Table Of ContentCONTRIBUTORS TO THIS VOLUME
DAVID M. BISHOP
s. BRATOZ
R. DAUDEL
FRANK E. HARRIS
PER-OLOV LOWDIN
H. MARGENAU
R. K. NESBET
KIM10 OHNO
J. STAMPER
ADVANCES IN
QUANTUM CHEMISTRY
EDITED BY
PER-OLOV LOWDIN
DEPARTMENT OF QUANTUM CHEMISTRY
UPPSALA UNIVERSITY
UPPSALA, SWEDEN
AND
QUANTUM THEORY PROJECT
UNIVERSITY OF FLORIDA
GAINESVILLE, FLORIDA
VOLUME 3-1967
ACADEMIC PRESS New York London
COPYRIGH0T 1967, BY ACADEMIPCR ESSIN C.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT
WRITTEN PERMISSION FROM THE PUBLISHERS.
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PRINTED IN THE UNITED STATES OF AMERICA
LIST OF CONTRIBUTORS
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
DAVIDM . BISHOP (25), University of Ottawa and National Research
Council, Ottawa, Canada
S. BRA TO^ (209), Centre de MCcanique Ondulatoire ApliquCe, Paris,
France
R. DAUDEL(1 61), Sorbonne and Centre de MCcanique Ondulatoire
ApliquCe, Paris, France
FRANKE. HARRIS(6 1), Department of Chemistry, Stanford University,
Stanford, California
PER-OLOVLO WDIN(3 23), Quantum Theory Project for Research in Atomic,
Molecular, and Solid State Chemistry and Physics, University of
Florida, Gainesville, Florida and Quantum Chemistry Group for
Research in Atomic, Molecular, and Solid-state Theory, Uppsala
University, Uppsala, Sweden
H. MARGENA(1U2 9), Yale University, New Haven, Connecticut
R. K. NESBET(l ), IBM Research Laboratory, San Jose, California
KIMIOO HNO(2 39), Department of Chemistry, Faculty of Science,Hokkaido
University, Sapporo, Japan
J. STAMPE(R12 9), Yale University, New Haven, Connecticut
V
PREFACE
In investigating the highly different phenomena in nature, scientists have
always tried to find some fundamental principles that can explain the
variety from a basic unity. Today they have not only shown that all the
various kinds of matter are built up from a rather limited number of atoms,
but also that these atoms are constituted of a few basic elements or building
blocks. It seems possible to understand the innermost structure of matter
and its behavior in terms of a few elementary particles: electrons, protons,
neutrons, photons, etc., and their interactions. Since these particles obey
not the laws of classical physics but the rules of modern quantum theory or
wave mechanics established in 1925, there has developed a new field of
“quantum science” which deals with the explanation of nature on this
ground.
Quantum chemistry deals particularly with the electronic structure of
atoms, molecules, and crystalline matter and describes it in terms of
electronic wave patterns. It uses physical and chemical insight, sophisticated
mathematics, and high-speed computers to solve the wave equations and
achieve its results. Its goals are great, but perhaps the new field can better
boast of its conceptual framework than of its numerical accomplishments.
It provides a unification of the natural sciences that was previously in-
conceivable, and the modern development of cellular biology shows that
the life sciences are now, in turn, using the same basis. “Quantum biology”
is a new field which describes the life processes and the functioning of the
cell on a molecular and submolecular level.
Quantum chemistry is hence a rapidly developing field which falls be-
tween the historically established areas of mathematics, physics, chemistry,
and biology. As a result there is a wide diversity of backgrounds among
those interested in quantum chemistry. Since the results of the research
are reported in periodicals of many different types, it has become increas-
ingly difficult both for the expert and the nonexpert to follow the rapid
development in this new borderline area.
The purpose of this serial publication is to try to present a survey of the
current development of quantum chemistry as it is seen by a number of the
internationallly leading research workers in various countries. The authors
vii
viii Preface
have been invited to give their personal points of view of the subject freely
and without severe space limitations. No attempts have been made to avoid
overlap-on the contrary, it has seemed desirable to have certain important
research areas reviewed from different points of view. The response from
the authors has been so encouraging that a fourth volume is now being
prepared.
The editor would like to thank the authors for their contributions which
give an interesting picture of the current status of selected parts of quan-
tum chemistry.
It is our hope that the collection of surveys of various parts of quantum
chemistry and its advances presented here will prove to be valuable and
stimulating, not only to the active research workers but also to the scientists
in neighboring fields of physics, chemistry, and biology, who are turning
to the elementary particles and their behavior to explain the details and
innermost structure of their experimental phenomena.
January, 1967 PER-OLOVLO WDIN
CONTENTS OF PREVIOUS VOLUMES
Volume I
The Schrodinger Two-Electron Atomic Problem
Egil A. Hylleraas
Energy Band Calculations by the Augmented Plane Wave Method
J. C. Slater
Spin-Free Quantum Chemistry
F. A. Matsen
On the Basis of the Main Methods of Calculating Molecular Electronic
Wave Functions
R. Daudel
Theory of Solvent Effects on Molecular Electronic Spectra
Sadhan Basu
The Pi-Electron Approximation
Peter G. Lykos
Recent Developments in the Generalized Huckel Method
Y. Z’Haya
Accuracy of Calculated Atomic and Molecular Properties
G. G. Hall
Recent Developments in Perturbation Theory
Joseph 0. Hirschfelder, W. Byers Brown, and Saul T. Epstein
AUTHOR INDEX-SUBJECT INDEX
Volume 2
Quantum Calculations, Which Are Accumulative in Accuracy, Unrestricted
in Expansion Functions, and Economical in Computation
S. F. Boys and P. Rajagopal
Zero Differential Overlap in n-Electron Theories
Inga Fischer- Hjalmars
Theory of Atomic Hyperfine Structure
S. M. Blinder
xiii
xiv Contents of Previous Volumes
The Theory of Pair-Correlated Wave Functions
R. Mc Weeny and E. Steiner
Quantum Chemistry and Crystal Physics, Stability of Crystals of Rare Gas
Atoms and Alkali Halides in Terms of Three-Atom and Three-Ion
Exchange Interactions
Lmrens Jansen
Charge Fluctuation Interactions in Molecular Biophysics
Herbert Jehle
Quantum Genetics and the Aperiodic Solid. Some Aspects on the Biological
Problems of Heredity, Mutations, Aging, and Tumors in View of the
Quantum Theory of the DNA Molecule
Per-Olov Lowdin
AUTHOR INDEX-SUBJECT INDEX
Approximate Hartree-Fock Calculations
on Small Molecules
R. K. NESBET
IBM Research Laboratory
Sun Jose, California
1. Introduction . . . . 1
The Molecular Hartree-Fock Approximation . . I
11. Computational Technique . . . . 3
A. Choice of Orbital Basis . . . . . 3
B. Computation of One-Electron Properties . . . 6
C. Molecular Energy Levels . . . . . . 8
111. Diatomic Molecules . . 9
A. Hydrides . . . . . . 9
B. Nz,CO, and BF . . . 15
C. Cz, BN, BeO, and LiF . . . . . . . 11
D. Other Diatomics . . . . . 17
IV. Polyatomic Molecules . . . . . . 18
A. Central Hydrides . . . . . 18
B. Linear Polyatomic Molecules . . . . . 19
C. Other Molecules . . . . 20
V. Discussion . . . 21
References . . . . . . 22
I. Introduction
The Molecular Hartree-Fock Approximation
For singlet ground states, the molecular Hartree-Fock approximation
consists of a variational calculation to find a Slater determinant that
minimizes the electronic energy, expressed as the mean value of' the
molecular Hamiltonian with fixed internuclear coordinates. For other
states, the traditional Hartree-Fock method would use as a trial function
the linear combination of Slater determinants needed to form a wave
function with specified spin or spatial point group quantum numbers. It
can easily be shown (e.g., Nesbet, 1961) that the N orthonormal orbitals of
a Hartree-Fock single determinant must satisfy equations
%o+i = Ei+i (1)
2 R. K. Nesbet
where 2Bis oan integrodifferential operator that contains terms dependent
on the density function
Hence Eqs. (I), while formally resembling one-particle Schrodinger equa-
tions, are actually nonlinear. For molecules, the Hartree-Fock equations
involve partial differential and integral operators with several independent
variables. As such, the equations are intractable to the standard methods
of numerical analysis.
As a consequence of developments of computational technique, it has
been possible in recent years to compute molecular Hartree-Fock wave
functions that compare well in accuracy with the corresponding functions
for atoms. There are three important elements in this work. First, the
matrix Hartree-Fock method, originally proposed by Roothaan (1 95 1) and
by Hall (1951) in the context of the molecular LCAO approximation,
replaces Eqs. (1) by the corresponding linear equations, with 2ore placed
by its matrix elements in a specified basis of atomic orbitals, and 4 replaced
by the vector representing the linear expansion of a molecular orbital
function in the same atomic orbital basis. The partial differential equations
are replaced by matrix equations whose elements are constructed from
multidimensional definite integrals. Second, it was recognised by Boys
(1 950a) that accurate molecular wave functions require the use of orbital
basis sets much more general than those considered in the LCAO approxi-
mation, where only a single basis orbital is used for each independent
occupied orbital in the component atoms. Third, the introduction of
electronic digital computers has made possible the enormously complex
calculations required to implement the matrix Hartree-Fock method in
an orbital basis sufficiently complete to produce quantitative results. The
efficient organization and programming of these calculations has been an
essential prerequisite to the results discussed here.
This article will review results obtained by this method for molecules
containing more than two or three electrons. This excludes the smallest
molecules, for which special methods can be used to give results of high
accuracy. The improvement in results obtained in going from simple
LCAO calculations to a reasonable approximation to a true Hartree-
Fock calculation has been substantial. For this reason, LCAO results will
not be discussed here, but the reader is referred to a review by Allen and
Karo (1960) and to papers by Rand (1960) and Mulliken (1962).
