Page 252 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.3 Non-degenerate perturbation theory 243
X (0) (0) (0) (1)
(0)
^
(0)
(0)
(E ÿ E ) a nj hø jø iˆ (E ÿ E )a nk ˆÿH
k n k j k n kn
j(6ˆn)
where again equation (9.26) is utilized. If the eigenvalue E (0) is non-degen-
n
(0) (0)
erate, then E k cannot equal E n for all k and n and we can divide by
(0) (0)
(E ÿ E ) to solve for a nk
k n
^
ÿH (1)
a nk ˆ (0) kn (9:32)
E ÿ E (0)
k n
The situation where E (0) is degenerate requires a more complex treatment,
n
which is presented in Section 9.5. The ®rst-order correction ø (1) is obtained by
n
combining equations (9.30) and (9.32)
^
X H (1)
ø (1) ˆÿ kn ø (0) (9:33)
n (0) (0) k
k(6ˆn) E k ÿ E n
Second-order corrections
The second-order correction E (2) to the eigenvalue E n is obtained by multi-
n
plying equation (9.23) by ø (0) and integrating over all space
n
^
^
(0)
(0)
(1)
(2)
(1)
(1)
(0)
(0)
(1)
hø jH (0) ÿ E jø i‡hø jH jø iÿ E hø jø i
n n n n n n n n
(0)
^
(2)
(0)
ˆÿhø jH jø i‡ E (2)
n
n
n
where the normalization of ø (0) has been noted. Application of the hermitian
n
^ (0)
property of H cancels the ®rst term on the left-hand side. The third term on
^ (2)
the left-hand side vanishes according to equation (9.31). Writing H for
nn
(0) ^ (2)
(0)
hø jH jø i and substituting equation (9.33) then give
n n
^
^
(1)
(1)
(0)
E (2) ˆ H (2) ‡hø jH jø i
n nn n n
^
X ^ (1) H (1) X jH j
^
(1) 2
H
^
^
ˆ H (2) ÿ nk kn ˆ H (2) ÿ kn (9:34)
nn (0) (0) nn (0) (0)
k(6ˆn) E k ÿ E n k(6ˆn) E k ÿ E n
^
^
^ (1)
where we have also noted that H (1) equals H (1) because H is hermitian.
nk kn
In many applications there is no second-order term in the perturbed
^ (2)
Hamiltonian operator so that H is zero. In such cases each unperturbed
nn
eigenvalue E (0) is raised by the terms in the summation corresponding to
n
eigenvalues E (0) less than E (0) and lowered by the terms with eigenvalues E (0)
k n k
(0)
greater than E . The eigenvalue E (0) is perturbed to the greatest extent by the
n
n
(0)
terms with eigenvalues E (0) close to E . The contribution to the second-order
k
n
correction E (2) of terms with eigenvalues far removed from E (0) is small. For
n
n
(0)
the lowest eigenvalue E , all of the terms are negative so that E (2) is negative.
0
0