Physical Chemistry , 1st ed.

where ^2 p 1 and ^2 p 2 are the Laplacian operators for the two nuclei (which are

just protons) and Eel(R) is the electronic potential energy from the electronic

Schrödinger equation, 12.37. These two equations must be solved simultane-

ously in order to get a complete wavefunction for the molecule.

In the application of the Born-Oppenheimer approximation to diatomic

molecules, often the kinetic energy of the nuclei is neglected and the internu-

clear repulsion is estimated from classical considerations for a particular in-

ternuclear distance R(for example, the equilibrium bond distance). This re-

pulsion is included in the potential energy of the electronic Schrödinger part

of equation 12.37, which is solved using perturbation or variation techniques.

A more complete treatment calculates the internuclear potential at a series of

R’s and then calculates (numerically) the electronic energy at each R.From

this, a plot of electronic energy versus internuclear distance can be constructed,

like the one in Figure 12.14. Such a plot is called a potential energy curvefor

the molecule. Figure 12.14 shows a potential energy curve for the ground elec-

tronic state, where the energy of the molecule is lowest at the equilibrium bond

distance. Each electronic wavefunction, which has its own characteristic en-

ergy, will have its own potential energy curve as the internuclear distance

changes. Figure 12.15 shows potential energy curves for ground and excited

states of a simple diatomic system.

12.11 Introduction to LCAO-MO Theory

The previous section points out that the Born-Oppenheimer approximation is

useful in that electronic parts of wavefunctions can be separated from nuclear

parts of wavefunctions. However, it does not assist us in determining what the

12.11 Introduction to LCAO-MO Theory 405

A  A

Internuclear separation

Potential energy

Re (A A)

Internuclear separation

Potential energy

Re Re*

Figure 12.14 A simple potential energy curve
for a diatomic molecule A 2. When the nuclei get
too close, nuclear repulsion increases quickly.
When the nuclei get too far apart, the bond breaks
and the molecule separates into two higher-
energy A atoms. The minimum-energy internu-
clear distance, labeled Re, represents the equilib-
rium bond distance of the A–A bond in the stable
molecule.

Figure 12.15 A typical set of potential energy

curves for the ground state and two excited states of

a hypothetical diatomic molecule. The minimum-

energy internuclear distance for the excited state,

Re*, is not necessarily the same as the Refor the

ground state. Potential energy diagrams for real

molecules are much more complicated than this.