VALIDITY OF WKB EIGENVALUE CRITERION FOR DIATOMIC MOLECULES

Abstract

The first-order Integral in the WKB semiclassical eigenvalue criterion has generally been used as the starting point for calculating RKR potential curves and centrifugal distortion constants of diatomic molecules. Extensions to include the second-order integral have also been reported. However, the contributions to the WKB wave function form an asymptotic series for which the appropriate truncation point may vary with vibrational energy. Therefore, this paper examines the region of validity of the higher order forms of the WKB eigenvalue condition. For a (Lennard-Jones) model potential, accurate first-, second-, and fourth-order semiclassical eigenvalues are compared with quantum mechanical results. These comparisons show that errors in first-order energies monotonically decrease with increasing vibrational quantum number (v). They also demonstrate that while the higher order results are usually much more accurate than those obtained in first order, their errors monotonically increase with v. Moreover, in the near-dissociation region the higher order integrals diverge, while the first-order results converge to the quantum mechanical values. Analytic expressions relating the higher order terms to the long-range tail of an arbitrary potential are derived and used to provide a simple method for predicting when they should be excluded from the eigenvalue criterion.