Physical Chemistry Third Edition

G The Perturbation Method 1287

where we define thematrix elementofĤ′

Hmj′ 

∫

Ψm(0),init∗ Ĥ′Ψ(0)j,initdq (G-29)

We can rewrite Eq. (G-27) as

0 

∑g

j 1

(Hmj′ −En(1)δmj)cnj (m1, 2,...,g) (G-30)

Equation (G-29) represents a set of simultaneous equations, one for each value ofm.

The number of equations isg, and the number of unknown coefficientscnjfor which

we can solve isg. These are homogeneous linear equations, which can be satisfied by

setting each of thecnjcoefficients equal to zero. This is called thetrivial solution.If

we divide all of thecnjcoefficients byc 11 it is clear that we really have onlyg− 1

independent coefficients. We have one equation too many and the set ofgequations is

overdetermined, which means that unless some condition is satisfied, the trivial solution

is the only solution.

The condition that must be satisfied is that the determinant of the factors multiplying

thecnjcoefficients must vanish. We illustrate this condition with the case thatg2.

The simultaneous equations are

(H 11 ′ −E(1)n)cn 1 +H 12 ′cn 2  0 (G-31a)

H 21 ′cn 1 +(H 22 ′ −E(1)n)cn 2  0 (G-31b)

and the condition that must be satisfied is

∣

∣

∣

∣

∣

(H 11 ′ −E(1)n ) H 12 ′

H 21 ′ H′ 22 −E(1)n

∣

∣

∣

∣

∣

 0 (G-32)

Equation (G-31) is called asecular equation. Since the matrix elements contain the

zero-order functions, they can be calculated. When the determinant is multiplied out,

we obtain a quadratic equation forE(1)n. This equation can be solved to give two values

ofE(1)n. Each of these values is substituted into the set of equations in Eq. (G-30), and

two different sets ofcnjcoefficients can be obtained, producing two different correct

zero-order wave functions, one corresponding to each value ofE(1)n.

For values ofglarger than 2, the secular equation is agbygdeterminant, yielding an

algebraic equation of degreeg. The solution of the secular equation providesgvalues

ofE(1)n, some of which might be equal to each other. If two or more of the values of

E(1)n are equal to each other, we say that the perturbation has not completely lifted

the degeneracy of the level. In any case, there is a different correct zero-order wave

function for each of thegvalues ofE(1)n. The coefficients for one of the zero-order

wave functions are found by solving the simultaneous equations after substituting one

of the values ofE

(1)

n into the equations. As in the nondegenerate case, formulas for

the first-order corrections to the wave functions and second-order corrections to the

energies can be found. We do not discuss these corrections.