Physical Chemistry , 1st ed.

Using equation 12.21, the nth real wavefunction n,realis written as

n,realn(0)

m^



(

E

m

(

n

(

0

0

)

)

)Hˆ





E



m

(0

n

(

)

0)d

 m(0) mn (12.22)

Note the ordering of the terms having mand nindices in the above equation;

it is important to keep them straight.n,realis still very similar to the nth ideal

wavefunction, but now it is corrected in terms of the other wavefunctions (0)m

that define the complete set of wavefunctions for the model system. Real wave-

functions defined in this way are not normalized. They must be normalized in-

dependently, once the proper set of expansion coefficients has been determined.

The presence of the term En(0)Em(0)in the denominator of equation 12.22

is very useful. Although this is just a first correction to the wavefunction, in

principle equation 12.22 can add an infinitenumber of terms to the wave-

function. However, consider the denominator in the definition of the expan-

sion coefficient. When the difference is small, the value of the fraction—and

therefore am—is relatively large. On the other hand, if the difference En(0)Em(0)

is large, then the fraction and therefore amare small. Negligibly small, some-

times. Consider the four-term linear expansion:

0,real  0 (0)0.95 1 (0)0.33 2 (0)0.74 3 (0)0.01 4 (0)

The fourth term in the expansion,4,ideal, has a very small expansion coeffi-

cient. This suggests that either the integral in the numerator ofa 4 is very small

or that the denominator ofa 4 is very large (or both). Either way, little of the

approximation is usually lost if that term is simply neglected:

0,real  0 (0)0.95 1 (0)0.33 2 (0)0.74 3 (0)

There is little way of knowing beforehand how large the integral in the nu-

merator of the expression 12.21 will be. Although the ideal wavefunctions m(0)

and n(0)are orthogonal, the presence of the operator Hˆmay make the value

of the integral nonzero, perhaps even large. But the denominator is in terms

of only the energies of the model system,Em(0)and En(0). Since a model system

typically has known energy eigenvalues, a good (but not necessarily absolute)

rule of thumb is that if the eigenvalue energies of the wavefunctions are far

enough apart, the expansion coefficient will be small.What this implies is that

the most important corrections in the real wavefunction n,realwill be wave-

functions whose energies are close to the ideal wavefunction n(0)of the orig-

inal wavefunction approximation. So although the complete set of wavefunc-

tions may have an infinite number of ideal wavefunctions, only those that have

eigenvalues for energy that are close to the energy of the nth state will have a

noticeable impact on the wavefunction correction.

Example 12.10

Because of electronegativity differences, the pelectron in a bond between two

different atoms—say in the CNion—does not act exactly like a particle-

in-a-(flat)-box, but like a particle in a box that has a slightly higher potential

energy on one side than the other. Assume, then, a perturbation ofHˆkx

for the ground state  1 of a particle-in-a-box system.

a.Draw the perturbed system.

b.Assuming that the only correction to the real ground-state wavefunction is

the second particle-in-a-box wavefunction  2 , calculate the coefficient a 2 and

determine the first-order-corrected wavefunction.

392 CHAPTER 12 Atoms and Molecules