1/ η expansion for quasistationary states: complex energies and complex angular momenta

Abstract

The recently developed 1/ η expansion technique, where η is a dimensionless well depth parameter, for the study of bound states is extended to treat resonances. It is shown that the method, with a minimum of numerical computation, permits the accurate calculation of values of complex energies and of complex angular momentum quantum numbers (Regge pole positions) comparable with numerical and semiclassical results. It is shown that this technique can be considered as complementary to the dimensional scaling method and, in several cases studied in this article, allows even greater accuracy. For bound states the method involves minimizing an effective potential function and evaluating the exact energy when η→∞, then computing approximate results for η finite by performing a perturbation expansion in 1/ η about this limit. For resonances we obtain complex energies by allowing the coordinate to be complex. For energy resonances we study several systems which range from one-dimensional (symmetric and nonsymmetric Eckart barriers) to three-dimensional ones: inverted Kratzer potential, Lennard-Jones potential, and the H 2 molecule. For Regge poles we study a Lennard-Jones (12,6) and a Lennard-Jones (6,4) potentials, and a potential which describes the elastic scattering of an electron by Ar. For the Eckart potential barrier this method offers a simple way to evaluate Siegert complex eigenstates which are becoming important in Chemical Physics due to the recent association of transition states with resonances. In all the cases studied, comparison with results of numerical, exact, and semiclassical calculations show that the 1/ η expansion technique is capable of producing accurate values of complex energies and complex angular momentum quantum numbers.