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.