  The hot carriers generated by plasmon damping hold significant potential for
photoelectric conversion and photocatalysis. Despite numerous experiments and
theoretical analyses, the precise role of plasmonic hot carriers in such
dynamical processes has not been well understood. Here we present a theory of
plasmonic photocatalysis based on the microscopic model of electron-vibrational
coupling and the vibrational heating mechanism. The nonthermal hot carrier
distribution was derived and treated on equal footing with the thermal
Fermi-Dirac distribution. The inelastic rates of vibrational excitations were
calculated including the effect of multiple electronic transitions. As an
example of application, the O$_2$ dissociation on silver nanoparticles was
explored with focus on the temperature- and light-intensity dependences. The
dissociation rate evolves from a linear regime into a superlinear regime due to
the onset of vibrational heating induced by hot carriers. In the nonlinear
regime, nonthermal hot carriers greatly promote molecular dissociation. Our
findings provide insight into plasmonic photocatalysis, and paves the way for
harnessing light energies in the nonthermal regime.
