Page 316 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Legendre and associated Legendre polynomials 307
1

2
I l0 ˆ [P l (ì)] dì
ÿ1
We solve the recurrence relation (E.4) for P l (ì), multiply both sides by P l (ì), integrate
with respect to ì from ÿ1to ‡1, and note that one of the integrals vanishes according
to the orthogonality relation (E.18), so that
1 1
2 2l ÿ 1
[P l (ì)] dì ˆ ìP l (ì)P lÿ1 (ì)dì
ÿ1 l ÿ1
Replacing l by l ‡ 1 in equation (E.4), we can substitute for ìP l (ì) on the right-hand
side. Again applying equation (E.18), we ®nd that
1 2l ÿ 1 1

2 2
[P l (ì)] dì ˆ [P lÿ1 (ì)] dì
ÿ1 2l ‡ 1 ÿ1
This relationship can then be applied successively to obtain
1 1

2 (2l ÿ 1)(2l ÿ 3) [P lÿ2 (ì)] dì
2
[P l (ì)] dì ˆ
ÿ1 (2l ‡ 1)(2l ÿ 1) ÿ1
. . .
(2l ÿ 1)(2l ÿ 3) 1 1 2
ˆ [P 0 (ì)] dì
(2l ‡ 1)(2l ÿ 1)(2l ÿ 3) 3 ÿ1
1
1 2
ˆ [P 0 (ì)] dì
2l ‡ 1 ÿ1
Since P 0 (1) ˆ 1, the desired result is
1 1

2 1 2
[P l (ì)] dì ˆ dì ˆ (E:19)
ÿ1 2l ‡ 1 ÿ1 2l ‡ 1
We are now ready to evaluate I lm . From equation (E.10) we have
!
m
2
1 m 1 d P l d d mÿ1
P l
2 m
2 m
I lm ˆ (1 ÿ ì ) d P l dì ˆ (1 ÿ ì ) m mÿ1 dì
m
ÿ1 dì ÿ1 dì dì dì
Integration by parts gives
1 1 mÿ1 m
m
mÿ1
d P l d P l d P l d 2 m d P l
2 m
I lm ˆ (1 ÿ ì ) ÿ (1 ÿ ì ) dì (E:20)
dì m dì mÿ1 dì mÿ1 dì dì m
ÿ1 ÿ1
2
The integrated part vanishes because (1 ÿ ì ) ˆ 0at ì ˆÐ 1.
To evaluate the integral on the right-hand side of equation (E.20), we replace m by
2 mÿ1
m ÿ 1 in (E.16) and multiply by (1 ÿ ì ) to obtain
2 dw mÿ1
d w mÿ1
2 mÿ1
2 m
(1 ÿ ì ) ÿ 2mì(1 ÿ ì )
dì 2 dì
2 mÿ1
‡ [l(l ‡ 1) ÿ m(m ÿ 1)](1 ÿ ì ) w mÿ1 ˆ 0
which can be rewritten as

d 2 m dw mÿ1 2 mÿ1
(1 ÿ ì ) ˆÿ(l ‡ m)(l ÿ m ‡ 1)(1 ÿ ì ) w mÿ1 ˆ 0
dì dì
From equation (E.14) we see that