Page 324 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Laguerre and associated Laguerre polynomials 315

To evaluate the left-hand integral, we substitute the analytical forms of the generating
functions from equation (F.10) to give
(st) j 1 j‡í ÿar
I ˆ r e dr (F:18)
(1 ÿ s) j‡1 (1 ÿ t) j‡1 0
where
s t 1 ÿ st
a 1 ‡ ‡ ˆ
1 ÿ s 1 ÿ t (1 ÿ s)(1 ÿ t)
The integral in equation (F.18) is just the gamma function (A.26), so that

1 Ã(j ‡ í ‡ 1) (j ‡ í)!
r j‡í ÿar dr ˆ ˆ , j ‡ í . 0
e
0 a j‡í‡1 a j‡í‡1
where we have restricted í to integer values. Thus, I in equation (F.18) is
í
j
(j ‡ í)!(st) (1 ÿ s) (1 ÿ t) í
I ˆ
(1 ÿ st) j‡í‡1
Applying the expansion formula (A.3), we have
1
X (j ‡ í ‡ i)!
(1 ÿ st) ÿ( j‡í‡1) ˆ (st) i
(j ‡ í)!i!
iˆ0
If we replace the dummy index i by á, where á ˆ i ‡ j, then this expression becomes
1
X (á ‡ í)!
(1 ÿ st) ÿ( j‡í‡1) ˆ (st) áÿ j
(j ‡ í)!(á ÿ j)!
ሠj
and I takes the form
1 (á ‡ í)!
X
í
I ˆ (1 ÿ s) (1 ÿ t) í (st) á
(á ÿ j)!
ሠj
Combining this result with equation (F.17), we have
X X á â 1 X (á ‡ í)!
1
1
1
s t
í
j
j
r j‡í ÿr L (r)L (r)dr ˆ (1 ÿ s) (1 ÿ t) í (st) á (F:19)
e
á!â! á â (á ÿ j)!
ሠj ∠j 0 ሠj
We now equate coef®cients of like powers of s and t on each side of this equation.
Since the integer í appears as an exponent of both s and t on the right-hand side, the
effect of equating coef®cients depends on the value of í. Accordingly, we shall ®rst
have to select a value for í.
For í ˆ 0, equation (F.19) becomes
X X á â 1 j X á!
1
1
1
s t
j ÿr
j
r e L (r)L (r)dr ˆ (st) á
á
á!â! â (á ÿ j)!
ሠj ∠j 0 ሠj
Since the exponent of s on the right-hand side is always the same as the exponent of t,
á â
the coef®cients of s t for á 6ˆ â on the left-hand side must vanish, i.e.

1
j ÿr
j
j
r e L (r)L (r)dr ˆ 0; á 6ˆ â (F:20)
â
á
0
Thus, the associated Laguerre polynomials form an orthogonal set over the range
j ÿr
0 < r < 1 with a weighting factor r e . For the case where s and t on the left-hand
side have the same exponent, we pick out the term ⠈ á in the summation over â,
giving