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Table 2. Summary of DHNNs, including model name, model type, code repository, data source and link, and a brief description of their usage
Model Type Code repository Data source Usage Data link
DeepH [4] Message-passing DFT Hamiltonian https://github.com/ Zenodo Uses DFT Hamiltonian matrices & energies generated on Materials Project (via https://zenodo.org/
network; dense invariant GNN mzjb/DeepH-pack ABACUS, OpenMX, FHI-aims, SIESTA) to train a direct mapping from crystal records/6555484
structure to Hamiltonian
DeepH- E(3)-equivariant attention https://github.com/ Provided with paper Uses Materials Project DFT structures & energies released alongside the https://www.nature.com/
[67]
E3 transformer Xiaoxun-Gong/DeepH-E3 publication to train an equivariant attention/Transformer architecture - achieving articles/s41467-023-
strict rotational & translational equivariance 38468-8
[77]
HamGNN E(3)-equivariant convolutional https://github.com/ Zenodo (pretrained Trained on DFT-generated tight-binding Hamiltonian matrices for QM9 Pretrained models:
GNN for tight-binding Hamiltonian QuantumLab-ZY/ models and molecules, carbon & silicon allotropes, SiO polymorphs, and Bi Se compounds - DOI: 10.5281/zenodo.
γ
x
2
HamGNN datasets) enabling high-accuracy transfer to large-scale systems (e.g., Moiré bilayer MoS , 8147631
2
Si dislocation supercells) Training data:
DOI: 10.5281/zenodo.
8157128
DeepH- Hybrid-functional DFT Hamiltonian https://github.com/ Zenodo Uses hybrid-functional Hamiltonian & frequency-response (χ ) data covering https://zenodo.org/
[78] xx
hybrid predictor aaaashanghai/DeepH- various twist angles of Moiré bilayer MoS ; bypasses SCF iterations to directly records/13444159
2
hybrid predict hybrid-functional Hamiltonians
DeepH- DFPT-enhanced Hamiltonian interface code + datasets Zenodo Includes FHI-aims computed DFPT phonon spectra & force-constant data to https://zenodo.org/
DFPT [79] network on Zenodo introduce phononic corrections into Hamiltonian predictions - improving accuracy records/13943187
for phonon-response properties
HarmoSE [80] Two-stage SO(3)-equivariance + N/A Materials Project Stage 1: group-theory neural layers extract SO(3)-equivariant baseline N/A
expressiveness framework Hamiltonians; stage 2: non-linear 3D graph Transformer refines them for high
accuracy
[81]
xDeepH E(3)×{I,T}-equivariant spin-orbital https://github.com/ Zenodo Uses constrained-DFT (OpenMX/DeepH-pack) Hamiltonian & overlap matrices https://zenodo.org/
GNN mzjb/xDeepH for magnetic superstructures (e.g., CrI , skyrmion lattices) to train a mapping from records/7669862
3
structure + spin to Hamiltonian
DeepH- Real-space reconstruction of plane- https://github.com/ Zenodo Provides reconstructed AO Hamiltonian datasets converted from PW DFT outputs DOI: 10.5281/zenodo.
PW [82] wave DFT → atomic-orbital Xiaoxun-Gong/HPRO for 300 bilayer-graphene and 256 MoS supercells - used to train the real-space 13377497
2
Hamiltonian reconstruction network
WANet [74] Scalable Hamiltonian prediction via N/A PubChemQH Builds on eSCN convolutions (reducing complexity), sparse experts for different N/A
SO(2) convolutions, Mixture-of- dataset length scales, and many-body density trick to predict full Hamiltonian matrices for
Experts & MACE density trick very large molecules
DeepH- Universal Materials Model for N/A N/A Built on > 10,000 Materials Project DFT Hamiltonian matrices spanning 89 N/A
[83]
UMM Hamiltonian elements to create a universal model transferable across element combinations
and crystal structures
[84]
MEHnet Multi-task CCSD(T)-trained https://github.com/ FigShare Includes scripts to build the CCSD(T) training dataset, train the multi-task DOI: 10.6084/m9.figshare.
molecular Hamiltonian + properties htang113/Multi-task- network on energies, dipoles, orbitals, etc., and apply it to both small hydrocarbons 25762212
predictor electronic and QM9 benchmarks
HamGNN- Electron–phonon coupling workflow https://github.com/ N/A Uses HamGNN-predicted Hamiltonian matrices & gradients together with N/A
[85]
EPC leveraging HamGNN QuantumLab-ZY/ Phonopy/DFPT data to accelerate computation of electron–phonon coupling
HamEPC matrices, carrier mobilities, superconducting transition temperatures, etc.
U- Unsupervised pretrained E(3)- https://github.com/ Zenodo Conducts unsupervised pretraining on a large unlabeled crystal-structure https://zenodo.org/
HamGNN [61] equivariant HamGNN QuantumLab-ZY/ database to learn E(3)-equivariant representations, then fine-tunes on small records/10827117
HamGNN labeled DFT Hamiltonian sets to boost initialization and generalization