Vibrational line shapes at surfaces

This work deals with the vibrational line shape of a molecule adsorbed on a substrate surface due to anharmonicity in the molecule-substrate bond. In particular, we consider the line shape of the mode, which consists predominantly of the whole molecule vibrating normal to the surface in the top position, at a frequency Ω which is substantially higher than other normal modes of relevance. It is shown that other modes vibrating perpendicular to the surface have negligible effect on the line shape, thus generalizing a result of B. Persson and Ryberg [Phys. Rev. B 40, 10273 (1989)], which had been only shown to apply to the case of large substrate to adsorbate mass ratio. We then derive a set of equations for the effect of the modes vibrating parallel to the surface which become the exact solution for interaction Hamiltonians of the type ${\mathit{H}}_{\mathrm{int}}$=K(${\mathit{x}}^2$+${\mathit{y}}^2$)${\mathit{z}}^2$ in the limit of large Ω for arbitrary K, where (x,y,z) is the excursion of the molecule relative to the surface atom to which it is bound. These equations are solved exactly for the special case of a projected density of parallel vibrational states which is a Lorentzian of width much smaller than its central frequency. All previous solutions of the dephasing problem of which we are aware are special cases of our exact solution, which we apply to existing experimental data, for example, CO on Pt(111), as well as to make predictions of where some of the new features of the unusual line shapes that we predict might be seen.