Neural Wave Functions for High-Pressure Atomic Hydrogen

TL;DR

The study advances neural quantum states for electron–proton systems by integrating a backflow-enhanced orbital structure with a two-graph message-passing neural network (MPNN) that couples electrons and protons, enabling simultaneous sampling of electronic and nuclear degrees of freedom. The approach yields Born–Oppenheimer energies at up to $N=128$ atoms that match or surpass prior DMC/RMC benchmarks using VMC, and it remains accurate when incorporating nuclear quantum effects beyond the BOA by explicitly sampling proton dynamics. It preserves translational invariance without enforcing crystalline symmetry, and employs twist-averaged boundary conditions to control finite-size shell effects, enabling application to ultra-high-density hydrogen and the onset of crystal formation and pressure-induced melting. Overall, the method opens a path to accurate, symmetry-free exploration of high-pressure phase diagrams, isotope effects, and nuclear-spin phenomena in hydrogen and related systems.

Abstract

We leverage the power of neural quantum states to describe the ground state wave function of solid and liquid atomic hydrogen, including both electronic and protonic degrees of freedom. For static protons, the resulting Born-Oppenheimer energies are consistently comparable to or lower than all previous projector Monte Carlo results for systems containing up to $128$ hydrogen atoms. The same level of accuracy is preserved upon inclusion of nuclear quantum effects, thus going beyond the Born-Oppenheimer approximation. In addition, our description overcomes major limitations of current wave functions, notably by avoiding any explicit symmetry assumption on the expected quantum crystal, and sidestepping efficiency issues of imaginary time evolution with disparate mass scales. As a first application, we examine crystal formation in an extremely high-density region up to pressure-induced melting.

Figures (2)

• Figure 1: Four pair correlation functions are shown for $N=54$ between three different types of particles: "e", "p" and "e$_{\uparrow}$", corresponding to electrons, protons and spin-up electrons respectively. An inset with 300 Monte Carlo configurations in the BCC crystal is depicted (for $N=16$, for clarity), where protons form the localized red balls while electrons form the delocalized blue cloud.
• Figure 2: Pair correlation function for $N=8$ hydrogen atoms at different densities under PBC. The same color scale is used in all subplots, globally adjusted with a power law normalization to enhance visibility of the structural features. A fixed threshold is manually set to prevent plotting areas where the pair correlation function has low values. At $r_s=0.2$, the structure is drastically reduced suggesting that the system turned liquid.