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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 $χ(R)$ and an induced vector potential $A_μ(R)$ and quantum geometric tensor $\mathcal{T}_{μν}(R)$ derived from the conditional electronic wavefunction $Φ_R(r)$ over nuclear configuration space, where $r=\mathbf{r}_1,\mathbf{r}_2,\ldots$ are electronic coordinates and $R=\mathbf{R}_1,\mathbf{R}_2,\ldots$ are nuclear coordinates. The ground state $(n_R,χ,A_μ,\mathcal{T}_{μν})$ can be calculated by solving self-consistently (i) conditional Kohn-Sham equations containing an effective potential $v_{\rm s}(\mathbf{r})$ that depends parametrically on $R$, (ii) the Schrödinger equation for $χ(R)$ and (iii) Euler-Lagrange equations that determine $\mathcal{T}_{μν}$. The theory is applied to the $E\otimes e$ Jahn-Teller model.