Exact factorization-based density functional theory of electrons and nuclei

The ground state energy of a system of electrons and nuclei is proven to be a variational functional of the conditional electronic density $n_R(\mathbf{r})$, the nuclear wavefunction $\chi (R)$ and an induced vector potential $A_{\mu }(R)$ and quantum geometric tensor ${\mathcal{T}}_{\mu \nu }(R)$ derived from the conditional electronic wavefunction ${\mathrm{\Phi }}_R(r)$ over nuclear configuration space, where $r={\mathbf{r}}_1,{\mathbf{r}}_2,\dots$ are electronic coordinates and $R={\mathbf{R}}_1,{\mathbf{R}}_2,\dots$ are nuclear coordinates. The ground state $(n_R,\chi ,A_{\mu },{\mathcal{T}}_{\mu \nu })$ can be calculated by solving self-consistently (i) conditional Kohn-Sham equations containing an effective potential $v_{\mathrm{s}}(\mathbf{r})$ that depends parametrically on $R$, (ii) the Schr"odinger equation for $\chi (R)$ and (iii) Euler-Lagrange equations that determine ${\mathcal{T}}_{\mu \nu }$. The theory is applied to the $E\otimes e$ Jahn-Teller model.