A Chemist's Guide to Density Functional Theory

306 Pages·2001·3.066 MB·English

by  Wolfram Koch, Max C. Holthausen

A Chemist’s Guide to Density Functional Theory. Second Edition
Wolfram Koch, Max C. Holthausen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)
Wolfram Koch, Max C. Holthausen
A Chemist’s Guide to
Density Functional Theory
Second Edition
I
A Chemist’s Guide to Density Functional Theory. Second Edition
Wolfram Koch, Max C. Holthausen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)
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II
A Chemist’s Guide to Density Functional Theory. Second Edition
Wolfram Koch, Max C. Holthausen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)
Wolfram Koch, Max C. Holthausen
A Chemist’s Guide to
Density Functional Theory
Second Edition
Weinheim · New York · Chichester · Brisbane · Singapore · Toronto
III
A Chemist’s Guide to Density Functional Theory. Second Edition
Wolfram Koch, Max C. Holthausen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)
Prof. Dr. Wolfram Koch Dr. Max C. Holthausen
Gesellschaft Deutscher Chemiker Fachbereich Chemie
(German Chemical Society) Philipps-Universität Marburg
Varrentrappstraße 40–42 Hans-Meerwein-Straße
D-60486 Frankfurt D-35032 Marburg
Germany Germany
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IV
A Chemist’s Guide to Density Functional Theory. Second Edition
Wolfram Koch, Max C. Holthausen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)
Foreword
It is a truism that in the past decade density functional theory has made its way from a
peripheral position in quantum chemistry to center stage. Of course the often excellent
accuracy of the DFT based methods has provided the primary driving force of this develop-
ment. When one adds to this the computational economy of the calculations, the choice for
DFT appears natural and practical. So DFT has conquered the rational minds of the quan-
tum chemists and computational chemists, but has it also won their hearts? To many, the
success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified.
Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave
function based methods. And unjustified it appeared to those who doubted the soundness of
the theoretical foundations. There has been misunderstanding concerning the status of the
one-determinantal approach of Kohn and Sham, which superficially appeared to preclude
the incorporation of correlation effects. There has been uneasiness about the molecular
orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have
used orbitals but which in the physics literature were sometimes denoted as mathematical
constructs devoid of physical (let alone chemical) meaning.
Against this background the Chemist’s Guide to DFT is very timely. It brings in the
second part of the book the reader up to date with the most recent successes and failures of
the density functionals currently in use. The literature in this field is exploding in such a
manner that it is extremely useful to have a comprehensive overview available. In particu-
lar the extensive coverage of property evaluation, which has very recently been enormously
stimulated by the time-dependent DFT methods, will be of great benefit to many (compu-
tational) chemists. But I wish to emphasize in particular the good service the authors have
done to the chemistry community by elaborating in the first part of the book on the ap-
proach that DFT takes to the physics of electron correlation. A full appreciation of DFT is
only gained through an understanding of how the theory, in spite of working with an orbital
model and a single determinantal wave function for a model system of noninteracting elec-
trons, still achieves to incorporate electron correlation. The authors justly put emphasis on
the pictorial approach, by way of Fermi and Coulomb correlation holes, to understanding
exchange and correlation. The present success of DFT proves that modelling of these holes,
even if rather crudely, can provide very good energetics. It is also in the simple physical
language of shape and extent (localized or delocalized) of these holes that we can under-
stand where the problems of that modelling with only local input (local density, gradient,
Laplacian, etc.) arise. It is because of the well equilibrated treatment of physical principles
and chemical applications that this book does a good and very timely service to the compu-
tational and quantum chemists as well as to the chemistry community at large. I am happy
to recommend it to this audience.
EVERT JAN BAERENDS, Amsterdam
October 1999
V
A Chemist’s Guide to Density Functional Theory. Second Edition
Wolfram Koch, Max C. Holthausen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)
Preface
This book has been written by chemists for chemists. In particular, it has not been written
by genuine theoretical chemists but by chemists who are primarily interested in solving
chemical problems and in using computational methods for addressing the many exciting
questions that arise in modern chemistry. This is important to realize right from the start
because our background of course determined how we approached this project. Density
functional theory is a fairly recent player in the computational chemistry arena. WK, the
senior author of this book remembers very well his first encounter with this new approach
to tackle electronic structure problems. It was only some ten years back, when he got a
paper to review for the Journal of Chemical Physics where the authors employed this method
for solving some chemical problems. He had a pretty hard time to understand what the
authors really did and how much the results were worth, because the paper used a language
so different from conventional wave function based ab initio theory that he was used to. A
few years later we became interested in transition-metal chemistry, the reactivity of
coordinatively unsaturated open-shell species in mind. During a stay with Margareta
Blomberg and Per Siegbahn at the University of Stockholm, leading researchers in this
field then already for a decade, MCH was supposed to learn the tricks essential to cope with
the application of highly correlated multireference wave function based methods to tackle
such systems. So he did – yet, what he took home was the feeling that our problems could
not be solved for the next decade with this methodology, but that there might be something
to learn about density functional theory (DFT) instead. It did not take long and DFT be-
came the major computational workhorse in our group. We share this kind of experience
with many fellow computational chemists around the globe. Starting from the late eighties
and early nineties approximate density functional theory enjoyed a meteoric rise in compu-
tational chemistry, a success story without precedent in this area. In the Figure below we
show the number of publications where the phrases ‘DFT’ or ‘density functional theory’
appear in the title or abstract from a Chemical Abstracts search covering the years from
1990 to 1999. The graph speaks for itself.
(cid:5)(cid:1)(cid:1)(cid:1)
(cid:4)(cid:2)(cid:1)(cid:1)
(cid:4)(cid:1)(cid:1)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:7)
(cid:3)(cid:2)(cid:1)(cid:1)
(cid:10)(cid:2)(cid:4)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:16)(cid:17)
(cid:3)(cid:1)(cid:1)(cid:1)
(cid:2)(cid:1)(cid:1)
(cid:1)
(cid:3)(cid:6)(cid:6)(cid:1) (cid:3)(cid:6)(cid:6)(cid:3) (cid:3)(cid:6)(cid:6)(cid:4) (cid:3)(cid:6)(cid:6)(cid:5) (cid:3)(cid:6)(cid:6)(cid:7) (cid:3)(cid:6)(cid:6)(cid:2) (cid:3)(cid:6)(cid:6)(cid:8) (cid:3)(cid:6)(cid:6)(cid:9) (cid:3)(cid:6)(cid:6)(cid:10) (cid:3)(cid:6)(cid:6)(cid:6)
(cid:18)(cid:5)(cid:14)(cid:6)
VII
This stunning progress was mainly fueled by the development of new functionals –
gradient-corrected functionals and most notably hybrid functionals such as B3LYP – which
cured many of the deficiencies that had plagued the major model functional used back then,
i. e., the local density approximation. Their subsequent implementation in the popular quan-
tum chemistry codes additionally catalyzed this process, which is steadily gaining momen-
tum. The most visible documentation that computational methods in general and density
functional theory in particular finally lost their ‘new kid on the block’ image is the award of
the 1998 Noble Prize in chemistry to two exceptional protagonists of this genre, John Pople
and Walter Kohn.
Many experimental chemists use sophisticated spectroscopic techniques on a regular
basis, even though they are not experts in the field, and probably never need to be. In a
similar manner, more and more chemists start to use approximate density functional theory
and take advantage of black box implementations in modern programs without caring too
much about the theoretical foundations and – more critically – limitations of the method. In
the case of spectroscopy, this partial unawareness is probably just due to a lack of time or
motivation since almost any level of education required seems to be well covered by text-
books. In computational chemistry, however, the lack of digestible sources tailored for the
needs of chemists is serious. Everyone trying to supplement a course in computational
chemistry with pointers to the literature well suited for amateurs in density functional theory
has probably had this experience. Certainly, there is a vast and fast growing literature on
density functional theory including many review articles, monographs, books containing
collections of high-level contributions and also text books. Indeed, some of these were very
influential in advancing density functional theory in chemistry and we just mention what is
probably the most prominent example, namely Parr’s and Yang’s ‘Density-Functional Theory
of Atoms and Molecules’ which appeared in 1989, just when density functional theory
started to lift off. Still, many of these are either addressing primarily the physics commu-
nity or present only specific aspects of the theory. What is not available is a text book,
something like Tim Clark’s ‘A Handbook of Computational Chemistry’, which takes a
chemist, who is interested but new to the field, by the hand and guides him or her through
basic theoretical and related technical aspects at an easy to understand level. This is pre-
cisely the gap we are attempting to fill with the present book. Our main motivation to
embark on the endeavor of this project was to provide the many users of standard codes
with the kind of background knowledge necessary to master the many possibilities and to
critically assess the quality obtained from such applications. Consequently, we are neither
concentrating on all the important theoretical difficulties still related to density functional
theory nor do we attempt to exhaustively review all the literature of important applications.
Intentionally we sacrifice the purists’ theoretical standpoint and a broad coverage of fields
of applications in favor of a pragmatic point of view. However, we did our best to include as
many theoretical aspects and relevant examples from the literature as possible to encourage
the interested readers to catch up with the progress in this rapidly developing field. In
collecting the references we tried to be as up-to-date as possible, with the consequence that
older studies are not always cited but can be traced back through the more recent investiga-
tions included in the bibliography. The literature was covered through the fall of 1999.
VIII
However, due to the huge amount of relevant papers appearing in a large variety of jour-
nals, certainly not all papers that should have come to our attention actually did and we
apologize at this point to anyone whose contribution we might have missed. One more
point: we have written this book dwelling from our own background. Hence, the subjects
covered in this book, particularly in the second part, mirror to some extent the areas of
interest of the authors. As a consequence, some chemically relevant domains of density
functional theory are not mentioned in the following chapters. We want to make clear that
this does not imply that we assign a reduced importance to these fields, rather it reflects our
own lack of experience in these areas. The reader will, for example, search in vain for an
exposition of density functional based ab initio molecular dynamics (Car-Parrinello) meth-
ods, for an assessment of the use of DFT as a basis for qualitative models such as soft- and
hardness or Fukui functions, an introduction into the treatment of solvent effects or the
rapidly growing field of combining density functional methods with empirical force fields,
i. e., QM/MM hybrid techniques and probably many more areas.
The book is organized as follows. In the first part, consisting of Chapters 1 through 7, we
give a systematic introduction to the theoretical background and the technical aspects of
density functional theory. Even though we have attempted to give a mostly self-contained
exposition, we assume the reader has at least some basic knowledge of molecular quantum
mechanics and the related mathematical concepts. The second part, Chapters 8 to 13 presents
a careful evaluation of the predictive power that can be expected from today’s density
functional techniques for important atomic and molecular properties as well as examples of
some selected areas of application. Of course, also the selection of these examples was
governed by our own preferences and cannot cover all important areas where density func-
tional methods are being successfully applied. The main thrust here is to convey a general
feeling about the versatility but also the limitations of current density functional theory.
For any comments, hints, corrections, or questions, or to receive a list of misprints and
corrections please drop a message at [email protected].
Many colleagues and friends contributed important input at various stages of the prepa-
ration of this book, by making available preprints prior to publication, by discussions about
several subjects over the internet, or by critically reading parts of the manuscript. In par-
ticular we express our thanks to V. Barone, M. Bühl, C. J. Cramer, A. Fiedler, M. Filatov, F.
Haase, J. N. Harvey, V. G. Malkin, P. Nachtigall, G. Schreckenbach, D. Schröder, G. E.
Scuseria, Philipp Spuhler, M. Vener, and R. Windiks. Further, we would like to thank
Margareta Blomberg and Per Siegbahn for their warm hospitality and patience as open
minded experts and their early inspiring encouragement to explore the pragmatic alterna-
tives to rigorous conventional ab initio theory. WK also wants to thank his former and
present diploma and doctoral students who helped to clarify many of the concepts by ask-
ing challenging questions and always created a stimulating atmosphere. In particular we
are grateful to A. Pfletschinger and N. Sändig for performing some of the calculations used
in this book. Brian Yates went through the exercise of reading the whole manuscript and
helped to clarify the discussion and to correct some of our ‘Germish’. He did a great job –
thanks a lot, Brian – of course any remaining errors are our sole responsibility. Last but
certainly not least we are greatly indebted to Evert Jan Baerends who not only contributed
IX
many enlightening discussions on the theoretical aspects and provided preprints, but who
also volunteered to write the Foreword for this book and to Paul von Ragué Schleyer for
providing thoughtful comments. MCH is grateful to Joachim Sauer and Walter Thiel for
support, and to the Fonds der Chemischen Industrie for a Liebig fellowship, which allowed
him to concentrate on this enterprise free of financial concerns. At Wiley-VCH we thank R.
Wengenmayr for his competent assistance in all technical questions and his patience. The
victims that suffered most from sacrificing our weekends and spare time to the progress of
this book were certainly our families and we owe our wives Christina and Sophia, and
WK’s daughters Juliana and Leora a deep thank you for their endurance and understanding.
WOLFRAM KOCH, Frankfurt am Main
MAX C. HOLTHAUSEN, Berlin
November 1999
Preface to the second edition
Due to the large demand, a second edition of this book had to be prepared only about one
year after the original text appeared. In the present edition we have corrected all errors that
came to our attention and we have included new references where appropriate. The discus-
sion has been brought up-to-date at various places in order to document significant recent
developments.
WOLFRAM KOCH, Frankfurt am Main
MAX C. HOLTHAUSEN, Marburg
April 2001
X
A Chemist’s Guide to Density Functional Theory. Second Edition
Wolfram Koch, Max C. Holthausen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)
Contents
Foreword .....................................................................................................................V
Preface ....................................................................................................................... VII
Preface to the second edition ..............................................................................X
Part A The Definition of the Model ................................................................1
1 Elementary Quantum Chemistry .................................................................3
1.1 The Schrödinger Equation ...............................................................................3
1.2 The Variational Principle .................................................................................6
1.3 The Hartree-Fock Approximation ...................................................................8
1.4 The Restricted and Unrestricted Hartree-Fock Models .................................13
1.5 Electron Correlation ......................................................................................14
2 Electron Density and Hole Functions ........................................................19
2.1 The Electron Density .....................................................................................19
2.2 The Pair Density ............................................................................................20
2.3 Fermi and Coulomb Holes .............................................................................24
2.3.1 The Fermi Hole ..............................................................................................25
2.3.2 The Coulomb Hole ........................................................................................27
3 The Electron Density as Basic Variable: Early Attempts .........................29
3.1 Does it Make Sense? ......................................................................................29
3.2 The Thomas-Fermi Model .............................................................................30
3.3 Slater’s Approximation of Hartree-Fock Exchange .......................................31
4 The Hohenberg-Kohn Theorems ................................................................33
4.1 The First Hohenberg-Kohn Theorem: Proof of Existence .............................33
4.2 The Second Hohenberg-Kohn Theorem: Variational Principle .....................36
4.3 The Constrained-Search Approach ................................................................37
4.4 Do We Know the Ground State Wave Function in Density
Functional Theory? ........................................................................................39
4.5 Discussion ......................................................................................................39
XI