Transferable equivariant graph neural networks for the Hamiltonians of molecules and solids (分子和固体哈密顿量的可转移等变图神经网络)

Yang Zhong, Hongyu Yu, Mao Su, Xingao Gong & Hongjun Xiang

npj Computational Materials 9: 182 (2023)

doi.org/10.1038/s41524-023-01130-4

Published online: 06 October 2023

编辑概述

大规模电子结构计算：可迁移的机器学习加速方法

密度泛函理论（DFT）是研究分子和材料电子结构的强大工具，它能够揭示许多物质性质的内在机制。然而，由于DFT在大型系统上计算时所需的高昂计算成本和运行时间，使得在此类系统中成功实施DFT计算仍然受到诸多限制。该研究提出了基于图神经网络实现的电子哈密顿矩阵的等变参数化方法，该方法可以实现从原子位置到电子哈密顿量的直接映射，从而绕过DFT方法中昂贵的自洽迭代过程。复旦大学的向红军教授等人设计了HamGNN图神经网络模型，该网络显式地考虑了哈密顿矩阵在三维实空间中的旋转等变性和宇称对称性，并在训练时以倒空间中随机采样的k点处的能带误差作为正则化项，使得该模型对训练集之外的分子和固体的电子结构具有很高的拟合能力和可迁移性。在碳同素异形体、硅同素异形体和SiO₂异构体的哈密顿矩阵上分别进行训练后的HamGNN模型对训练集之外的同类结构预测的能带与DFT计算得到的能带高度一致。在硅同素异形体结构上训练后的HamGNN模型对包含4,284个原子的硅位错缺陷的能带和缺陷波函数进行了预测，揭示了硅位错引起的缺陷能级的高度局域性。在无转角的双层MoS₂结构上训练之后的HamGNN模型准确预测出含1626个原子的Moiré双层MoS₂中的Dirac锥能带色散和价带顶波函数的空间分布。HamGNN还在测试中准确拟合了不同化学计量比的Bi_xSe_y族材料的含自旋轨道耦合效应（SOC）的哈密顿矩阵。这些实际测试证明该研究提出的机器学习模型对电子哈密顿量的预测具有很高的精度和可迁移性，可以替代DFT用于高效计算大型系统的电子结构。

Editorial Summary

Large-scale electronic structure calculations: acceleration through a transferable machine learning approach

Density functional theory (DFT) is a powerful tool for studying the electronic structure of molecules and materials to unveil the underlying mechanisms behind various material properties. However, the broad-scale application of DFT calculation on large-scale systems remains circumscribed by the substantial computational overhead and running time required. This study proposes an equivariant parameterization framework based on a graph neural network for the electronic Hamiltonian matrix, which enables direct mapping from atomic positions to electronic Hamiltonians, thus bypassing the expensive self-consistent iterative process in DFT methods. A group led by Prof. Hongjun Xiang from Fudan University has designed a graph neural network called HamGNN. This network explicitly considers the rotational equivariance and parity symmetry of the Hamiltonian matrix in the real space, and uses the energy band error at randomly sampled k points in the reciprocal space as the regularization term during training. The proposed model has high fitting ability and transferability to the electronic structures of molecules and solids beyond the training set. After being trained on the Hamiltonian matrices of carbon allotropes, silicon allotropes, and SiO₂ isomers, the HamGNN model demonstrates a high degree of consistency with the band structures obtained from DFT calculations for the configurations beyond the training set. The HamGNN model trained on the silicon allotropes successfully predicts the energy bands and defect wave functions of a silicon dislocation model containing 4,284 atoms, revealing the highly localized nature of the defect energy levels caused by edge dislocation. The HamGNN model trained on the untwisted bilayer MoS₂ structure accurately predicts the Dirac cone band dispersion and spatial distribution of wave function at the valence band maximum (VBM) for the Moiré angle twisted bilayer MoS₂ with 1626 atoms. HamGNN also accurately fits the Hamiltonian matrices with spin-orbit coupling effects (SOC) for the Bi_xSe_y family with different stoichiometric ratios. The practical tests demonstrate that the machine learning model proposed in this study exhibits high accuracy and transferability for predicting the electronic Hamiltonians of various materials, making it a viable alternative to DFT for efficiently calculating the electronic structure of large systems.