We formulate a time-dependent density functional theory for the coupled dynamics of electrons and nuclei that goes beyond the Born-Oppenheimer (BO) approximation. We prove that the time-dependent marginal nuclear probability density $|\chi (\underset{\_}{\underset{\_}{\mathbf{R}}},t){|}^2$, the conditional electronic density $n_{\underset{\_}{\underset{\_}{\mathbf{R}}}}(\mathbf{r},t)$ and the current density ${\mathit{J}}_{\underset{\_}{\underset{\_}{\mathbf{R}}}}(\mathbf{r},t)$ are sufficient to uniquely determine the full time-evolving electron-nuclear wave function, and thus the dynamics of all observables. Moreover, we propose two time-dependent Kohn-Sham (KS) schemes which reproduce the exact conditional electronic density and current density and the exact $N$-body nuclear density. The remaining task is to look for functional approximations for the Kohn-Sham exchange-correlation scalar and vector potentials. Using a model driven proton transfer system, we numerically demonstrate that the adiabatic extension of a beyond-BO ground state functional captures the dominant nonadiabatic effects in the regime of slow driving.

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• Density functional theory
• Molecular dynamics