  In bulk systems, molecules are routinely identified by their vibrational
spectrum using Raman or infrared spectroscopy. In recent years, vibrational
excitation lines have been observed in low-temperature conductance measurements
on single molecule junctions and they can provide a similar means of
identification. We present a method to efficiently calculate these excitation
lines in weakly coupled, gateable single-molecule junctions, using a
combination of ab initio density functional theory and rate equations. Our
method takes transitions from excited to excited vibrational state into account
by evaluating the Franck-Condon factors for an arbitrary number of vibrational
quanta, and is therefore able to predict qualitatively different behaviour from
calculations limited to transitions from ground state to excited vibrational
state. We find that the vibrational spectrum is sensitive to the molecular
contact geometry and the charge state, and that it is generally necessary to
take more than one vibrational quantum into account. Quantitative comparison to
previously reported measurements on pi-conjugated molecules reveals that our
method is able to characterize the vibrational excitations and can be used to
identify single molecules in a junction. The method is computationally feasible
on commodity hardware.
