TUNNELING TRAJECTORIES OF 2-PROTON TRANSFER

Two-proton transfer is described in the coordinate basis consisting of symmetric(Q) and antisymmetric (q) proton coordinates and a low-frequency in-plane skeleton displacement (x) that promotes the transfer. A model three-dimensional potential energy surface (PES) is proposed, which is characterized by two transition states, and the tunneling trajectories on it are calculated using the instanton formalism. For fixed x-values, the extremum two-dimensional (Q, q) trajectories are shown to be of three distinct types, alternating one another at bifurcation values of PES parameters and beta = 1/k(B)T: (1) the doubly degenerate minimum energy path (MEP) associated with the step-wise transfer and passing through the saddle points occurs in the temperature region of thermally activated transitions (beta < beta(c)); (2) two-dimensional instanton; (3) one-dimensional instanton corresponding to the synchronous motion of the both protons. The two-dimensional instanton region is characterized by an apparent activation energy smaller than the barrier height along the MEP. The transition between one- and two-dimensional instantons occurs when the stability parameter of the tunneling trajectory vanishes. The transverse prefactor for the rate constant as a function of temperature is numerically calculated. In the three-dimensional case the rate constant found using the sudden approximation for the slow x-coordinate contains contributions from both channels (2) and (3). The transfer in formic acid dimer and porphyrine base are considered as examples.