The Fundamentals of Density Functional Theory.

224 Pages·2003·1.12 MB·English

by  Helmut Eschrig

The Fundamentals of
Density Functional Theory
(revised and extended version)
H. Eschrig
Institute for Solid State and Materials Research Dresden
and
University of Technology Dresden

Preface
Density functional methods form the basis of a diversified and very active
area of present days computational atomic, molecular, solid state and even
nuclear physics. A large number of computational physicists use these meth-
ods merely as a recipe, not reflecting too much upon their logical basis. One
also observes, despite of their tremendous success, a certain reservation in
their acceptance on the part of the more theoretically oriented researchers
in the above mentioned fields. On the other hand, in the seventies (Thomas-
Fermi theory) and in the eighties (Hohenberg-Kohn theory), density func-
tional concepts became subjects of mathematical physics.
In 1994 a number of activities took place to celebrate the thirtieth an-
niversary of Hohenberg-Kohn-Sham theory. I took this an occasion to give
lectures on density functional theory to senior students andpostgraduates in
the winter term of 1994, particularly focusing on the logical basis of the the-
ory. Preparing these lectures, the impression grew that, although there is a
wealth of monographs and reviews in the literature devoted to density func-
tional theory, the focus is nearly always placed upon extending the practical
applications of the theory and on the development of improved approxima-
tions. The logical foundation of the theory is found somewhat scattered in
the existing literature, and is not always satisfactorily presented. This situ-
ation led to the idea to prepare a printed version of the lecture notes, which
resulted in the present text.
It is my intention to provide a thorough introduction to the theoretical
basis of density functional methods in a form which is both rigorous and yet
concise. It is aimed not only for those who are already or are going to be
active in the field, but also for those who just want to get a deeper insight
into the meaning of the results of practical calculations, and last but not
least to provide the interested mathematician with the physicist’s view on
the logical roots of the theory. High value is put on the self-containment of
the text, so that it should be accessible to anybody who has been through
the standard course in quantum mechanics.
6 Preface
The shorter Part II of these notes deals with the relativistic theory. This
part of the theory is less rigorous logically, due to the unsolved basic prob-
lems inquantum fieldtheory, but it isnevertheless physically very important
because heavier atoms and many problems of magnetism need a relativistic
treatment.
Finally, as is always the case, many things had to be omitted. I decided
in particular to omit everything which seemed to me to not yet have its final
form or at least be in some stabilized shape. This does not exclude that
important points are missing, just because I was unaware of them.
The author’s views on the subject have been sharpened by discussions
with many colleagues, but particularly during the course of scientific coop-
eration with M. Richter and P. M. Oppeneer on the applied side of the story.
Dresden, July 1996 Helmut Eschrig
In the present updated and extended version, errors and misprints have
been corrected in the original text. Throughout, an effort has been made
additionally to improve overall clarity. This includes in particular major
changes in Chapter 4. Here, also the non-collinear spin case was added as
Section 4.8, a very important issue presently. Chapter 6 and Section 9.2,
however, have been completely rewritten, since in the opinion of the author
the material gains much in systematics and clarity, if one against tradition
starts with the dependence on the particle number and afterwards treats the
density-potential interrelation. Although not yet completely developed, the
presently most common versions of LDA+U have been included as Section
7.5. The last section of the first edition on the other hand has been omitted
as it is not relevant any more: with the drastic increase of computer data
storage capacities, radial basis functions are stored numerically now and the
direct projection onto the electron sector of the Dirac-Fock space is not a
problem any more.
Dresden, August 2003 Helmut Eschrig
Contents
Introduction 9
Part I: NON-RELATIVISTIC THEORY 12
1 Many-Body Systems 13
1.1 The Schro¨dinger Representation, N Fixed . . . . . . . . . . . . . . 14
1.2 The Momentum Representation, N Fixed . . . . . . . . . . . . . . 17
1.3 The Heisenberg Representation, N Fixed . . . . . . . . . . . . . . 24
1.4 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 The Occupation Number Representation, N Varying . . . . . . . . 29
1.6 Field Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Density Matrices and Density Operators 35
2.1 Single-Particle Density Matrices . . . . . . . . . . . . . . . . . . . 35
2.2 Two-Particle Density Matrices . . . . . . . . . . . . . . . . . . . . 38
2.3 Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Expectation Values and Density Matrices . . . . . . . . . . . . . . 43
2.5 The Exchange and Correlation Hole . . . . . . . . . . . . . . . . . 47
2.6 The Adiabatic Principle . . . . . . . . . . . . . . . . . . . . . . . . 52
2.7 Coulomb Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Thomas-Fermi Theory 61
3.1 The Thomas-Fermi Functional and Thomas-Fermi Equation . . . . 62
3.2 The Thomas-Fermi Atom . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 The Thomas-Fermi Screening Length . . . . . . . . . . . . . . . . 69
3.4 Scaling Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Correction Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Hohenberg-Kohn Theory 76
4.1 The Basic Theorem by Hohenberg and Kohn . . . . . . . . . . . . 77
4.2 The Kohn-Sham Equation . . . . . . . . . . . . . . . . . . . . . . 81
4.3 The Link to the Hartree-Fock-Slater Approximation . . . . . . . . 86
4.4 Constrained Search Density Functionals . . . . . . . . . . . . . . . 87
4.5 Ensemble State Density Functionals . . . . . . . . . . . . . . . . . 89
4.6 Dependence on Particle Number N . . . . . . . . . . . . . . . . . 95
4.7 Spin Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.8 Non-Collinear Spin Configurations . . . . . . . . . . . . . . . . . . 105
8 Contents
5 Legendre Transformation 109
5.1 Elementary Introduction . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Prelude on Topology . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Prelude on Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . 121
5.4 Banach Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.5 Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6 Conjugate Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.7 The Functional Derivative . . . . . . . . . . . . . . . . . . . . . . 132
5.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . 135
6 Density Functional Theory by Lieb 138
6.1 The Ground State Energy . . . . . . . . . . . . . . . . . . . . . . 139
6.2 The Hohenberg-Kohn Variational Principle . . . . . . . . . . . . . 142
6.3 The Functionals F, G, and H . . . . . . . . . . . . . . . . . . . . 144
6.4 The Kohn-Sham Equation . . . . . . . . . . . . . . . . . . . . . . 151
7 Approximative Variants 155
7.1 The Homogeneous Electron Liquid . . . . . . . . . . . . . . . . . . 156
7.2 The Local Density Approximation . . . . . . . . . . . . . . . . . . 161
7.3 Generations of Kohn-Sham Type Equations . . . . . . . . . . . . . 165
7.4 The Self-Interaction Correction . . . . . . . . . . . . . . . . . . . . 168
7.5 The LDA+U Approach . . . . . . . . . . . . . . . . . . . . . . . . 171
Part II: RELATIVISTIC THEORY 178
8 A Brief Introduction to Quantum Electrodynamics 179
8.1 Classical Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 180
8.2 Lorentz Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.3 Lagrange Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.4 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 186
8.5 Relativistic Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.6 The Principles of Relativistic Quantum Theory . . . . . . . . . . . 188
8.7 The Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9 Current Density Functional Theory 196
9.1 QED Ground State in a Static External Field . . . . . . . . . . . 197
9.2 Current Density Functionals and Kohn-Sham-Dirac Equation . . . 202
9.3 The Gordon Decomposition and Spin Density . . . . . . . . . . . 207
9.4 Approximative Variants . . . . . . . . . . . . . . . . . . . . . . . . 210
Bibliography 214
Index 223
Introduction
Density functional theory provides a powerful tool for computations of the
quantum state of atoms, molecules and solids, and of ab-initio molecular
dynamics. It was conceived in its initial na¨ıve and approximative version
by Thomas and Fermi immediately after the foundation of quantum me-
chanics, in 1927. In the middle of the sixties, Hohenberg, Kohn and Sham
on the one hand established a logically rigorous density functional theory
of the quantum ground state on the basis of quantum mechanics, and on
the other hand, guided by this construction, introduced an approximative
explicit theory called the local-density approximation, which for computa-
tions of the quantum ground state of many-particle systems proved to be
superior to both Thomas-Fermi and Hartree-Fock theories. From that time
on, density functional theory has grown vastly in popularity, and a flood
of computational work in molecular and solid state physics has been the
result. Motivated by its success, there has been always a tendency to widen
the fields of application of density functional theory, and in these develop-
ments, some points which were left somewhat obscure in the basic theory,
were brought into focus from time to time. This led in the early eighties to
a deepening of the logical basis, essentially by Levy and Lieb, and finally
Lieb gave the basic theory a form of final mathematical rigor. Since that
treatment, however, is based on the tools of modern convex functional anal-
ysis, its implications only gradually became known to the many people who
apply density functional theory. A thorough treatment of the dependence
on particle number on the basis of Lieb’s theory is given for the first time
in the present text.
While quite a number of high quality and up-to-date surveys and mono-
graphs on the variants and applications of density functional theory exist,
the aim of these lecture notes is to provide a careful introduction into the
safe basis of this type of theory for both beginners and those working in
the field who want to deepen their understanding of its logical basis. To
this end, references are not only given to further introductory texts, surveys
and some key original papers regarding the physics, but also to literature
treating the underlying mathematics. Only Chapter 5 of this text provides
a more systematic and self-explanatory introduction to a piece of modern
10 Introduction
mathematics appropriate in this context, namely convex (non-linear) func-
tional analysis, and in particular duality theory. This chapter was included
for those physicists who either are for one reason or another not willing to
dive into comprehensive treatments of modern mathematics or who want
some guidance into those parts of mathematics necessary for a deeper un-
derstanding of the logics of density functional theory. Even the material of
this chapter is presented on an introductory level: proofs are only sketched
if they are essentially enlightening as regards the logical interrelations.
The material has been divided into two parts. Part I contains the non-
relativistic theory, and, because this part has to a larger extent achieved
some final logical state, it contains a discussion of the central ideas and
constructions of density functional theory, including some relevant mathe-
matical aspects. The shorter Part II is devoted to the relativistic extensions,
which have not yet reached the same level of rigor, mainly because realistic
quantum field theory is in a much less explicit state than quantum mechan-
ics.
Chapters 1, 2 andat least partof 3 areelementary andprovide a physical
introduction for students having been through the standard first course in
quantum mechanics. Those who are already to some extent familiar with
many-body quantum theory may immediately start with Chapter 4 and use
the first chapters only for reference (for instance to explain the notation
used). Care has been taken to present the material of Part II in such a way
that it should be readable also for somebody who is not very familiar with
quantum fieldtheory, yet without goingsmoothlyaroundthereefs. Somere-
marksscattered throughoutthetext address themorephysics-minded math-
ematician.
Notation is systematic throughout the text. Bold math (r) denotes a
usually three-dimensional vector, z∗ is the complex conjugate to the com-
plex number z, Aˆ is the operator corresponding to the quantum observable
A (for brevity the same symbol is used to denote an operator in different
representation spaces), Aˆ† is the Hermitian conjugate to Aˆ, det A means
ik
|| ||
the determinant of the matrix A , while A or i A k is the matrix el-
ik ik
|| || h | | i
ement. In cross-references, (2.1) refers to Formula(e) 1 in Chapter 2. (In
somecasestheformulanumber referstoacouple oflines ofmathematicalex-
pressions not interrupted by text lines; the formula number mostly appears
on or below the last of those lines.) Citations are given as the author(s) and
year in square brackets. In case the reader is confused by some notion he
has no definite idea of, it can most probably be found in the index at the
end of the book with a reference to places in the text, where the notion is
Introduction 11
explained or put into context.
Density functional theory can be built up in several versions: (i) as
a theory with particle densities (summed over spin variables) and spin-
independent externalpotentialsonly,irrespectivewhetherthequantumstate
is spin-polarized or not; (ii) as a theory with spin-up and spin-down den-
sities and external potentials which possibly act differently on spin-up and
spin-down particles for collinear polarization situations with one global spin
quantization direction (generalization to more than two eigenvalues of the
z-component of the spin is straightforward); and finally, (iii) as a general
theory with (spatially diagonal) spin-density matrices and general doubly
indexed spin-dependent potentials. All three versions are treated in the
present text, the two former cases are considered in parallel throughout by
consequently using a combined variable x = (r,s) of spatial position r and
z-component of spin s. Most expressions given refer immediately to the
spin-independent cases, if x is read as synonymous with r. The reason for
presenting the material in this way is that practically all formulae thereby
refer simultaneously to both cases, whereas the most general case of spin-
density matrices does not lead to essentially new aspects in many questions
addressed in large parts of the text.
These are lecture notes, hence neither an attempt in any respect is made
to give a complete reference list of existing literature, nor are questions
of priority addressed. The cited literature was selected exclusively (and
subjectively of course) to facilitate the access to the subject, and to give
advice for delving deeper into related subjects. Nevertheless, key original
papers which have introduced a new approach or a new theoretical tool are
cited, since the author is convinced that in particular students will benefit
from reading the key original papers.
Part I:
NON-RELATIVISTIC THEORY