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

Yang Zhong; Xiwen Li; Xingao Gong; Hongjun Xiang

physics.comp-phMar 20, 2026

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

Authors: Yang Zhong, Xiwen Li, Xingao Gong, Hongjun Xiang

Organizations: Key Laboratory of Computational Physical Sciences (Ministry of Education), Institute of Computational Physical Sciences, State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai, 200433, China.

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.

Figures & tables

Figure 1: Architecture of HamGNN-LR for predicting

H_{\text{total}}=H_{\text{sr}}+H_{\text{lr}}

. Short-range E(3)-equivariant message passing constructs

H_{\text{sr}}

from local environments. An Ewald attention module captures long-range correlations in reciprocal space with

\mathcal{O}(N|\mathcal{K}|)

cost, linear in system size

N

for fixed sampled wave vectors

|\mathcal{K}|

, and decodes ionic charges

Q_{i}

and weight matrices

W_{i,\mu\nu}

. The decoded charges are neutralized (

Q_{j}\!\mapsto\!\tilde{Q}_{j}

) and, together with the weight matrices

W_{i,\mu\nu}

, substituted into the analytic correction

H_{\text{lr}}

[Eq. ( 3 )].

System	Split	SR	LR-loc	LR-EA	EA
ZnO slab	Train	1.278	0.685	0.596	1.099
ZnO slab	Test	1.336	0.747	0.635	1.153
GaN/AlN	Train	3.623	1.462	1.184	3.556
GaN/AlN	Test	3.646	1.313	1.162	3.598
CdSe/ZnS	Train	3.380	1.311	1.057	3.231
CdSe/ZnS	Test	3.279	1.292	1.058	3.158

Table 1: Hamiltonian MAE (meV) for single-layer models on three polar/heterostructure systems. SR: short-range baseline; LR-loc: long-range correction [Eq. ( 3 )] with local decoding; LR-EA: full HamGNN-LR (correction + Ewald attention); EA: Ewald attention only, without physics-based

H_{\text{lr}}

Figure 2: Hamiltonian MAE of single-layer SR (red circles) and LR (blue squares) models versus layer thickness for (a) GaN/AlN superlattices and (b) polar ZnO slabs. In (a), SR error grows with superlattice period and saturates at

\sim

140 Å; in (b), SR error rises with slab thickness before leveling off. The LR model [Eq. ( 3 )] maintains uniformly low errors across all thicknesses.

Figure 3: Real-space onsite energy and band structure of the 56-layer polar ZnO slab. (a) Structural schematic; red arrow denotes spontaneous polarization $\mathbf{P}$ . (b)–(e) Spatially resolved average onsite Hamiltonian matrix elements along $z$ for (b) DeepH-E3, (c) DeePTB-E3, (d) short-range HamGNN, and (e) HamGNN-LR. Open symbols: DFT reference; solid lines: model predictions. Shaded regions in (b)–(d) highlight staircase discontinuities arising from finite receptive fields; these artifacts are absent in (e) owing to the long-range correction. (f)–(h) Band structures along $\Gamma$ – $K$ – $M$ – $\Gamma$ for (f) short-range HamGNN, (g) HamGNN-LR, and (h) DFT reference. Dashed red lines mark the Fermi level. HamGNN-LR closely reproduces DFT bands, whereas the short-range model exhibits visible deviations near the band gap.

Split	DeepH-E3	DeePTB-E3	HamGNN	HamGNN-LR
Train	0.462	1.940	0.397	0.291
Test	0.578	2.450	0.538	0.349
ZnO (thick)	1.525	2.383	1.565	0.473

Table 2: Hamiltonian MAE (meV) for three-layer models on ZnO slabs. “ZnO (thick)” denotes size extrapolation to a 56-layer (147.25 Å) slab unseen during training.

Computational Materials Science

Computational materials science uses computation and data-driven models to predict, explain, and design materials and their properties.

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Scalable long-range machine learning interatomic potentials with electrical response for atomistic material modeling

Yicheng Chen, Xiaoyun Yu, Hyejeong Song, Xintian Wang+4

Department of Materials Science and Engineering, National University of Singapore, Singapore · Department of Physics, National University of Singapore, Singapore

Machine Learning Interatomic Potentials Computational Materials Science Materials Property Prediction High-Performance Computing

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Accurate atomistic simulations of materials under electric fields require long-range electrostatics and electrical-response modeling at length scales beyond the reach of conventional first-principles methods. Yet these capabilities remain difficult to reconcile with the scalability of machine-learning interatomic potentials (MLIPs). Strictly local models enable efficient spatial parallelization but neglect long-range interactions, whereas existing electrostatic extensions introduce global dependencies that hinder distributed computation. Here, we overcome this limitation by analytically separating the long-range electrostatic forces and Born effective charges (BECs) into a global electrostatics calculation and strictly local derivatives of learned latent charges. This formulation preserves the spatial decomposability of equivariant MLIPs while enabling distributed evaluation of energies, forces, stresses, and BECs across different boundary conditions. Implemented as DPA4C-LR on the nvalchemi platform, the framework scales to 5.2 million atoms on four GPUs at approximately twice the computational cost of its short-range backbone. Trained on energies and forces, DPA4C-LR accurately infers BECs and reproduces the field-dependent infrared spectra of water. We further demonstrate its capabilities by recovering the characteristic slim polarization hysteresis of relaxor \ce{Pb(Mg_{1/3}Nb_{2/3})O3}, which is obscured by finite-size effects in smaller simulations; predicting a room-temperature field-driven ionic conductivity of

6.8 \pm 0.7

~mS/cm in amorphous \ce{Li_{0.9}NbO_{0.9}Cl_{4.1}}; and modeling interfacial thermal transport across Si/diamond interfaces with slab-corrected electrostatics. By combining long-range physics, electrical-response inference, and distributed computation, this framework extends predictive atomistic modeling toward device-relevant length scales.

Abstract

Computational Materials Science

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