Page 326 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Laguerre and associated Laguerre polynomials 317
d kÿ j k ÿr k! d kÿ j 1 e irs i kÿ j k! 1 e irs kÿ j
(r e ) ˆ ds ˆ s ds
dr kÿ j 2ð dr kÿ j (1 ‡ is) k‡1 2ð (1 ‡ is) k‡1
ÿ1 ÿ1
so that ÷ kj (r) in integral form is
ÿ j r 1=2
j kÿ j
(ÿ1) i k!r e 1 e irs kÿ j
÷ kj (r) ˆ s ds (F:28)
2ð (k ÿ j)! (1 ‡ is) k‡1
ÿ1
To demonstrate that the set ÷ kj (r) is complete, we need to evaluate the sum
1
X
÷ kj (r)÷ kj (r9)
kˆ j
Expressing (F.28) in terms of the dummy variable of integration s for r and in terms of
t for r9, we obtain for the summation
1
X
÷ kj (r)÷ kj (r9)
kˆ j
" #
1 kÿ j kÿ j
1 ÿ j=2 (r‡r9)=2 1 1 i(rs‡r9t) X (ÿ1) k! (st)
ˆ (rr9) e e ds dt
4ð 2 (k ÿ j)! [1 ‡ i(s ‡ t) ÿ st] k‡1
ÿ1 ÿ1
kˆ j
By letting á ˆ k ÿ j, we may express the sum on the right-hand side as
1 á
X (ÿ1) (á ‡ j)! ÿ(ᇠj‡1) á ÿ( j‡1)
[1 ‡ i(s ‡ t) ÿ st] (st) ˆ j![1 ‡ i(s ‡ t)]
á!
áˆ0
where we have applied equation (A.3) to evaluate the sum over á. We now have

1 1 1 i(rs‡r9t)
X j! ÿ j=2 (r‡r9)=2 e
÷ kj (r)÷ kj (r9) ˆ (rr9) e ds dt (F:29)
4ð 2 [1 ‡ i(s ‡ t)] j‡1
kˆ j ÿ1 ÿ1
To evaluate the double integral, we introduce the variables u and v
s ‡ t s ÿ t
u ˆ , v ˆ or s ˆ u ‡ v, t ˆ u ÿ v
2 2
ds dt ˆ 2du dv
The double integral then factors into

i(r‡r9)u j
1 e 1 2ð r ‡ r9
2 du e i(rÿr9)v dv ˆ e ÿ(r‡r9)=2 [2ðä(r ÿ r9)]
(1 ‡ 2iu) j‡1 j! 2
ÿ1 ÿ1
where the ®rst integral is evaluated by equation (A.11) and the second by (C.6).
Equation (F.29) becomes
j
1
X r ‡ r9
÷ kj (r)÷ kj (r9) ˆ ä(r ÿ r9)
2(rr9) 1=2
kˆ j
By applying equation (C.5e), we obtain the completeness relation
1
X
÷ kj (r)÷ kj (r9) ˆ ä(r ÿ r9) (F:30)
kˆ j
demonstrating according to equation (3.31) that the set ÷ kj (r) is complete.