Uniform instanton theory for bridge-mediated nonadiabatic tunneling
Instanton theory provides a semiclassical approximation to nonadiabatic rate constants within a path-integral framework and captures nuclear quantum effects such as tunneling and zero-point energy. Although the simplest reactions proceed directly from reactants to products, a number of important nonadiabatic processes involve three electronic states and react via a concerted mechanism through a virtual bridge state, also referred to as bridge-mediated tunneling or superexchange. In previous work, we have derived an instanton theory to describe such reactions using a dominant tunneling trajectory that propagates on each of the three potential energy surfaces. However, this approach is only valid when the bridge state is sufficiently low in energy. In contrast, in the high-bridge regime, the trajectories on the bridge surface become infinitesimally short, which causes the previous theory to break down. In this work, we derive the appropriate asymptotic approximation for the high-bridge regime and find that the resulting theory reduces to a two-state golden-rule instanton with an effective coupling that describes a nonresonant hop to the virtual bridge state. We further develop a uniform asymptotic instanton theory that eliminates divergences and smoothly connects the high- and low-bridge regimes. In a companion paper, we apply the method to study the quintet-to-singlet transition (via a triplet bridge) of an iron(II) complex.