# Anharmonic Oscillation

The approximation of the potential energy to a parabola cannot be correct at all extensions, as it does not permit dissociation of the bond. At high vibrational excitations (i.e. in states with high values of the quantum number ν), the parabolic approximation is particularly poor. The motion at such a position is described as anharmonic, as the restoring force is no longer proportional to the square of the displacement from the equilibrium position.

One way to accommodate this problem is to use, rather than a parabola, a function that more closely resembles the true form of the potential energy. One function that is commonly used is the Morse potential:

(Recall ω = (k / μ)½ ). R is the bond length and Re is the bond length at equilibrium. De is the depth of the minimum in the curve:

The vibrational energy levels are labelled with the value of the quantum number ν. The bond dissociation energy, Do, is included for comparison with De.

Near its potential minimum, the curve does indeed approximate to a parabolic shape, but unlike the parabola, the Morse curve does allow for dissociation at high enough excitations. This means that the number of vibrational levels of a Morse oscillator is finite (there is a value νmax beyond which the energy of the oscillator is not quantised but continuous.) It is possible to solve the Schrodinger equation when the Morse curve is used for the potential, and the permitted energy levels turn out to be as follows:

The quantity xe is known as the anharmonicity constant.

For a Morse oscillator, the wavenumbers of transitions with Δν = +1 are given by:

When anharmonicities are present, weak absorption lines corresponding to transitions such as 2¬0, 3¬0 etc may be observed, even though such transitions as these are formally forbidden by the Δν = ±1 selection rule.

Such transitions are known as overtones, and arise because the selection rule is derived assuming that the oscillator is harmonic. If it does not behave perfectly harmonically, then the selection rule does not have to be obeyed completely – for an anharmonic oscillator, transitions with any value of Δν may be observed, but only weakly.

Another way of approaching the problem of anharmonicity, that is used more often in practice than the Morse oscillator, is to write the permitted energy levels as a series of terms:

where xe, ye etc are empirical constants which give the best fit to experimental data.