  Within the nuclear-electronic orbital (NEO) framework, the real-time NEO
time-dependent density functional theory (RT-NEO-TDDFT) approach enables the
simulation of coupled electronic-nuclear dynamics. In this approach, the
electrons and quantum nuclei are propagated in time on the same footing. A
relatively small time step is required to propagate the much faster electronic
dynamics, thereby prohibiting the simulation of long-time nuclear quantum
dynamics. Herein, the electronic Born--Oppenheimer (BO) approximation within
the NEO framework is presented. In this approach, the electronic density is
quenched to the ground state at each time step, and the real-time nuclear
quantum dynamics is propagated on an instantaneous electronic ground state
defined by both the classical nuclear geometry and the nonequilibrium quantum
nuclear density. Because the electronic dynamics is no longer propagated, this
approximation enables the use of an order-of-magnitude larger time step, thus
greatly reducing the computational cost. Moreover, invoking the electronic BO
approximation also fixes the unphysical asymmetric Rabi splitting observed in
previous semiclassical RT-NEO-TDDFT simulations of vibrational polaritons even
for small Rabi splitting, instead yielding a stable, symmetric Rabi splitting.
For intramolecular proton transfer in malonaldehyde, both RT-NEO-Ehrenfest
dynamics and its BO counterpart can describe proton delocalization during the
real-time nuclear quantum dynamics. Thus, the BO RT-NEO approach provides the
foundation for a wide range of chemical and biological applications.
