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Page 14 of 23 Li et al. J. Mater. Inf. 2025, 5, 43 https://dx.doi.org/10.20517/jmi.2025.17

HamGNN- E(3)-equivariant GNN for charged- N/A OpenMX Trained on TB Hamiltonian matrices for GaAs charged defects (vacancies, N/A
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Q defect tight-binding Hamiltonian interstitials, substitutions at Q = 0, ±3) to learn structure + background-charge →
Hamiltonian mapping with reciprocal-space band-energy regularization
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SchNOrb SchNet-based molecular orbital & https://github.com/ Quantum- Uses QM9 dataset [134 k small molecules at B3LYP/6-31G(2df,p)] to predict http://www.quantum-
wavefunction predictor atomistic-machine- machine.org molecular orbital coefficients and electron densities machine.org/datasets/
learning/SchNOrb
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DeepTB Deep-learning Slater–Koster tight- https://github.com/ DeePTB examples Provides Slater–Koster TB parameters for graphene, MoS and bulk phases https://github.com/
2
binding Hamiltonian; supports SOC deepmodeling/DeePTB (VASP/Wannier90) to train deep TB models for large-scale, near-DFT-level deepmodeling/DeePTB/
simulations tree/main/examples
DHNNs: Deep Hamiltonian neural networks; GNN: graph neural network; DFT: density functional theory; SCF: self-consistent field; DFPT: density functional perturbation theory; EPC: electron-phonon coupling; SOC:
spin-orbit coupling; VASP: Vienna Ab initio Simulation Package.

Scalability poses a fundamental bottleneck for DHNNs. As the number of atoms grows, both the dimensionality of the Hamiltonian matrix and the complexity
of interatomic couplings increase combinatorially. The strictly localized equivariant message-passing (SLEM) framework [Figure 6C] overcomes this by
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enforcing a hard cutoff on each node’s receptive field, so that messages propagate only within a fixed local neighborhood. By confining interactions in this way,
SLEM reduces the overall computational cost to scale linearly with system size, while its inherently local structure lends itself to straightforward parallelization.
Thereby, it enables accurate Hamiltonian predictions for truly large and complex materials.

Regarding the gold-standard accuracy, two complementary strategies have emerged in DHNNs: delta learning and multi-task training. In the delta learning
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paradigm exemplified by the DeepKS framework , the network is trained to predict only the difference between a low-cost GGA (PBE) Hamiltonian and the
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corresponding high-accuracy hybrid (HSE06) result, thereby attaining hybrid-functional precision at a computational cost comparable to PBE . In contrast,
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the multi-task electronic Hamiltonian network (MEHnet) learns to map atomic geometries simultaneously to Kohn-Sham Hamiltonians across multiple
levels of theory and basis sets, endowing the model with robust transferability. Although initially trained on small molecules, MEHnet has been shown to
generalize seamlessly to much larger systems, such as polycyclic aromatics and semiconducting polymers, where even single-point CCSD(T) calculations are
impossible.

Ultimately, the true impact of a universal DHNN will be measured by its seamless integration into end-to-end application workflows from quantum-transport
and optical-response simulations to high-throughput materials screening. Realizing this vision demands the development of robust, modular interfaces and
pipelines that permit straightforward fine-tuning and deployment of pretrained Hamiltonians across diverse computational tasks. By assembling such an
interoperable software ecosystem, we can democratize access to first-principles accuracy and unlock new avenues for rapid, large-scale quantum mechanical
discovery across the materials science community.