Quantization function for deep potentials with attractive tails

The interactions between atoms and molecules with each other and with surfaces are typically deep potential wells with attractive tails behaving asymptotically as an inverse power of the distance. In such potential wells, bound state energies En are determined by a quantization function F (E) according to nth -n=F (En), and F (E) is dominantly determined by the singular potential tail for near-threshold states. In this paper we formulate a general theory for the contribution Ftail (E) of the singular potential tail to the quantization function. The general expression for Ftail (E) contains two terms: a difference of action integrals and a difference of outer reflection phases, taken at threshold and at energy E. Close to threshold, E=0, strongly energy dependent and nonanalytic contributions of both terms cancel, so Ftail (E) acquires a universal form determined by a threshold length and an effective length which is related to a subthreshold effective range. For a homogeneous potential tail proportional to -1 rα, one universal expression for Ftail (E) caters for all potential strengths. We give an explicit analytical expression for the important case α=6 and present applications involving the derivation of atom-atom scattering lengths from the binding energies of high-lying bound states of the associated diatomic molecule. We also demonstrate how the dissociation energy of a diatomic molecule can be determined from spectroscopic energies of high-lying states, and we make a quantitative comparison with the performance of the LeRoy-Bernstein formula, which fails near threshold, because the strongly energy dependent and nonanalytic contribution from the action integrals is not, as it should be, compensated by terms coming from the corresponding energy dependence of the outer reflection phase.