Damping in the vibrational spectroscopy of adsorbates with STM

The damping of vibrationally excited adsorbates on metal surfaces may be an important issue in the observability of their vibrational spectra with a scanning tunneling microscope. In the present paper, we discuss these effects in the framework of two-particle Green's functions, following a method originally employed to describe inelastic scattering in heterostructures [N. S. Wingreen, K. W. Jacobson, and J. W. Wilkins, Phys. Rev. B 40, Il 834 (1989)]. By dressing the phonon lines, one can take into account the finite lifetime of the oscillations and obtain an expression for the total transition probability, through all the possible phonon channels. We also show, in the no-damping limit, and for a single vibrating coordinate, the analytical correspondence between the present method and the one presented in a previous related work [M. A. Gata and P. R. Antoniewicz, Phys. Rev. B 47, 13 797 (1993)].