The recently developed real-time nuclear–electronic orbital (RT-NEO) approach provides an elegant framework for treating electrons and selected nuclei, typically protons, quantum mechanically in nonequilibrium dynamical processes. However, the RT-NEO approach neglects the motion of the other nuclei, preventing a complete description of the coupled nuclear–electronic dynamics and spectroscopy. In this work, the dynamical interactions between the other nuclei and the electron–proton subsystem are described with the mixed quantum–classical Ehrenfest dynamics method. The NEO-Ehrenfest approach propagates the electrons and quantum protons in a time-dependent variational framework, while the remaining nuclei move classically on the corresponding average electron–proton vibronic surface. This approach includes the non-Born–Oppenheimer effects between the electrons and the quantum protons with RT-NEO and between the classical nuclei and the electron–proton subsystem with Ehrenfest dynamics. Spectral features for vibrational modes involving both quantum and classical nuclei are resolved from the time-dependent dipole moments. This work shows that the NEO-Ehrenfest method is a powerful tool to study dynamical processes with coupled electronic and nuclear degrees of freedom.
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