Page 350 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Appendix J

Evaluation of the two-electron interaction integral

In the application of quantum mechanics to the helium atom, the following integral I
arises and needs to be evaluated
ÿ(r 1 ‡r 2 )
e
I ˆ drr 1 drr 2
r 12
ÿ(r 1 ‡r 2 )
e 2 2
ˆ r r sin è 1 sin è 2 dr 1 dè 1 dj 1 dr 2 dè 2 dj 2 (J:1)
1 2
r 12
where the position vectors rr i (i ˆ 1, 2) have components r i , è i , j i in spherical polar
coordinates and where
r 12 ˆjrr 2 ÿ rr 1 j
The distance r 12 is related to r 1 and r 2 by the law of cosines
2
2
r 2 12 ˆ r ‡ r ÿ 2r 1 r 2 cos ã (J:2)
2
1
where ã is the angle between rr 1 and rr 2 as shown in Figure J.1. The integration is
taken over all space for each position vector.
The integral I may be evaluated more easily if we orient the coordinate axes so that
the vector rr 1 lies along the positive z-axis as shown in Figure J.2. In that case, the
angle ã between rr 1 and rr 2 is equal to the angle è 2 . If we de®ne r . as the larger and
r , as the smaller of r 1 and r 2 and de®ne s by the ratio
r ,
s
r .
so that s < 1, then equation (J.2) may be expressed in the form
1 1 2 ÿ1=2
ˆ (1 ‡ s ÿ 2s cos è 2 ) (J:3)
r 12 r .
At this point, we may proceed in one of two ways, which are mathematically
equivalent. In the ®rst procedure, we note that from the generating function (E.1) for
Legendre polynomials P l , equation (J.3) may be written as
1
1 1 X l
ˆ P l (cos è 2 )s
r 12 r .
lˆ0
The integral I in equation (J.1) then becomes

341