Simulation of homogeneous electrochemical proton-coupled electron transfer using the hybrid-bath hierarchical equations of motion

The dynamics of homogeneous electrochemical proton-coupled electron transfer (PCET) are governed by the complex interactions among the continuous electronic states of the electrode, molecular vibrational modes, and the solvent environment. Here, we study this process within a Newns-Anderson model using the hierarchical equations of motion (HEOM) method combined with matrix product states (MPS) for hybrid fermionic and bosonic baths. The simulations reveal how the reaction dynamics depend on a variety of parameters, including the proton-transfer distance, electrode chemical potential, molecule-electrode coupling strength, and solvent reorganization energy. Comparison with Fermi's Golden Rule shows that the perturbative rate theory is reliable in the weak-coupling regime, but may become inaccurate at strong molecule-electrode coupling. Rates extracted from population dynamics yield Tafel plots whose shapes depend on both solvent and electrode couplings. The calculations also reproduce a primary kinetic isotope effect, with hydrogen transfer faster than deuterium transfer and with a larger effective transfer coefficient. These results highlight the capability of the hybrid-bath MPS-HEOM method to provide a unified description of electrochemical PCET in a wide range of parameter regimes.