An independent-electron formalism is developed to describe the energetic distributions of hot electrons and holes excited in the interaction between an adsorbate and a metal surface. The formalism encompasses both a fully nonadiabatic treatment and a perturbation expansion in the adsorbate velocity that can be taken to arbitrary order. The widely used electronic friction and forced oscillator models are shown to be approximations of the second order perturbation result. A simple tight binding model of an atomic adsorbate interacting with a metal surface is used to demonstrate the formalism. It is shown that many orders (>10) of perturbation theory are required for quantitative agreement with fully nonadiabatic calculations of the electron and hole distribution functions. However, lower order approximations can provide a useful, semiquantitative picture of the distribution functions, and they are in good agreement with nonperturbative results for the total rate of nonadiabatic energy dissipation.
Mizielinski, M. S., & Bird, D. M. (2010). Accuracy of perturbation theory for nonadiabatic effects in adsorbate-surface dynamics. Journal of Chemical Physics, 132(18), . https://doi.org/10.1063/1.3424765