DeepHartree: A Poisson-Coupled Neural Field for One-Shot Density Functional Theory

Abstract

Linear-combination-of-atomic-orbital (LCAO) density functional theory (DFT) incurs steep costs when it constructs Coulomb terms and iterates the self-consistent field (SCF) equations. Matrix-learning approaches can bypass parts of this workflow, but their outputs inherit the dimensions and conventions of a fixed orbital basis. We introduce DeepHartree, a Poisson-coupled neural field that connects continuous real-space prediction to LCAO electronic structure. An E(3)-equivariant network predicts the Hartree potential, and the Poisson equation converts this potential into electron density. Atom-centred Gaussian fields resolve the near-nuclear region, while a molecule-level correction enforces the electron count. Numerical quadrature assembles the Kohn--Sham matrix. One diagonalization recovers energy components, frontier levels, and occupied subspaces and produces the density matrix used for SCF initialization. The molecular mean weighted normalized mean absolute error (wNMAE) is 0.361% on QM9 and 1.397% on the chemically broader VQM24 dataset. Our Hybrid7 scheme uses a seven-point finite difference to obtain the learned local density and first-order automatic differentiation to obtain its gradient. It avoids higher-order differentiation graphs, runs 1.33--1.37 times faster than full automatic differentiation, and remains stable for systems approaching 1,000 atoms. Without fine-tuning, the QM9 model attains a total-energy MAE of 15.601~meV atom${}^{-1}$ on 1,000 larger OE62 molecules. DeepHartree initial guesses reduce SCF iterations for 88.62% of QM9 test molecules, with a 14.5% mean reduction. These results establish continuous electrostatic fields as an accurate and scalable interface between machine learning and LCAO DFT.