A uniform approximation for one-dimensional matrix elements

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Abstract
A general expression for one-dimensional perturbation integrals is derived, which requires no knowledge of the radial wavefunctions. It is shown to generalize and to unify all available semi-classical approximations, and, further, to reduce to a number of known results for the curve-crossing problem and to the Jackson-Mott for the Landau-Teller vibrational relaxation model.