Physical Chemistry Third Edition

G The Perturbation Method 1285

This integral is called amatrix elementofĤ′, and is also denoted byHnj′. The second-

order correction for the energy is given by

E(2)n 

∑∞

j 1

<n|H′|j><j|H′|n>

E

(0)

n −E

(0)

j

(G-14)

These corrections cannot generally be computed exactly since they contain an infinite

number of terms, but various approximation schemes have been worked out. The higher-

order corrections are even more difficult to work with, but a number of approximate

calculations of higher-order corrections to the energy of the helium atom have been

published.

G. 2 The Degenerate Case

We must modify perturbation theory to apply it to an energy level that is degenerate

in zero order. For example, the two configurations (1s)(2s) and (1s)(2p) of the helium

atom correspond to several states that have the same energy in zero order. We assume

the same kind of Hamiltonian as in Eq. (G-1) and consider a zero-order energy level

with a degeneracy equal tog. We havegzero-order wave functions. We call them the

“initial” zero-order functions. Unfortunately, there is no guarantee that each of these

functions would turn smoothly into one of the exact wave functions ifλis increased

from0to1.

The first task is to find the “correct” zero-order wave functions, which means that

there is a one-to-one correspondence between each of these functions and one of the

exact wave functions. Since there are no other zero-order wave functions corresponding

to the energy eigenvalue being considered, it must be possible to express each correct

zero-order wave functions as a linear combination of theginitial zero-order wave

functions:

Ψ

(0)

n,corr

∑g

j 1

cnjΨ

(0)

j,init (n1, 2,...,g) (G-15)

where we denote thenth correct zero-order wave function byΨ(0)n,corrand thejth initial

zero-order wave function byΨ

(0)

j,init.

Since the initial zero-order functions all correspond to the same zero-order energy

eigenvalue, the correct zero-order functions correspond to the same zero-order energy

eigenvalue. We assume that the exact energy eigenfunctions and energy eigenvalues

can be represented by power series inλas in Eqs. (19.3-5) and (19.3-6). For state

numbern,

ΨnΨ

(0)

n,corr+Ψ

(1)

n λ+Ψ

(2)

n λ

(^2) + ··· (G-16)
EnE(0)+E(1)nλ+E(2)n λ^2 + ··· (G-17)
where we must include the correct zero-order function as the first term in Eq. (G-16),
not the initial function.
We now substitute the expressions of Eqs. (G-16) and (G-17) into the time-
independent Schrödinger equation:
Ĥ(λ)Ψn(λ)En(λ)Ψn(λ) (G-18)