Page 260 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.5 Degenerate perturbation theory 251
g n
X (1)
^
(1)
c áã (H ÿ E ä âã ) ˆ 0, á, ⠈ 1, 2, ... , g n (9:64)
nâ,nã ná
ãˆ1
^ (1)
Note that the integrals H are evaluated with the known initial set of
ná,nã
unperturbed eigenfunctions, in contrast to the integrals in equation (9.63),
(1)
(0)
which require the unknown functions ö . For a given eigenvalue E , the
ná ná
expression (9.64) is a set of g n linear homogeneous simultaneous equations,
one for each value of â (⠈ 1, 2, ... , g n )
^
(1)
^
^
^
⠈ 1: c á1 (H (1) ÿ E ) ‡ c á2 H (1) ‡ c á3 H (1) ‡ ‡ c á g n H (1) ˆ 0
n1,n1 ná n1,n2 n1,n3 n1,ng n
^
^
^
^
(1)
⠈ 2: c á1 H (1) ‡ c á2 (H (1) ÿ E ) ‡ c á3 H (1) ‡ ‡ c á g n H (1) ˆ 0
ná
n2,n3
n2,n1
n2,n2
n2,ng n
.
. .
(1)
^
^
^
^
⠈ g n : c á1 H (1) ‡ c á2 H (1) ‡ c á3 H (1) ‡ ‡ c á g n (H (1) ÿ E )
ná
ng n ,n3
ng n ,n2
ng n ,n1
ng n ,ng n
ˆ 0
Equation (9.64) has the form of (9.13) with the coef®cients c áã correspond-
^
(1)
ing to the unknown quantities x i and the terms (H (1) ÿ E ä âã ) correspond-
ná
nâ,nã
ing to the coef®cients a ki . Thus, a non-trivial solution for the g n coef®cients
^
c áã (㠈 1, 2, ... , g n ) exists only if the determinant with elements (H (1) ÿ
nâ,nã
(1)
E ä âã ) vanishes
ná

H (1) ÿ E (1) H (1) H (1)
n1,n1 ná n1,n2 n1,ng n
(1) (1) (1) (1)
H n2,n1 H n2,n2 ÿ E ná H ˆ 0 (9:65)
n2,ng n

(1) (1) (1) (1)
H H H ÿ E

ng n ,n1 ng n ,n2 ng n ,ng n ná
Only for some values of the ®rst-order correction term E (1) is the secular
ná
(1)
equation (9.65) satis®ed. This secular equation is of degree g n in E , giving
ná
g n roots
(1) (1) (1)
E , E , ... , E ng n
n1
n2
^ (1)
all of which are real because H is hermitian. The perturbed eigenvalues to
®rst order are, then
E n1 ˆ E (0) ‡ ëE (1)
n1
n
.
. .
ˆ E (0) ‡ ëE (1)
E ng n n ng n
If the g n roots are all different, then in ®rst order the g n -fold degenerate
unperturbed eigenvalue E (0) is split into g n different perturbed eigenvalues. In
n
this case, the degeneracy is removed in ®rst order by the perturbation. We