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.