Reliability of high-order phase-integral eigenvalues for single and double minimum potentials

Abstract

For single and double minimum model potentials, energy eigenvalues are calculated using the Fröman and Fröman phase-integral approximations and compared with exact (numerical) quantum mechanical results. For the double minimum potential, results are obtained both from the correct phase-integral quantization condition including quantum effects near the barrier maximum (the σ term) up to and including the fifth-order approximation, and from the quantization condition with the σ term neglected up to and including the 13th-order approximation. When the σ term is included, the third- and fifth-order eigenvalues are accurate enough for all practical purposes, even close to the barrier maximum, but if the σ term is neglected, the phase-integral quantization condition breaks down near the barrier maximum, and this breakdown becomes more dramatic with increasing order. This property is used to discuss the question of which order of phase-integral approximation will give the optimum result. For a single minimum LJ(12,6) potential, phase-integral eigenvalues are calculated up to and including the 13th-order approximation. In this case, the phase-integral approximations appear able to yield much higher accuracy than can be practically obtained by quantal calculations using existing numerical methods. The reliability of phase-integral eigenvalues at energies near the dissociation limit is discussed, and a generalization of an earlier criterion for the onset of the breakdown of higher-order phase-integral quantization conditions near the asymptote of a potential with an attractive inverse-power r−ν long range tail (where ν≳2) is given.