Computational Chemistry

whereg 1 r 1 s¼g 1 r'g 1 sand so on. Thus with contracted Gaussians as basis functions,
each two-electron integral becomes a sum of easily calculated two-center two-
electron integrals. Gaussian integrals can be evaluated so much faster than Slater
integrals that the use of contracted Gaussians instead of Slater functions speeds up the
calculation of the integrals enormously, despite the larger number of integrals.
Discussions of the number of integrals in an ab initio calculation usually refer to
those at the contracted Gaussian level, rather than the greater number engendered by
the use of primitive Gaussians; thus the program Gaussian 92 [ 29 ] says that both an
STO-1G and an STO-3G calculation on water use the same number (144) of two-
electron integrals, although the latter clearly involves more “primitive integrals.” The
fruitful suggestion to use Gaussians in molecular calculations came from Boys (1950
[ 35 ]); it played a major role in making ab initio calculations practical, and this is
epitomized in the names of the Gaussian series of programs, which are primarily
devoted to ab initio and DFT (Chapter 7) and are among the most widely-used
quantum mechanics-oriented computational chemistry programs [ 36 ].
Fast calculation of integrals is particularly important for the two-electron inte-
grals, as their number increases rapidly with the size of the molecule and the basis
set (basis sets are discussed inSection 5.3.3). Consider a calculation on water with
an STO-1G basis set (and bear in mind that the smallest basis set normally used in
ab initio calculations is the STO-3G set). In a standard ab initio calculation we use
at least one basis function for each core orbital and each valence-shell orbital. Thus
the oxygen requires five basis functions, for the 1s,2s,2px,2pyand 2pzorbitals; we
can designate these functionsf 1 ,f 2 ,...f 5 , and denote the 1s hydrogen functions,
one for each H,f 6 andf 7. In computational chemistry atoms beyond hydrogen and
helium in the periodic table are called “heavy atoms”, and the computational “first
row” is lithium–neon. With experience, the number of heavy atoms in a molecule
gives a quick indication of about how many basis functions will be invoked by a
specified basis set. Following the procedure for HHe+in Eq.5.106:

G 11 ¼

X^7

t¼ 1

X^7

u¼ 1

Ptu ð 11 jtuÞ#

1

2

ð 1 ujt 1 Þ



Nowuruns from 1 to 7 andtfrom 1 to 7, soG 11 will consist of 49 terms, each
containing two two-electron integrals for aG 11 total of 98 integrals. The Fock
matrix for seven basis functions is a 7'7 matrix with 49 elements,G 11 ,G 12 ,...,
G 17 ,...G 77 , so apparently there are 49' 98 ¼4,802 two-electron integrals.
Actually, many of these are duplicates (Gij¼Gji, so ann'nFock matrix has
only aboutn^2 /2differentelements), differ from other integrals only in sign, or are
very small, and the number of unique nonvanishing two-electron integrals is 119
(calculated with Gaussian 92 [ 29 ]). For an STO-1G calculation on hydrogen
peroxide (12 basis functions), there are ca. 700 unique nonvanishing two-electron
integrals (cf. a naive theoretical maximum of 41,472). The usual formula for
estimating the maximum number of unique two-electron integrals for a set ofm

236 5 Ab initio Calculations