Computational Chemistry

where

jr#rAj^2 ¼ðx#xAÞ^2 þðy#yAÞ^2 þðz#zAÞ^2

and jr#rBj^2 ¼ðx#xBÞ^2 þðy#yBÞ^2 þðz#zBÞ^2

ð 5 : 154 Þ

with the nuclear and electron positions in Cartesian coordinates (if these were nots-
type functions, the preexponential factor would contain one or more cartesian
variables to give the function – the “orbital” – nonspherical shape). It is not hard
to show that

gAgB¼aCe#aCjr#rCj

2

¼gC ð 5 : 155 Þ

The product ofgAandgBis the GaussiangC, centered atrC. Now consider the
general electron-repulsion integral

ðrsjtuÞ¼

ZZ

f$rð 1 Þfsð 1 Þf$tð 2 Þfuð 2 Þ

r 12

dv 1 dv 2 ð 5 : 156 ¼ 5 : 73 Þ

If each basis functionfwere a single, real Gaussian, then from Eq.5.155this
would reduce to

ðv=wÞ¼

ZZ

fvð 1 Þfwð 2 Þ

r 12

dv 1 dv 2 ð 5 : 157 Þ

i.e. three- and four-center two-electron integrals with four basis functions would
immediately simplify to tractable two-center integrals with two functions.
Actually, things are a little more complicated. A single Gaussian is a poor
approximation to the nearly ideal description of an atomic wavefunction that a Slater
function provides. Figure5.12shows that a Gaussian (designated STO-1G) is
rounded nearr¼0 while a Slater function has a cusp there (zero slope vs a finite
slope atr¼0); the Gaussian also decays somewhat faster than the Slater function at
larger. The solution to the problem of this poor functional behaviour is to use several
Gaussians to approximate a Slater function. In Fig.5.12a single Gaussian and a linear
combination of three Gaussians have been used to approximate the Slater function
shown; the nomenclature STO-1G and STO-3G mean “Slater-type orbital (approxi-
mated by) one Gaussian” and “Slater-type orbital (approximated by) three Gaus-
sians”, respectively. The Slater function shown is one suitable for a hydrogen atom in
a molecule (z¼1.24 [ 31 ]) and the Gaussians are the best fit to this Slater function.
STO-1G functions were used in our illustrative Hartree–Fock calculation on HHe+
(Section 5.2.3.6.5), and the STO-3G function is the smallest basis function used in
standard ab initio calculations by commercial programs. Three Gaussians are a good
speed versus accuracy compromise between two and four or more [ 31 ].
The STO-3G basis function in Fig.5.12is acontracted Gaussianconsisting of
threeprimitive Gaussianseach of which has acontraction coefficient(0.4446,

234 5 Ab initio Calculations