General Expression for Vibronic Coupling in Proton-Coupled Energy Transfer

Proton-coupled energy transfer (PCEnT) is a recently discovered process in which electronic excitation energy transfer is coupled to proton transfer. In nonadiabatic PCEnT theory, the reaction is described in terms of nonadiabatic transitions between pairs of electron-proton vibronic states, and the PCEnT rate constant associated with each pair depends on the square of the coupling between the reactant and product vibronic states. Herein, we derive an analytical expression for the diabatic vibronic coupling in PCEnT processes. The resulting total vibronic coupling encompasses both direct and indirect coupling between the reactant and product vibronic states. The direct coupling includes both Coulomb and exchange interactions between the reactant and product vibronic states, whereas the indirect coupling describes the interaction mediated by virtual intermediate vibronic states. The relative magnitudes of these interactions determine whether the Förster or Dexter mechanism for energy transfer is dominant, or whether a more general mechanism prevails. Moreover, the direct and indirect coupling terms can interfere constructively or destructively, facilitating or inhibiting the PCEnT process. We provide a general algorithm and procedure to calculate the input quantities for this vibronic coupling expression and apply it to the PCEnT process occurring in an anthracene-phenol-pyridine triad. We find that the indirect coupling is comparable in magnitude to the direct coupling at most proton donor-acceptor distances. The theory and methods developed in this work will enable the calculation of vibronic couplings and therefore rate constants for a wide range of PCEnT processes.