The ground state energy of a system of electrons (r = r₁, r₂, ...) and nuclei (R = R₁, R₂), ... is proven to be a variational functional of the electronic density n(r,R) and paramagnetic current density j_p(r,R) conditional on R, the nuclear wave function χ(R), an induced vector potential A_μ(R) and a quantum geometric tensor T_μι. n, j_p, A_μ and T_μι are defined in terms of the conditional electronic wave function ΦR(r). The ground state (n, j_p, χ, A_mu, T_μι ) can be calculated by solving self-consistently (i) conditional Kohn-Sham equations containing effective scalar and vector potentials v_s(r) and Axc(r) that depend parametrically on R, (ii) the Schrödinger equation for χ(R), and (iii) Euler-Lagrange equations that determine T_μι. The theory is applied to the E ⊗ e Jahn-Teller model.