Page 254 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.3 Non-degenerate perturbation theory 245
2 3
^ (1) 2
X jH j
^
^
2
E n E (0) ‡ ëH (1) ‡ ë 4 H (2) ÿ (0) kn (0) 5 (9:40)
nn
nn
n
k(6ˆn) E k ÿ E n
The corresponding eigenfunction ø n to second order is obtained by combining
equations (9.19), (9.33), and (9.39)
^
X H (1)
ø n ø (0) ÿ ë kn ø (0)
n (0) (0) k
k(6ˆn) E k ÿ E n
2 (1) (1) 3
^
^
^
^
^ (1)
X ÿH (2) X H kj H jn H (1) H
‡ ë 2 4 kn ‡ ÿ kn nn 5 ø (0)
(0)
(0)
(0) 2
E (0) ÿ E (0) (E (0) ÿ E )(E (0) ÿ E ) (E (0) ÿ E ) k
k(6ˆn) k n j(6ˆn) k n j n k n
(9:41)
While the eigenvalue E (0) for the unperturbed system must be non-degen-
n
erate for these expansions to be valid, some or all of the other eigenvalues E (0)
k
for k 6ˆ n may be degenerate. The summations in equations (9.40) and (9.41)
are to be taken over all states of the unperturbed system other than the state
(0)
ø . If an eigenvalue E (0) is g i -fold degenerate, then it is included g i times in
n i
the summations. If the unperturbed eigenfunctions have a continuous range,
then the summations in equations (9.40) and (9.41) must include an integration
over those states as well.
Relation to variation method
If we use the wave function ø (0) for the unperturbed ground state as a trial
0
^
function ö in the variation method of Section 9.1 and set H equal to
^ (1)
^ (0)
H ‡ ëH , then we have from equations (9.2), (9.18), and (9.24)
(0)
(0)
^
^
^
(1)
E ˆhöjHjöiˆhø jH (0) ‡ ëH jø iˆ E (0) ‡ ëE (1)
0 0 0 0
and E is equal to the ®rst-order energy as determined by perturbation theory. If
we instead use a trial function ö which contains some parameters and which
equals ø (0) for some set of parameter values, then the corresponding energy E
0
from equation (9.2) is at least as good an approximation as E (0) ‡ ëE (1) to the
0
0
true ground-state energy.
Moreover, if the wave function ø (0) ‡ ëø (1) is used as a trial function ö, then
0
0
the quantity E from equation (9.2) is equal to the second-order energy
determined by perturbation theory. Any trial function ö with parameters which
reduces to ø (0) ‡ ëø (1) for some set of parameter values yields an approximate
0
0
energy E from equation (9.2) which is no less accurate than the second-order
perturbation value.