## Molecular Vibrations

A typical potential energy curve for a diatomic molecule has the following form:

R_{Eqm} is the internuclear separation between the atoms at equilibrium – the equilibrium bond length. At smaller separations the potential energy rises rapidly as a result of repulsion between the outermost electrons. At larger separations, the potential rises more slowly until it eventually levels out. At this point, the molecule has dissociated into two atoms. |

At separations close to R_{Eqm}, the potential energy curve can be approximated quite well by a parabola, allowing us to write:

where x is the displacement from the equilibrium position;

and k is the force constant of the bond. The larger the force constant, the steeper the potential walls and the stiffer the bond.

The Schrodinger equation for the motion of two particles relative to each other when they are confined to a parabolic potential is:

where μ is the reduced mass of the two particles:

This may be used to model the vibration of a chemical bond, which does consist of two molecules moving relative to each other, confined in a near-parabolic potential.

The above Schrodinger equation is the same as that for a particle of mass μ undergoing harmonic motion, which was covered under the quantum mechanical treatment of vibration, here.

From this we can write down the permitted energy levels of vibrational motion:

and the quantum number ν can take values 0, 1, 2, 3….

The energies of vibrational states are commonly expressed as vibrational terms, G(ν). These are merely the energies expressed as wavenumbers, and are obtained by division of the energies by hc: