  Non-adiabatic molecular phenomena, arising from the breakdown of the
Born-Oppenheimer approximation, govern the fate of virtually all photo-physical
and photochemical processes and limit the quantum efficiency of molecules and
other solid-state embedded quantum emitters. A simple and elegant description,
the energy gap law, was derived five decades ago, predicting that the
non-adiabatic coupling between the excited and ground potential landscapes lead
to non-radiative decay with a quasi-exponential dependence on the energy gap.
We revisit and extend this theory to account for crucial aspects such as
vibrational relaxation, dephasing, and radiative loss. We find a closed
analytical solution with general validity which indicates a direct
proportionality of the non-radiative rate with the vibrational relaxation rate
at low temperatures, and with the dephasing rate of the electronic transition
at high temperatures. Our work establishes a connection between nanoscale
quantum optics, open quantum system dynamics and non-adiabatic molecular
physics.
