Page 339 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

330 Appendix H

1 1 1
í
í
r uu9 dr ˆÿí r íÿ1 2 r uu9 dr
u dr ÿ
0 0 0
Combining the integral on the left-hand side with the last one on the right-hand side,
we obtain the desired result

1 í
í
r uu9 dr ˆÿ hr íÿ1 i nl (H:5)
0 2
To obtain the recurrence relation (H.2), we multiply equation (H.4) by r k‡1 u9 and
integrate over r
2
1 1 2Z 1 Z 1
k
r k‡1 u9u0 dr ˆ l(l ‡ 1) r kÿ1 uu9 dr ÿ r uu9 dr ‡ 2 2 r k‡1 uu9 dr
0 0 a 0 0 n a 0 0
l(l ‡ 1)(k ÿ 1) kÿ2 kZ kÿ1 (k ‡ 1)Z 2 k
ˆÿ hr i nl ‡ hr i nl ÿ hr i nl (H:6)
2 2
2 a 0 2n a
0
where equation (H.5) was applied to the right-hand side. The integral on the left-hand
side of (H.6) may be integrated by parts twice to give

1 1 d
r k‡1 u9u0 dr ˆÿ u9 (r k‡1 u9)dr
0 0 dr

1 1
k
ˆÿ(k ‡ 1) r u9u9 dr ÿ r k‡1 u9u0 dr
0 0

1 d 1
k
ˆ (k ‡ 1) u (r u9)dr ÿ r k‡1 u9u0 dr
0 dr 0

1 1 1
k
ˆ k(k ‡ 1) r kÿ1 uu9 dr ‡ (k ‡ 1) r uu0 dr ÿ r k‡1 u9u0 dr
0 0 0
The integral on the left-hand side and the last integral on the right-hand side may be
combined to give

1 (k ÿ 1)k(k ‡ 1) (k ‡ 1) 1
k‡1 kÿ2 k
r u9u0 dr ˆÿ hr i nl ‡ r uu0 dr (H:7)
0 4 2 0
where equation (H.5) has been used for the ®rst integral on the right-hand side.
Substitution of equation (H.4) for u0 in the last integral on the right-hand side of (H.7)
yields

1 (k ÿ 1)k(k ‡ 1) (k ‡ 1)l(l ‡ 1)
r k‡1 u9u0 dr ˆÿ hr kÿ2 i nl ‡ hr kÿ2 i nl
0 4 2
(k ‡ 1)Z kÿ1 (k ‡ 1)Z 2 k
ÿ hr i nl ‡ 2 2 hr i nl (H:8)
a 0 2n a 0
Combining equations (H.6) and (H.8), we obtain the recurrence relation (H.2).