Hartree-Fock Ab Initio Treatment of Crystalline Systems

201 Pages·1988·11.01 MB·English

by  C. Pisani

Editors
Prof. Dr. Gaston Berthier Prof. Dr. Hans H. Jaffe
Universite de Paris Department of Chemistry
Institut de Biologie University of Cincinnati
Physico-Chimique Cincinnati, Ohio 452211USA
Fondation Edmond de Rothschild
13, rue Pierre et Marie Curie
F-75005 Paris
Prof. Dr. Joshua Jortner
Institute of Chemistry
Prof. Dr. Michael J. S. Dewar
Tel-Aviv University
Department of Chemistry
IL-61390 Ramat-Aviv
The University of Texas
Tel-Aviv/Israel
Austin, Texas 78712/USA
Prof. Dr. Hanns Fischer
Physikalisch-Chemisches Institut
Prof. Dr. Werner Kutzelnigg
der Universitat ZOrich
Lehrstuhl fOr Theoretische Chemie
Ramistr.76
der Universitat Bochum
CH-8001 ZOrich
Postfach 102148
0-4630 Bochum 1
Prof. Dr. Kenichi Fukui
Kyoto University
Dept. of Hydrocarbon Chemistry
Kyoto/Japan Prof. Dr. Klaus Ruedenberg
Department of Chemistry
Prof. Dr. George G. Hall
Iowa State University
4 Westgate Court
Ames, Iowa 50010/USA
Highroad - Chilwell
Nottingham NG9 4BT
UK
Prof. Dr. Jacopo Tomasi
Prof. Dr. JOrgen Hinze Dipartimento di Chi mica e
Fakultat fOr Chemie Chimica Industriale
Universitat Bielefeld Universita di Pisa
Postfach 8640 Via Risorgimento, 35
0-4800 Bielefeld 1-56100 Pisa
Lecture Notes in
Chemistry
Edited by G. Berthier M.J.S. Dewar H. Fischer
K. Fukui G. G. Hall J. Hinze H. H.J affe J. Jortner
W. Kutzelnigg K. Ruedenberg J. Tomasi
48
C. Pisani R. Dovesi
C. Roetti
Hartree-Fock
Ab Initio Treatment
of Crystalline Systems
Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo
Authors
C. Pisani
R. Dovesi
C. Roetti
Dipartimento di Chimica Inorganica, Chimica Fisica e Chimica dei Materiali
Universita di Torino
Via Giuria 5,1-10125 Torino, Italy
ISBN-13: 978-3-540-19317-3 e-ISBN-13: 978-3-642-93385-1
001: 10.1007/978-3-642-93385-1
This work is subject to copyright. All rights are reserved, whether the whole or part of the matenal
is concerned, speCifically the rights of translation, reprinting, re-use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication
of this publication or parts thereof is only permitted under the proviSIOns of the German Copyright
Law of September 9, 1965, In its version of June 24, 1985, and acopyright fee must always be
paid. Violations fall under the prosecution act of the German Copyright Law
© Springer-Verlag Berlin Heidelberg 1988
Foreword
This book presents a computational scheme for calculating the
electronic properties of crystalline systems at an ab-ini tio
Hartree-Fock level of approximation.
The first chapter is devoted to discussing in general terms the limits
and capabilities of this approximation in solid state studies, and to
examining the various options that are open for its implementation.
The second chapter illustrates in detail the algorithms adopted in one
specific computer program, CRYSTAL, to be submitted to QCPE. Special
care is given to illustrating the role and in:fluence of computational
parameters, because a delicate compromise must always be reached
between accuracy and costs.
The third chapter describes a number of applications, in order to
clarify the possible use of this kind of programs in solid state
physics and chemistry.
Appendices A, B, and C contain various standard expressions, formulae,
and definitions that may be useful for reference purposes; appendix D
is intended to facilitate the interpretations of symbols, conventions,
and acronyms that occur in the book.
Thanks are due to all those who have contributed to the implementation
and test of the CRYSTAL program, especially to V.R. Saunders and M.
Causal, and to F. Ricca, E. Ferrero, R. Or lando, E. Ermondi, G.
Angonoa, P. Dellarole, G. Baracco.
The authors want to express their warmest gratitude to V.R. Saunders,
one of the coauthors of CRYSTAL, for useful suggestions and the
revision of part of the manuscript; to M. Causal, for his help in
preparing a number of data tables, and for his contribution in the
analysis of correlation corrections; to J. Tomasi, R. Orlando, and L.
Salasco for the careful reading of the text.
Many lengthy calculations have been expressly performed for this book,
particularly for testing the influence of computational parameters on
numerical accuracy and computer times: this has been made possible by
generous contributions from the Consorzio per il Sistema Informativo
Piemonte (CSI Piemonte).
TABLE OF CONTENTS
CHAPTER I. DIFFERENT APPROACHES TO THE STUDY OF THE ELECTRONIC
PROPERTIES OF PERIODIC SYSTEMS
1.1 Many-electron systems: the viewpoint of theoretical chemists
and physicists 1
1.2 A mosaic of options for the ab initio treatment of
crystalline systems 5
a) Introduction
b) Hamiltonians
c) Basis sets
d) Solution techniques
e) An example: beryllium
1.3 Specific features of crystalline with respect to molecular
Hartree-Fock computational schemes 13
a) HF schemes in solid state physics
b) Basis set selection
c) The treatment of Coulomb interactions
d) The treatment of exchange interactions
e) Sampling and integration in reciprocal space
f) Exploitation of local symmetry
1.4 Overcoming the limitations of the all-electron HF
perfect-crystal model 28
a) Introduction
b) Pseudopotentials
c) Correlation corrections
d) Local defects in crystals
CHAPTER II. IMPLEMENTATION OF THE HARTREE-FOCK EQUATIONS FOR
PERIODIC SYSTEMS
11.1 Introductory remarks 34
11.2 Basis functions and charge distributions 35
a) Atomic orbitals, Bloch functions, and representation of
operators
b) Compact expressions for charge distributions
11.3 Basic equations 40
11.4 The Coulomb series and the Madelung problem
a) General considerations
b) Rearrangement of the Coulomb series: partition of the h
space in a "bielectronic" and a "monoelectronic" zone
c) Further partition of the bielectronic zone: the bipolar
expansion
v
d) Further partition of the monoelectronic zone: the two
center expansion
e) Remarks and comments on the "partition and multipolar
expansion scheme"
f) On the evaluation of the integrals
g) Evaluation of field and multipole integrals
h) Integrals involving infinite summations
i) Numerical examples on the Coulomb series
11.5 The exchange series 74
a) Introduction
b) Long range behavior of the bond order matrix and influence
on total energy and wave function
c) Truncation criteria of the exchange series
11.6 Interpolation and integration in reciprocal space 86
a) Theory
b) Examples of application
11.7 Symmetry properties 96
a) Local symmetry operators
b) Transformation properties of the AOs
c) Symmetry properties of the Bloch functions and of the COs
d) Transformation properties of direct space integrals and
matrices
e) The irreducible set
f) Sums of two electron integrals
g) Multipole and field integrals
h) Symmetry and SCF procedure. Fock matrix and energy
expression
11.8 Choice of basis set and related problems 111
11.9 The CRYSTAL program 119
a) General scheme of the program
b) Further considerations about the computational parameters
c) SCF convergence
d) Limits of applicability
CHAPTER III. CALCULATION OF OBSERVABLE QUANTITIES IN THE HF
APPROXIMATION
111.1 Energy and energy related quantities 125
a) General considerations
b) Polyacetylene, as a test of accuracy of computational
techniques
c) Basis set and numerical accuracy effects: different
crystalline phases of MgO
d) The slab model in the study of surfaces and their
adsorption properties: MgO (001) and (011) and Be (0001)
crystal faces
e) Formation energy of crystals: MgO, diamond, graphite and
lithium
VI
III.2 Band structure and density of states (DOS) 144
a) One-electron HF levels and single particle excitation
spectrum of crystals
b) Projected DOSs and Mulliken populations
c) Chemical information from band structure and DOSs:
3D and 2D systems
III.3 Electron charge density and related quantities 150
a) Computational techniques and general considerations
b) Total and difference electron charge density maps:
beryllium, a-alumina
c) Comparison of calculated and experimental x-ray structure
factors: beryllium
III.4 Electron Momentum Distribution (EMe) and related quantities 157
a) Computational techniques: Momentum density, Compton
profiles (CP) and Autocorrelation function (AF)
b) Momentum distribution properties: lithium and MgO
c) Mulliken analysis of the AF: the case of MgO
APPENDICES
A. Solid harmonics and multi120lar eX12ansion of Coulomb
interactions 169
B. Definition of Brillouin Zone and related quantities 172
C. The McMurchie-Davidson technique for the evaluation of
the molecular integrals 174
Cl. Hermite gaussian type functions (HGTF)
C2. The Gaussian product theorem and the expansion in HGTF
C3. The expansion coefficients E
a) Recursion in 2
b) Recursion in 2 and Iml = 2
c) Method of generation
C4. Overlap and kinetic integrals
C5. Nuclear attraction integrals
C6. Two electron repulsion integrals
D. Symbols and notations 182
REFERENCES 184
CHAPTER I. DIFFERENT APPROACHES TO THE STUDY OF THE ELECTRONIC
PROPERTIES OF PERIODIC SYSTEMS
1.1 MANY-ELECTRON SYSTEMS THE VIEWPOINT OF THEORETICAL CHEMISTS
AND PHYSICISTS
It is the purpose of this book to present and discuss an ab initio
Hartree-Fock (HF) scheme for the calculation of the electronic
structure of crystalline systems. The techniques adopted are in a
sense midway between those currently used in molecular quantum
chemistry and those traditionally employed in solid state physics.
In past years, the implementation of computer programs for the study
of molecules and crystals has followed almost independent patterns,
and the state of the art in the two fields of research presents
markedly distinctive features, although the basic problem is the same:
the study of a many-electron system in the Coulomb field of fixed
nuclei.
Theoretical chemists have succeeded in developing powerful, general
purpose, universally adopted ab initio programs for molecules.
Consider for instance GAUSSIAN-82 (Binkleyet al 1981), the newborn of
a family of high-quality molecular programs developed by Pople and
coworkers during almost twenty years' work, and which may be obtained
on request from Prof. Pople at the Carnegie-Mellon University,
Pittsburgh, Pennsylvania. It is primarily designed for users
interested in applications of quantum chemistry methods rather than in
the calculation itself; the input is therefore extremely easy, and the
output is self-explanatory. Its capabilities include:
1. calculation of one- and two-electron integrals over atomic
orbitals;
2. Self-Consistent Field (SCF)-HF calculations for closed- and
open-shell molecular systems;
3. evaluation of one-electron properties of the HF wavefunction;
4. automated geometry optimization of the molecule, and calculation of
force constants;
5. harmonic vibrational analysis;
6. correlation energy calculated by using one of the following
choices: configuration interaction, M~ller-Plesset perturbation
theory or coupled cluster techniques.
GAUSSIAN-82 is by no means tbe only ab initio program in current use
for the calculation of molecular properties. However, the
computational schemes that have proved most successful use essentially
the same ingredients: the linearized HF-Roothaan equations are solved,
by using as a basis set a small number of atomic orbitals (AO) per
atom; in turn, the AOs are expressed as a linear combination of
gaussian type orbitals (GTO), or primitives, with appropriate exponent
coefficients and "contraction" coefficients; after obtaining the
molecular orbitals (MO), eigenvectors of the Fock matriX, through a
SCF procedure, the correlation correction to the ground state
wavefunction and properties and the description of excited states is
usually performed by means of configuration interaction (CI) or
2
perturbation techniques. In all cases much ingenuity has been spent in
making the use of these programs as easy and transparent as possible
in spite of their enormous complexity.
The sequence of steps that has finally led to these accurate and
reliable machines for the solution of the molecular problem is
brilliantly reconstructed in a recent booklet by Schaefer (1985). A
posteriori, theoretical quantum chemistry appears to have followed
quite a linear path, with a substantial concentration of efforts, from
the original formulation of the HF equations (Fock 1930, Slater 1930)
to their linearized expression by Roothaan (1951), from Boys' (1950 a,
1950 b, 1956) anticipation of present day techniques, to their full
realization; this has implied, e.g., the selection of standard basis
sets of assessed efficiency (Clementi and Roetti 1974, Huzinaga 1984),
the development of powerful integration techniques (Shavitt 1963,
McMurchie and Davidson 1978), the efficient handling of hundreds of
thousand configurations in CI calculations (Nesbet 1963, Yoshimine
1973, Saunders and van Lenthe 1983) and the appearance of new fruitful
ideas for the treatment of the correlation problem (Cizek 1966,
Krishnan et al 1980, Guest and Wilson 1980).
In his comment on Barnett's ( 1963) paper introducing the POLYATOM
program, Schaefer (1985) writes:
"Visionaries have for some years imagined a futuristic 'black
box' computer program, to which the bench chemist specifies a
desired molecule and a series of properties of interest.
After a few moments of cogitation, the computer politely
returns the answers, reliable to specified tolerances".
With all the risks of improper use or sheer misuse of these powerful
instruments, the present situation is not too far from that vision,
wi th regard to the HF solution and the corresponding one-electron
properties, and if small molecules not containing heavy atoms are
considered.
In comparison with the situation described above, a much greater
variety of techniques are currently employed for the description of
the electronic properties of crystals, resulting in much less
standardization. This is so for both intrinsic and historical reasons.
Much of the theory of molecules and molecular reactivity can be
constructed from information concerning the ground state and its total
energy. The primary aim of molecular quantum chemistry is therefore
understanding the reasons for the stability of certain aggregates of
atoms, and predicting how the binding energy depends on the
geometrical arrangement of the nuclei. In this respect, the preference
of theoretical chemists for the HF approximation is easily understood,
since, it provides, in a variational sense, the best monoelectronic
approximation for the ground state, that is the single determinant
wave function with the lowest expectation value for energy.
On the contrary, the problem of crystal stability is not of primary
importance in solid state physics, at least when chemically simple
crystalline compounds are considered, such as elemental or binary
compounds. As a rule, these kinds of crystal are well characterized
and they can generally be classified rather accurately, according to
one of a few selected categories. For each of them, well established
empirical or semi-empirical schemes exist for correlating crystal
structure and stability with the chemical nature of the atoms involved
(O'Keeffe and Navrotsky 1981); in many instances, especially in ionic
3
systems, the binding energy can be calculated with high accuracy using
sui tably optimized two-body potentials (Catlow and Mackrodt 1982).
Only recently has attention been given to complicated crystalline
systems where structural predictions are difficult, and where
interesting solid-solid phase transitions or chemical reactions can
occur.
For these reasons, the problem of describing the ground state of real
crystals and calculating its energy has historically not been the
object of much concern for solid state physicists. This attitude is
well expressed by the following excerpt from a recent book by Inkson
(1984):
" When considering how to describe an interacting system, we
should be very clear what it is we require. First of all, it
will not be useful to consider the ground state; it will
almost certainly be a highly complicated object and, apart
perhaps from its formation energy from some other state, of
very little interest. What we are normally interested in is
what happens to the system when it is disturbed, by some
experimental probe for instance. What is required then, are
expectation values of operators, transitions between states,
time development in particular circumstances, etc.".
These kind of phenomena are however very difficult to account for
satisfactorily ab initio, that is, only from the knowledge of the
atomic number and geometrical arrangement of the nuclei, and using the
basic quantum mechanical equations. An extraordinary variety of
approaches has therefore flourished in solid state physics: a wide
spectrum of sophisticated mathematical tools are employed, specific
hamiltonians are used for the different problems, and the real systems
under investigation are described according to different models. Two
prototype schemes emerge due to their simplicity and importance: the
homogeneous electron gas and the tight binding (TB) model.
The homogeneous electron gas has attracted enormous attention, as a
prototype of condensed many-electron systems, as a good model for the
response properties of simple metals, and as a necessary reference
point for the treatment of periodic non-homogeneous systems (Lundqvist
and March 1983). Some of the peculiarities of present day solid state
physics with respect to molecular quantum mechanics can probably be
traced back to the central role assumed by the electron gas problem in
the former case. First, the HF approximation performs unsatisfactorily
here. The binding energy of the system is grossly underestimated;
still more important, the distribution of the one-electron energies
associated with the HF canonical orbitals (plane waves, in this case)
does not provide reliable information on the spectrum of single
electron excitation energies. In particular, the density of states of
the electron gas as obtained from HF calculations is zero at the Fermi
energy: this unphysical feature, related to the nature of the "bare"
exchange operator is preserved in all cases, when rigorous HF
calculations for metallic systems are performed (Monkhorst,1979).
These failures are probably at the root of the deep mistrust of solid
state physicists with regard to HF approaches. Second, the use of
plane waves (PW) or augmented plane waves (APW) as a flexible and
universal basis set appears nearly mandatory if the electron gas has
to be included as a limiting case among the systems to be treated;
localized basis functions, such as the AOs used in molecular quantum
chemistry seem quite inadequate for this purpose. Third, the
homogeneous electron gas is the natural reference point and the exact
limiting case for density-functional (DF) theories {Hohenberg and Kohn