  Recent experiments showed that the chemical reaction rate is modified, either
increased or decreased, by strongly coupling a nuclear vibration mode to the
single mode of an optical cavity. Herein we investigate how the rate of an
electron-transfer reaction depends on the molecule-cavity coupling in the
ultrastrong coupling regime, where the coupling strength is comparable in
magnitude with both the vibrational and the cavity frequencies. We found two
main factors that determine the modification of the reaction rate: the relative
shifts of the energy levels induced by the coupling and the mixing of the
ground and excited states of molecular vibration in the ground state of the
hybrid molecule-plus-cavity system through which the Franck-Condon factor
between the initial and final states of the transition is altered. The former
is the dominant factor if the molecule-cavity coupling strengths for the
reactant and product states differ significantly from each other and gives rise
to an increase in the reaction rate over a wide range of system's parameters.
The latter dominates if the coupling strengths and energy levels of the
reactant and product states are close to each other and it leads to a decrease
in the reaction rate. The effect of the mixing of molecular vibrational states
on the reaction rate is, however, suppressed in a system containing a large
number of molecules due to the collective nature of the resulting polariton,
and thus should be observed in a system containing a small number of molecules.
In contrast, the effect of the relative shifts of the energy levels should be
essentially independent of the number of molecules coupled to the cavity.
