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Spin-symmetry-enforced solution of the many-body Schrödinger equation with a deep neural network

Nature Computational Science volume 4, pages 910–919 (2024)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

The integration of deep neural networks with the variational Monte Carlo (VMC) method has marked a substantial advancement in solving the Schrödinger equation. In this work we enforce spin symmetry in the neural-network-based VMC calculation using a modified optimization target. Our method is designed to solve for the ground state and multiple excited states with target spin symmetry at a low computational cost. It predicts accurate energies while maintaining the correct symmetry in strongly correlated systems, even in cases in which different spin states are nearly degenerate. Our approach also excels at spin–gap calculations, including the singlet–triplet gap in biradical systems, which is of high interest in photochemistry. Overall, this work establishes a robust framework for efficiently calculating various quantum states with specific spin symmetry in correlated systems.

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Fig. 1: Framework overview.
Fig. 2: Improved ground-state estimation.
Fig. 3: Improved excited-state calculation.
Fig. 4: The S–T gap for a collection of biradical systems.

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Data availability

The statistical estimations shown in the figures are provided in Supplementary Information. The dataset is available at Zenodo68. Source Data are provided with this paper.

Code availability

We open-source the implementation of our spin symmetry-enforced solution in the JaQMC repository on Github (https://github.com/bytedance/jaqmc). The sample code to reproduce the results in this work is also available on CodeOcean: https://codeocean.com/capsule/8440915/tree (ref. 69).

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Acknowledgements

We would like to thank H. Chen for helpful discussions. We also want to thank ByteDance Research Group for inspiration and encouragement. This work is directed and supported by H. Li and ByteDance Research. J. Chen is supported by the National Key R&D Program of China (grant no. 2021YFA1400500) and National Science Foundation of China (grant no. 12334003). Liwei Wang is supported by National Science Foundation of China (grant no. NSFC62276005).

Author information

Author notes

  1. These authors contributed equally: Zhe Li, Zixiang Lu, Ruichen Li.

Authors and Affiliations

  1. ByteDance Research, Fangheng Fashion Center, Beijing, P. R. China

    Zhe Li, Zixiang Lu, Ruichen Li, Xuelan Wen, Xiang Li & Weiluo Ren

  2. School of Physics, Peking University, Beijing, P. R. China

    Zixiang Lu & Ji Chen

  3. National Key Laboratory of General Artificial Intelligence, School of Intelligence Science and Technology, Peking University, Beijing, P. R. China

    Ruichen Li & Liwei Wang

  4. Center for Machine Learning Research, Peking University, Beijing, P. R. China

    Liwei Wang

  5. Interdisciplinary Institute of Light-Element Quantum Materials, Frontiers Science Center for Nano-Optoelectronics, Peking University, Beijing, P. R. China

    Ji Chen

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Contributions

Z. Li, Z. Lu, R.L, L.W., J.C. and W.R. conceived the study. Z. Li, Z. Lu and R.L. proposed the algorithms. Z. Li, Z. Lu and X.L. performed implementations. Z. Li, Z. Lu, R.L. and X.W. performed simulations and analysis. Z. Li, L.W., J.C. and W.R. supervised the project. Z. Li, Z. Lu, R.L, X.L., X.W., L.W., J.C. and W.R. wrote the paper.

Corresponding authors

Correspondence to Zhe Li, Liwei Wang, Ji Chen or Weiluo Ren.

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Peer review

Peer review information

Nature Computational Science thanks Xiao He, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Kaitlin McCardle, in collaboration with the Nature Computational Science team.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–3 and Tables 1–7.

Source data

Source Data Fig. 1

Raw data of time cost each step of different penalty terms.

Source Data Fig. 2

The zip file contains several files. Subfolder ‘subfigure_a’ contains two data files to plot Fig. 2a: ‘singlet_no_penalty.csv’ and ‘singelt_with_penalty.csv’, which correspond to the orange and red curves, respectively. Subfolder ‘subfigure_b’ contains three data files to plot Fig. 2b: 'overlap.csv', the raw data to compute the overlap between the contaminated state and the singlet and triplet states in Fig. 2b; 'singlet_nature_orb_occupy_num.csv', the occupation number of cc-pVTZ atomic orbitals for all of the occupied nature orbitals in the singlet state visualized in Fig. 2b; and 'triplet_nature_orb_occupy_num.csv', the occupation number of cc-pVTZ atomic orbitals for all of the occupied nature orbitals in the triplet state visualized in Fig. 2b.

Source Data Fig. 3

There are three subfolders/file under the root directory. Estimation values and error bars in Fig. 3a,c are derived by reblock analysis using the raw data from subfolders ‘subfigure_a’ and ‘subfigure_c’; ‘subfigure_d’ comprises the raw data to plot the curves in Fig. 3d.

Source Data Fig. 4

All raw data to derive the estimation values and error bars of our penalty method in Fig. 4.

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Li, Z., Lu, Z., Li, R. et al. Spin-symmetry-enforced solution of the many-body Schrödinger equation with a deep neural network. Nat Comput Sci 4, 910–919 (2024). https://doi.org/10.1038/s43588-024-00730-4

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