# Advancing Universal Deep Learning for Electronic-Structure Hamiltonian Prediction of Materials

arXiv: https://arxiv.org/abs/2509.19877 · HTML: https://mlsanity.com/paper/2509.19877/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 51 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 24 of 24)

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction](https://mlsanity.com/paper/2502.12147/) (2025)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Spherical Channels for Modeling Atomic Interactions](https://mlsanity.com/paper/2206.14331/) (2022)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [QH9: A Quantum Hamiltonian Prediction Benchmark for QM9 Molecules](https://mlsanity.com/paper/2306.09549/) (2023)
- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [Efficient and Scalable Density Functional Theory Hamiltonian Prediction through Adaptive Sparsity](https://mlsanity.com/paper/2502.01171/) (2025)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction](https://mlsanity.com/paper/2405.05722/) (2024)
- [Universal Machine Learning Kohn-Sham Hamiltonian for Materials](https://mlsanity.com/paper/2402.09251/) (2024)
- [Towards Harmonization of SO(3)-Equivariance and Expressiveness: a Hybrid Deep Learning Framework for Electronic-Structure Hamiltonian Prediction](https://mlsanity.com/paper/2401.00744/) (2024)

## Built on by (most cited first, 2 of 2)

- [NextCrystal: a Symmetry-Driven Generative Framework for Crystal Structure Prediction](https://mlsanity.com/paper/2602.17176/) (2026)
- [Physics-Aligned Electronic Ground-State Learning Improves Generalization](https://mlsanity.com/paper/2610.10298/) (2026)

## Abstract (arXiv)

Deep learning methods for electronic-structure Hamiltonian prediction has offered significant computational efficiency advantages over traditional DFT methods, yet the diversity of atomic types, structural patterns, and the high-dimensional complexity of Hamiltonians pose substantial challenges to the generalization performance. In this work, we contribute on both the methodology and dataset sides to advance universal deep learning paradigm for Hamiltonian prediction. On the method side, we propose NextHAM, a neural E(3)-symmetry and expressive correction method for efficient and generalizable materials electronic-structure Hamiltonian prediction. First, we introduce the zeroth-step Hamiltonians, which can be efficiently constructed by the initial charge density of DFT, as informative descriptors of neural regression model in the input level and initial estimates of the target Hamiltonian in the output level, so that the regression model directly predicts the correction terms to the target ground truths, thereby significantly simplifying the input-output mapping for learning. Second, we present a neural Transformer architecture with strict E(3)-Symmetry and high non-linear expressiveness for Hamiltonian prediction. Third, we propose a novel training objective to ensure the accuracy performance of Hamiltonians in both real space and reciprocal space, preventing error amplification and the occurrence of "ghost states" caused by the large condition number of the overlap matrix. On the dataset side, we curate a high-quality broad-coverage large benchmark, namely Materials-HAM-SOC, comprising 17,000 material structures spanning 68 elements from six rows of the periodic table and explicitly incorporating SOC effects. Experimental results on Materials-HAM-SOC demonstrate that NextHAM achieves excellent accuracy and efficiency in predicting Hamiltonians and band structures.