Unified quantum framework for electrons and ions: The self-consistent harmonic approximation on a neural network curved manifold

The numerical solution of the many-body problem, which involves interacting electrons and ions, is a key challenge in condensed matter physics, chemistry, and materials science. Traditional methods to solve the multicomponent quantum Hamiltonian are usually specialized for one kind of particles - electrons or ions - and can suffer from a methodological gap when applied to the other ones. This work extends the self-consistent harmonic approximation, a proven successful technique for simulating quantum ions at finite temperatures in anharmonic crystals, to electrons. The approach minimizes the total free energy by optimizing an ansatz density matrix, solving a fermionic self-consistent harmonic Hamiltonian on a curved manifold parameterized through a neural network. This approach preserves an analytical expression for entropy, enabling the direct computation of free energies and phase diagrams of materials. By benchmarking this technique across several prototypical cases - a double-well potential, the hydrogen atom, and the H2 dissociation - we demonstrate that it can address both the ground- and excited-state properties of electronic systems, capture quantum tunneling and static electronic correlations, and thereby provide a unified quantum framework of electrons and atomic nuclei.