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Density functional theory (DFT) serves as the primary method of calculating electronic structures in physics, chemistry, and materials science. However, its practical application is fundamentally limited by a computational cost that scales cubically with system size, making high-precision studies of complex or large-scale materials prohibitively expensive. This review addresses the key challenge by examining the rapidly evolving paradigm of integrating machine learning (ML) with first-principles calculations to significantly accelerate and expand electronic structure prediction. Our primary objective is to provide a comprehensive and critical overview of the methodological advances, physical outcomes, and transformative potential of this interdisciplinary field. The core methodological progress involves a shift from black-box property predictors to symmetry-preserving, transferable models that learn the fundamental Hamiltonian—the central quantity from which diverse electronic properties are derived. We detail this evolution, beginning with pioneering applications in molecular systems by using graph neural networks (e.g., SchNOrb, DimeNet) to predict energies, wavefunctions, and Hamiltonian matrices with meV-level accuracy. This review then focuses on the critical extension to periodic solids, where maintaining symmetries such as E(3)-equivariance and handling vast configurational spaces are of utmost importance. We systematically analyze three leading model families that define the state-of-the-art: the DeepH series, which uses local coordinate message passing and E(3)-equivariant networks to achieve sub-meV accuracy and linear scaling; the HamGNN framework, built on rigorous equivariant tensor decomposition, which excels in modeling systems with spin-orbit coupling and charged defects; and the DeePTB approach, which leverages deep learning for tight-binding Hamiltonian parameterization, enabling quantum-accurate simulations of millions of atoms. These methods yield significant physical results and computational breakthroughs. Key outcomes include: 1) unprecedented accuracy and speed. Models consistently achieve Hamiltonian prediction mean absolute errors (MAE) below 1 meV (e.g., DeepH-E3: ~0.4 meV in graphene; HamGNN: ~1.5 meV in QM9 molecules), along with computational speedups of 3 to 5 orders of magnitude compared with traditional DFT. 2) Scale bridging. Successful applications now range from small molecules to defect-containing supercells with over 10000 atoms (e.g., HamGNN-Q on a 13,824-atom GaAs defect) and even to millions of atoms for optoelectronic property simulations (DeePTB). 3) Expanded application scope. This review highlights how these ML-accelerated tools are revolutionizing research in previously intractable areas: predicting spectroscopic properties of molecules (e.g., DetaNet for NMR/UV-Vis spectra), elucidating electronic structures of topological materials and magnetic moiré systems, computing electron-phonon coupling and carrier mobility with DFT-level accuracy but far greater efficiency (HamEPC framework), and enabling high-throughput screening for materials design. In conclusion, ML-accelerated electronic structure calculation has matured into a powerful paradigm, transitioning from a proof-of-concept to a tool capable of delivering DFT-fidelity results at dramatically reduced cost for systems of realistic scale and complexity. However, challenges remain, including model interpretability (“black-box” nature), transferability to unseen elements, and seamless integration with existing plane-wave DFT databases. Future directions include physics-constrained unsupervised learning (e.g., DeepH-zero), developing more universal and element-agnostic architectures, and creating closed-loop, artificial intelligence (AI)-driven discovery pipelines. By overcoming current limitations, these methods have the potential to fundamentally change the field of materials research, accelerating the process from atomistic simulation to rational material design and discovery. 
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Keywords:
- machine learning /
- graph neural networks /
- first-principles /
- electronic structure
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图 1 ML加速电子结构计算模型在分子领域发展时间轴(2017—2025), 来源于Hegde和Bowen[22] (2017)、Schütt等[11,28] (2017, 2019)、Gasteiger等[29] (2020)等文献[26–41]并整理. 图中E为轨道能量, H为哈密顿矩阵, MAE为平均绝对误差, GNN为图神经网络, DTNNs为深度张量神经网络, GPR为高斯过程回归, HOMO/LUMO为最高占据分子轨道/最低未占据分子轨道, LHC为光捕获复合物, KRR为核岭回归, GTO 为高斯轨道
Fig. 1. Development timeline for machine learning-accelerated electronic structure calculation models (2017—2025), with sources compiled from references such as Hegde, Bowen[22] (2017), Schütt et al.[11,28] (2017, 2019), Gasteiger et al.[29] (2020), and so on[26–41]. In the figure, E, orbital energy; H, Hamiltonian matrix; MAE, mean absolute error; GNN, graph neural network; DTNNs, deep tensor neural networks; GPR, Gaussian process regression; HOMO/LUMO, highest occupied molecular orbital/lowest unoccupied molecular orbital; LHC, light-harvesting complex; KRR, kernel ridge regression; GTO, Gaussian orbital.
图 2 ML加速固体材料电子结构计算的三大模型发展历程图(2022—2025), 来源根据徐勇(2022—2025)、向红军(2023—2025)、顾强强(2024, 2025)等文献[45–56]整理
Fig. 2. Timeline (2022—2025) of machine-learning-accelerated electronic-structure models in solid systems, compiled from the works of Xu Yong (2022—2025), Xiang Hongjun (2023—2025), Gu Qiangqiang (2024, 2025) and Refs. [45–56].
图 3 DeepH深度电子结构预测模型架构 (a) 通用信息处理框架, 展示了从结构输入→特征嵌入→核心网络(消息传递)→哈密顿矩阵→性质输出的完整流程; (b) DeepH架构, 基于局部坐标的LCMP消息传递; (c) xDeepH架构, 同时处理晶体结构和磁结构的双通道扩展网络; (d) DeepH-E3架构, 基于E3等变图神经网络; (e) DeepH-2神经网络架构图; (f) DeepH-zero神经网络架构和实现; (g) 发展历程与技术突破, 来源根据徐勇(2022—2025)等文献[45–50]整理
Fig. 3. Architecture evolution of deep electronic structure prediction models of DeepH: (a) Universal framework: structure → feature embedding → message passing → Hamiltonian matrix → property output; (b) DeepH: LCNN-based LCMP message passing; (c) XDeepH: dual-input extension (crystal + magnetic structures); (d) neural network architecture of DeepH-E3; (e) network architecture of DeepH-2; (f) architecture and implementation of neural-network DFT of DeepH-zero; (g) development history and technological breakthroughs; Compiled from the works of Xu Yong (2022—2025)[45–50].
图 4 HamGNN深度电子结构预测模型架构对比 (a) HamGNN和HamGNN-Q通用E(3)等变架构框架[51]; (b) HamGNN的通用电子结构预测架构[51]; (c) HamGNN-Q带电缺陷预测专用架构[54]
Fig. 4. Architecture evolution of deep electronic structure prediction models of HamGNN: (a) Common E(3)-equivariant framework of HamGNN and HanGNN-Q[51]; (b) general electronic structure prediction architecture of HamGNN[51]; (c) charged defect prediction specialist of HamGNN-Q[54].
图 6 ML加速电子结构计算在预测分子光谱中的应用实例[84] (a) 从左到右依次为环己酮、2-甲基吡嗪、庚–3, 5-二炔–2-酮、苯胺和5-甲氧基–1, 3-噁唑–2-甲醛的分子结构示意图; (b) 对比DetaNet预测的紫外-可见光谱(红色)与图(a)所示分子的参考数据(蓝色)DetaNet预测的吸收强度误差随波长的变化; (c) 对比了DetaNet预测的13CNMR谱与图(a)所示分子的参考数据; (d) 对比了DetaNet预测的1HNMR谱与图(a)所示分子的参考数据
Fig. 6. Application examples of machine learning accelerating electronic structure computation in predicting molecular spectroscopy, reprinted from[84]: (a) Left to right, schematic structures of cyclohexanone, 2-methylpyrazine, hepta3, 5-diyn-2-one, aniline and 5-methoxy-1, 3-oxazole-2-carbaldehyde; (b) comparison of the DetaNet-predicted (red) UV-Vis spectra with reference data (blue) for the molecules shown in panel (a); (c) comparison of the DetaNet-predicted 13C NMR spectra with reference data for the molecules shown in panel (a); (d) comparison of the DetaNetpredicted 1HNMR spectra with reference data for the molecules shown in panel (a).
图 7 ML加速电子结构方法预测缺陷体系的示例 (a) HamGNN模型预测的硅位错的能带结构和电荷密度[51]; (b) HamGNN-Q模型预测砷化镓带电缺陷结构的能带结构和电荷密度[54]
Fig. 7. Examples of defect system predictions using machine learning-accelerated electronic structure methods: (a) Band structure and charge density of a silicon dislocation predicted by using the HamGNN model[51]; (b) band structure and charge density of a charged defect structure in gallium arsenide (GaAs) by using the HamGNN-Q model[54].
图 8 ML加速电子结构计算应用于量子材料领域内的实例 (a) 采用DeepH-E3模型预测转角为1.08°, 11164个原子数的双层转角石墨烯结构的能带结构与DFT和连续介质模型计算的能带结构的对比图[47]; (b) HamGNN模型预测的Bi2Se3薄膜的能带结构与DFT计算的能带结构的对比图, 以及导带底附近未占据态的自旋-动量锁定特征[51]; (c) DeepH-E3模型预测在SOC强度变化导致的带隙闭合后重新打开, 拓扑量子相变从ℤ2 = 0到ℤ2 = 1 [47]
Fig. 8. Several examples of machine-learning-accelerated electronic-structure calculations applied to quantum materials: (a) Comparison of the band structure predicted by using the DeepH-E3 model with the DFT-computed band structure for a twisted bilayer graphene supercell containing 11164 atoms at a twist angle of 1.08°[47]; (b) band structure of Bi2Se3 thin film with the HamGNN model compared to DFT results, together with the spin–momentum locking of the unoccupied states near the conduction-band minimum[51]; (c) DeepH-E3 of a topological quantum phase transition driven by varying SOC strength: after gap closure, the gap re-opens, changing the ℤ2 invariant from 0 to 1[47].
图 9 ML加速电子结果计算在光电与输运性质领域内的应用实例 (a) DeePTB模型预测超百万个原子的GaP体系的复折射率结果与Aspnes等[91]的实验结果的结果对比[55]; (b) HamEPC框架预测的碳化硅(SiC)EPC与DFT(OpenMX)计算结果的对比[53]; (c) HamEPC框架在杂化泛函HSE级别下预测的GaAs迁移率在PBE级别下(Perturbo软件)的计算结果对比[53]; (d) HamEPC框架预测CsV3Sb5 CDW超晶格的声子谱并标定了每个声子模式对应的电子-声子耦合强度(γqv)[53]
Fig. 9. Several examples of machine-learning-accelerated electronic-structure calculations applied to optoelectronic and transport properties: (a) Complex refractive index of a GaP system with over one million atoms using the DeePTB model, compared with the experimental data of Aspnes et al[91]. Reproduced from Ref. [55]; (b) EPC matrix of SiC computed by Xiang Hongjun et al. with the HamEPC framework, benchmarked against DFT (OpenMX) results[53]; (c) GaAs mobility predicted at the HSE level by using HamEPC, compared with Perturbo calculations at the PBE level[53]; (d) complete phonon spectrum of the CsV3Sb5 CDW supercell and mode and moment-resolved electron–phonon coupling strengths (γqv) quantitatively characterized via the HamEPC framework[53]
表 1 三大主流ML方法的特征比较
Table 1. Characteristics across of capillary of different kind of machine-learning models.
模型 哈密顿精度
(MAE)计算效率 计算极限 优势体系 接口软件 资源链接 DeepH 2.1 meV
(石墨烯)
1.3 meV
(MoS2)对比DFT
达103倍
加速11164 原子 (魔角石墨烯) 常规大尺寸材料、
无/弱SOC体系OpenMX
Abacus
FHI-aims
SIESTA代码: https://github.com/mzjb/DeepH-pack
数据:https://doi.org/10.5281/zenodo.6555484 xDeepH 0.36 meV (CrI3)
0.56 meV (NiBr2)对比DFT
达103倍
加速4336 原子
(CrI3斯格明子)磁性、拓扑材料
体系OpenMX
Abacus
FHI-aims
SIESTA代码: https://github.com/mzjb/xDeepH
数据:https://doi.org/10.5281/zenodo.7561013 DeepH-E3 0.40 meV
(石墨烯)
0.37 meV (MoS2)GPU
分钟级
推理11164 原子 (魔角石墨烯) 常规大尺寸材料、强SOC
体系OpenMX
Abacus
FHI-aims
SIESTA代码: https://github.com/Xiaoxun-Gong/DeepH-E3
数据:https://doi.org/10.5281/zenodo.7553640 HamGNN 1.49 meV
(QM9)
0.89 meV
(MoS2)对比DFT
达3500倍加速多元的复杂体系、缺陷体系 OpenMX
Abacus
SIESTA
HONPAS代码: https://github.com/QuantumLab-ZY/HamGNN
数据:https://zenodo.org/records/11171478 HamGNN-Q 1.013 meV (GaAs) 极化子
秒级计算13824原子($ \text{As}_{\text{Ga}}^{6+} $
体系)带电缺陷体系 OpenMX
Abacus
SIESTA
HONPAS代码: https://github.com/QuantumLab-Z7/HamGNN
数据:https://doi.org/10.5281/zenodo.8147631 DeepTB 40 meV(Si)
26 meV
(106原子 GaP)对比DFT
达105倍
加速106原子
(GaP)超大型体系(106) Abacus 代码: https://github.com/deepmodeling/DeePTB
数据:https://www.aissquare.com/datasets/detail?name=DeePTBDataSet&id=255&pageType=datasets DeePTB-E3 0.34 meV
(MoS2)
0.21 meV
(GaN)张量积计算快100倍(相比DeepH-E3) 103原子
(HfO2)含重元素体系 Abacus 代码: https://github.com/deepmodeling/DeePTB
数据:https://www.aissquare.com/datasets/detail?pageType=datasets&name=Quantum_Operator_Dataset&id=286 备注: 本表所列模型的训练与测试数据集均已公开, 主要存储于Zenodo知识库和AIS Square中. 数据集多采用HDF5、npz等适用于科学计算的格式, 便于下载、验证与二次开发. 表格仅针对已开源的模型, 因此DeepH-2及DeepH-zero未包含. 
下载: 导出CSV
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