  A first principles theory of inelastic tunneling between a model probe tip
and an atom adsorbed on a surface is presented, extending the elastic tunneling
theory of Tersoff and Hamann. The inelastic current is proportional to the
change in the local density of states at the center of the tip due to the
addition of the adsorbate. We use the theory to investigate the vibrational
heating of an adsorbate below an STM tip. We calculate the desorption rate of H
from Si(100)-H(2$\times$1) as function of the sample bias and tunnel current,
and find excellent agreement with recent experimental data.
