Page 259 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

250 Approximation methods
2
E ná ˆ E (0) ‡ ëE (1) ‡ ë E (2) ‡ , á ˆ 1, 2, ... , g n (9:59)
n ná ná
2
ø ná ˆ ø (0) ‡ ëø (1) ‡ ë ø (2) ‡ , á ˆ 1, 2, ... , g n (9:60)
ná
ná
ná
Note that in equation (9.59) the zero-order term is the same for all values of á.
^
In the limit ë ! 0, the Hamiltonian operator H approaches the unperturbed
^ (0)
operator H and the perturbed eigenvalue E ná approaches the degenerate
(0)
unperturbed eigenvalue E . The perturbed eigenfunction ø ná approaches a
n
function which satis®es equation (9.53), but this limiting eigenfunction may
(0)
not be any one of the initial functions ø . In general, this limiting function is
ná
(0)
some linear combination of the initial unperturbed eigenfunctions ø ,as
ná
expressed in equation (9.54). Thus, along with the determination of the ®rst-
(1)
order correction terms E (1) and ø , we must ®nd the set of unperturbed
ná
ná
eigenfunctions ö (0) to which the perturbed eigenfunctions reduce in the limit
ná
ë ! 0. In other words, we need to evaluate the coef®cients c áâ in the linear
combinations (9.54) which transform the initial set of unperturbed eigenfunc-
tions ø (0) into the `correct'set ö . Equation (9.60) is then replaced by
(0)
ná ná
2
ø ná ˆ ö (0) ‡ ëø (1) ‡ ë ø (2) ‡ , á ˆ 1, 2, ... , g n (9:61)
ná
ná
ná
The ®rst-order equations (9.22) and (9.24) apply here provided the additional
index á and the `correct' unperturbed eigenfunctions are used
^
^
(0)
(1)
(H (0) ÿ E )ø (1) ˆÿ(H (1) ÿ E )ö (0) (9:62)
ná
ná
ná
n
^
(0)
(1)
(0)
E (1) ˆhö jH jö i (9:63)
ná
ná
ná
However, equation (9.63) for the ®rst-order corrections to the eigenvalues
cannot be used directly at this point because the functions ö (0) are not known.
ná
To ®nd E (1) we multiply equation (9.62) by ø (0) , the complex conjugate of
ná
nâ
one of the initial unperturbed eigenfunctions belonging to the degenerate
(0)
eigenvalue E , and integrate over all space to obtain
n
^
^
(0)
(0)
(0)
(1)
(1)
(0)
hø jH (0) ÿ E jø iˆ ÿhø jH (1) ÿ E jö i
nâ n ná nâ ná ná
^ (0)
Applying the hermitian property of H , we see that the left-hand side
vanishes. Substitution of the expansion (9.54) for ö (0) using ã as the dummy
ná
expansion index gives
g n
X (0) ^ (1) (1) (0)
c áã hø jH ÿ E jø iˆ 0, á, ⠈ 1, 2, ... , g n
nâ ná nã
ãˆ1
If we introduce the abbreviation
(0)
^
^
(1)
(0)
H (1) hø jH jø i, â, 㠈 1, 2, ... , g n
nâ,nã nâ nã
and apply the orthonormality condition (9.55), this equation takes the form