A Unified Langevin Framework for Bosonic and Fermionic Dissipation in Nonadiabatic Electrochemical Proton Transfer

We present an influence functional path integral framework for treating the coupled dynamics of solvated proton and electron transfer within a nonequilibrium open system. This method formulates a generalized Langevin equation describing dynamics in systems where proton is simultaneously coupled to fermionic (metal electrons) and bosonic (solvent phonons) reservoirs. It accounts for multiple dissipative channels without relying on phenomenological assumptions. With this scheme, we capture the relaxation of oscillations associated with large quantum zero-point fluctuations when protons are trapped in a harmonic potential. When the proton's translational motion is slow, the dynamics become effectively Markovian. In this regime, dissipation to the electronic reservoir is characterized by a position-dependent electronic friction. Using an effective electronic model Hamiltonian, we demonstrate that electronic friction introduces a sharp, localized resistance when the proton level crosses the Fermi level, effectively delaying the reaction. In contrast, solvent friction arising from assumed Caldeira-Leggett-type coupling, exerts a uniform, position-independent drag. Both mechanisms contribute comparable amounts to the overall energy dissipation. This framework offers a computationally efficient route to simulate complex electrochemical environments involving multiple dissipative baths.