Page 352 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Evaluation of two-electron interaction interval 343
ð 1

P l (cos è 2 ) sin è 2 dè 2 ˆ P l (ì)P 0 (ì)dì ˆ 2ä l0
0 ÿ1
where equations (E.18) and (E.19) have been introduced. Thus, only the term with
l ˆ 0 in the summation does not vanish and we have
e ÿ(r 1 ‡r 2 )
2 2
I ˆ 16ð 2 r r dr 1 dr 2 (J:4)
1 2
r .
In the second procedure, we substitute equation (J.3) directly into (J.1) and evaluate
the integral over è 2
ð
ð
sin è 2 1 2
dè 2 ˆ (1 ‡ s ÿ 2s cos è 2 ) 1=2
2
0 (1 ‡ s ÿ 2s cos è 2 ) 1=2 s 0
1 2 1=2 2 1=2
ˆ [(1 ‡ s ‡ 2s) ÿ (1 ‡ s ÿ 2s) ]
s
1
ˆ [(1 ‡ s) ÿ (1 ÿ s)] ˆ 2
s
The integrals over è 1 , j 1 , and j 2 are the same as before and equation (J.4) is obtained.
Since r . is the larger of r 1 and r 2, the integral I in equation (J.4) may be written in
the form
" #

1 1 r 1 1
I ˆ 16ð 2 e ÿr 1 2 1 e ÿr 2 2 2 e ÿr 2 r 2 dr 2 dr 1
r
r dr 2 ‡
0 r 1 0 r 1

1
2
ˆ 16ð 2 e ÿr 1 r 1 f[2 ÿ (r ‡ 2r 1 ‡ 2)e ÿr 1 ] ‡ r 1 (r 1 ‡ 1)e ÿr 1 g dr 1 ˆ 16ð ( ‡ )
5
2 5
1
8
8
0
Accordingly, the ®nal result is
ÿ(r 1 ‡r 2 )
e 2
I ˆ drr 1 drr 2 ˆ 20ð (J:5)
r 12