On the application, breakdown, and near-dissociation behavior of the higher-order JWKB quantization condition

Abstract

Numerical studies and analytic results show that singularities in the appropriate contour integrals cause the higher-order JWKB eigenvalue criteria to break down for energies near the dissociation limit of any potential with an attractive inverse power (Cn/Rn) tail for which n≳2. However, the expressions derived for the characteristic near-dissociation behavior of these contour integrals provide a simple way of predicting the range of binding energies over which this occurs from a knowledge of only n, the potential constant Cn, and the reduced mass μ. This analysis also shows that for more than 70% of all diatomic molecule electronic states, no vibrational levels lie in the very narrow region near dissociation where this breakdown occurs. A related near-dissociation analysis shows that for levels in the upper part of a potential well, applying the Langer–Kemble correction to the first-order quantization condition reduces the errors in calculated eigenvalues by the fraction 3/(n+1), for n?2. In contrast, for the (n=6) model problem studied numerically, use of the two-term (third-order) or three-term (fifth-order) quantization conditions reduces these errors by four or eight orders of magnitude, respectively.