Page 351 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

342 Appendix J
z

2
r
12
1
r
1 r
2
γ

y

x

Figure J.1 Distance between two particles 1 and 2 and their respective distances from
the origin.

z

r 2
12
1
r
2
r 1 θ 2

y

j 2

x

Figure J.2 Rotation of the coordinate axes in Figure J.1 so that the z-axis lies along r 1 .

ð ð 2ð 2ð
1 ÿ(r 1 ‡r 2 )
X e l 2 2
I ˆ s r r dr 1 dr 2 P l (cos è 2 ) sin è 2 dè 2 sin è 1 dè 1 dj 1 dj 2
1 2
r .
lˆ0 0 0 0 0
The integrals over è 1 , j 1 , and j 2 are readily evaluated. Since P 0 (ì) ˆ 1, we may write
the integral over è 2 as