Page 251 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

242 Approximation methods
^ (0)
applying the hermitian property of H to the ®rst term on the left-hand side,
(1) (0) (0)
^ (1)
and writing H kn for hø jH jø i, we may express equation (9.25) as
n
k
(0)
^
(0)
(1)
(E (0) ÿ E )hø jø iˆÿH (1) (9:27)
k n k n kn
The orthonormal eigenfunctions ø (0) for the unperturbed system are as-
j
sumed to form a complete set. Thus, the perturbation corrections ø (1) may be
n
expanded in terms of the set ø (0)
j
X X
ø (1) ˆ a nj ø (0) ˆ a nn ø (0) ‡ a nj ø (0)
n j j j
j j(6ˆn)
where a nj are complex constants given by
(0) (1)
a nj ˆhø jø i (9:28)
j
n
If the complete set of eigenfunctions for the unperturbed system includes a
continuous range of functions, then the expansion of ø (1) must include these
n
functions. The inclusion of this continuous range is implied in the summation
notation. The total eigenfunction ø n for the perturbed system to ®rst order in ë
is, then
X
ø n ˆ (1 ‡ ëa nn )ø (0) ‡ ë a nj ø (0) (9:29)
n j
j(6ˆn)
Since the function ø (0) is already included in zero order in the expansion of
n
ø n , we may, without loss of generality, set a nn equal to zero, so that
X
ø (1) ˆ a nj ø (0) (9:30)
n j
j(6ˆn)
This choice affects the normalization constant of ø n , but has no other
consequence. Furthermore, equation (9.28) for j ˆ n becomes
(0)
(1)
hø jø iˆ 0 (9:31)
n
n
showing that with a nn ˆ 0, the ®rst-order correction ø (1) is orthogonal to the
n
(0)
unperturbed eigenfunction ø .
n
With the choice a nn ˆ 0, the total eigenfunction ø n to ®rst order is normal-
ized. To show this, we form the scalar product hø n jø n i using equation (9.29)
and retain only zero-order and ®rst-order terms to obtain
X (0) (0)
(0)
(0)

(0)
(0)
hø n jø n iˆhø jø i‡ ë (a nj hø jø i‡ a hø jø i)
n n n j nj j n
j(6ˆn)
X
ˆ 1 ‡ ë (a nj ‡ a )ä nj ˆ 1

nj
j(6ˆn)
where equation (9.26) has been used.
Substitution of equation (9.30) into (9.27) gives