Computational Chemistry

in Eq.5.103the variable is the distance of the electron from the helium nucleus;ZH
andZHeare 1 and 2, respectively.
From Eq.5.100the two-electron contribution to each Fock matrix element is

Grs¼

Xm

t¼ 1

Xm

u¼ 1

Ptu ðrsjtsÞ#

1

2

ðrujtsÞ



ð 5 : 104 Þ

Each elementGrsis calculated from a density matrix elementPtu(Eqs.5.80
and5.81) and two two-electron integrals (rs|tu) and (ru|ts) (Eqs.5.73and5.77).
The required one-electron integrals for calculating the Fock matrixFare

T 11 ¼ 0 : 6249 T 12 ¼T 21 ¼ 0 : 2395 T 22 ¼ 1 : 1609

V 11 ðHÞ¼# 1 : 0300 V 12 ðHÞ¼V 21 ðHÞ¼# 0 : 4445 V 22 ðHÞ¼# 0 : 6563

V 11 ðHeÞ¼# 1 : 2555 V 12 ðHeÞ¼V 21 ðHeÞ¼# 1 : 1110 V 22 ðHeÞ¼# 2 : 8076

ð 5 : 105 Þ

To see which two-electron integrals are needed we evaluate the summation in
Eq.5.104for each of the matrix elements (G 11 ,G 12 ,G 21 ,G 22 ):

G 11 ¼

X^2

t¼ 1

X^2

u¼ 1

Ptu ð 11 jtuÞ#

1

2

ð 1 ujt 1 Þ



i.e: G 11 ¼

X^2

t¼ 1

Pt 1 ð 11 jt 1 Þ#

1

2

ð 11 jt 1 Þ



þPt 2 ð 11 jt 2 Þ#

1

2

ð 12 jt 1 Þ



¼P 11 ð 11 j 11 Þ#

1

2

ð 11 j 11 Þ



þP 12 ð 11 j 12 Þ#

1

2

ð 12 j 11 Þ



þP 21 ð 11 j 21 Þ#

1

2

ð 11 j 21 Þ



þP 22 ð 11 j 22 Þ#

1

2

ð 12 j 21 Þ



ð 5 : 106 Þ

G 12 ¼G 21 ¼

X^2

t¼ 1

X^2

u¼ 1

Ptu ð 12 jtuÞ#

1

2

ð 1 ujt 2 Þ



i:e: G 12 ¼G 21 ¼

X^2

t¼ 1

Pt 1 ð 12 jt 1 Þ#

1

2

ð 11 jt 2 Þ



þPt 2 ð 12 jt 2 Þ#

1

2

ð 12 jt 2 Þ



¼P 11 ð 12 j 11 Þ#

1

2

ð 11 j 12 Þ



þP 12 ð 12 j 12 Þ#

1

2

ð 12 j 12 Þ



þP 21 ð 12 j 21 Þ#

1

2

ð 11 j 22 Þ



þP 22 ð 12 j 22 Þ#

1

2

ð 12 j 22 Þ



ð 5 : 107 Þ

218 5 Ab initio Calculations