Page 315 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

306 Appendix E

We then substitute equation (E.15) for w m and take the ®rst and second derivatives as
indicated to obtain
" #
2
d P m dP m m 2 m
2
l
l
(1 ÿ ì ) ÿ 2ì ‡ l(l ‡ 1) ÿ P (ì) ˆ 0 (E:17)
l
dì 2 dì 1 ÿ ì 2
Equation (E.17) is the associated Legendre differential equation.
Orthogonality
m
m
Equation (E.17) as satis®ed by P (ì) and by P (ì) may be written as
l l9
" #
m 2
d 2 dP l m m
(1 ÿ ì ) ‡ l(l ‡ 1) ÿ P (ì) ˆ 0
dì dì 1 ÿ ì 2 l
and
" #
m 2
d 2 dP l9 m m
(1 ÿ ì ) ‡ l9(l9 ‡ 1) ÿ P (ì) ˆ 0
dì dì 1 ÿ ì 2 l9
m
m
If we multiply the ®rst by P (ì) and the second by P (ì) and then subtract, we have
l9 l
m m
d 2 dP l m d 2 dP l9 m m
m
P l9 (1 ÿ ì ) ÿ P l (1 ÿ ì ) ˆ [l9(l9 ‡ 1) ÿ l(l ‡ 1)]P P l9
l
dì dì dì dì
We then add to and subtract from the left-hand side the term
m
dP dP m
l9
l
2
(1 ÿ ì )
dì dì
so as to obtain
m m
d 2 m dP l m dP l9 m m
(1 ÿ ì ) P ÿ P ˆ [l9(l9 ‡ 1) ÿ l(l ‡ 1)]P P
dì l9 dì l dì l l9
We next integrate with respect to ì from ÿ1to ‡1 and note that
m m 1
dP l m dP l9
2
m
(1 ÿ ì ) P l9 ÿ P l ˆ 0
dì
dì
ÿ1
giving
1

m
m
[l9(l9 ‡ 1) ÿ l(l ‡ 1)] P P dì ˆ 0
l l9
ÿ1
If l9 6ˆ l, then the integral must vanish

1
m
m
P (ì)P (ì)dì ˆ 0 (E:18)
l l9
ÿ1
m
so that the associated Legendre polynomials P (ì) with ®xed m form an orthogonal
l
set of functions. Since equation (E.18) is valid for m ˆ 0, the Legendre polynomials
P l (ì) are also an orthogonal set.
Normalization
We next wish to evaluate the integral I lm
1

m
2
I lm [P (ì)] dì
l
ÿ1
As a ®rst step, we evaluate I l0