On the Use of the WBK Method for Obtaining Energy Eigenvalues

Abstract

A general method is proposed for evaluation of the second‐order WBK energy integral. The domain of the potential is extended into the complex plane by expansion in Taylor's series about the classical turning points. The method is illustrated by application to the one‐dimensional harmonic oscillator for which the second‐order integral is exactly zero. Application is also made to a typical molecular potential, the Heitler—London potential for H₂. In the latter problem ``exact'' eigenenergies, accurate to about 0.2 cm^—1, are obtained by a Runge—Kutta integration of the Schrödinger equation. These are compared with first‐order WBK energies obtained with and without use of the Langer—Kemble radial correction, and with second‐order WBK energies obtained with and without use of the Langer—Kemble correction. The uncorrected first‐order WBK energies are accurate to within 2 cm^—1 and first differences to within 0.6 cm^—1. Uncorrected second‐order WBK energies agree with the Runge—Kutta values and, in fact, are probably more accurate than the latter. The first‐order Langer—Kemble corrected energies are up to 9 cm^—1 in error; corrected first differences, however, are more accurate than uncorrected values at low quantum numbers. The poorest set of energies and first differences are the corrected second‐order values, indicating that the Langer—Kemble correction is not applicable when the second‐order WBK approximation is employed. Computing time requirements and advantages of the WBK and Runge—Kutta methods are briefly discussed.