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The exact time-dependent potential energy surface driving the nucleardynamics was recently shown to be a useful tool to understand and interpret thecoupling of nuclei, electrons, and photons, in cavity settings. Here we providea detailed analysis of its structure for exactly-solvable systems that modeltwo phenomena: cavity-induced suppression of proton-coupled electron transferand its dependence on the initial state, and cavity-induced electronicexcitation. We demonstrate the inadequacy of simply using a weighted average ofpolaritonic surfaces to determine the dynamics. Such a weighted average missesa crucial term that redistributes energy between the nuclear and thepolaritonic systems, and this term can in fact become a predominant term indetermining the nuclear dynamics when several polaritonic surfaces areinvolved. Evolving an ensemble of classical trajectories on the exact potentialenergy surface reproduces the nuclear wavepacket quite accurately whileevolving on the weighted polaritonic surface fails after a short period oftime. The implications and prospects for application of mixed quantum-classicalmethods based on this surface are discussed.

Case Studies of the Time-Dependent Potential Energy Surface for Dynamics in Cavities