Analytical correction to the WKB quantization condition for the highest levels in a molecular potential

Abstract

We have recently shown that the highest vibrational level in a molecular potential is not well described by the semiclassical JWKB quantization formula and that there is a complete analogy between this effect and the quantum threshold behaviour for continuum levels. The goal of the present paper is to complete the description of this effect. In a first step, we verify that the quantization condition for zero-energy bound states deduced from the work of Gribakin and Flambaum on the scattering length is very accurate. Then, we discuss the correction term to the JWKB quantization condition, using numerical calculations as well as various analytic approximations. In particular, the second order JWKB formula for this phase correction term and a new resonant expression are introduced, and we also improve our previous numerical integral method. The discussion is centered mostly on the case of long range potentials behaving like r−6 at long range but the case of a r−3 behavior is also considered and most of the formulas developed here are valid for any . Finally, we show how to deduce the scattering length from the binding energy of the highest vibrational level.