  Theoretical analysis of the effect of an external vibration on proton
transfer (PT) in a hydrogen bond (HB) is carried out. It is based on the
two-dimensional Schr\"odinger equation with trigonometric double-well
potential. Its solution obtained within the framework of the standard adiabatic
approximation is available. An analytic formula is derived that provides the
calculation of PT rate with the help of elements implemented in {\sl
{Mathematica}}. We exemplify the general theory by calculating PT rate constant
for the intermolecular HB in the Zundel ion ${\rm{H_5O_2^{+}}}$ (oxonium
hydrate). This object enables one to explore a wide range of the HB lengths.
Below some critical value of the frequency of the external vibration the
calculated PT rate yields extremely rich resonant behavior (multiple
manifestations of bell-shaped peaks). It takes place at symmetric coupling of
the external vibration to the proton coordinate. This phenomenon is absent for
anti-symmetric and squeezed mode couplings.

Chemical Physics;Materials Science