Tunnelling paths for proton transfer reactions in solution. A discussion over a simplified model

Tunnelling of proton transfer reactions in solution is analysed through a molecular dynamics study based on a classical trajectories plus semiclassical tunnelling methodology. The potential energy surface is modelled as a proton moving on a symmetric double well interacting with a couple of dipoles, the whole system being kept collinear. Three different tunnelling paths are considered: (a) only motion of the proton through a symmetric potential, (b) only motion of the proton through an asymmetric potential, and (c) motion of both the proton and the solvent through a symmetric potential. Strictly speaking, motions of type (a) are never found. By comparing the tunnelling rates obtained through paths (b) and (c) it is found that at low energies (c) is dominant whereas (b) rate constants are slightly greater at high energies. It is predicted that for a more realistic model, with more solvent molecules included, paths of type (c) will dramatically lower their probability so that tunnelling will be dominated by paths of type (b). Finally, analysis of the (b) tunnelling paths reveals that the total tunnelling rate is dominated by the few paths which have the smallest asymmetry. This result agrees with the extended idea that proton transfer rates in solution depend on the difficulty in reaching a fluctuation of the solvent which yields a symmetric energy profile for the motion of the proton.