  We report first-principles calculations of local heating in nanoscale
junctions formed by a single molecule and a gold point contact. Due to a larger
heat dissipation, the single molecule heats up less than the gold point
contact. We also find, at zero temperature, a threshold bias $V_{onset}$ of
about 6 mV and 11 mV for the molecule and the point contact, respectively, is
required to excite the smallest vibrational mode and generate heat. The latter
estimate is in very good agreement with recent experimental results on the same
system. At a given external bias $V$ below $V_{onset}$, heating becomes
noticeable when the background temperature is on the order of $\sim
e(V_{onset}-V)/k_{B}$. Above $V_{onset}$, local heating increases dramatically
with increasing bias but is also considerably suppressed by thermal dissipation
into the electrodes. The results provide a microscopic picture of
current-induced heat generation in atomic-scale structures.
