Page 249 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

240 Approximation methods

where N is the number of particles in the system. We suppose that the
È
Schrodinger equation for this Hamiltonian operator
^
Hø n ˆ E n ø n (9:15)
cannot be solved exactly by known mathematical techniques.
^
In perturbation theory we assume that H may be expanded in terms of a
small parameter ë
^
^
^
^
^
^
2
H ˆ H (0) ‡ ëH (1) ‡ ë H (2) ‡ ˆ H (0) ‡ H9 (9:16)
where
^
^
^
2
H9 ˆ ëH (1) ‡ ë H (2) ‡ (9:17)
^ (0)
The quantity H is the unperturbed Hamiltonian operator whose orthonormal
eigenfunctions ø (0) and eigenvalues E (0) are known exactly, so that
n n
^
(0)
(0)
H ø (0) ˆ E ø (0) (9:18)
n n n
^
^
The operator H9 is called the perturbation and is small. Thus, the operator H
^
^ (0)
differs only slightly from H and the eigenfunctions and eigenvalues of H do
^ (0)
not differ greatly from those of the unperturbed Hamiltonian operator H .
The parameter ë is introduced to facilitate the comparison of the orders of
magnitude of various terms. In the limit ë ! 0, the perturbed system reduces
to the unperturbed system. For many systems there are no terms in the
^ (1)
perturbed Hamiltonian operator higher than H and for convenience the
parameter ë in equations (9.16) and (9.17) may then be set equal to unity.
The mathematical procedure that we present here for solving equation (9.15)
is known as Rayleigh±Schro Èdinger perturbation theory. There are other
È
procedures, but they are seldom used. In the Rayleigh±Schrodinger method,
the eigenfunctions ø n and the eigenvalues E n are expanded as power series
in ë
2
ø n ˆ ø (0) ‡ ëø (1) ‡ ë ø (2) ‡ (9:19)
n
n
n
2
E n ˆ E (0) ‡ ëE (1) ‡ ë E (2) ‡ (9:20)
n n n
The quantities ø (1) and E (1) are the ®rst-order corrections to ø n and E n , the
n n
quantities ø (2) and E (2) are the second-order corrections, and so forth. If the
n n
^
perturbation H9 is small, then equations (9.19) and (9.20) converge rapidly for
all values of ë where 0 < ë < 1.
We next substitute the expansions (9.16), (9.19), and (9.20) into equation
(9.15) and collect coef®cients of like powers of ë to obtain
^
^
^
^
^
^
(0)
(1)
(2)
(0)
(0)
(1)
(1)
(2)
2
H ø (0) ‡ ë(H ø (0) ‡ H ø ) ‡ ë (H ø (0) ‡ H ø (1) ‡ H ø ) ‡
n n n n n n
(2)
(0)
(0)
(1)
2
(1)
(2)
(1)
(0)
ˆ E ø (0) ‡ ë(E ø (0) ‡ E ø ) ‡ ë (E ø (0) ‡ E ø (1) ‡ E ø ) ‡
n
n
n
n
n
n
n
n
n
n
n
n
(9:21)