Physical Chemistry Third Edition

1284 G The Perturbation Method

The coefficients of the linear (first-degree) terms obey the relation:

Ĥ′Ψ(0)n +Ĥ(0)Ψ(1)n E(1)nΨ(0)n +E(0)nΨ(1)n (G-6)

We represent the first-order correction to the wave function by a linear combination

of the unperturbed (zero-order) wave functions:

Ψ(1)n 

∑∞

j 1

anjΨ(0)j (G-7)

This representation is substituted into Eq. (G-6), giving

Ĥ′Ψ(0)n +Ĥ(0)

∑∞

j 1

anjΨ

(0)

j E

(1)

nΨ

(0)

n +E

(0)

n

∑∞

j 1

anjΨ

(0)

j (G-8)

SinceΨ

(0)

j is an eigenfunction of

Ĥ(0)with eigenvalueE(0)

j ,

̂H′Ψ(0)n +

∑∞

j 1

anjE

(0)

j Ψ

(0)

j E

(1)

n Ψ

(0)

n +E

(0)

n

∑∞

j 1

anjΨ

(0)

j (G-9)

We now multiply each term of Eq. (G-9) byΨn(0)∗and integrate over all values

of the coordinates on which the wave function depends. Since the zero-order energy

eigenfunctions are orthogonal to each other if belonging to different eigenvalues, every

integral in the sums vanishes except the one for whichnj. We assume that the zero-

order wave functions are normalized, so that this integral equals unity. There is only

one term surviving in each sum, and we now have

∫

Ψ(0)n∗Ĥ′Ψ(0)ndq+annE(0)n E(1)n +annEn(0) (G-10)

where we abbreviate the coordinates of the system byq. The second term on each side

cancels so that we obtain the result given in Eq. (19.3-8):

E(1)n 

∫

Ψ(0)n∗̂H′Ψ(0)ndq (G-11)

The first-order correction to the wave function and the second-order correction to the

energy eigenvalue involve sums over all of the unperturbed wave functions and energy

eigenvalues. We do not derive these formulas, but present them here. The formula for

the coefficientanjin Eq. (G-7) is

Ψ(1)n −

∑∞

j 1

<n|H′|j>

E

(0)

n −E

(0)

j

Ψj(0) (jn) (G-12)

where we introduce the “bracket” notation for the integral

<n|H′|j>

∫

Ψn(0)∗Ĥ′Ψ(0)j dq (G-13)