Beyond the Harmonic Approximation

The harmonic approximation has to break down somewhere. A diatomic molecule with its two nuclei harmonically bound would never be able to dissociate into two individual atoms or ions.

One way is to introduce phenomenological potentials with fitting parameters, e.g. the Morse potential

$$U_\alpha(r) \to U_\alpha^{\rm Morse}(r)\equiv D_\alpha \left[ 1- e^{-\beta_\alpha(r-r_\alpha)}\right]^2,$$ (1.36)

where $D_\alpha$ is the depth of the minimum below the asymptote and represents the dissociation energy of the molecule.