Physical Chemistry , 1st ed.

part of the overall wavefunction, so the complete wavefunction is the sum of

many simple parts. Computers are also useful in perturbation theory, espe-

cially where the number of perturbations included in a calculation gets large.

However, since many of the perturbations are selected because their mathe-

matical behavior is understood, calculation of perturbed energies by hand is

not inherently difficult.

Researchers in quantum mechanical calculations should understand the

limitations and strengths of each method. Typically, the method used is the one

that provides the information a particular researcher wants about a real sys-

tem. If a well-defined Hamiltonian and wavefunction are desired, then pertur-

bation theory provides that. If the absolute energy is important, variation the-

ory provides a way to get better and better results. The calculational cost is also

a factor. Those with access to supercomputers can work with a lot of equations

in a relatively short time. Those without may find themselves limited to a small

number of corrections to ideal wavefunctions.

An important thing to understand about both of these theories is that when

properly applied, they can be used to understand any atomic or molecular sys-

tem. By using more and more terms in a perturbation-theory treatment or

more and better trial functions in variation-theory treatments, one can do ap-

proximation calculations that yield virtually exact results. So even though the

Schrödinger equation cannot be solved analyticallyfor multielectron systems,

it can be solved numericallyusing these techniques. The lack of analytic solu-

tions does not mean that quantum mechanics is wrong or incorrect or in-

complete; it just means that analytic solutions are not available.Quantum

mechanics does provide toolsfor understanding any atomic or molecular system

and so it replaces classical mechanics as a way to properly describe electron

behavior.

12.10 Simple Molecules and the

Born-Oppenheimer Approximation

Since most chemical systems are molecules, it is important to understand

how quantum mechanics is applied to molecules. When we use the word

molecule,we are usually speaking of some chemically bonded system that ex-

ists as discrete collections of atoms bonded to each other in some specific

way. This contrasts with ionic compounds, which are atoms (or groups of co-

valently bonded atoms, the so-called polyatomic ions) held together by their

opposing charge; that is, they are composed of cations and anions. As one

might expect from the previous discussions about wavefunctions in multi-

electron atoms, wavefunctions of molecules get even more complicated. In

reality there are some useful simplifications, which we will get to in the next

chapter, but a general consideration of a simple diatomic molecule is useful

at this point.

The simplest diatomic molecule is H 2 , the diatomic hydrogen molecule

cation. This system has two nuclei and a single electron. It is illustrated in

Figure 12.11, along with definitions of the coordinates used to describe the po-

sitions of the particles. Because two nuclei are present, we must consider not

only the interaction of the electron with the two nuclei, but also the interac-

tion of the two nuclei with each other. The kinetic energy part of the complete

Hamiltonian will have three terms, one for each particle. The potential energy

part will also have three terms: an attractive electrostatic potential between

the electron and nucleus 1, an attractive electrostatic potential between the

12.10 Simple Molecules and the Born-Oppenheimer Approximation 403

R

r 1 r 2

H+ H+

e–

Figure 12.11 Definitions of the coordinates
for the H 2 molecule.