Extended Lagrangian molecular dynamics on vibronic surfaces in the nuclear-electronic orbital framework

Proton transfer is central to many processes of chemical interest. The simulation of proton transfer dynamics requires the inclusion of nuclear quantum effects, such as zero-point energy, nuclear delocalization, and tunneling. Herein, we introduce methods within the nuclear-electronic orbital (NEO) framework, where specified nuclei are treated quantum mechanically on the same level as the electrons, for the simulation of proton transfer dynamics. In particular, NEO density functional theory (NEO-DFT) is used to treat the transferring protons quantum mechanically, and the other nuclei are propagated classically on the adiabatic vibronic ground-state surface. We formulate a NEO extended Lagrangian molecular dynamics (NEO-ELMD) approach to incorporate the motion of the nuclear basis function centers during such simulations. Density matrix extrapolation and purification are introduced as a means to accelerate the NEO self-consistent field procedure at each time step by reducing the number of iterations required for convergence. We demonstrate the fidelity and efficiency of NEO-ELMD through comparisons with related dynamical methods for intramolecular proton transfer in malonaldehyde. We also use these accelerated techniques to simulate the nonequilibrium single and double proton transfer dynamics of proton-coupled electron transfer in much larger benzimidazole-phenol systems. This work provides a foundation for future methodologies to efficiently simulate proton transfer dynamics within the NEO-DFT framework while incorporating nonadiabatic effects between adiabatic vibronic states.