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A Chemist’s Guide to Density Functional Theory

Second Edition

Wolfram Koch

Max C. Holthausen


Further Reading from Wiley-VCH and John Wiley & Sons

P. Comba/T. W. Hambley

Molecular Modeling of Inorganic Compounds, Second Edition

2000, approx. 250 pages with approx. 200 figures and a CD-ROM with an interactive

tutorial. Wiley-VCH.

ISBN 3-527-29915-7

H.-D. Höltje/G. Folkers

Molcular Modeling. Basic Priniciples and Applications

1997, 206 pages. Wiley-VCH.

ISBN 3-527-29384-1

F. Jensen

Introduction to Computational Chemistry

1998, 454 pages. Wiley.

ISBN 0-471-98425-6

K. B. Lipkowitz/D. B. Boyd (Eds.)

Reviews in Computational Chemistry, Vol. 13

1999, 384 pages. Wiley.

ISBN 0-471-33135-X

M. F. Schlecht

Molecular Modeling on the PC

1998, 763 pages. Wiley-VCH.

ISBN 0-471-18467-1

P. von Schleyer (Ed.)

Encyclopedia of Computational Chemistry

1998, 3580 pages. Wiley.

ISBN 0-471-96588-X

J. Zupan/J. Gasteiger

Neural Networks in Chemistry and Drug Design

1999, 400 pages. Wiley-VCH.

ISBNs 3-527-29779-0 (Softcover), 3-527-29778-2 (Hardcover)


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 development. 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 quantum 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 particular 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 (computational) 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 approach 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 electrons, 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 understand 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 computational and quantum chemists as well as to the chemistry community at large. I am happy to recommend it to this audience.


October 1999


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 became 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 computational 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.


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 quantum chemistry codes additionally catalyzed this process, which is steadily gaining momentum. 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 textbooks. 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 community 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 precisely 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 investigations included in the bibliography. The literature was covered through the fall of 1999.

However, due to the huge amount of relevant papers appearing in a large variety of journals, 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) methods, 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 functional 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

Many colleagues and friends contributed important input at various stages of the preparation 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 particular 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 alternatives 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 asking 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 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


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 discussion has been brought up-to-date at various places in order to document significant recent developments.

WOLFRAM KOCH, Frankfurt am Main


April 2001

The Definition of the Model

What is density functional theory? The first part of this book is devoted to this question and we will try in the following seven chapters to give the reader a guided tour through the current state of the art of approximate density functional theory. We will try to lift some of the secrets veiling that magic black box, which, after being fed with only the charge density of a system somewhat miraculously cranks out its energy and other ground state properties. Density functional theory is rooted in quantum mechanics and we will therefore start by introducing or better refreshing some elementary concepts from basic molecular quantum mechanics, centered around the classical Hartree-Fock approximation. Since modern density functional theory is often discussed in relation to the Hartree-Fock model and the corresponding extensions to it, a solid appreciation of the related physics is a crucial ingredient for a deeper understanding of the things to come. We then comment on the very early contributions of Thomas and Fermi as well as Slater, who used the electron density as a basic variable more out of intuition than out of solid physical arguments. We go on and develop the red line that connects the seminal theorems of Hohenberg and Kohn through the realization of this concept by Kohn and Sham to the currently popular approximate exchange-correlation functionals. The concept of the exchange-correlation hole, which is rarely discussed in detail in standard quantum chemical textbooks holds a prominent place in our exposition. We believe that grasping its characteristics helps a lot in order to acquire a more pictorial and less abstract comprehension of the theory. This intellectual exercise is therefore well worth the effort. Next to the theory, which – according to our credo – we present in a down-to-earth like fashion without going into all the many intricacies which theoretical physicists make a living of, we devote a large fraction of this part to very practical aspects of density functional theory, such as basis sets, numerical integration techniques, etc. While it is neither possible nor desirable for the average user of density functional methods to apprehend all the technicalities inherent to the implementation of the theory, the reader should nevertheless become aware of some of the problems and develop a feeling of how a solution can be realized.

Elementary Quantum Chemistry

In this introductory chapter we will review some of the fundamental aspects of electronic structure theory in order to lay the foundations for the theoretical discussion on density functional theory (DFT) presented in later parts of this book. Our exposition of the material will be kept as brief as possible and for a deeper understanding the reader is encouraged to consult any modern textbook on molecular quantum chemistry, such as Szabo and Ostlund, 1982, McWeeny, 1992, Atkins and Friedman, 1997, or Jensen, 1999. After introducing the Schrödinger equation with the molecular Hamilton operator, important concepts such as the antisymmetry of the electronic wave function and the resulting Fermi correlation, the Slater determinant as a wave function for non-interacting fermions and the Hartree-Fock approximation are presented. The exchange and correlation energies as emerging from the Hartree-Fock picture are defined, the concepts of dynamical and nondynamical electron correlation are discussed and the dissociating hydrogen molecule is introduced as a prototype example.

1.1 The Schrödinger Equation

The ultimate goal of most quantum chemical approaches is the – approximate – solution of the time-independent, non-relativistic Schrödinger equation

(1-1) image046.jpg

where hdash.jpg is the Hamilton operator for a molecular system consisting of M nuclei and N electrons in the absence of magnetic or electric fields. hdash.jpg is a differential operator representing the total energy:

(1-2) image049.jpg

Here, A and B run over the M nuclei while i and j denote the N electrons in the system. The first two terms describe the kinetic energy of the electrons and nuclei respectively, where the Laplacian operator underdelta.jpg is defined as a sum of differential operators (in cartesian coordinates)

(1-3) image050.jpg

and MA is the mass of nucleus A in multiples of the mass of an electron (atomic units, see below). The remaining three terms define the potential part of the Hamiltonian and represent the attractive electrostatic interaction between the nuclei and the electrons and the repulsive potential due to the electron-electron and nucleus-nucleus interactions, respectively. rpq (and similarly Rpq) is the distance between the particles p and q, i. e.,image054.gif image055.gif stands for the wave function of the i’th state of the system, which depends on the 3N spatial coordinates images, and the N spin coordinates1 {si} of the electrons, which are collectively termed images and the 3M spatial coordinates of the nuclei, images The wave function ψi contains all information that can possibly be known about the quantum system at hand. Finally, Ei is the numerical value of the energy of the state described by ψi

All equations given in this text appear in a very compact form, without any fundamental physical constants. We achieve this by employing the so-called system of atomic units, which is particularly adapted for working with atoms and molecules. In this system, physical quantities are expressed as multiples of fundamental constants and, if necessary, as combinations of such constants. The mass of an electron, me, the modulus of its charge, |e|, Planck’s constant h divided by image052.gif , and image053.gif the permittivity of the vacuum, are all set to unity. Mass, charge, action etc. are then expressed as multiples of these constants, which can therefore be dropped from all equations. The definitions of atomic units used in this book and their relations to the corresponding SI units are summarized in Table 1-1.

Table 1-1. Atomic units.

Quantity Atomic unit Value in SI units Symbol (name)
mass rest mass of electron 9.1094 x 10–31 kg me
charge elementary charge 1.6022 x 10–19 C e
action Planck’s constant/2π 1.0546 x 10–34 J s hcut.jpg
length 4πε0hcut.jpg/me e2 5.2918 x 10–11 m a0 (bohr)
energy hcut.jpg2/measupsub.jpg 4.3597 x 10–18 J Eh (hartree)

Note that the unit of energy, 1 hartree, corresponds to twice the ionization energy of a hydrogen atom, or, equivalently, that the exact total energy of an H atom equals –0.5 Eh. Thus, 1 hartree corresponds to 27.211 eV or 627.51 kcal/mol.2

The Schrödinger equation can be further simplified if we take advantage of the significant differences between the masses of nuclei and electrons. Even the lightest of all nuclei, the proton (1H), weighs roughly 1800 times more than an electron, and for a typical nucleus such as carbon the mass ratio well exceeds 20,000. Thus, the nuclei move much slower than the electrons. The practical consequence is that we can – at least to a good approximation – take the extreme point of view and consider the electrons as moving in the field of fixed nuclei. This is the famous Born-Oppenheimer or clamped-nuclei approximation. Of course, if the nuclei are fixed in space and do not move, their kinetic energy is zero and the potential energy due to nucleus-nucleus repulsion is merely a constant. Thus, the complete Hamiltonian given in equation (1-2) reduces to the so-called electronic Hamiltonian

(1-4) image063.jpg

The solution of the Schrödinger equation with hbase.jpg is the electronic wave function image070.gif and the electronic energy image071.gif depends on the electron coordinates, while the nuclear coordinates enter only parametrically and do not explicitly appear in image072.gif The total energy Etot is then the sum of Eelec and the constant nuclear repulsion term,

(1-5) image064.jpg


(1-6) image065.jpg

The attractive potential exerted on the electrons due to the nuclei – the expectation value of the second operator image in equation (1-4) – is also often termed the external potential, Vext, in density functional theory, even though the external potential is not necessarily limited to the nuclear field but may include external magnetic or electric fields etc. From now on we will only consider the electronic problem of equations (1-4) – (1-6) and the subscript ‘elec’ will be dropped.

The wave function ψ itself is not observable. A physical interpretation can only be associated with the square of the wave function in that

(1-7) image066.jpg

represents the probability that electrons 1, 2, …, N are found simultaneously in volume elements image062.gif. Since electrons are indistinguishable, this probability must not change if the coordinates of any two electrons (here i and j) are switched, viz.,

(1-8) image067.jpg

Thus, the two wave functions can at most differ by a unimodular complex number e. It can be shown that the only possibilities occurring in nature are that either the two functions are identical (symmetric wave function, applies to particles called bosons which have integer spin, including zero) or that the interchange leads to a sign change (antisymmetric wave function, applies to fermions, whose spin is half-integral). Electrons are fermions with spin = ½ and ψ must therefore be antisymmetric with respect to interchange of the spatial and spin coordinates of any two electrons:

(1-9) image075.jpg

We will soon encounter the enormous consequences of this antisymmetry principle, which represents the quantum-mechanical generalization of Pauli’s exclusion principle (‘no two electrons can occupy the same state’). A logical consequence of the probability interpretation of the wave function is that the integral of equation (1-7) over the full range of all variables equals one. In other words, the probability of finding the N electrons anywhere in space must be exactly unity,

(1-10) image076.jpg

A wave function which satisfies equation (1-10) is said to be normalized. In the following we will deal exclusively with normalized wave functions.

1.2 The Variational Principle

What we need to do in order to solve the Schrödinger equation (1-5) for an arbitrary molecule is first to set up the specific Hamilton operator of the target system. To this end we need to know those parts of the Hamiltonian Ĥ that are specific for the system at hand. Inspection of equation (1-4) reveals that the only information that depends on the actual molecule is the number of electrons in the system, N, and the external potential Vext. The latter is in our cases completely determined through the positions and charges of all nuclei in the molecule. All the remaining parts, such as the operators representing the kinetic energy or the electron-electron repulsion, are independent of the particular molecule we are looking at. In the second step we have to find the eigenfunctions image074.gif and corresponding eigenvalues image077.gif Once the ψi are determined, all properties of interest can be obtained by applying the appropriate operators to the wave function. Unfortunately, this simple and innocuous-looking program is of hardly any practical relevance, since apart from a few, trivial exceptions, no strategy to solve the Schrödinger equation exactly for atomic and molecular systems is known.

Nevertheless, the situation is not completely hopeless. There is a recipe for systematically approaching the wave function of the ground state image073.gif i. e., the state which delivers the lowest energy E0. This is the variational principle, which holds a very prominent place in all quantum-chemical applications. We recall from standard quantum mechanics that the expectation value of a particular observable represented by the appropriate operator Ô using any, possibly complex, wave function ψtrial that is normalized according to equation (1-10) is given by

(1-11) image087.jpg

where we introduce the very convenient bracket notation for integrals first used by Dirac, 1958, and often used in quantum chemistry. The star in image089.gif indicates the complex-conjugate of ψtrial.

where we introduce the very convenient bracket notation for integrals first used by Dirac, 1958, and often used in quantum chemistry. The star in image089.gif indicates the complex-conjugate of ψtrial.

The variational principle now states that the energy computed via equation (1-11) as the expectation value of the Hamilton operator image092.gif from any guessed image091.gif will be an upper bound to the true energy of the ground state, i. e.,

(1-12) image088.jpg

where the equality holds if and only if image079.gif is identical to image080.gif The proof of equation (1-12) is straightforward and can be found in almost any quantum chemistry textbook.

Before we continue let us briefly pause, because in equations (1-11) and (1-12) we encounter for the first time the main mathematical concept of density functional theory. A rule such as that given through (1-11) or (1-12), which assigns a number, e. g., Etrial, to a function, e. g., image081.gif is called a functional. This is to be contrasted with the much more familiar concept of a function, which is the mapping of one number onto another number. Phrased differently, we can say that a functional is a function whose argument is itself a function. To distinguish a functional from a function in writing, one usually employs square brackets for the argument. Hence, f(x) is a function of the variable x while F[f] is a functional of the function f. Recall that a function needs a number as input and also delivers a number:


For example, image082.gif Then, for x = 2, the function delivers y = 5. On the other hand, a functional needs a function as input, but again delivers a number:


For example, if we define image083.gif dx and use f(x) as defined above as input, this functional delivers F [f(x) = x2 + 1] = 28/15. If, instead we choose f(x) = 2x2+1, the result is F [f(x) = 2x2 + 1] = 47/15.

Expectation values such as odash.jpg in equation (1-11) are obviously functionals, since the value of image086.gif depends on the function image084.gif inserted.

Coming back to the variational principle, the strategy for finding the ground state energy and wave function should be clear by now: we need to minimize the functional E[ψ] by searching through all acceptable N-electron wave functions. Acceptable means in this context that the trial functions must fulfill certain requirements which ensure that these functions make physical sense. For example, to be eligible as a wave function, image098.gif must be continuous everywhere and be quadratic integrable. If these conditions are not fulfilled the normalization of equation (1-10) would be impossible. The function3 which gives the lowest energy will be image097.gif and the energy will be the true ground state energy E0. This recipe can be compactly expressed as

(1-13) image095.jpg

where image099.gif indicates that ψ is an allowed N-electron wave function. While such a search over all eligible functions is obviously not possible, we can apply the variational principle as well to subsets of all possible functions. One usually chooses these subsets such that the minimization in equation (1-13) can be done in some algebraic scheme. The result will be the best approximation to the exact wave function that can be obtained from this particular subset. It is important to realize that by restricting the search to a subset the exact wave function itself cannot be identified (unless the exact wave function is included in the subset, which is rather improbable). A typical example is the Hartree-Fock approximation discussed below, where the subset consists of all antisymmetric products (Slater determinants) composed of N spin orbitals.

Let us summarize what we have shown so far: once N and Vext (uniquely determined by ZA and RA) are known, we can construct image100.gif. Through the prescription given in equation (1-13)