Table Of ContentMethods in Computational
Molecular Physics
NATO ASI Series
Advanced Science Institutes Series
A series presenting the results of activities sponsored by the NATO Science Committee,
which aims at the dissemation of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
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A Life Sciences Plenum Publishing Corporation
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C Mathematical D. Reidel Publishing Company
and Physical Sciences Dordrecht, Boston and Lancaster
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Series C: Mathematical and Physical Sciences Vol. 113
Methods in Computational
Molecular Physics
edited by
G. H. F. Diercksen
Max-Planck-Institute for Physics and Astrophysics,
Garching, West Germany
and
S.Wilson
University of Oxford, Theoretical Chemistry Department,
Oxford, U.K.
D. Reidel Publishing Company
Dordrecht / Boston / Lancaster
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on
Methods in Computational Molecular Physics
Bad Windsheim, West Germany,
August 1982
Library of Congress Cataloging in Publication Data
Main entry under title:
Methods in computational molecular physics.
(NATO ASI series. Series C, Mathematical and physical sciences; no. 113)
"Published in cooperation with NATO Scientific Affairs Division."
Includes index.
1. Molecules-Measurement-Congresses. 2. Molecular structure-Congresses.
3. Mathematical physics-Congresses. I. Diercksen, G. H. F. II. Wilson, S. (Stephen),
1950- III. NATO Advanced Study Institute. IV. North Atlantic Treaty Organi-
zation. Scientific Affairs Division. V. Series.
QC173.M4748 1983 539'.6'0151 83-13929
ISBN-13: 978-94-009-7202-5 e-ISBN-13: 978-94-009-7200-1
DOl: 10.1007/978-94-009-7200-1
Published by D. Reidel Publishing Company
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CONTENTS
Preface vii
MOLECULAR INTEGRALS FOR GAUSSIAN TYPE FUNCTIONS
V.R. Saunders
ON THE EVALUATION OF EXPONENTIAL (SLATER) TYPE INTEGRALS 37
E. Otto Steinborn
BASIS SETS 7I
S. Wilson
MATRIX EIGENVECTOR METHODS 95
Ernest R. Davidson
GROUP THEORY APPLIED TO CI METHODS 115
B.T. Sutcliffe
THE MULTI CONFIGURATIONAL (MC) SCF METHOD 161
Bjorn O. Roos
THE DIRECT CI METHOD 189
Per E.M. Siegbahn
PAIR CORRELATION THEORIES 209
R. Ahlrichs
ON A GREEN'S FUNCTION METHOD FOR THE CALCULATION OF
IONIZATION SPECTRA IN THE OUTER AND INNER VALENCE REGION 227
W. von Niessen, J. Schirmer, and L.S. Cederbaum
INTRODUCTORY POLARIZATION PROPAGATOR THEORY 249
Jens Oddershede
DIAGRAMMATIC MANY-BODY PERTURBATION THEORY 273
S. Wilson
SCHRODINGER SPECTRA 299
P.W. Langhoff
vi CONTENTS
COMPUTERS AND COMPUTATION IN MOLECULAR PHYSICS 335
G.H.F. Diercksen, N.E. Grliner, and J. Steuerwald
PARTICIPANTS 351
INDEX 361
PREFACE
This NATO Advanced Study Institute was concerned with modern
ab initio methods for the determination of the electronic structure
of molecules. Recent years have seen considerable progress in
computer technology and computer science and these developments
have had a very significant influence on computational molecular
physics. Progress in computer technology has led to increasingly
larger and faster systems as well as powerful minicomputers.
Simultaneous research in computer science has explored new methods
for the optimal use of these resources. To a large extent develop
ments in computer technology, computer science and computational
molecular physics have been mutually dependent. The availability
of new computational resources, particularly minicomputers and,
more recently, vector processors, has stimulat'ed a great deal of
research in molecular physics. Well established techniques have
been reformulated to make more efficient use of the new computer
technology and algorithms which were previously computationally
intractable have now been successfully implemented. This research
has given a new and exciting insight into molecular structure and
molecular processes by enabling smaller systems to be studied in
greater detail and larger systems to be studied for the first time.
Although significant computational advances have been made over
the past few years, perhaps the most exciting developments have
resulted from theoretical innovations. The application of the
unitary group and the symmetric group to the molecular electronic
structure problem has been very fruitful. The direct configuration
interaction method continues to be popular. The Green's function
and polarization propagator methods have given new physical in
sight. The many-body perturbation theory has been shown to be an
efficient and accurate approach to the electron correlation
problem.
In all of the courses given the interplay between theoretical
innovations and the development of efficient computational schemes
were emphasized.
This NATO Advanced Study Institute was financially sponsored
by the NATO Scientific Affairs Division. The organizers wish to
express their gratitude for this support.
G.H.F. Diercksen
Dr. S. Wilson
MOLECULAR INTEGRALS FOR GAUSSIAN TYPE FUNCTIONS
V.R. Saunders
Science & Engineering Research Council, Daresbury
Laboratory, Daresbury, Warrington WA4 4AD, UK
1. INTRODUCTION
The Gaussian type function (GTF) was introduced into quantum
chemistry by Boys [1], who showed that the molecular integrals in
volving GTFs take a reasonably simple form. The field has been re
viewed by Shavitt [2], and the present author [3], the latter in
1975, and it is our present purpose to summarize the considerable
progress made since 1975.
The scope of the work will cover the evaluation of all the in
tegrals required to compute the molecular electronic energy using
the SchrBdinger spin-free and Breit-Pauli spin dependent Hamilton
ians, both in the Born-Oppenheimer approximation. The evaluation
of the derivatives of the SchrBdinger integrals with respect to
nuclear displacements is briefly described, as are integrals which
arise when one wishes to evaluate one electron properties; for ex
ample, dipole, electric field and electric field gradient.
2 BASIS FUNCTIONS
2.1 The 'Simple' GTF.
The unnormalized three dimensional 'simple' GTF is defined by:
(1)
where a, the orbital exponent, is real and positive, and where rA
denotes the distance between a fixed point, A, with co-ordinates
(A ,A ,A ), and a variable point, normally i;terpreted as the co-
x y z
1
G. H. F. Dierck~en and S. Wil~on (ed~.), Method~ in Computational Molecular Phy~ic~, 1-36.
© 1983 by D. Reidel Publi~hing Company.
2 V. R. SAUNDERS
ordinates of an electron, ~ = (x,y,z) so that:
(2a)
(2b)
The most immediately obvious property of G[a,A] is that it is sepa
rable in the co-ordinates x, y and z. A further, and extremely im
portant property concerns the 'Gaussian product theorem' [1-3],
which states that the product of the two 'simple' GTF, located a
centres A and B, is itself a simple GTF multiplied by a constant
factor, KAB, l~cated at a point P along the line segment AB, with
exponent p, where -
p a + b (3a)
P (~ + b~)/p (3b)
KAB exp[-ab AB2/p] (3c)
AB A- - B (3d)
2.2 The Gaussian Lobe Orbitals.
The Gaussian lobe orbitals (GLOs) [4] may be constructed by
taking linear combinations of the simple GTF, these GTF being loca
ted on different centres. The orbital exponents, linear coeffici
ents and centres of the GTF are normally chosen so that a good ap
proximation to some other category of GTF (for example'the Spheri
calor Cartesian GTF, to be. described below) is given. The major
disadvantage of the GLOs is that the molecular integrals involving
them are expressed as small differences of potentially very large
quantities, so that some care is needed if adequate numerical pre
cision is to be obtained. A further disadvantage of the GLOs is
that they cannot be made to transform under symmetry operations ex
actly as the target GTF they are supposed to be simulating, except
in the limit that the numerical precision problems become over
whelming, so that special care must be taken when symmetry adapt
ing the basis set. Associated with this symmetry problem is the
fact that calculations using GLOs are not strictly invariant to
changes in the rotational orientation chosen for the molecule.
The greatest advantage of the GLO is that the formalism for
computing the molecular integrals is very simple, but we refer to
a large literature [4] for further details.
MOLECULAR INTEGRALS FOR GAUSSIAN TYPE FUNCTIONS 3
2.3 The "Trigonometric" GTF.
The trigonometric GTF (TGTF) are defined [5] through:
(4)
with T* the complex conjugate of T. Allison et al. [5] showed that
the TGTF could be expressed as the product of a constant factor and
a 'simple' GTF, with the latter being of complex origin. Thus let:
A" ~/2a (5a)
A A' + iA" (5b)
when it may be shown directly that:
T[a,A',k] = G[a,A]exp[a(~2 _ ~'2)] (6)
It has further been shown that molecular integrals derived for
the 'simple' GTF of real origin may be easily generalized to the
complex origin case, but we refer to [5] for the details, merely
noting that the 'Gaussian product theorem' eqn.(3), also applies
to GTF of complex origin.
Perhaps the major potential use of the TGTF (and generaliza
tions thereof) is in the treatment of certain magnetic properties
(for example, magnetic shielding constants) in a gauge invariant
manner [6] within a finite perturbation theory approach. The avail
ability of formulae for the molecular integrals over the TGTF im
plies that mixed integrals between GTF and plane wave functions are
also available, and this may prove useful in the treatment of cer
tain continuum problems [7], such as charge exchange in heavy par
ticle collisions.
2.4 The Cartesian GTF.
A more general class of function than the 'simple' GTF was
first proposed by Boys [1], the cartesian GTF (CGTF), which take
the form:
G[a,A,i,j,k] = xi yj zk exp[-a r2] (7)
A A A A
where i,j and k (the 'quantum numbers') are integers ~ O. In com
mon with the 'simple' GTF, the CGTF are separable in the co-ordin
ates x, y and Z:
G[a,A,i,j,k] (8a)
(8b)
