Consider an
s
type Gaussian centered on nucleus
A
and one on nucleus
B
; we are
considering real functions, which is what basis functions normally are:
g
A
¼
a
A
e
±
a
A
j
r
±
r
A
j
2
;
g
B
¼
a
B
e
±
a
B
j
r
±
r
B
j
2
ð
5
:
153
Þ
5.3 Basis Sets
233
where
j
r
±
r
A
j
2
¼ ð
x
±
x
A
Þ
2
þ ð
y
±
y
A
Þ
2
þ ð
z
±
z
A
Þ
2
and
j
r
±
r
B
j
2
¼ ð
x
±
x
B
Þ
2
þ ð
y
±
y
B
Þ
2
þ ð
z
±
z
B
Þ
2
ð
5
:
154
Þ
with the nuclear and electron positions in Cartesian coordinates (if these were not
s

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
g
A
g
B
¼
a
C
e
±
a
C
j
r
±
r
C
j
2
¼
g
C
ð
5
:
155
Þ
The product of
g
A
and
g
B
is the Gaussian
g
C
, centered at
r
C
. Now consider the
general electronrepulsion integral
ð
rs
j
tu
Þ ¼
Z Z
f
²
r
ð
1
Þ
f
s
ð
1
Þ
f
²
t
ð
2
Þ
f
u
ð
2
Þ
r
12
dv
1
dv
2
ð
5
:
156
¼
5
:
73
Þ
If each basis function
f
were a single, real Gaussian, then from Eq.
5.155
this
would reduce to
ð
v
=
w
Þ ¼
Z Z
f
v
ð
1
Þ
f
w
ð
2
Þ
r
12
dv
1
dv
2
ð
5
:
157
Þ
i.e. three and fourcenter twoelectron integrals with four basis functions would
immediately simplify to tractable twocenter 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. Figure
5.12
shows that a Gaussian (designated STO1G) is
rounded near
r
¼
0 while a Slater function has a cusp there (zero slope vs a finite
slope at
r
¼
0); the Gaussian also decays somewhat faster than the Slater function at
large
r
. The solution to the problem of this poor functional behaviour is to use several
Gaussians to approximate a Slater function. In Fig.
5.12
a single Gaussian and a linear
combination of three Gaussians have been used to approximate the Slater function
shown; the nomenclature STO1G and STO3G mean “Slatertype orbital (approxi
mated by) one Gaussian” and “Slatertype orbital (approximated by) three Gaus
sians”, respectively. The Slater function shown is one suitable for a hydrogen atom in
a molecule (
z
¼
1.24 [
31
]) and the Gaussians are the best fit to this Slater function.
STO1G functions were used in our illustrative Hartree–Fock calculation on HHe
+
(Section 5.2.3.6.5), and the STO3G 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 STO3G basis function in Fig.
5.12
is a
contracted Gaussian
consisting of
three
primitive Gaussians
each of which has a
contraction coefficient
(0.4446,
234
5 Ab initio Calculations
0.5353 and 0.1543). Typically, an ab initio basis function consists of a set of
primitive Gaussians bundled together with a set of contraction coefficients. Now
consider the twoelectron integral (
rs

tu
) (Eq.
5.156
¼
5.73
). Suppose each basis
function is an STO3G contracted Gaussian, i.e.
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 Fall '19
 dr. ahmed