  The calculation of the nonlinear conductance of a single-molecule junction is
revisited. The self energy on the junction resulting from the electron-phonon
interaction has at low temperatures logarithmic singularities (in the real
part) and discontinuities (in the imaginary one) at the frequencies
corresponding to the opening of the inelastic channels. These singularities
generate discontinuities and logarithmic divergences (as a function of the bias
voltage) in the low-temperature differential conductance around the inelastic
thresholds. The self energy also depends on the population of the vibrational
modes. The case of a vibrating free junction (not coupled to a thermal bath),
where the phonon population is determined by the bias voltage is examined. We
compare the resulting zero-temperature differential conductance with the one
obtained for equilibrated phonons, and find that the difference is larger the
larger is the bare transmission of the junction and the product of the electron
dwell time on the junction with the phonon frequency.
