Connecting the Dunham Expansion to the Dissociation Limit for Interatomic Potentials: Application to Lennard-Jones m-...

Dunham generated the expansion for energy levels of a rotating, vibrating diatomic molecule from an expansion of the potential about the equilibrium position. For partition functions, however, the energy levels are needed all the way to dissociation. Analytic Morse oscillator energies are not very useful because the exponential decay of the Morse potential is much too short-ranged for any physical system. The longer-range Lennard-Jones 12-6 potential could be used, but quantum energies have not previously been conveniently fit. I show how Dunham coefficients begin a set of asymptotic functions for any interaction potential, one function arising from each successive term in the WKB expansion. I apply this to the family of Lennard-Jones m- n (LJ m- n) potentials with an R^-m repulsive term and R^-n attractive term ( m > n) and demonstrate how m can be used as a parameter to adjust either the equilibrium distance or harmonic frequency. I present an empirical parametrization of LJ m- n vibrotor energies starting with Dunham coefficients generated from four terms in the WKB expansion. This information is combined with data from numerically solved energies and asymptotic limits to fit the functions all the way to dissociation. One can also treat exp-6 and similar model potentials with different repulsive parts using the same method because the expansion form is controlled by the long-range part of the potential.