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.