Physical Chemistry Third Edition

1286 G The Perturbation Method

(Ĥ(0)+λ̂H′)(Ψ

(0)

n,corr+Ψ

(1)

n λ+Ψ

(2)

n λ

(^2) + ···)
(E(0)+E(1)nλ+E(2)n λ^2 + ···)(Ψ(0)n +Ψ(1)nλ+Ψ(2)nλ^2 + ···) (G-19)
The coefficients of any power ofλon the two sides of the equation must be equal. For
the terms proportional toλwe have
̂H(0)Ψn(1)+Ĥ′Ψ(0)n,corrE(0)Ψ(1)n +E(1)nΨ(0)n,corr (G-20)
or
(Ĥ(0)−E(0))Ψn(1)(E(1)n −Ĥ′)Ψ
(0)
n,corr (G-21)
We now multiply by the complex conjugate of one of the initial wave functions,Ψ(0)m,init∗ ,
and integrate:
∫
Ψ(0)m,init∗Ĥ(0)Ψ(1)n dq−E(0)

∫

Ψ(0)m,init∗ Ψ(1)n dq

E(1)n

∫

Ψ(0)m,init∗Ψ(0)n,corrdq−

∫

Ψ(0)m,init∗ ̂H′Ψn(0),corrdq (G-22)

We now apply the hermitian property to the first term on the left-hand side of this

equation and use the fact that the eigenvalues of a hermitian operator are real:

∫

Ψ(0)m,init∗ ̂H(0)Ψn(1)dq

∫

(Ĥ(0)∗Ψ(0)m,init∗)Ψn(1)dqE(0)

∫

Ψ(0)m,init∗Ψ(1)ndq (G-23)

We omit any subscript onE(0)since all of the zero-order wave functions correspond

to the same value ofE(0). The two terms on the left side of Eq. (G-22) cancel and we

have

0 E(1)n

∫

Ψ

(0)∗

m,initΨ

(0)

n,corrdq−

∫

Ψ

(0)∗

m,initĤ

′Ψ(0)n,corrdq (G-24)

We use the expression in Eq. (G-15):

0 E(1)n

∫

Ψ(0)m,init∗

∑g

j 1

cnjΨ(0)j,initdq−

∫

Ψ(0)m,init∗ ̂H′

∑g

j 1

cnjΨ(0)j,initdq (G-25)

We interchange the order of integration and summation to obtain

0 E(1)n

∑g

j 1

cnj

∫

Ψ(0)m,init∗ Ψ(0)j,initdq−

∑g

j 1

cnj

∫

Ψ(0)m,init∗ ̂H′Ψ(0)j,initdq (G-26)

Since the initial zero-order functions are assumed to be normalized and orthogonal to

each other, the integral in the first sum vanishes ifmjand equals unity ifmj:

∫

Ψ(0)m,init∗Ψ(0)j,initdqδmj

{

1ifmj

0ifmj

(G-27)

whereδmjis called the Kronecker delta. We now can write

0 E(1)n

∑g

j 1

cnjδmj−

∑g

j 1

cnjHmj′ (G-28)