Physics-Informed Long-Range Coulomb Correction for Machine-learning Hamiltonians

2026-03-20

AI Summary

This work addresses the inability of existing machine learning Hamiltonian models to accurately capture macroscopic electrostatic behavior in polar crystals and heterostructures due to their neglect of long-range Coulomb interactions. The authors propose HamGNN-LR, a novel model that, for the first time, rigorously derives long-range Hamiltonian matrix elements within a non-orthogonal atomic orbital basis by variationally decomposing the electrostatic energy and establishing a consistent mapping from the density matrix to effective atomic charges. By integrating E(3)-equivariant message passing with reciprocal-space Ewald summation, the model constructs a dual-channel architecture that fuses real- and reciprocal-space information under physical constraints. Evaluated on ZnO, CdSe/ZnS, and GaN/AlN systems, HamGNN-LR reduces prediction errors by 2–3×, effectively eliminates staircasing artifacts under built-in electric fields, and demonstrates excellent cross-scale transferability.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: (Large) Language Models
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Machine-learning electronic Hamiltonians achieve orders-of-magnitude speedups over density-functional theory, yet current models omit long-range Coulomb interactions that govern physics in polar crystals and heterostructures. We derive closed-form long-range Hamiltonian matrix elements in a nonorthogonal atomic-orbital basis through variational decomposition of the electrostatic energy, deriving a variationally consistent mapping from the electron density matrix to effective atomic charges. We implement this framework in HamGNN-LR, a dual-channel architecture combining E(3)-equivariant message passing with reciprocal-space Ewald summation. Benchmarks demonstrate that physics-based long-range corrections are essential: purely data-driven attention mechanisms fail to capture macroscopic electrostatic potentials. Benchmarks on polar ZnO slabs, CdSe/ZnS heterostructures, and GaN/AlN superlattices show two- to threefold error reductions and robust transferability to systems far beyond training sizes, eliminating the characteristic staircase artifacts that plague short-range models in the presence of built-in electric fields.

Problem

• long-range Coulomb interactions
• machine-learning Hamiltonians
• polar crystals
• heterostructures
• electrostatic potentials

Innovation

• Physics-informed machine learning
• Long-range Coulomb interaction
• Electronic Hamiltonian
• Ewald summation
• E(3)-equivariant GNN