Physical Chemistry , 1st ed.

is not a bad approximation.) For the energy of the ground state of H 2 ,one
needs to evaluate

EH 2 ^12 

6

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)]*

Hˆ[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] d 1 d 2

where the 6 on the integral sign means that it is a sixfold integration over the
three coordinates of electron 1 and the three coordinates of electron 2. This in-
tegral can be expanded by multiplying out the terms of the wavefunction. We
get four sextuple-integrals

EH 2 ^12 

6

[1sH1(1)  1 sH2(2)]*Hˆ[1sH1(1)  1 sH2(2)d 1 d 2

+ 

6

[1sH1(2)  1 sH2(1)]*Hˆ[1sH1(1)  1 sH2(2)] d 1 d 2

+ 

6

[1sH1(1)  1 sH2(2)]*Hˆ[1sH1(2)  1 sH2(1)] d 1 d 2

+ 

6

[1sH1(2)  1 sH2(1)]*Hˆ[1sH1(2)  1 sH2(1)] d 1 d 2

As complicated as this looks, each term can be broken down into a number of
one-electron parts that can be approximated using the known solution for the hy-
drogen atom. This is very reminiscent of the earlier treatment of the helium atom.
But just as for the helium atom, the terms that involve the repulsion between the
two electrons cannot be separated and so cannot be evaluated analytically. Those
terms, one from each of the integrals above, have the following forms:



6

[1sH1(1)  1 sH2(2)]*

4 

e



2

0 r^212

[1sH1(1)  1 sH2(2)] d 1 d



6

[1sH1(2)  1 sH2(1)]*

4 

e



2

0 r

2

12

[1sH1(2)  1 sH2(1)] d 1 d 2 (13.13)

J 12

as well as



6

[1sH1(2)  1 sH2(1)]*

4 

e



2

0 r^212

[1sH1(1)  1 sH2(2)] d 1 d 2



6

[1sH1(2)  1 sH2(1)]*

4 

e



2

0 r

2

12

[1sH1(1)  1 sH2(2)] d 1 d 2 (13.14)

K 12

Equation 13.13 represents two electrons in defined atomic orbitals. The first
electron is in H1’s atomic orbital and the second electron is in H2’s atomic or-
bital, or vice versa. Since the two hydrogen atoms are the same, the two inte-
grals are equal to each other. The operator is the repulsion due to both elec-
trons having the same, negative charge. Integrals of the form in equation 13.13
are called Coulomb integralsand are represented by the letter J. Since coulom-
bic forces were known by classical mechanics, an integral involving such
coulombic effects is not surprising for multielectron systems (indeed, they ap-
peared in the helium atom).
However, equation 13.14 is something different. The coulombic repulsion
between negatively charged electrons is still a part of the operator, but the

13.10 Valence Bond Theory 449