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-centered Gaussian fields resolve the near-nuclear region, while a molecule-level correction enforces the electron count. Analytical Gaussian integrals and numerical quadrature connect the predicted fields to the Kohn–Sham matrix. One diagonalization provides estimates of frontier orbital energies and the occupied subspace, and yields the density matrix used to evaluate energy components and initialize SCF. Mean molecular weighted normalized mean absolute errors (wNMAEs) are 0.110% for QM9 and 0.199% for VQM24 on PySCF level-0 grids, increasing to 1.366% and 2.218%, respectively, on level-3 grids. Models trained on def2-SVP densities transfer without retraining to def2-TZVP and def2-QZVP, retaining total-energy accuracy and occupied-subspace agreement. The Hybrid method combines finite differences and first-order automatic differentiation with analytical Gaussian integrals and multilevel quadrature to accelerate field-to-matrix conversion. It achieves near-linear density-matrix construction in both alkanes and PEG chains. For the 1004-atom PEG chain, this construction is 8.75-fold faster than full AD and 7.34-fold faster than converged GPU4PySCF SCF. Without fine-tuning, the QM9 model attains a total-energy MAE of 9.437 meV atom–1 on 1000 larger OE62 molecules. Across the three basis sets, DeepHartree initialization reduces mean QM9 SCF iteration counts by 15.3–15.6%. For a 620-atom organic cage, it reduces total SCF wall-clock time by 12.9%, including initialization. These results establish continuous electrostatic fields as an accurate and scalable interface between machine learning and LCAO DFT.