Testing exact-factorization-based density functional approximation on a continuous density model

In this work, we test the performance of an exact-factorization-based density functional approximation (DFA) for electron-nuclear correlation beyond the Born-Oppenheimer approximation that was derived in Li et al. [J. Chem. Phys.148, 084110 (2018)]. The present work extends beyond the Hubbard model that was used previously in a two-electron Shin-Metiu model with continuous electronic density in one dimension. Using new iteration techniques, we managed to solve coupled Kohn-Sham equations that embody nonadiabatic corrections with a nuclear Schrödinger equation. Our results show that the DFA can successfully capture the nonadiabatic effect caused by electron-nuclear correlation as manifested in the correct shift of the critical nuclear position where the proton-coupled electron transfer takes place. The nonadiabatic correction to the Kohn-Sham potential also leads to a shift of orbital energies, which can potentially be useful to study band renormalization induced by electron-phonon interactions in bulk materials.