Page 325 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

316 Appendix F
1 á 1 1
X (st) j ÿr j 2 X á! á
r e [L (r)] dr ˆ (st)
á
(á!) 2 (á ÿ j)!
ሠj 0 ሠj
á
Equating coef®cients of (st) on each side yields
3
1 (á!)
j ÿr j 2
r e [L (r)] dr ˆ (F:21)
á
0 (á ÿ j)!
Equations (F.20) and (F.21) may be combined into a single expression
3
1 (á!)
j
j
j ÿr
r e L (r)L (r)dr ˆ ä áâ (F:22)
â
á
0 (á ÿ j)!
For í ˆ 1, equation (F.19) becomes
X X á â 1 j
1
1
s t
j
r j‡1 ÿr L (r)L (r)dr
e
á!â! á â
ሠj ∠j 0
1
X (á ‡ 1)! á á‡1 á‡1 á á á‡1
ˆ [(st) ‡ (st) ÿ s t ÿ s t ]
(á ÿ j)!
ሠj
Equating coef®cients of like powers of s and t on both sides of this equation, we see
that

1
j
j
r j‡1 ÿr L (r)L (r)dr ˆ 0; â 6ˆ á, á 1 (F:23)
e
â
á
0
and that
2
1 á![(á ‡ 1)!]
j
r j‡1 ÿr L (r)L á‡1 (r)dr ˆÿ (F:24)
j
e
á
0 (á ÿ j)!

1 (á ‡ 1)! á! (2á ÿ j ‡ 1)(á!) 3
r j‡1 ÿr á j 2 2 ‡ ˆ (F:25)
e [L (r)] dr ˆ (á!)
0 (á ÿ j)! (á ÿ 1 ÿ j)! (á ÿ j)!
The term in which ⠈ á ÿ 1 is equivalent to the term in which ⠈ á ‡ 1 after the
dummy indices á and â are interchanged. Equations (F.23), (F.24), and (F.25) are
pertinent to the wave functions for the hydrogen atom.
Completeness
We de®ne the set of functions ÷ kj (r) by the relation
1=2
(k ÿ j)! j ÿr j
÷ kj (r) ˆ r e L (r) (F:26)
k
(k!) 3
According to equation (F.22), the functions ÷ kj (r) constitute an orthonormal set. We
1
now show that this set is complete.
Substitution of equation (F.15) into (F.26) gives
r e 1=2 d kÿ j
ÿ j r
k ÿr
÷ kj (r) ˆ (ÿ1) j (r e ) (F:27)
k!(k ÿ j)! dr kÿ j
If we apply equation (A.11), we may express the derivative in (F.27) as
1 D. Park, personal communication. This method parallels the procedure used to demonstrate the complete-
ness of the set of functions in equation (D.15).