The Variation Method - Methods of Molecular Quantum Mechanics - page 63

namely:

c 0

c 0

5
8 c 0 ¼ c 0 2 Z

5
16

« w ¼ 2

2 Zc 0

þ ð 4 :

27 Þ

In (4.26), the first term on the right is the one-electron matrix element
already found for H, while the second is the two-electron repulsion
integral between the two spherical 1s charge distributions written in
charge density notation:

ðð d r 1 d r 2 ½ 1s ðr 2 Þ 1s ðr 2 Þ
r 12

h 1s 1 1s 2 j r 1

1s ðr 1 Þ 1 s ðr 1 Þ

¼ ð 1s 2 j 1s 2

12 j 1s 1 1s 2 i¼

½ Þ

ð 4 :

28 Þ

This new two-electron integral is evaluated in terms of the purely radial
electrostatic potential J 1s (r 1 ):

ð d r 2 ½ 1s ðr 2 Þ 1s ðr 2 Þ
r 12

exp ð 2c 0 r 1 Þ
r 1

1
r 1

J 1s ð r 1 Þ¼ ¼ ð 1 þ c 0 r 1 Þ 4 : 29 Þ

when use is made of the one-centre Neumann expansion for 1 = r 12

(Magnasco, 2007). Then, integral (4.28) is readily obtained by integration
in spherical coordinates:

ð d r 1 J 1s ð r 1 Þ½ 1s ðr 1 Þ 1s ðr 1 Þ ¼

5
8 c 0

ð 1s 2 j 1s 2

Þ¼ ð 4 : 30 Þ

Optimization of (4.27) with respect to c 0 gives

5
16

c 0 ¼ Z ð 4 : 31 Þ

as best value for the orbital exponent and

2

5
16

« w ð best Þ¼ Z : ð 4 : 32 Þ

as best energy for the S(1s 2 ) configuration of the He-like atom.