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Deep-neural-network solution of the electronic Schrödinger equation

Nature Chemistry volume 12, pages 891–897 (2020)Cite this article

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Abstract

The electronic Schrödinger equation can only be solved analytically for the hydrogen atom, and the numerically exact full configuration-interaction method is exponentially expensive in the number of electrons. Quantum Monte Carlo methods are a possible way out: they scale well for large molecules, they can be parallelized and their accuracy has, as yet, been only limited by the flexibility of the wavefunction ansatz used. Here we propose PauliNet, a deep-learning wavefunction ansatz that achieves nearly exact solutions of the electronic Schrödinger equation for molecules with up to 30 electrons. PauliNet has a multireference Hartree–Fock solution built in as a baseline, incorporates the physics of valid wavefunctions and is trained using variational quantum Monte Carlo. PauliNet outperforms previous state-of-the-art variational ansatzes for atoms, diatomic molecules and a strongly correlated linear H10, and matches the accuracy of highly specialized quantum chemistry methods on the transition-state energy of cyclobutadiene, while being computationally efficient.

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Fig. 1: Combinatorial explosion of the number of Slater determinants in quantum chemistry.
Fig. 2: Architecture of the newly developed PauliNet wavefunction ansatz.
Fig. 3: Performance of PauliNet with one and six determinants on atoms and diatomic molecules.
Fig. 4: Roles of backflow and multiple determinants in training of PauliNet ansatz.
Fig. 5: Convergence of PauliNet with the number of determinants.
Fig. 6: PauliNet captures strong correlation in H10 along the dissociation curve.
Fig. 7: Calculating the transition barrier of cyclobutadiene automerization.

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Data availability

All raw data were generated with the accompanying code and are available in Figshare (https://doi.org/10.6084/m9.figshare.12720569.v2)51. Processed data used to generate figures are both included with the code and provided with this paper as source data.

Code availability

All computer code developed in this work is released either in the general DeepQMC package available on Zenodo (https://doi.org/10.5281/zenodo.3960827)52 and developed on Github (https://github.com/deepqmc/deepqmc), or in the project-specific repository (https://doi.org/10.6084/m9.figshare.12720833.v1)53, both under the MIT license.

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Acknowledgements

We thank C. Clementi (Rice, FU Berlin), J. Eisert (FU Berlin), G. Scuseria (Rice), J. Neugebauer (MPIE) and H. Wu (Tongji) for inspiring discussions. Funding is acknowledged from the European Commission (ERC CoG 772230 ‘Scale-Cell’), Deutsche Forschungsgemeinschaft (CRC1114/A04, GRK2433 DAEDALUS/P04) and the MATH+ Berlin Mathematics Research Center (AA1x6, EF1x2). J.H. thanks K.-R. Müller for support and acknowledges funding from TU Berlin.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, FU Berlin, Berlin, Germany

    Jan Hermann, Zeno Schätzle & Frank Noé

  2. Machine Learning Group, TU Berlin, Berlin, Germany

    Jan Hermann

  3. Department of Physics, FU Berlin, Berlin, Germany

    Frank Noé

  4. Department of Chemistry, Rice University, Houston, TX, USA

    Frank Noé

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Contributions

J.H. and F.N. designed the research. J.H. developed the method with contributions from F.N. and Z.S.; J.H. wrote the computer code with contributions from Z.S.; J.H. and Z.S. carried out the numerical calculations. All authors analysed the data. J.H. and F.N. wrote the manuscript.

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Correspondence to Jan Hermann or Frank Noé.

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Extended data

Extended Data Fig. 1

Variational correlation energy (%) of five test systems obtained with four types of trial wave functions.

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Hyperparameters used in numerical calculations.

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Hermann, J., Schätzle, Z. & Noé, F. Deep-neural-network solution of the electronic Schrödinger equation. Nat. Chem. 12, 891–897 (2020). https://doi.org/10.1038/s41557-020-0544-y

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