New insights into the nonadiabatic state population dynamics of model proton-coupled electron transfer reactions from the mixed quantum-classical Liouville approach

In a previous study [F. A. Shakib and G. Hanna, J. Chem. Phys. 141, 044122 (2014)], we investigated a model proton-coupled electron transfer (PCET) reaction via the mixed quantum-classical Liouville (MQCL) approach and found that the trajectories spend the majority of their time on the mean of two coherently coupled adiabatic potential energy surfaces. This suggested a need for mean surface evolution to accurately simulate observables related to ultrafast PCET processes. In this study, we simulate the time-dependent populations of the three lowest adiabatic states in the ET-PT (i.e., electron transfer preceding proton transfer) version of the same PCET model via the MQCL approach and compare them to the exact quantum results and those obtained via the fewest switches surface hopping (FSSH) approach. We find that the MQCL population profiles are in good agreement with the exact quantum results and show a significant improvement over the FSSH results. All of the mean surfaces are shown to play a direct role in the dynamics of the state populations. Interestingly, our results indicate that the population transfer to the second-excited state can be mediated by dynamics on the mean of the ground and second-excited state surfaces, as part of a sequence of nonadiabatic transitions that bypasses the first-excited state surface altogether. This is made possible through nonadiabatic transitions between different mean surfaces, which is the manifestation of coherence transfer in MQCL dynamics. We also investigate the effect of the strength of the coupling between the proton/electron and the solvent coordinate on the state population dynamics. Drastic changes in the population dynamics are observed, which can be understood in terms of the changes in the potential energy surfaces and the nonadiabatic couplings. Finally, we investigate the state population dynamics in the PT-ET (i.e., proton transfer preceding electron transfer) and concerted versions of the model. The PT-ET results confirm the participation of all of the mean surfaces, albeit in different proportions compared to the ET-PT case, while the concerted results indicate that the mean of the ground- and first-excited state surfaces only plays a role, due to the large energy gaps between the ground- and second-excited state surfaces.