One-electron matrix elements in the theory of ion–metal-surface scattering

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Abstract
A systematic evaluation of one-electron matrix elements appearing in the theory of charge-exchange processes in ion–metal-surface scattering is presented. The conduction-band electrons are described by jellium-type wave functions, while hydrogenic wave functions are employed for the electronic states of the ion. By applying Fourier-transform methods and complex integration techniques, exact closed-form expressions for overlap and Coulomb matrix elements are derived for arbitrary hydrogenic quantum numbers, momenta of the conduction-band electrons, and distances of the ion from the surface. The general case of arbitrary energy difference between initial and final state (including the strictly resonant case) is treated. For the near-resonant case, an expansion of the matrix elements about the resonant limit is derived. The effect of the motion of the ion is taken into account by including electronic translational factors. The actual computation of the matrix elements involves multiple summations, which can be accurately performed even for high-lying ionic Rydberg states. Structural properties of the matrix elements are revealed by studying, for typical cases, their dependence on the various parameters. Possible applications of our method are indicated.