  We study a simple model for photoinduced electron transfer reactions for the
case of many donor-acceptor pairs that are collectively and homogeneously
coupled to a photon mode of a cavity. We describe both coherent and dissipative
collective effects resulting from this coupling within the framework of a
quantum optics Lindblad master equation. We introduce a method to derive an
effective rate equation for electron transfer, by adiabatically eliminating
donor and acceptor states and the cavity mode. The resulting rate equation is
valid both for weak and strong coupling to the cavity mode, and describes
electronic transfer through both the cavity coupled bright states and the
uncoupled dark states. We derive an analytic expression for the instantaneous
electron transfer rate that depends non-trivially on the time-varying number of
pairs in the ground state. We find that under proper resonance conditions, and
in the presence of an incoherent drive, reaction rates can be enhanced by the
cavity. This enhancement persists, and can even be largest, in the weak
light-matter coupling regime. We discuss how the cavity effect is relevant for
realistic experiments.
