Analytical matrix elements for non-curve-crossing situations

Abstract

Available semiclassical approximations for one-dimensional matrix elements are discussed, with emphasis on their realms of validity. It is shown that the canonical form for non-curve-crossing situations is of exponential type, but that a well established exponential approximation involving a difference of semiclassical phase integrals may become inapplicable in certain cases. A new approximation dependent exponentially on a phase integral sum is derived and tested against two exactly soluble models. Errors of at most 10 per cent are obtained in realistic molecular cases. Useful accuracy, with a systematic underestimate of 20–25 per cent, is also obtained for distorted wave matrix elements, even when the de Broglie wavelength becomes long compared with the range of the distortion potential, provided certain normalization corrections are applied to the classically forbidden part of the semiclassical wavefunction. This points to a deficiency in the established semiclassical connection formulae. The relative merits of the present exponential approximation and of rival classical path approximations are examined in some detail.