Concerted proton–electron transfer via vibration correlation function approach with effective anharmonicity: Beyond classical approximation with Duschinsky effects

Proton-coupled electron transfer reactions play a central role in natural and artificial energy conversion processes. The concerted type (CPET) offers unique kinetic advantages but remains challenging to model quantitatively within theoretical and computational frameworks, owing to the interplay of multiple quantum mechanical (QM) factors such as proton-electron nonadiabatic dynamics, anharmonic proton transfer (PT) potentials, and multimode vibronic coupling. When applied to CPET, standard simulation frameworks such as the Multistate Continuum (MSC) and Marcus-Levich-Jortner (MLJ) theories typically treat all solute vibrations other than the transferring-proton mode classically and neglect Duschinsky (mode-mixing) effects. Here, we present the vibration correlation function with effective anharmonicity (VCF-H) method, which extends the VCF formalism to systematically incorporate multidimensional vibronic coupling and Duschinsky transformations together with effective anharmonic corrections. Under a harmonic-potential setting, applications to gas-phase CPET model systems-specifically, the phenoxyl/phenol and thymine-acrylamide complexes-reveal that Duschinsky coupling between the transferring-proton motion and the donor-acceptor stretching mode, which is neglected in MSC/MLJ treatments, can play a major role and increase the predicted rate constants by several orders of magnitude. Furthermore, comparison across hierarchical approximations to VCF-H demonstrates that classical treatments of non-PT vibrational modes can substantially underestimate rate-constant predictions, whereas the fully QM description retained in VCF-H exerts a substantial influence on them. These results-currently limited to the gas phase-establish the VCF-H method as a streamlined and physically transparent framework for modeling CPET reactions, clarify its theoretical relationship to MSC/MLJ formulations, and highlight the critical role of multidimensional vibronic and Duschinsky effects in modeling CPET dynamics.