, concomittantly, the resulting optimization
problem is highly complex. We have shown, however, that the overall geometric
gradient can be significantly decreased already with a comparatively small number
of point charges. Unfortunately, the reconstruction of a ligand sphere corresponding
to such an assembly of point charges is not straightforward, although the concept
of electronegativity can prove useful in this respect.
Finally, we investigated the possibility of representing the jacket potential on
a grid, corresponding to a numerical solution of the working equation mentioned
above. In density functional theory, the exchange–correlation energy is usually
integrated numerically, and concomittantly, a (quadrature) grid is already present
in these calculations. We have therefore tried to find a potential, discretized on
the DFT grid, minimizing the geometric gradient on the central fragment. Even
after several weeks of computational time, we could not achieve a solution for the
Mo–N2 fragment employed in our previous studies, since the optimization problem
is too complex in this case (already very coarse DFT grids include about 7000
grid points). Therefore, we turned to a simpler model system, namely H2 O where
the oxygen atom has been removed. The central frgament is thus stretched H2 ,
and the geometric gradient can be nullified by adding an oxygen atom at the right
position. This simple model system will be of great utility in future studies. With
this model system, we could find a jacket potential leading to a smaller geometric
gradient, but it is not immediately obvious how it could be related to an actual
ligand sphere.
This raises the concern that the solution for the jacket potential is not unique,
and that solutions may exist which cannot be represented by an assembly of atomic
nuclei and electrons. In order to solve this problem, one could follow an approach
proposed by Beratan and coworkers, namely the linear combination of atom-centered
potentials. By constructing special potentials representing a given atom (or even
an entire functional group), and then expanding the jacket potential in the basis of
these potentials, one necessarily obtains a potential which can be represented by a
molecular framework.
Only DFT is currently able to describe large transition metal complexes (such
as the Schrock catalyst for nitrogen fixation) at acceptable computational costs.
However, the accuracy of a given density functional for a certain molecule can a
priori not be known. Therefore, benchmark studies are important to assess the performance of exchange–correlation functionals. Unfortunately, reliable reference data
for large transition metal complexes are hardly available. We therefore assembled
the new WCCR10 dataset containing ten ligand dissociation reactions of different
large transition metal compounds for which accurate experimental gas-phase data
have been reported. Then, we assessed the performance of nine popular density
functionals, namely BP86, BP86-D3, B3LYP, B3LYP-D3, B97-D-D2, PBE, PBE0,
TPSS, and TPSSh, and found that the nonempirical functionals PBE and TPSS
perform very good, which is appealing from a theoretical point of view. The admixture of Hartree–Fock exchange to these two functionals further improves their
139
performance; PBE0 has the smallest mean absolute deviation with 22 kJ mol−1 and
a largest absolute deviation of 51 kJ mol−1 . These values lie well within the range of
chemically important energies, and therefore, the accuracy of contemporary density
functionals often cannot be expected to be sufficient for a given application. It is
therefore of utmost importance to further analyze the density functionals available
today in order to find possible cures to their weaknesses.
Focussing on BP86, we investigated the dependence of the WCCR10 reaction
energies on the two empirical parameters present in BP86. All ten reaction energies
depend qualitatively in the same way on the two parameters, even though the
complexes studied are rather different from a chemical point of view. Within the
parameter intervals analyzed, the ligand dissociation energies span a wide range.
With the data of the WCCR10 set, one can reoptimize the parameters of BP86,
However, this decreases the overall error of BP86 only by 2.5 kJ mol−1 (measured
on the WCCR10 set), which is due to the fact that the optimal parameter values are
rather different for different reactions. This suggests that the parameters of BP86
are not system-independent.
In the last part of this thesis, we presented vibrational Raman optical activity
calculations on large model systems of protein β-sheets. We could confirm the
experimental finding that a −/+ couplet in the amide I region is characteristic
for β-sheets. This signature is particularly well suited for the discrimination of
the β-sheet structure from other secondary structure elements. A second signature
proposed based on experimental results, namely a negative peak between ∼1240
and 1255 cm−1 , cannot be found in the all-(S)-alanine model systems, but is clearly
visible when glycine is present in the sheet structure. Therefore, our analysis gives
some evidence that this signature is rather dependent on the amino acid side chains.
In addition, we were able to propose another signature characteristic for parallel
as well as antiparallel β-sheets, namely a sharp negative peak at approximately
1350 cm−1 . The robustness of all these signatures has been tested with respect to
the exchange of a single amino acid residue, large-scale conformational changes,
as well as microsolvation effects.
Microsolvation effects have been studied by adding one and two water molecules
to one and two peptide units, respectively. Such an approach cannot account for
all effects of a full solvation, but it allows us to gain valuable insight into the
solvation procedure. With an analysis of the amide I region in terms of localized
modes, we found that the coupling pattern of the localized modes does not change
significantly upon microsolvation, but the localized C–O stretching frequency of
the microsolvated residue is shifted by roughly 30 cm−1 to lower wavenumbers.
Finally, we analyzed the theoretical spectra in order to find features allowing the
discrimination of parallel and antiparallel β-sheets. It was found that a peak in the
extended amide III region, namely at ∼1275 cm−1 allows for a reliable discrimination
of all analyzed spectra. This peak is negative for antiparallel sheets, but exhibits
positive ROA intensity in the case of parallel β-sheets. Further theoretical as well
as experimental work should focus on testing the validity of this signature.
These vibrational calculations do not only provide additional insight into the vibrational spectroscopy of β-sheets, but exemplify also the capabilities of M OV I PAC,
a software package for the massively parallel calculation of vibrational spectra. This
140
Conclusions and Outlook
software package has been constantly further developed in this research group, and
focusses especially on vibrational calculations for large molecules. With Modeand Intensity-Tracking, it includes also subspace iteration methods for the selective
calculation of specific spectral features. They demonstrate that such subspace iteration methods are a feasible way to iteratively solve an inverse problem — in this
case, to optimize normal modes from pre-defined molecular distortions. The calculations presented in the last part of this work can be used as reference data in the
further development of approaches for the inverse design of molecular vibrational
properties.
Appendix A
List of Important Abbreviations
Table A.1: List of important abbreviations used in this thesis.
abbreviation
meaning
DFT
ECP
GdMC
GGA
HIPT
HF
IPA
LAD
LCAP
LDA
MAD
MDC
PES
QED
QSAR
QSPR
RMSD
ROA
RPA
ZORA
ZPE
Density functional theory
Effective core potential
Gradient-driven molecule construction
Generalized gradient approximation
Hexa-iso-propyl terphenyl
Hartree–Fock
Inverse perturbation analysis
Largest absolute deviation
Linear combination of atomic potentials
Local density approximation
Mean absolute deviation
Multipole-derived charge
Potential energy surface
Quantum electrodynamics
Quantitative Structure–Activity Relationship
Quantitative Structure–Property Relationship
Root mean square deviation
Raman optical activity
Random phase approximation
Zero-order regular approximation
Zero-point energy
Appendix B
List of Important Symbols
Table B.1: List of important symbols used in this thesis.
symbol
Fundamental constants
c
0
h
~
i
Other symbols
A
α(i)
β
∆i
dr i
E
Eel
Exc
Φi
φ
ϕi
gi,j
H
Hel
H (m)
H (nr)
hi
mi
N
∇i
ν(r)
meaning
speed of light in vacuum
dielectric constant of the vacuum
Planck constant
h
reduced Planck constant,
~ = 2π
√
imaginary unit, i = −1
vector potential
i-th Dirac α-matrix
Dirac β-matrix
∇2i
infinitesimal volume element around r i
total energy of a system
electronic energy
exchange–correlation energy
i-th eigenfunction of an operator O
scalar potential
i-th orbital
quasi-relativistic or nonrelativistic two-electron operator
total Hamiltonian operator
nonrelativistic electronic Hamiltonian
mass-weighted Hessian matrix
total nonrelativistic Hamiltonian
nonrelativistic one-electron operator
mass of particle i
total number of particles
vector of all spatial first partial derivatives of particle i
one-particle potential energy term
Continued on next page
144
List of Important Symbols
Table B.1 – continued from previous page
symbol
meaning
hOi
pi
Ψ
Ψel
q
ri
ρ
σi
V
Zi
expectation value of operator O
momentum of particle i
total state function
electronic state function
(general) charge
spatial location of particle i
electron density
i-th Pauli spin matrix
(general) potential energy or volume
charge of nucleus i
Appendix C
List of Publications
The following publications have already originated from this work. They are reproduced in this thesis with permission from the respective publisher. The numbers in
parentheses give the sections in this thesis, in which (a part of) a given publication
is reproduced.
• T. Weymuth, M. Reiher, Characteristic Raman Optical Activity Signatures of
Protein β-Sheets, J. Phys. Chem. B, 2013, 117, 11943. (6)
• T. Weymuth, M. Reiher, Toward an Inverse Approach for the Design of
Small-Molecule Fixating Catalysts, MRS Proceedings, 2013, 1524, DOI:
10.1557/opl.2012.1764. (5)
• T. Weymuth, M. P. Haag, K. Kiewisch, S. Luber, S. Schenk, Ch. R. Jacob,
C. Herrmann, J. Neugebauer, M. Reiher, M OV I PAC: Vibrational Spectroscopy
with a Robust Meta-Program for Massively Parallel Standard and Inverse
Calculations, J. Comput. Chem., 2012, 33, 2186. (3.7, D)
• M. Podewitz, T. Weymuth, M. Reiher, Density Functional Theory for Transition Metal Chemistry: The Case of a Water-Splitting Ruthenium Cluster,
in Modeling of Molecular Properties, edited by P. Comba, 137. Wiley-VCH
Verlag GmbH & Co. KGaA, 2011. (2.5.3)
Additional publications:
• B. Simmen, T. Weymuth, M. Reiher, How Many Chiral Centers Can Raman
Optical Activity Spectroscopy Distinguish in a Molecule?, J. Phys. Chem. A,
2012, 116, 5410.
• T. Weymuth, Ch. R. Jacob, M. Reiher, Identifying Protein β-Turns with
Vibrational Raman Optical Activity, ChemPhysChem, 2011, 12, 1165.
• T. Weymuth, Ch. R. Jacob, M. Reiher, A Local-Mode Model for Understanding
the Dependence of the Extended Amide III Vibrations on Protein Secondary
Structure, J. Phys. Chem. B, 2010, 114, 10649.
Appendix D
Standard and Inverse Vibrational
Calculations with M OV I PAC
D.1
Introduction into M OV I PAC
Theoretical vibrational spectroscopy on large molecules usually relies on the harmonic approximation to the potential energy surface [50], i.e., one approximates
in the nuclear Schrödinger equation (within the Born–Oppenheimer approximation)
the total electronic energy as a quadratic potential (see also section 2.4.2). The
equation to solve is then
1 (m)† (m)
(m)†
(m) (m)
∇ +Q
H Q
χtot = Etot χtot ,
(D.1)
− ∇
2
where χtot is the total nuclear wave function, Etot its associated energy, the vector
∇(m) contains the mass-weighted first derivatives with respect to the nuclear coordinates, Q(m) are the mass-weighted normal modes, and H (m) is the mass-weighted
Hessian matrix, the elements of which are defined as
!
(m)
2
∂
E
(Q
)
el
(m)
Hi,j =
,
(D.2)
(m) (m)
∂Qi Qj
eq
in which Eel is the total electronic energy and we explicitly stated the parametric
dependence of the latter on the nuclear coordinates. The subscript “eq” indicates
that the second derivative has to be taken at an equilibirum geometry, i.e., at a
stationary point on the potential energy surface.
More accurate methods for the variational calculation of molecular vibrations
including anharmonic effects [439–444] are under constant development but can
hardly be applied to truly large molecules containing on the order of 100 atoms and
more. Hence, molecular dynamics approaches are often employed as a means to
include anharmonic effects on the motion of atomic nuclei [445,446] but suffer from
the neglect of quantum effects on the nuclear motion and force-field approximations
or require tremendous resources for the generation of trajectories if the potential
energy surface is calculated with first-principles methods [447].
148
Standard and Inverse Vibrational Calculations with M OV I PAC
As a consequence, the harmonic approximation remains the standard approach in
quantum chemistry [448]. It has the important advantage that intensity expressions
for numerous spectroscopic techniques can be easily evaluated within the doubleharmonic approximation [415]. Anharmonic corrections are usually considered
perturbatively by calculating cubic and quartic force constants for elongations along
the normal modes obtained within the harmonic approximation [337, 449]. This
method is then often referred to as VPT2; note that the first calculations of second
order perturbations to harmonic frequencies were done as early as 1933 [450].
Often one observes a very good agreement of calculated harmonic frequencies
and measured fundamental ones (see, e.g., Refs. [389, 451–453]), which is due
to a fortunate error compensation of some density-functional approximations and
sufficiently large basis sets [337]. If this is not the case, scaling factors may be
employed; note that the so called scaled quantum mechanical (SQM) force fields
are one particular “flavour” of these techniques. [337, 454–459].
In 2002, we presented a semi-numerical, massively parallel implementation for
quantum chemical calculations of molecular vibrations in the harmonic approximation which has become known as the S NF program [275]. S NF was based on
earlier work in the 1990s by Grimme and Marian [460] and its structure has been
under constant development. Moreover, offsprings have been developed with special capabilities. One is the A KIRA program which implements the Mode-Tracking
algorithm [266] for the efficient and targeted calculation of selected molecular
vibrations in large molecules and molecular aggregates [265]. Also A KIRA has
been continuously developed and supplemented by new algorithms like IntensityTracking [267–270]. Other offsprings are the anharmonicity program A NF [337]
and the resonance Raman program K KTRANS [461].
Here, we present the latest developments of S NF and A KIRA which are now united
into one single program package for vibrational spectroscopy called M OV I PAC.
D.2
The M OV I PAC Philosophy
M OV I PAC sacrifices the possibility to analytically evaluate the quantum chemical
force field (i.e., second partial derivatives of the electronic energy with respect to nuclear coordinates) and the spectroscopic intensities (i.e., first derivatives of property
tensors with respect to nuclear coordinates), but instead uses a (semi-)numerical
differentiation scheme (i.e., the Hessian matrix is evaluated as the numeric first
derivative of analytic energy gradients, while the spectroscopic intensities are obtained as numeric first derivatives of the respective property tensors). This has two
disadvantages. First, it is less efficient by a constant, but small factor than the
fully analytical evaluation and second, it introduces a numerical error. The latter
disadvantage, namely the numerical error, turns out to be negligible in view of the
harmonic approximation employed (about 1 cm−1 ) and can be easily reduced by
taking advantage of a Bickley finite difference formula with more grid points [462].
The former disadvantage is not severe and turns into more than one advantage: The
fact that one needs only analytic geometry gradients and molecular property tensors
as raw data, which are provided by almost any multi-purpose quantum chemistry
D.3 Performance of Numerical and Analytical Derivatives
149
program, allows us to calculate spectra in a parallel fashion for many spectroscopic techniques employing basically any electronic structure method. We thus
circumvent the implementation of analytic property gradients and second geometry
derivatives for non-standard quantum chemical methods and their parallelization at
a negligible price. The reduced efficiency resulting in longer calculation times is
made up for by the huge number of computing cores available to research groups
nowadays. In order to shed more light onto this issue, we shall provide a reliable
comparison of analytical vs. numerical derivatives in the next section.
Special features of M OV I PAC are
1. its trivial and massive parallelism with load-balancing capabilities that provides automatic parallelization for electronic structure methods for which no
parallelized analytic second derivatives of the electronic energy and property
gradients are available,
2. its modular structure that has already been interfaced to five standard quantum
chemistry program packages (A DF [38], DALTON [463], G AUSSIAN [116],
M OLPRO [464], T URBOMOLE [117]) and that can be easily extended to other
programs for the production of the raw data,
3. its stable and efficient restart capabilities,
4. its advanced, inverse-quantum-chemical algorithms like Mode- and IntensityTracking, and
5. its specialized spectroscopic techniques like Raman Optical Activity, in particular in combination with our local version [426] of Turbomole [117] for
the calculation of ROA property tensors, or the constitution of spectra using
CTTM [465].
Another feature of the program convenient for research groups is that it can be run
on desktop computers and allows one to exploit a heterogeneous computer cluster
in an efficient way. Moreover, the program structure allows one to easily extend
the machinery in order to account for additional spectroscopic intensity expressions
and new electronic structure methods implemented in quantum chemistry program
packages that have not yet been interfaced.
We plan to continuously develop the M OV I PAC package and shall provide anharmonic corrections based on perturbation theory as well as new interfaces (e.g.,
to the M OLCAS environment [466]) in the future.
D.3
Performance of Numerical and Analytical
Derivatives
The Raman spectrum of the C60 molecule has been calculated with S NF (using the
so-called message passing interface, MPI, as parallelization scheme) in conjunction
with a new version of T URBOMOLE [117], i.e., T URBOMOLE 6.3.1 and with T UR BOMOLE 6.3.1 alone. For the S NF calculations, a serial version of T URBOMOLE
150
Standard and Inverse Vibrational Calculations with M OV I PAC
was employed, while for the T URBOMOLE-only calculations, both an MPI- and a
shared-memory parallelized (SMP) version were considered. In all these calculations, density functional theory was employed using the BP86 exchange–correlation
functional [81, 82] and Ahlrichs’ def2-TZVP basis set [332] at all atoms. Advantage was taken of the resolution-of-the-identity technique with the corresponding
auxiliary basis set [334]. The high symmetry of Buckminsterfullerene (point group
Ih ) can be conveniently exploited with both programs, S NF as well as T URBO MOLE . All calculations have been carried out in the same computing environment,
namely a blade system featuring two dodeca-core AMD Opteron 6174 processors
(i.e., a total of 24 cores) and 48 GB of memory. Therefore, these calculations are
truly comparable to each other in terms of timings, as the very same properties
have been calculated with the same methods (note also that the keywords in the
T URBOMOLE control file were the same in all calculations, except the one defining
the symmetry — the individual steps during the S NF calculation are carried out in
C1 symmetry since the distorted structure do no longer exhibit point group Ih ) on
the same computers.
The individual timings obtained are reported in Table D.1. We note that S NF
already reaches its maximum performance on seven cores; this is because for C60 we
only need to calculate six individual steps in the icosahedral point group (namely,
two distortions in each spatial dimension for one atom, as all 60 atoms are related
to each other by symmetry; on the seventh core a master task is running which does
no calculations but only distributes the individual steps and collects the results).
Note that one could of course link a parallelized version of T URBOMOLE (or any
other quantum mechanical backend) with S NF in order to speed up the calculation
of the individual steps. In fact, when resorting to a shared-memory parallelized
(SMP) version of T URBOMOLE (see below), the entire IR spectrum of C60 can be
calculated in only one hour and 26 minutes (when each step is running on 3 cores),
compared to the two hours and 54 minutes (see Table D.1) in the case of a serial
T URBOMOLE version.
Table D.1: Comparison of the total wall-time needed to calculate the Raman spectrum
of C60 with S NF in conjunction with T URBOMOLE 6.3.1 and T URBOMOLE 6.3.1 alone.
All timings are reported in the format hrs:min.
program (parallelization)
7 cores
IR
S NF (MPI) 2:54
T URBOMOLE (MPI) 6:33
T URBOMOLE (SMP) 2:24
Raman
12 cores
18 cores
IR
IR
Raman
n/a
n/a
6:58 6:34
2:21 2:03
n/a
7:01
2:19
6:26
n/a
6:59 6:32
2:40 2:04
Raman
Next, by looking at the results in Table D.1, we note that MPI-parallelized
T URBOMOLE is significantly slower than S NF. This is due to the fact that in
this T URBOMOLE version, the AOFORCE module, required to calculate the Hessian
matrix, is not parallelized. As more than 90% of the total calculation time is spent
D.3 Performance of Numerical and Analytical Derivatives
151
in the module AOFORCE, there is of course also no speedup of the calculation
when additional cores are employed. The results in Table D.1 nicely illustrate this.
When employing the shared-memory parallelized version of T URBOMOLE (which
uses MPI and shared memory versions of the modules RIDFT and RDGRAD, and
multi-threaded versions of AOFORCE, and EGRAD) however, the Hessian matrix can
be calculated in a parallel fashion, which significantly speeds up the calculation.
In this case, T URBOMOLE is faster than S NF.
In this context it is interesting to note that the calculation of the IR spectrum
takes only roughly 30 minutes longer on seven cores with S NF as compared to
T URBOMOLE, while the Raman spectrum needs almost 4 hours more to be calculated. This may be attributed to the fact that the calculation of the Hessian matrix
scales in the same fashion for both analytical and numerical implementations (only
the prefactor is somewhat larger in the case of a numerical implementation). For
the calculation of the Raman intensities, however, an analytical implementation has
a better scaling behavior. In fact, in a recent paper by Rappoport and Furche, it was
claimed [467] that analytic gradients of the electric-dipole–electric-dipole polarizability (in short often called polarizability) important for Raman intensities are by
a factor of 100 more efficient than the numerical first derivatives as implemented
in S NF [275]. Unfortunately, this conclusion is misleading as the authors of that
paper compared calculations which are not comparable for the following reason. In
our original S NF paper [275] we studied Buckminsterfullerene (C60 ) with the T UR BOMOLE 5.1 program. However, that early version of T URBOMOLE was not able
to calculate polarizability tensors with the resolution-of-the-identity density-fitting
technique, which increases the efficiency by a factor of about ten. By contrast,
the authors of Ref. [467] employed this technique. Of course, since more recent
T URBOMOLE versions always invoke the resolution-of-the-identity technique [467],
also S NF benefits from it. Indeed, the data in Table D.1 clearly shows that the
analytical polarizability gradients are by no means hundred times more efficient.
The difference between the time needed to obtain the Raman spectra and the IR
spectra is 16 minutes in the case of the SMP version of T URBOMOLE on 7 cores.
This is precisely the time required by the T URBOMOLE module EGRAD, which is
responsible for the analytical derivatives of the polarizability. Compared to this, we
may estimate the time needed by S NF for the numerical derivatives to be roughly
386 minutes. Thus, it can be said that the analytic derivatives are about a factor of
24 more efficient than their numerical counterparts. Furthermore, one should not
forget that the calculation of these derivatives is only one part required to obtain
the entire Raman spectrum. If one takes into consideration also the time needed to
compute the Hessian matrix, one understands that the semi-numeric S NF calculation
as a whole is only a factor of roughly 2.5 slower than the fully analytic calculation
employing T URBOMOLE alone.
From the timings in Table D.1 we can also see that the maximum performance
of the SMP version of T URBOMOLE is reached at roughly twelve cores; employing
six additional cores does lower the required computing time only very little.
Due to the different scaling behaviors of the calculation of the Hessian matrix and
the property derivatives, the former often becomes the bottleneck in a vibrational
calculation for large molecules. This is precisely the background for techniques
152
Standard and Inverse Vibrational Calculations with M OV I PAC
such as Mode- and Intensity-Tracking (see below).
D.4
Technical Aspects of M OV I PAC
After having outlined the fundamental philosophy behind M OV I PAC, some remarks
on the technical progress of the program package are appropriate. The entire source
code base of the S NF program has been reviewed. Numerous modifications have
been incorporated. Code duplications and dead code paths have been eliminated.
The parser of the output generated by the quantum chemical back-end programs is
now more robust due to the use of Regular Expressions. Especially the C code parts
have seen large changes in order to adhere to the coding standards of contemporary
Linux distributions. The code now relies on the compiler and operating system (OS)
to support the C99 and POSIX-1.2001 standards, respectively, that are fulfilled, e.g.,
by any reasonably recent Linux distribution [468].
external quantum chemistry programs providing raw
data (energy gradients and property tensors)
Add-on packages: CTTM, LOCVIB
ADF
SNFDEFINE:
setup program
AKIRADEFINE:
setup program
DALTON
SNF: evaluation
of raw data for
full vibrational
spectra
restart file
AKIRA:
iterative evaluation
of raw data for
selected vibrations
(Mode-Tracking,
Intensity-Tracking)
input files
output files
restart file
SNFDC: raw
data collector
parallel (MPI)
or serial
GAUSSIAN
MOLPRO
TURBOMOLE
input files
output files
Figure D.1: Schematic overview of the meta-structure of M OV I PAC. The dashed boxes
on the right hand side illustrate the possibility to straightforwardly implement interfaces
to additional quantum chemical programs.
The build system has been revamped from scratch and now uses the GNU build
system (autotools) [469, 470] that is used by the majority of GNU software. The
“autoconf” tool from this collection is used to create a “configure” script that automatically detects features of the OS, e.g., 64 vs. 32 bit, compiler, libraries (including
BLAS/LAPACK) or header files. The “automake” tool is used to create the different
Makefiles. This combination of tools allows for parallel compilation, out-of-tree
builds or installation into non-default subdirectories. Furthermore, it provides the
D.4 Technical Aspects of M OV I PAC
153
means for a clean implementation of the Fortran/C interface by automatic detection
of the name mangling scheme of the compiler combination used.
By default, a serial version of M OV I PAC is built. A parallel version relying on
MPI can be built by passing an appropriate command line option to “configure”.
The MPI version has received a lot of testing and turned out to work well. It uses
a master-slave architecture where one master process is used to distribute tasks
to the slave nodes. All MPI implementations that provide compiler/linker wrapper
scripts (e.g., “mpif77”) are supported.
While the two main parts of M OV I PAC, S NF and A KIRA, were originally independent programs they are now united in a single meta-program. A schematic
overview of M OV I PAC’s structure is shown in Fig. D.1. As can bee seen, M OV I PAC
features a modular structure, which simplifies additions to it (like, e.g., a new interface to another quantum mechanical program). There are programs to interactively
set up the input files necessary for the calculations (S NFDEFINE for common frequency analyses; A KIRADEFINE for Mode- and Intensity-Tracking), and programs
to evaluate the resulting output (S NF and A KIRA, respectively). In the case of
S NF, there is a dedicated program controlling the actual calculation of all necessary data (S NFDC) by steering the quantum mechanical backend programs via
simple file-based communication. For Mode- and Intensity-Tracking calculations,
this task is done by the program A KIRA. We would like to put special emphasis
on the central restart file, in which the data computed so far is stored. In the
case of a computer crash, this data is not lost, and, thus, the calculation can be
straightforwardly restarted at the point it crashed.
The M OV I PAC source code is made available on the internet at www.reiher.
ethz.ch/software/movipac.
Appendix E
Description of Program
Implementations
E.1
Program for the Optimization of the Jacket Potential
The trial studies reported in chapter 5 required the development of programs able to
minimize the overall geometry gradient. Luckily, the geometry gradient itself can
readily be obtained from almost any quantum chemical program such as A DF [38]
and T URBOMOLE [117]. Moreover, libraries such as the GNU Scientific Library
(G SL) [471] provide optimization algorithms. Therefore, our task consists of linking
these functionalities together.
Our group is currently developing a library of functions which can be used
to access quantum chemical software packages from within other programs. The
working title of this library is T NP; currently, it provides interfaces to A DF [38],
NWC HEM [472], and T URBOMOLE [117]. A simple example program calculating
the overall gradient for a predefined molecular structure with A DF can read as
follows:
#include <TNP>
int main(int argc, char *argv[]){
string commandLineParameters[1] = {argv[1]};
// read in structure
structure::Structure structure = structure::Structure();
structure.read(commandLineParameters[0], "xyz");
// calculate gradient
calculateGradient(structure);
return 0;
}
The function calculateGradient may be written as
double calculateGradient(Structure structure){
156
Description of Program Implementations
// initialize ADF interface
ADF adfCalc = ADF();
// define settings for the calculation
ParameterCollection parameters;
parameters.insert(Parameter("CHARGE", "0 1"));
parameters.insert(Parameter("BASIS", "type TZP"));
parameters.insert(Parameter("BASIS", "core None"));
...
adfCalc.setParameters(&parameters);
Singlepoint singlepoint = Singlepoint(&adfCalc, structure);
// calculate geometry gradient
singlepoint.calculateGradient();
GradientCollection gradients = singlepoint.getGradients();
// calculate overall gradient
...
return absoluteGradient;
}
The T NP library makes use of the E IGEN [473] library as well as the B OOST
libraries “Filesystem” [474] and “Regex” [475].
After the geometry gradient is available in our custom program, the algorithms
provided by the G SL may be used for the optimization of the overall gradient. The
G SL and its usage is explained in great detail in the official documentation [471].
E.2
Interfacing A DF with M ATHEMATICA
The computer algebra system M ATHEMATICA [362] provides highly sophisticated
mathematical algorithms, combined with an intuitive, easy-to-use graphical user
interface. Of particular interest for this work is the fact that M ATHEMATICA features
algorithms for unconstrained and constrained local and global optimizations. For
example, attempting to find the global minimum of a given function, f (x, y) under
the constraint that y > 0 by means of the differential evolution algorithm is done
with the simple input
NMinimize[{f[x,y], y>0}, {x,y},
Method -> "DifferentialEvolution"]
Further options of the command NMinimize are explained in great detail in the
official documentation of M ATHEMATICA. In this work, the NMinimize function is
used to find the global minimum of the overall geometry gradient with respect to
some point charge arrangement. The function f therefore yields the overall geometry
gradient, while the number and exact meaning of the independent variables of f
depend on the exact case studied. For example, in the case where only one point
charge is employed (see section 5.4.2), f features four independent variables, namely
three for the spatial position and one for the charge of the point charge. The main
problem we face is that the calculation of the function f , that is the geometry
E.2 Interfacing A DF with M ATHEMATICA
157
gradient, is not trivial at all, but requires special quantum chemical software. It is
therefore not straightforward to implement the calculation of the geometry gradient
in M ATHEMATICA itself, but it is possible to interface an external quantum chemical
program (in our case A DF) with M ATHEMATICA. The function f can thus be defined
externally, and then called from within M ATHEMATICA.
The definition of f can be done in a simple C program (called f.c here), as,
for example
#include "mathlink.h"
double f(double x, double y) {
// write ADF input file
...
// calculate gradient
system("module load adf/hpmpi/2012.01b && export NSCM=4 &&
$ADFBIN/adf < /home/eth/tweymuth/prog/iqc/mathematica_test/
adf.in > /home/eth/tweymuth/prog/iqc/mathematica_test/adf.out
2> /home/eth/tweymuth/prog/iqc/mathematica_test/adf.err");
// read in gradient and calculate overall gradient
...
return overallGradient;
}
int main(int argc, char *argv[]){
return MLMain(argc, argv);
}
In addition, one needs a so-called MathLink template file (called f.tm here), which
specifies the properties of the external function f (x, y),
:Begin:
:Function:
:Pattern:
:Arguments:
:ArgumentTypes:
:ReturnType:
:End:
f
f[x_?NumberQ, y_?NumberQ]
{x, y}
{Real, Real}
Real
The two files f.c and f.tm have to be compiled together to produce an executable.
This has do be done with a special compiler (mcc1 ) provided with M ATHEMATICA,
mcc -o f.exe f.tm f.c
Then, one can import this external function into M ATHEMATICA, and use it like
any function defined within M ATHEMATICA, for example
Install["f.exe"]
f[0.01,0.02]
1
This compiler is not installed in any of the usual locations but instead located in the M ATHE directory hierarchy. A possible path would be <path to M ATHEMATICA>SystemFiles/
Links/MathLink/DeveloperKit/Linux-x86-64/CompilerAdditions/mcc.
MATICA
Acknowledgments
“We change here gar nix.”
M. T. Stiebritz, author, philosopher, thinker
During the last years, I had the pleasure to work together with many different
people, and it is appropriate to point them out in the following.
First of all, I would like to thank my supervisor Prof. Markus Reiher for giving
me the opportunity to work in his research group. He remains to be an example
for me in many ways.
Every research group lives from the spirit, ideas and energy of its members. I
am therefore very much indebted to all past and present members of our research
group, which I was pleased to meet and work with during the last four years:
Maike Bergeler, Dr. Katharina Boguslawski, Marta Bruska, Dr. Steven Donald,
Arndt Finkelmann, Dr. Samuel Fux, Moritz Haag, Romy Isenegger, Dr. Christoph
Jacob, Sebastian Keller, Dr. Karin Kiewisch, Dr. Stefan Knecht, Florian Krausbeck,
Dr. Vincent Liégeois, Dr. Sandra Luber, Dr. Hans Peter Lüthi, Halua Pinto de Magalhães, Dr. Konrad Marti, Dr. Remigius Mastalerz, Dr. Edit Mátyus, Dr. Villö Pálfi,
Dr. Daoling Peng, Dr. Maren Podewitz, Oliver Sala, Dr. Stephan Schenk, Benjamin
Simmen, Dr. Martin Stiebritz, Dr. Pawel Tecmer, and Dr. Elizabeth Chirackal Varkey.
I am going to miss the friendly and collaborative atmosphere of our group — let
alone our numerous hikes.
I would also like to give thanks to Prof. Carmen Herrman, Dr. Christoph Jacob, and Prof. Johannes Neugebauer for their active support in the new release
of M OV I PAC. Especially warm thanks go to Dr. Christoph Jacob for his hospitality during my stay in his group in Karlsruhe. Moreover, I would like to thank
Prof. Johannes Neugebauer for many discussions on DFT and its implementation
in A DF.
Prof. Peter Chen and Dr. Erik Couzijn are gratefully acknowledged for providing
me with the experimental data on the transition metal reactions studed in this thesis,
and for the helpful discussions about the DFT benchmarks.
During my PhD, I had the chance to supervise several semester and Master
students, namely Denitsa Baykusheva, Anna-Lena Deppenmeier, Florian Hodel,
Martina Minges, and Benjamin Simmen. I enjoyed teaching these students and, in
turn, could also learn a lot from them.
After Stephan Schenk and after him Villö and Steven left our group, Arndt,
Moritz, and me were presented with the challenge of taking care of our entire IT
infrastructure. I cannot overemphasize how much the entire group ows to Moritz,
who, besides numerous other things, migrated our entire data once around the
universe, and to Arndt, who energetically helped in every possible situation. I am
also very thankful that Sebastian is highly motivated to take over the important job
of a system administrator after me.
I continuously tried to bring our IT system to a higher level of professionality,
such that we get the maximum performance and availability with a minimum of
maintenance. Needless to say that this was only possible with the immediate and
competent help of our IT services. I should especially mention Eric Müller and
Dr. Vladislav Nespor, who helped me with our new LDAP authentication, Marcus
Möller, who takes care of the student computer rooms, Dr. Tilo Steiger, who assists
us with our new storage share, and last but not least Daniel Freund and Markus
Traber who were always good contact points for all my questions and needs.
“Das Gremium” granted me with the privilege of sitting in the only two-person
office of our group. What is even better, I was extraordinarily pleased to have Martin
as my office mate — never before there were such fancy, quirky, and magniloquent
discussions and dialogues on this planet. Together, we enjoyed many hiking and
climbing tours, and achieved a major scientific breakthrough with the discovery that
camera memory cards always become full in the middle of the worst escarpments.
I would also like to thank Martin and Florian for our many coffee breaks, during
which we had many discussions about quantum chemistry, science, and the life in
general.
Irene Bräunlich carefully proofread the entire thesis manuscript. I am therefore
very much indebted to her.
My last — but also biggest — thank finally goes to my family. They constantly
support me in every aspect of my life.
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Curriculum Vitae
Name:
Thomas Weymuth
Date of birth: November 26, 1987
Place of birth: Zurich, Switzerland
Nationality:
Swiss
Education
2000 – 2006
2006 – 2009
2009 – 2010
2010 – 2013
Matur (equivalent of US american high school diploma)
Kantonsschule Rychenberg, Winterthur, Switzerland
Bachelor of Science ETH in Chemistry
Eidgenössische Technische Hochschule (ETH), Zurich, Switzerland
Master of Science ETH in Chemistry (with distinction)
Eidgenössische Technische Hochschule (ETH), Zurich, Switzerland
Master thesis: Can Raman Optical Activity Discriminate Between
Different Types of Protein β-Turns? (supervision: Prof. M. Reiher)
Doctoral studies in theoretical chemistry
Prof. M. Reiher, Eidgenössische Technische Hochschule (ETH),
Zurich, Switzerland
Awards
2011
IBM Research Forschungspreis
Award for an outstanding Master thesis in computational modeling
and simulation in chemistry, biology, and materials science
Talks
02/2011
05/2012
09/2012
Identifying Protein β-Turns with Vibrational Raman Optical
Activity
Theoretical Chemistry Seminar KIT, Karlsruhe, Germany
Identifying Protein β-Turns with Vibrational Raman Optical
Activity
Competence Center in Computational Chemistry (C4) Seminar,
IBM Research Switzerland, Rüschlikon, Switzerland
Theoretical Raman Optical Activity Spectroscopy for Large
Molecules
Centre Européen de Calcul Atomique et Moléculaire (CECAM)
Workshop 2012: Vibrational Optical Activity: Interplay of Theory
and Experiment, Pisa, Italy