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

We have recently shown that the highest vibrational level in a molecular po tential is not well described by the semiclassical JWKB quantization formul a 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 tha t the quantization condition for zero-energy bound states deduced from the work of Gribakin and Flambaum on the scattering length is very accurate. Th en, we discuss the correction term to the JWKB quantization condition, usin g numerical calculations as well as various analytic approximations. In par ticular, the second order JWKB formula for this phase correction term and a new resonant expression are introduced, and we also improve our previous n umerical 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 ar e valid for any n greater than or equal to 3. Finally, we show how to deduc e the scattering length from the binding energy of the highest vibrational level.