Lagrangian Formulation of Nuclear-Electronic Orbital Ehrenfest Dynamics with Real-time TDDFT for Extended Periodic Systems

We present a novel Lagrangian-based implementation of Ehrenfest dynamics using nuclear-electronic orbital (NEO) theory in combination with real-time time-dependent density functional theory (RT-TDDFT) to study condensed matter systems. This approach enables the quantum mechanical treatment of both electrons and selected protons while accounting for the classical motion of all other nuclei. In addition, we introduce a Lagrangian formulation for the traveling proton basis method, accompanied by new schemes to enhance its applicability to periodic systems. As a proof-of-concept, we apply this framework to study electronically excited proton transfer in o-hydroxybenzaldehyde with explicit solvation by water molecules. Our results highlight the critical role of solvation dynamics and the necessity of treating transferring protons quantum mechanically. This work extends the applicability of NEO Ehrenfest dynamics to complex heterogeneous environments, providing new insights into the quantum nature of proton-related processes in the condensed phase.