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Deep-learning density functional theory Hamiltonian for efficient ab initio electronic-structure calculation

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The marriage of density functional theory (DFT) and deep-learning methods has the potential to revolutionize modern computational materials science. Here we develop a deep neural network approach to represent the DFT Hamiltonian (DeepH) of crystalline materials, aiming to bypass the computationally demanding self-consistent field iterations of DFT and substantially improve the efficiency of ab initio electronic-structure calculations. A general framework is proposed to deal with the large dimensionality and gauge (or rotation) covariance of the DFT Hamiltonian matrix by virtue of locality, and this is realized by a message-passing neural network for deep learning. High accuracy, high efficiency and good transferability of the DeepH method are generally demonstrated for various kinds of material system and physical property. The method provides a solution to the accuracy–efficiency dilemma of DFT and opens opportunities to explore large-scale material systems, as evidenced by a promising application in the study of twisted van der Waals materials.

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Fig. 1: Learning the DFT Hamiltonian \({\hat{H}}_{{{{\rm{DFT}}}}}\) by virtue of locality.
Fig. 2: Crystal graph and MPNN including L layers employed by DeepH.
Fig. 3: Performance of DeepH on studying graphene.
Fig. 4: Performance of DeepH on studying monolayer MoS2.
Fig. 5: Generalization ability of DeepH, from flat sheets to curved nanotubes.
Fig. 6: Application of the DeepH method to study moiré-twisted materials.

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Data availability

Source data are provided with this Paper. The dataset used to train the deep-learning model is available at Zenodo61.

Code availability

The code used in the current study is available at GitHub ( and Zenodo62.


  1. Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964).

    MathSciNet  Google Scholar 

  2. Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).

    MathSciNet  Google Scholar 

  3. Jones, R. O. Density functional theory: its origins, rise to prominence and future. Rev. Mod. Phys. 87, 897–923 (2015).

    MathSciNet  Google Scholar 

  4. LeCun, Y., Bengio, Y. & Hinton, G. Deep learning. Nature 521, 436–444 (2015).

    Google Scholar 

  5. Jordan, M. I. & Mitchell, T. M. Machine learning: trends, perspectives and prospects. Science 349, 255–260 (2015).

    MathSciNet  MATH  Google Scholar 

  6. Carleo, G. et al. Machine learning and the physical sciences. Rev. Mod. Phys. 91, 045002 (2019).

    Google Scholar 

  7. Behler, J. & Parrinello, M. Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 98, 146401 (2007).

    Google Scholar 

  8. Schütt, K. T., Sauceda, H. E., Kindermans, P.-J., Tkatchenko, A. & Müller, K.-R. Schnet—a deep learning architecture for molecules and materials. J. Chem. Phys. 148, 241722 (2018).

    Google Scholar 

  9. Zhang, L., Han, J., Wang, H., Car, R. & E, W. Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics. Phys. Rev. Lett. 120, 143001 (2018).

    Google Scholar 

  10. Gasteiger, J., Groß, J. & Günnemann, S. Directional message passing for molecular graphs. Proc. International Conference on Learning Representations (ICLR, 2020);

  11. Unke, O. T. et al. SpookyNet: learning force fields with electronic degrees of freedom and nonlocal effects. Nat. Commun. 12, 7273 (2021).

    Google Scholar 

  12. Brockherde, F. et al. Bypassing the Kohn-Sham equations with machine learning. Nat. Commun. 8, 872 (2017).

    Google Scholar 

  13. Grisafi, A. et al. Transferable machine-learning model of the electron density. ACS Cent. Sci. 5, 57–64 (2019).

    Google Scholar 

  14. Chandrasekaran, A. et al. Solving the electronic structure problem with machine learning. npj Comput. Mater. 5, 22 (2019).

    Google Scholar 

  15. Tsubaki, M. & Mizoguchi, T. Quantum deep field: data-driven wave function, electron density generation, and atomization energy prediction and extrapolation with machine learning. Phys. Rev. Lett. 125, 206401 (2020).

    Google Scholar 

  16. Grisafi, A., Wilkins, D. M., Csányi, G. & Ceriotti, M. Symmetry-adapted machine learning for tensorial properties of atomistic systems. Phys. Rev. Lett. 120, 036002 (2018).

    Google Scholar 

  17. Gu, Q., Zhang, L. & Feng, J. Neural network representation of electronic structure from ab initio molecular dynamics. Sci. Bull. 67, 29–37 (2022).

    Google Scholar 

  18. Schütt, K. T., Gastegger, M., Tkatchenko, A., Müller, K.-R. & Maurer, R. J. Unifying machine learning and quantum chemistry with a deep neural network for molecular wavefunctions. Nat. Commun. 10, 5024 (2019).

    Google Scholar 

  19. Unke, O. T. et al. SE(3)-equivariant prediction of molecular wavefunctions and electronic densities. In Proc. Advances in Neural Information Processing Systems (eds. Beygelzimer, A., Dauphin, Y., Liang, P. & Vaughan, J. W.) 14434–14447 (Curran Associates, 2021);

  20. Nagai, R., Akashi, R. & Sugino, O. Completing density functional theory by machine learning hidden messages from molecules. npj Comput. Mater. 6, 43 (2020).

    Google Scholar 

  21. Dick, S. & Fernandez-Serra, M. Machine learning accurate exchange and correlation functionals of the electronic density. Nat. Commun. 11, 3509 (2020).

    Google Scholar 

  22. Kirkpatrick, J. et al. Pushing the frontiers of density functionals by solving the fractional electron problem. Science 374, 1385–1389 (2021).

    Google Scholar 

  23. Mills, K. et al. Extensive deep neural networks for transferring small scale learning to large scale systems. Chem. Sci. 10, 4129–4140 (2019).

    Google Scholar 

  24. Zubatiuk, T. & Isayev, O. Development of multimodal machine learning potentials: toward a physics-aware artificial intelligence. Acc. Chem. Res. 54, 1575–1585 (2021).

    Google Scholar 

  25. Goedecker, S. Linear scaling electronic structure methods. Rev. Mod. Phys. 71, 1085–1123 (1999).

    Google Scholar 

  26. Hegde, G. & Bowen, R. C. Machine-learned approximations to density functional theory Hamiltonians. Sci. Rep. 7, 42669 (2017).

    Google Scholar 

  27. Thomas, N. et al. Tensor field networks: rotation- and translation-equivariant neural networks for 3D point clouds. Preprint at (2018).

  28. Anderson, B., Hy, T. S. & Kondor, R. Cormorant: covariant molecular neural networks. In Proc. Advances in Neural Information Processing Systems Vol. 32 (eds. Wallach, H. et al.) 14537–14546 (Curran Associates, 2019);

  29. Fuchs, F. B., Worrall, D. E., Fischer, V. & Welling, M. SE(3)-transformers: 3D roto-translation equivariant attention networks. In Proc. Advances in Neural Information Processing Systems Vol. 33 (eds. Larochelle, H., Ranzato M., Hadsell, R., Balcan, M. F. & Lin, H.) 1970–1981 (Curran Associates, 2020);

  30. Martin, R. M. Electronic Structure: Basic Theory and Practical Methods (Cambridge Univ. Press, 2004);

  31. Kohn, W. Density functional and density matrix method scaling linearly with the number of atoms. Phys. Rev. Lett. 76, 3168–3171 (1996).

    Google Scholar 

  32. Prodan, E. & Kohn, W. Nearsightedness of electronic matter. Proc. Natl. Acad. Sci. USA 102, 11635 (2005).

    Google Scholar 

  33. Wang, C. et al. First-principles calculation of optical responses based on nonorthogonal localized orbitals. New J. Phys. 21, 093001 (2019).

    Google Scholar 

  34. Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).

    Google Scholar 

  35. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O. & Dahl, G. E. Neural message passing for quantum chemistry. In Proc. 34th International Conference on Machine Learning (ICML) PMLR 70 (eds. Precup, D. & Teh, Y. W.) 1263–1272 (2017);

  36. Schütt, K. T., Arbabzadah, F., Chmiela, S., Müller, K.-R. & Tkatchenko, A. Quantum-chemical insights from deep tensor neural networks. Nat. Commun. 8, 13890 (2017).

    Google Scholar 

  37. Xie, T. & Grossman, J. C. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. Phys. Rev. Lett. 120, 145301 (2018).

    Google Scholar 

  38. Wang, Z. et al. Symmetry-adapted graph neural networks for constructing molecular dynamics force fields. Sci. China Phys. Mech. Astron. 64, 117211 (2021).

    Google Scholar 

  39. Ba, J. L., Kiros, J. R. & Hinton, G. E. Layer normalization. Preprint at (2016).

  40. Morimoto, T. & Nagaosa, N. Topological nature of nonlinear optical effects in solids. Sci. Adv. 2, e1501524 (2016).

    Google Scholar 

  41. Wang, C. et al. First-principles calculation of nonlinear optical responses by Wannier interpolation. Phys. Rev. B 96, 115147 (2017).

    Google Scholar 

  42. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl. Acad. Sci. USA 108, 12233 (2011).

    Google Scholar 

  43. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Google Scholar 

  44. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Google Scholar 

  45. Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    Google Scholar 

  46. Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600, 439–443 (2021).

    Google Scholar 

  47. Carr, S., Fang, S. & Kaxiras, E. Electronic-structure methods for twisted moiré layers. Nat. Rev. Mater. 5, 748–763 (2020).

    Google Scholar 

  48. Jeong, W., Yoo, D., Lee, K., Jung, J. & Han, S. Efficient atomic-resolution uncertainty estimation for neural network potentials using a replica ensemble. J. Phys. Chem. Lett. 11, 6090–6096 (2020).

    Google Scholar 

  49. Lucignano, P., Alfè, D., Cataudella, V., Ninno, D. & Cantele, G. Crucial role of atomic corrugation on the flat bands and energy gaps of twisted bilayer graphene at the magic angle θ ~ 1.08°. Phys. Rev. B 99, 195419 (2019).

    Google Scholar 

  50. David, A., Rakyta, P., Kormányos, A. & Burkard, G. Induced spin-orbit coupling in twisted graphene-transition metal dichalcogenide heterobilayers: twistronics meets spintronics. Phys. Rev. B 100, 085412 (2019).

    Google Scholar 

  51. Gou, J. et al. The effect of moiré superstructures on topological edge states in twisted bismuthene homojunctions. Sci. Adv. 6, eaba2773 (2020).

    Google Scholar 

  52. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    Article  Google Scholar 

  53. Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

    Google Scholar 

  54. Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).

    Article  Google Scholar 

  55. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  Google Scholar 

  56. Ozaki, T. Variationally optimized atomic orbitals for large-scale electronic structures. Phys. Rev. B 67, 155108 (2003).

    Google Scholar 

  57. Ozaki, T. & Kino, H. Numerical atomic basis orbitals from H to Kr. Phys. Rev. B 69, 195113 (2004).

    Google Scholar 

  58. Morrison, I., Bylander, D. M. & Kleinman, L. Nonlocal Hermitian norm-conserving Vanderbilt pseudopotential. Phys. Rev. B 47, 6728–6731 (1993).

    Google Scholar 

  59. Sipe, J. E. & Shkrebtii, A. I. Second-order optical response in semiconductors. Phys. Rev. B 61, 5337–5352 (2000).

    Google Scholar 

  60. Fey, M. & Lenssen, J. E. Fast graph representation learning with PyTorch Geometric. In Proc. ICLR Workshop on Representation Learning on Graphs and Manifolds (ICLR, 2019);

  61. Li, H. Dataset for deep-learning density functional theory Hamiltonian for efficient ab initio electronic-structure calculation (Zenodo, 2022);

  62. Li, H. Code for deep-learning density functional theory Hamiltonian for efficient ab initio electronic-structure calculation (Zenodo, 2022);

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This work was supported by the Basic Science Center Project of NSFC (grant no. 51788104), the National Science Fund for Distinguished Young Scholars (grant no. 12025405), the National Natural Science Foundation of China (grant no. 11874035), the Ministry of Science and Technology of China (grant nos. 2018YFA0307100 and 2018YFA0305603), the Beijing Advanced Innovation Center for Future Chip (ICFC) and the Beijing Advanced Innovation Center for Materials Genome Engineering. M.Y. was supported by the Shuimu Tsinghua Scholar Program and Postdoctoral International Exchange Program. R.X. was funded by the China Postdoctoral Science Foundation (grant no. 2021TQ0187).

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Authors and Affiliations



Y.X. and W.D. proposed the project and supervised H.L., Z.W. and N.Z. in carrying out the research, with the help of M.Y., R.X. and X.G. All authors discussed the results. Y.X. and H.L. prepared the manuscript with input from the other co-authors.

Corresponding authors

Correspondence to Wenhui Duan or Yong Xu.

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Nature Computational Science thanks the anonymous reviewers for their contribution to the peer review of this work. Primary Handling Editor: Kaitlin McCardle, in collaboration with the Nature Computational Science team. Peer reviewer reports are available.

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Supplementary Information

Details of computational methods and results, Supplementary Figs. 1–21 and Tables 1–5.

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Supplementary Data 1

Atomic structures of the three distorted graphene supercells in crystallographic information file (CIF) format.

Supplementary Data 2

Atomic structures of the three distorted MoS2 supercells in CIF format.

Supplementary Data 3

The atomic structure of the distorted silicon supercell in CIF format.

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Source Data Fig. 5

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Li, H., Wang, Z., Zou, N. et al. Deep-learning density functional theory Hamiltonian for efficient ab initio electronic-structure calculation. Nat Comput Sci 2, 367–377 (2022).

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