  We investigate the effects of phonon scattering on the electronic current
noise through nanojunctions using the non-equilibrium Green's functions
formalism extended to include the counting field. In the case of weak
electron-phonon coupling and a single broad electronic level we derive an
analytic expression for the current noise at arbitrary temperature and identify
physically distinct contributions based on their voltage dependence. We apply
our theory to the experimentally relevant case of a deuterium molecule placed
in a break-junction and predict a significant inelastic contribution to the
current noise.
