Vibrational Transitions

The second term in the expansion of the dipole moment as a function of nuclear coordinate

gives rise to the vibrational transitions.

The vibrational wave functions

form an orthonormal set of wave functions, c_n. The first few vibrational wave functions are listed below.

The orthonormality can be illustrated using the first two wave functions in the set:

The last integral is an odd function over even limits. If we now consider the second term in the dipole expansion as a operator we have:

Thus, the vibrational transition moment for the transition from v = 0 to v = 1 is:

Vibrational selection rules arise from the fact that the integral for transition from v = 0 to v > 1 is zero. For example,

Both of the integrals are odd functions over even limits. While we do not prove the general expression here it can be shown that transitions occur from v à v’ where v’ = v ± 1. This is depicted below.