Correlated double-proton transfer. I. Theory.

The dynamics of double-proton transfer reactions is studied on a model of transfer along two identical hydrogen bonds represented by quartic double-minimum potentials. Correlation between the proton motions is introduced by a coupling term that is bilinear in the two proton coordinates; it is shown that this form properly accounts for the polarity and symmetry of the interaction and correctly reproduces the observed transfer behavior in the strong- and weak-coupling limits. The model allows a universal description of double-proton transfer mechanisms in symmetric systems in terms of the variation of a single parameter, the (dimensionless) coupling between the two hydrogen bonds. The corresponding two-dimensional (2D) transfer potential has up to nine stationary points, depending on the coupling strength. The resulting dynamics and its dependence on temperature and isotopic substitution are studied analytically by instanton techniques for the full range of the correlation parameter whereby the potential has multiple saddle points. For any coupling, the dynamics at high temperatures is dominated by classical transitions over the saddle point of lowest barrier. Strong coupling leads exclusively to synchronous transfer along a single collective coordinate, weak coupling to competition between this synchronous transfer, and stepwise transfer along local coordinates, the relative contributions of these mechanisms being governed by the temperature. Below a certain crossover temperature, transfer dynamics is dominated by the instanton, i.e., the trajectory with maximum tunneling probability. Two types of instanton are found on the 2D potential. The well-known one-dimensional instanton, corresponding to synchronous motion, exists for any coupling. It dominates at low temperatures and is responsible for any observed tunneling splittings, independent of the number of saddle points of the symmetric potential. An alternative 2D instanton, corresponding to asynchronous motion, exists for weak coupling. It is shown that under conditions where 2D tunneling dominates, it is much slower than stepwise transfer. Therefore 2D tunneling trajectories do not contribute significantly to the rate of transfer and can be ignored. The favorable quantitative aspects of the model are illustrated by an application to double-proton rate constants in porphine, which have been measured in a wide range of temperatures.