Physical Chemistry , 1st ed.

That does not mean that there is no understanding of, or recourse for, such

systems. Nor does it imply that quantum mechanics is a useless theory for

these systems. There are two main tools for applying quantum mechanics to

systems whose Schrödinger equations cannot be solved exactly. Use of either

depends on the type of system under study as well as what information you

want to determine.

The first of these tools is called perturbation theory.Perturbation theory as-

sumes that a system can be approximated as a known, solvable system and that

any difference between the system of interest and the known system is a small,

additive perturbation that can be calculated separately and added on. We will

assume that all of the energy levels under discussion are singly degenerate, so

this tool is more appropriately named nondegenerate perturbation theory.Also,

perturbation theory can be taken to very complex levels. Here, we focus on the

first level of approximation, which is called first-orderperturbation theory.

Perturbation theory assumes that the Hamiltonian for a real system can be

written as

HˆsystemHˆidealHˆperturbHˆ°Hˆ (12.14)

where Hˆsystemis the Hamiltonian of the system of interest that is being ap-

proximated,Hˆ° is the Hamiltonian of an ideal or model system, and Hˆrep-

resents the small, additive perturbation. For example, in the case of the helium

atom, the ideal part of the Hamiltonian can represent two hydrogen-like atoms.

The perturbation part of the Hamiltonian can represent the coulombic repul-

sion between the electrons:

HˆHe(HˆH-likeHˆH-like) 

4

e



2

0 r 12



(In this case, there are two hydrogen-like Hamiltonians because there are two

electrons. Despite this rewriting, the Schrödinger equation for He has not re-

ally changed and is still analytically unsolvable.) Any number of additive per-

turbations can be combined with an ideal Hamiltonian. It is, of course, easier

to keep the number of terms in the Hamiltonian as small as possible. What is

usually found is that there is a trade-off between the number of terms and the

accuracy of the solution to the Schrödinger equation.

If we assume that the wavefunction of the real system is similar to the

wavefunction of the ideal system, denoted (0), then one can say that,approx-

imately,

Hˆsystem(0) Esystem(0) (12.15)

where Esystemis the eigenvalue for the energy of the real system. Over the

course of many observations, one eventually determines an average value of

the observable energy,E. By using one of the postulates of quantum me-

chanics Ecan be approximated by the expression

E ((0))*Hˆsystem(0)d (12.16)

Given the form ofHˆsystem, one can substitute into equation 12.16 and partially

evaluate:

E ((0))*(Hˆ°Hˆ)(0)d

((0))Hˆ°(0)d((0))Hˆ(0)d

E(0)((0))*Hˆ^0 dE(0)E(1) (12.17)

388 CHAPTER 12 Atoms and Molecules