Computational Chemistry

G 22 ¼

X^2

t¼ 1

X^2

u¼ 1

Ptu ð 22 jtuÞ#

1

2

ð 2 ujt 2 Þ



i.e: G 22 ¼

X^2

t¼ 1

Pt 1 ð 22 jt 1 Þ#

1

2

ð 21 jt 2 Þ



þPt 2 ð 22 jt 2 Þ#

1

2

ð 22 jt 2 Þ



¼P 11 ð 22 j 11 Þ#

1

2

ð 21 j 12 Þ



þP 12 ð 22 j 12 Þ#

1

2

ð 22 j 12 Þ



þP 21 ð 22 j 21 Þ#

1

2

ð 21 j 22 Þ



þP 22 ð 22 j 22 Þ#

1

2

ð 22 j 22 Þ



ð 5 : 108 Þ

Each element of the electron repulsion matrixGhas eight 2-electron repulsion
integrals, and of these 32 there appear to be 14 different ones:

fromG 11 : (11|11), (11|12), 12|11), (11|21), (11|22), (12|21)

new withG 12 ¼G 21 : (12|12), (12|22)

new withG 22 : (22|11), (21|12), (22|12), (22|21), (21|22), (22|22)

However, examination of Eq.5.73shows that many of these are the same. It is
easy to see that if the basis functions are real (as is almost always the case) then

ðrsjtuÞ¼ðrsjutÞ¼ðsrjtuÞ¼ðsrjutÞ¼ðtujrsÞ¼ðtujsrÞ¼ðutjrsÞ

¼ðutjsrÞð 5 : 109 Þ

Taking this into account, there are only six unique two-electron repulsion inte-
grals, whose values are:

ð 11 j 11 Þ¼ 0 : 7283 ð 21 j 21 Þ¼ 0 : 2192

ð 21 j 11 Þ¼ 0 : 3418 ð 22 j 21 Þ¼ 0 : 4368

ð 22 j 11 Þ¼ 0 : 5850 ð 22 j 22 Þ¼ 0 : 9927

ð 5 : 110 Þ

The integrals (11|11) and (22|22) represent repulsion between two electrons
both in the same orbital (f 1 orf 2 , respectively), while (22|11) represents repulsion
between an electron inf 2 and one inf 1 ; (21|11) could be regarded as representing
the repulsion between an electron associated withf 2 andf 1 and one confined tof 1 ,
and analogously for (22|21), while (21|21) can be thought of as the repulsion
between two electrons both of which are associated withf 2 andf 1 (Fig.5.10).
Note that in theTandVterms of the Fock matrix elements, the operator in the
integrals is –ð 1 = 2 Þr^2 and ZH=rH1or ZHe=rHe1, while in theGterms it is 1/r 12
(Eqs.5.101–5.103and5.73).
The overlap integrals are

S 11 ¼ 1 : 0000 S 12 ¼S 21 ¼ 0 : 5017 S 22 ¼ 1 : 0000 ð 5 : 111 Þ

5.2 The Basic Principles of the ab initio Method 219