Comparison of quantum and semiclassical time delay resonance energies and widths: the effect of nearby inner turning points

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Abstract
Semiclassical three turning point time delay calculations of resonance energies and widths have been carried out for the long lived resonance states that lie well below a barrier maximum in the effective potential energy curve of a Lennard-Jones (12, 6) potential. The calculations are based on an alternative uniform semiclassical phase shift and time delay function that allow for the proximity of the inner pair of turning points, unlike previous semiclassical calculations which have allowed for the coalescence of the outer pair of turning points. For narrow resonance states, it is shown that the new semiclassical time delay calculations are in better agreement with accurate quantum mechanical time delay resonance energies and widths than are the previous semiclassical calculations. In addition, some simple approximate semiclassical expressions for the resonance energy and width of a long lived state have been derived, and their accuracy tested.