Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF

We perform model calculations for a stretched LiF molecule, demonstrating that nonadiabatic charge transfer effects can be accurately and seamlessly described within a density functional framework. In alkali halides like LiF, there is an abrupt change in the ground state electronic distribution due to an electron transfer at a critical bond length $R=R_c$, where a barely avoided crossing of the lowest adiabatic potential energy surfaces calls the validity of the Born-Oppenheimer approximation into doubt. Modeling the $R$-dependent electronic structure of LiF within a two-site Hubbard model, we find that nonadiabatic electron-nuclear coupling produces a sizable elongation of the critical $R_c$ by 0.5 Bohr. This effect is very accurately captured by a simple and rigorously-derived correction, with an $M^{-1}$ prefactor, to the exchange-correlation potential in density functional theory; $M=$ reduced nuclear mass. Since this nonadiabatic term depends on gradients of the nuclear wavefunction and conditional electronic density, $\nabla_R \chi(R)$ and $\nabla_R n(\mathbf{r},R)$, it couples the Kohn-Sham equations at neighboring $R$ points. Motivated by an observed localization of nonadiabatic effects in nuclear configuration space, we propose a local conditional density approximation -- an approximation that reduces the search for nonadiabatic density functionals to the search for a single function $y(n)$.