Physical Chemistry , 1st ed.

Figure 12.9 shows a representation of the energies of the initial basis func-

tions compared to the approximate energies for the linear combination as de-

termined in Example 12.12. The process of taking linear combinations of ideal

wavefunctions is sometimes referred to as mixingwavefunctions. Note that

there has been a slight energy change as the two basis functions combine to

yield an approximate wavefunction for a real system. In this case, the energy

levels spread further apart when going from the ideal system to the real system

approximated by the linear combination of ideal wavefunctions. In all cases of

the mixing of two basis functions, the energies spread apart more. The closer

in energy the ideal levels, the farther apart they spread. Figure 12.10 illustrates

qualitatively how basis functions that are nearly degenerate mix and yield ap-

proximate wavefunctions that are now relatively far apart in energy. However,

the sum of the energies of the two levels—ideal and approximate—remains the

same. (This is true only if basis functions themselves are orthogonal; other-

wise, this is not the case.)

Two-level examples are relatively straightforward. In modern computational

quantum mechanics, dozens or even hundreds of levels can be calculated us-

ing means similar to those described above, although more complex.

12.9 Comparison of Variation and Perturbation Theories

Of the two approximation theories, which one is “better”? As with many such

questions, the answer is, it depends. In both cases, the energy can be deter-

mined more accurately than the wavefunction. In variation theory, the basis

functions can be anything,as long as the proposed function fits the require-

ments of wavefunctions in general and satisfies whatever boundary condition

exists. In applications to many large systems, researchers typically use ideal

wavefunctions that bear only a slight resemblance to true wavefunctions but

that can be integrated easily by computer. [For example, Gaussian-type func-

tions (that is, functions based on ex

2

) are common in variation-theory appli-

cations, even though atomic orbitals are not Gaussian functions.] The idea “the

lower the energy, the better the energy and the wavefunction” provides a ma-

jor yardstick for critical evaluation of the wavefunctions by solving the secular

determinant. But the functions used in the trial wavefunctions may not make

sense in terms of having the form of a true atomic orbital, or may not be eas-

ily visualized. Computers are almost irreplaceable in variation-theory calcula-

tions, because in order to manipulate a large number of equations, the speed

of a computer becomes necessary.

Perturbation theory lacks the guarantee of variation theory. Results from

perturbation-theory calculations may yield an energy that can be either

higher or lower than the true energy. As such, to a certain extent the pre-

dicted energy for a perturbation-theory calculation is always suspect. But the

perturbation Hamiltonians Hˆcan usually be defined so as to make sense in

that their mathematical forms and behaviors are usually well known. For ex-

ample, common perturbations include electric and magnetic interactions,

two- and three-body interactions, dipole-dipole or dipole-induced dipole in-

teractions, or crystal-field interactions—all of which have known mathe-

matical forms and so can easily be included as part of a complete

Hamiltonian. Usually, the perturbation wavefunctions also make sense, since

many of them are simply corrections to ideal, well-known wavefunctions.

Each of the common perturbations listed above can be treated as a separate

402 CHAPTER 12 Atoms and Molecules

Ideal

wavefunction

energies

Mixed

wavefunction

energies

H 22

Energy

E 2

H 11

E 1

Ideal

wavefunction

energies

Mixed

wavefunction

energies

H 22

Energy

E 2

H 11

E 1

Figure 12.10 When the wavefunctions being
mixed are almost degenerate, the separation of
the energies upon mixing is much larger than
when the energy eigenvalues are far apart.
Compare this with Figure 12.9.

Figure 12.9 A representation of the change in
energy values when wavefunctions are mixed. The
isolated wavefunction energies,H 11 and H 22 ,are
on the left, and the calculated values for the mixed
wavefunction energies,E 1 and E 2 , are on the right.
The lower ideal energy has gone down slightly,
while the higher ideal energy increases.