  We present a Lagrangian-based implementation of Ehrenfest dynamics with
nuclear-electronic orbital (NEO) theory and real-time time-dependent density
functional theory (RT-TDDFT) for extended periodic systems. In addition to a
quantum dynamical treatment of electrons and selected protons, this approach
allows for the classical movement of all other nuclei to be taken into account
in simulations of condensed matter systems. Furthermore, we introduce a
Lagrangian formulation for the traveling proton basis approach and propose new
schemes to enhance its application for extended periodic systems. Validation
and proof-of-principle applications are performed on electronically excited
proton transfer in the o-hydroxybenzaldehyde molecule with explicit solvating
water molecules. These simulations demonstrate the importance of solvation
dynamics and a quantum treatment of transferring protons. This work broadens
the applicability of the NEO Ehrenfest dynamics approach for studying complex
heterogeneous systems in the condensed phase.
