Physical Chemistry , 1st ed.

where E(0)is the average energy of the ideal or model system (that is, an
eigenvalue energy, usually) and E(1)is the first-order correctionto the energy.
Thus, the first approximation to the energy of a real system is equal to the ideal
energy plus some additional amount given by ((0))*Hˆ(0)d. If this in-
tegral can be evaluated or approximated, then a correction to the energy can
be determined. What equation 12.17 means is that when we write a Hamiltonian
as a perturbed ideal operator, the energy—the observable associated with the
Hamiltonian—is also perturbed from the ideal.

Example 12.8

What is the correction to the energy of the helium atom, assuming that the

perturbation can be approximated as a coulombic repulsion of the two

electrons?

Solution

According to equation 12.17, the perturbation is

E(1)((0))*

4

e



2

0 r 12

(0)d

If some way can be found to evaluate this integral, a correction to the total

energy—and thus a perturbation-theory approximation to the energy of a He

atom—can be approximated.

The integral in the above example can be approximated by mathematical
techniques and substitutions that we will not go into. (A discussion of its so-
lution can be found in more advanced texts.) Approximations and substitu-
tions are possible and the above integral can be estimated as

E(1)^5

4



4

e



2

0 a 0



where eis the charge on the electron, 0 is the permittivity of free space, and
a 0 is the first Bohr radius (0.529 Å). When we substitute the values of the con-
stants into this expression, we get

E(1)5.450    10 ^18 J

Combining this result with the “ideal” energy, which was determined by as-
suming the sum of two hydrogen electron energies (see Example 12.3), we get
for the total energy of helium

EHe1.743  10 ^17 J 5.450    10 ^18 J 1.198   10 ^17 J

which, when compared to the experimentally determined energy of the helium
atom (given in Example 12.3 as 1.265 10 ^17 J), is found to be off by only
5.3%. Compared to the hydrogen-like approximation of helium, this is a big
improvement. It points out the usefulness of perturbation theory.

Example 12.9

In a particle-in-a-box having length a, instead of being zero the potential en-

ergy in the box is a linear function of the position. That is,

Vkx

12.6 Perturbation Theory 389