Page 253 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

244 Approximation methods
We also see that, in these cases, the ®rst-order correction ø (1) to the eigenfunc-
n
tion determines the second-order correction E (2) to the eigenvalue.
n
To obtain the second-order perturbation correction ø (2) to the eigenfunction,
n
we multiply equation (9.23) by ø (0) for k 6ˆ n and integrate over all space
kn
(0) ^ (0) (0) (2) (0) ^ (1) (1) (1) (0) (1)
hø jH ÿ E jø i‡hø jH jø iÿ E hø jø i
k n n k n n k n
(0) ^ (2) (0) (2) (0) (0)
ˆÿhø jH jø i‡ E hø jø i
k n n k n
^ (0)
As before, we apply the hermitian property of H , introduce the abbreviation
(2)
^
H , and use the orthogonality relation (9.26) to obtain
kn
(0) (0) (0) (2) (0) ^ (1) (1) (1) (0) (1) ^ (2)
(E ÿ E )hø jø i‡hø jH jø iÿ E hø jø iˆÿH (9:35)
k n k n k n n k n kn
We next expand the function ø (2) in terms of the complete set of unperturbed
n
eigenfunctions ø (0)
j
X
ø (2) ˆ b nj ø (0) (9:36)
n j
j(6ˆn)
where, without loss of generality, the term j ˆ n may be omitted for the same
reason that ø (0) is omitted in equation (9.30). The coef®cients b nj are complex
n
constants given by
(0) (2)
b nj ˆhø jø i (9:37)
n
j
Substitution of equations (9.24), (9.28), (9.30), and (9.37) into (9.35) gives
X
(0) (0) ^ (1) ^ (1) ^ (2)
(E ÿ E )b nk ‡ a nj H ÿ a nk H ˆÿH
k n kj nn kn
j(6ˆn)
or
X (1)
^
^
(2)
^
H ‡ a nj H ÿ a nk H (1)
kn kj nn
j(6ˆn)
b nk ˆÿ (0) (9:38)
(0)
(E ÿ E )
k n
Combining equations (9.32), (9.36), and (9.38), we obtain the ®nal result
2 (1) (1) 3
^
^
^
^
^ (1)
X ÿH (2) X H kj H jn H (1) H
ø (2) ˆ 4 kn ‡ ÿ kn nn 5 ø (0)
n (0) (0) (0) (0) (0) (0) (0) (0) 2 k
k(6ˆn) E k ÿ E n j(6ˆn) (E k ÿ E )(E j ÿ E ) (E k ÿ E )
n
n
n
(9:39)
Summary
The non-degenerate eigenvalue E n for the perturbed system to second order is
obtained by substituting equations (9.24) and (9.34) into (9.20) to give