The Variation Method - Methods of Molecular Quantum Mechanics - page 71

calculation of the excited P state is larger than that resulting for the
excited S state. The splitting between the P levels is about one order of
magnitude smaller than that observed for the S levels.

APPENDIX: THE INTEGRALS J, K, J 0 AND K 0

The two-electron one-centre integrals J, K, J 0 , and K 0 occurring in the
calculation of the first excited states of He can be evaluated as one-electron
integrals in spherical coordinates once the appropriate electrostatic
potentials are known. With reference to the 1s, s, and 2p z STOs defined
by Equations 3.61-3.63, the electrostatic potentials are evaluated using
the one-centre Neumann expansion for 1 = r 12 , giving

ð d r 2 ½ 1s ðr 2 Þ

2
r 12 ¼

1
r 1 1 exp ð 2c 0 Þð 1 þ c 0 r 1 Þ

J 1s 2 ð r 1 Þ¼ ½ ð 4 : 65 Þ

ð d r 2 ½ s ðr 2 Þ 1s ðr 2 Þ
r 12

J s1s ð r 1 Þ¼

1 = 2 1
r

" ! #

c 0 3 c s 5
3

12
ð c 0 þ c s Þ

4
3 r þ

2
3 r

2

¼ 1 exp ð 2 rÞ 1 þ

3

ð 4 : 66 Þ

ð d r 2 ½ 1s ðr 2 Þ 2p z ðr 2 Þ
r 12

J 1s2p z ð r 1 ; uÞ¼

ð 4

: 67 Þ

8c 3 = 0 c 5 = 2

1
r

p
ð c 0 þ c p Þ

2 3

¼

3 cos

u 1 exp ð 2 rÞð 1 þ 2 r þ 2 r þr Þ

2

where 2 r ¼ð c 0 þ c s ; p Þ r 1 . It is important to note that the radial part of

the potentials can be also evaluated in spheroidal coordinates by
choosing r 1 ¼ fixed as Roothaan (1951a) did. Once the potentials are
known, the basic two-electron integrals needed are easily evaluated in
spherical coordinates. The results are simple functions of the orbital
exponents:

5
8 c 0

ð 1s 2 j 1s 2 Þ¼ ð 4 : 68 Þ