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

TL;DR

This work delivers a unified quantum framework for electrons and ions by extending the self-consistent harmonic approximation (SCHA) to fermionic systems and embedding the density matrix in a curved manifold defined by a neural-network transformation. The key idea is to replace the Gaussian density with a nonlinear deformation (Gaussian stretcher) implemented as stacked layers, while preserving an analytically tractable entropy and the harmonic auxiliary Hamiltonian. The resulting nonlinear SCHA (NLSCHA) captures non-Gaussian features, quantum tunneling, and static electronic correlations, demonstrated through benchmark problems such as a deep 1D double-well, the hydrogen atom cusp, and H$_2$ dissociation, including excited states. This approach enables direct computation of free energies and phase diagrams without relying on adiabatic approximations, potentially enabling more accurate simulations of correlated quantum materials and excitonic effects in a scalable, first-principles framework.

Abstract

The numerical solution of the many-body problem of interacting electrons and ions is a key challenge in condensed matter physics, chemistry, and materials science. Traditional methods to solve the multi-component 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 parametrized 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 H$_2$ dissociation -- we demonstrate it can address both the ground- and excited-state properties of electronic systems, capture quantum tunneling and static electronic correlations, thereby providing a unified quantum framework of electrons and atomic nuclei.

Figures (8)

• Figure 1: Schematic representation of the SCHA on a neural network curved manifold (nonlinear SCHA: NLSCHA). The Gaussian density matrix $\tilde{\rho}$, solution of the trial harmonic Hamiltonian $\tilde{\mathcal{H}}$ in an auxiliary space $\boldsymbol{q}$ (the curved manifold) is transformed into real space thanks to the transformation ${\bm r}(\boldsymbol{q})$ defined through a neural network. The nonlinearity of the transformation deforms the Gaussian shape and introduces new features on the final density matrix, including correlations beyond the original distribution.
• Figure 2: States of the noninteracting fermionic oscillator in 3D. Electrons populate the excited states of the harmonic Hamiltonian $\ket {n_1n_2 n_3}$ according to the Fermi-Dirac statistics and the spin multiplicity.
• Figure 3: Comparison between the SCHA, NLSCHA, and exact (numerical) solution for a double well potential. (a) Probability density of the ground state wavefunction at $b = 6.0\hartree\per\bohr^2$ by varying the number of layers of the nonlinear transformation. (b) Potential energy landscape of the double well. (c) Comparison between the nonlinear transformations $q(r)$ encoded by the neural network and the one that exactly maps the SCHA Gaussian wavefunction to the true ground state of the double well potential. (d) The energy error at several values of $b$ and varying the number of layers. All the values are in the deep quantum regime $b >2\hartree\per\bohr^2$. The wavefunction is constrained to keep the inversion symmetry (inhibiting the transition into the localized state). The exact ground state of the double well potential has been evaluated numerically with the implicitly restarted Lanczos algorithmLanczos as implemented in Scipyscipy.
• Figure 4: Comparison between the exact and NLSCHA (2 layers) excited state density. The potential is a double well of the form $V(r) = a r^4 - br^2$, where $a = 2\hartree\per\bohr^4$ and $b = -6\hartree\per\bohr^2$, as in Fig. \ref{['fig:double:well']}(c). We highlight the position of the node of the wavefunction in the origin.
• Figure 5: Hydrogen atom, comparison between the SCHA, nonlinear SCHA (NLSCHA), and exact (analytical) solution. (a) Comparison between the radial wavefunction of the $1s$ orbital around the electron-ion cusp in the origin. (b) Error on the ground state energy compared to the analytical result (log scale).