Page 287 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

278 Molecular structure
2 2
b 2 b "
(2) 1 3b 1 b 3 " ÿ 1  3 2
E nJ ˆÿ 2ìù 2 ÿ ì ù 3 n ‡ 2 ÿ 8ì ù 4 (30n ‡ 30n ‡ 11)
2
3
Substitution of equations (10.40), (10.45a), and (10.45c) leads to
2
(2) 4B 2 e 2 2 2B R e U (3) (0) ÿ 1 
e
E nJ ˆÿ " ù 2 J (J ‡ 1) ‡ ì"ù 3 n ‡ 2 J(J ‡ 1)
2
2
(3)
" [U (0)] 2 h ÿ  2 i
ÿ 30 n ‡ 1 ‡ 7 (10:49)
288ì ù 4 2 2
3
The nuclear energy levels in this higher-order approximation are given to
second order in the perturbation by combining equations (10.41), (10.47), and
(10.49) to give
(0) (1) (2)
E nJ E nJ ‡ E nJ ‡ E nJ
ÿ 1  ÿ 1  2
ˆ U(0) ‡ "ù n ‡ ÿ "ùx e n ‡ ‡B e J(J ‡ 1)
2 2
2 2 ÿ 1 
ÿ DJ (J ‡ 1) ÿ á e n ‡ J(J ‡ 1) (10:50)
2
where we have de®ned
!
" 5[U (3) (0)] 2
(4)
x e ÿ U (0) (10:51a)
16ì ù 3 3ìù 2
2
4B 2 e
D (10:51b)
" ù 2
2
!
(3)
ÿ6B 2 e R e U (0)
á e 1 ‡ (10:51c)
"ù 3ìù 2
!
2 (3) 2
1 " (4) 7[U (0)]
U(0) U(0) ‡ U (0) ÿ (10:51d)
64 ìù 9ìù 2
The approximate expression (10.50) for the nuclear energy levels E nJ is
1
observed to contain the initial terms of a power series expansion in (n ‡ ) and
2
2
1 2
J(J ‡ 1). Only terms up to (n ‡ ) and [J(J ‡ 1)] and the cross term in
2
1
(n ‡ )J(J ‡ 1) are included. Higher-order terms in the expansion may be
2
found from higher-order perturbation corrections.
The second term on the right-hand side of equation (10.50) is the energy of a
harmonic oscillator. Since the factor x e in equation (10.51a) depends on the
third and fourth derivatives of the internuclear potential at R e , the third term in
equation (10.50) gives the change in energy due to the anharmonicity of that
potential. The fourth term is the energy of a rigid rotor with moment of inertia
I. The ®fth term is the correction to the energy due to centrifugal distortion in