Computational Chemistry

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 two-electron integral (rs|tu) (Eq.5.156¼5.73). Suppose each basis
function is an STO-3G contracted Gaussian, i.e.

fr¼d 1 rg 1 rþd 2 rg 2 rþd 3 rg 3 r ð 5 : 158 Þ

and analogously forfs,ft, andfu. Then it is easy to see that

ðrsjtuÞ¼

ZZ

d 1 rd 1 sg 1 r 1 s

1

r 12

d 1 td 1 ug 1 t 1 udv 1 dv 2

þ

ZZ

d 1 rd 1 sg 1 r 1 s

1

r 12

d 1 td 2 ug 1 t 2 udv 1 dv 2 þ(((

þ

ZZ

d 3 rd 3 sg 3 r 3 s

1

r 12

d 3 td 3 ug 3 t 3 udv 1 dv 2

ð 5 : 159 Þ

123

0.5

1.0

Slater

STO-3G

STO-1G

0

r, Å

f (STO– 3G) = 0.4446 e–ar^2 + 0.5353 e–ar^2 + 0.1543 e–ar^2

= 0.0835 e–0.1689r^2 + 0.2678 e–0.6239r^2 + 0.2769 e–3.4253r^2

2 a

p

3/4 2 a

p

3/4 2 a

p

3/4

2 a

p

3/4

f (STO– 1G) = e–ar^2 = 0.3696e–0.4166r^2

z^3

p

1/2

f (Slater) = e–zr^ = 0.7790e–1.24r

f

Fig. 5.12 Comparison of Slater, STO-1G and STO-3G functions for hydrogen. The Slater
function shown is the most appropriate one for hydrogen in a molecular environment, and the
Gaussians are the best 1-G and 3-G fits to this Slater function. Slater and Gaussian functions are
usually characterized by parameters designatedz(zeta) anda, respectively, as shown [ 31 ]

5.3 Basis Sets 235