Page 261 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

252 Approximation methods

assume in the continuing presentation that all the roots are indeed different
from each other.

`Correct' zero-order eigenfunctions
The determination of the coef®cients c áã is not necessary for ®nding the ®rst-
order perturbation corrections to the eigenvalues, but is required to obtain the
`correct' zero-order eigenfunctions and their ®rst-order corrections. The coef®-
cients c áã for each value of á (á ˆ 1, 2, ... , g n ) are obtained by substituting
the value found for E (1) from the secular equation (9.65) into the set of
ná
in
simultaneous equations (9.64) and solving for the coef®cients c á2 , ... , c á g n
terms of c á1 . The normalization condition (9.57) is then used to determine c á1 .
This procedure uniquely determines the complete set of coef®cients c áã (á,
㠈 1, 2, ... , g n ) because we have assumed that all the roots E (1) are different.
ná
If by accident or by clever choice, the initial set of unperturbed eigenfunc-
tions ø (0) is actually the `correct' set, i.e., if in the limit ë ! 0 the perturbed
ná
eigenfunction ø ná reduces to ø (0) for all values of á, then the coef®cients c áã
ná
are given by c á㠈 ä áã and the secular determinant is diagonal

(1) (1)
H ÿ E 0 0
n1,n1 ná
(1) (1)
0 H ÿ E 0

n2,n2 ná ˆ 0

(1) (1)
0 0 H ÿ E ná
ng n ,ng n
The ®rst-order corrections to the eigenvalues are then given by
^
E (1) ˆ H (1) , á ˆ 1, 2, ... , g n (9:66)
ná
ná,ná
It is obviously a great advantage to select the `correct' set of unperturbed
eigenfunctions as the initial set, so that the simpler equation (9.66) may be
used. A general procedure for achieving this goal is to ®nd a hermitian operator
^ ^ (0) and H and has eigenfunctions ÷ á with non-
^ (1)
A that commutes with both H
degenerate eigenvalues ì á , so that
^ ^
(0)
(1)
^ ^
[A, H ] ˆ [A, H ] ˆ 0 (9:67)
and
^
A÷ á ˆ ì á ÷ á
^
^ (0)
Since A and H commute, they have simultaneous eigenfunctions. Therefore,
as the initial set of unperturbed eigenfunctions
we may select ÷ 1 , ÷ 2 , ... , ÷ g n
ø (0) ˆ ÷ á , á ˆ 1, 2, ... , g n
ná
^ ^ (1)
We next form the integral h÷ â j[A, H ]j÷ á i (â 6ˆ á), which of course vanishes
according to equation (9.67),