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

Abstract

Density functional theory (DFT) with linear combinations of atomic orbitals (LCAO) is costly because of Coulomb-matrix construction and self-consistent-field (SCF) iterations. 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 yields the electron density. Atom-centred Gaussian fields resolve near-nuclear structure, and a molecular correction enforces the electron count. Analytical Gaussian integrals and numerical quadrature assemble the Kohn-Sham matrix; one diagonalization yields orbital estimates and a density matrix for energy evaluation and SCF initialization. Mean molecular weighted normalized mean absolute density errors (wNMAEs) are 0.110% for QM9 and 0.199% for VQM24 on PySCF level-0 grids, rising to 1.366% and 2.218% on level-3 grids. Models trained on def2-SVP densities transfer to def2-TZVP and def2-QZVP without retraining, retaining total-energy accuracy and occupied-subspace agreement. A Hybrid method combines finite differences, first-order automatic differentiation, analytical integrals, and multilevel quadrature for near-linear density-matrix construction in alkanes and PEG chains. Excluding grid construction, the density-matrix build for a 1004-atom PEG chain is 8.75-fold faster than full automatic differentiation and 7.34-fold faster than converged GPU4PySCF SCF. Without fine-tuning, the QM9 model achieves a total-energy MAE of 9.437 meV per atom on 1,000 larger OE62 molecules. DeepHartree initialization reduces mean QM9 SCF iterations by 15.3-15.6% across the three basis sets and total SCF time by 12.9% for a 620-atom organic cage, including initialization. These results support continuous electrostatic fields as an accurate, scalable interface between machine learning and LCAO DFT.

arXiv commentaccepted by Journal of Chemical Theory and Computation