Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks

A preprint version of the article is available at arXiv.


The Schrödinger equation describes the quantum-mechanical behaviour of particles, making it the most fundamental equation in chemistry. A solution for a given molecule allows computation of any of its properties. Finding accurate solutions for many different molecules and geometries is thus crucial to the discovery of new materials such as drugs or catalysts. Despite its importance, the Schrödinger equation is notoriously difficult to solve even for single molecules, as established methods scale exponentially with the number of particles. Combining Monte Carlo techniques with unsupervised optimization of neural networks was recently discovered as a promising approach to overcome this curse of dimensionality, but the corresponding methods do not exploit synergies that arise when considering multiple geometries. Here we show that sharing the vast majority of weights across neural network models for different geometries substantially accelerates optimization. Furthermore, weight-sharing yields pretrained models that require only a small number of additional optimization steps to obtain high-accuracy solutions for new geometries.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Overview of the DeepErwin framework.
Fig. 2: Results using weight-sharing during optimization for four different sets of molecules.
Fig. 3: Results when reusing pretrained weights for four different sets of molecules.
Fig. 4: Transition barriers for H\({}_{4}^{+}\) and ethene.
Fig. 5: Validation of nuclear forces.

Data availability

All data in this manuscript were generated using the Python package DeepErwin or the quantum-chemistry code MOLPRO as described in Methods. All data required to perform the reported calculations as well as the processed data that was used to generate figures are available on Code Ocean42. Source data are provided with this paper.

Code availability

The DeepErwin package alongside a detailed documentation is available on the Python Package Index (PyPI) and GitHub ( under the MIT license. All codes and configuration files that were used to perform the reported calculations are also available on Code Ocean42.


  1. Han, J., Zhang, L. & E, W. Solving many-electron Schrödinger equation using deep neural networks. J. Comput. Phys. 399, 108929 (2019).

    MathSciNet  Article  Google Scholar 

  2. Hermann, J., Schätzle, Z. & Noé, F. Deep-neural-network solution of the electronic Schrödinger equation. Nat. Chem. 12, 891–897 (2020).

    Article  Google Scholar 

  3. Manzhos, S. Machine learning for the solution of the Schrödinger equation. Mach. Learn. Sci. Technol. 1, 013002 (2020).

    Article  Google Scholar 

  4. Pfau, D., Spencer, J. S., Matthews, A. G. D. G. & Foulkes, W. M. C. Ab initio solution of the many-electron Schrödinger equation with deep neural networks. Phys. Rev. Res. 2, 033429 (2020).

    Article  Google Scholar 

  5. Wilson, M., Gao, N., Wudarski, F., Rieffel, E. & Tubman, N. M. Simulations of state-of-the-art fermionic neural network wave functions with diffusion Monte Carlo. Phys. Rev. Res. 4, 013021 (2021).

  6. Bartlett, R. J. & Musiał, M. Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79, 291–352 (2007).

    Article  Google Scholar 

  7. Spencer, J. S., Pfau, D., Botev, A. & Foulkes, W. M. C. Better, faster fermionic neural networks. Preprint at (2020).

  8. Unke, O. T. et al. Machine learning force fields. Chem. Rev. 121, 10142–10186 (2021).

    Article  Google Scholar 

  9. Behler, J. Four generations of high-dimensional neural network potentials. Chem. Rev. 121, 10037–10072 (2021).

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. Westermayr, J. & Marquetand, P. Machine learning for electronically excited states of molecules. Chem. Rev. 121, 9873–9926 (2020).

    Article  Google Scholar 

  12. 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).

    Article  Google Scholar 

  13. Bogojeski, M., Vogt-Maranto, L., Tuckerman, M. E., Müller, K.-R. & Burke, K. Quantum chemical accuracy from density functional approximations via machine learning. Nat. Commun. 11, 5223 (2020).

    Article  Google Scholar 

  14. Faber, F. A. et al. Prediction errors of molecular machine learning models lower than hybrid DFT error. J. Chem. Theory Comput. 13, 5255–5264 (2017).

    Article  Google Scholar 

  15. Devlin, J., Chang, M.-W., Lee, K. & Toutanova, K. Bert: pre-training of deep bidirectional transformers for language understanding. Preprint at (2018).

  16. Tan, C. et al. A survey on deep transfer learning. In International Conference on Artificial Neural Networks 270–279 (Springer, 2018).

  17. Matthews, D. A. Analytic gradients of approximate coupled cluster methods with quadruple excitations. J. Chem. Theory Comput. 16, 6195–6206 (2020).

    Article  Google Scholar 

  18. Schütt, K. et al. SchNet: a continuous-filter convolutional neural network for modeling quantum interactions. In Proc. 31st Conference on Neural Information Processing Systems (eds Guyon, I. et al.) 992–1002 (Curran Associates, 2017).

  19. Ma, A., Towler, M. D., Drummond, N. D. & Needs, R. J. Scheme for adding electron–nucleus cusps to Gaussian orbitals. J. Chem. Phys. 122, 224322 (2005).

    Article  Google Scholar 

  20. Martens, J. & Grosse, R. Optimizing neural networks with kronecker-factored approximate curvature. In International Conference on Machine Learning 2408–2417 (PMLR, 2015).

  21. Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. Preprint at (2014).

  22. Alijah, A. & Varandas, A. J. C. H4+: what do we know about it? J. Chem. Phys. 129, 034303 (2008).

    Article  Google Scholar 

  23. Feynman, R. P. Forces in molecules. Phys. Rev. 56, 340–343 (1939).

    Article  Google Scholar 

  24. Peierls, R. E. & Peierls, R. S. Quantum Theory of Solids (Oxford Univ. Press, 1955).

  25. Pulay, P. Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. Mol. Phys. 17, 197–204 (1969).

    Article  Google Scholar 

  26. Gao, N. & Günnemann, S. Ab-initio potential energy surfaces by pairing GNNs with neural wave functions. In International Conference on Learning Representations (2022).

  27. Ríos, P. L., Ma, A., Drummond, N. D., Towler, M. D. & Needs, R. J. Inhomogeneous backflow transformations in quantum Monte Carlo calculations. Phys. Rev. E 74, 066701 (2006).

    Article  Google Scholar 

  28. Kato, T. On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10, 151–177 (1957).

    MathSciNet  Article  Google Scholar 

  29. Liu, D. C. & Nocedal, J. On the limited memory bfgs method for large scale optimization. Math. Prog. 45, 503–528 (1989).

    MathSciNet  Article  Google Scholar 

  30. Ceperley, D., Chester, G. V. & Kalos, M. H. Monte Carlo simulation of a many-fermion study. Phys. Rev. B 16, 3081–3099 (1977).

    Article  Google Scholar 

  31. Hastings, W. K. Monte carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970).

    MathSciNet  Article  Google Scholar 

  32. Chiesa, S., Ceperley, D. M. & Zhang, S. Accurate, efficient, and simple forces computed with quantum monte carlo methods. Phys. Rev. Lett. 94, 036404 (2005).

    Article  Google Scholar 

  33. Kalos, M. H. & Whitlock, P. A. Monte Carlo Methods (Wiley, 1986);

  34. Werner, H.-J., Knowles, P. J., Knizia, G., Manby, F. R. & Schütz, M. MOLPRO: a general-purpose quantum chemistry program package. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2, 242–253 (2012).

    Article  Google Scholar 

  35. Werner, H.-J. et al. MOLPRO, version 2012.1. A package of ab initio programs (MOLPRO, 2012);

  36. Adler, T. B., Knizia, G. & Werner, H.-J. A simple and efficient CCSD(T)-F12 approximation. J. Chem. Phys. 127, 221106 (2007).

    Article  Google Scholar 

  37. Shiozaki, T., Knizia, G. & Werner, H.-J. Explicitly correlated multireference configuration interaction: MRCI-F12. J. Chem. Phys. 134, 034113 (2011).

    Article  Google Scholar 

  38. Peterson, K. A., Adler, T. B. & Werner, H.-J. Systematically convergent basis sets for explicitly correlated wavefunctions: the atoms H, He, B–Ne, and Al–Ar. J. Chem. Phys. 128, 084102 (2008).

    Article  Google Scholar 

  39. Hill, J. G., Mazumder, S. & Peterson, K. A. Correlation consistent basis sets for molecular core-valence effects with explicitly correlated wave functions: the atoms B–Ne and Al–Ar. J. Chem. Phys. 132, 054108 (2010).

    Article  Google Scholar 

  40. Langhoff, S. R. & Davidson, E. R. Configuration interaction calculations on the nitrogen molecule. Int. J. Quantum Chem. 8, 61–72 (1974).

    Article  Google Scholar 

  41. Sun, Q. et al. PySCF: the Python-based simulations of chemistry framework. Wiley Interdiscip. Rev. Comput. Mol. Sci. 8, e1340 (2018).

    Article  Google Scholar 

  42. Scherbela, M., Reisenhofer, R., Gerard, L., Marquetand, P. & Grohs, P. DeepErwin—a framework for solving the Schrödinger equation with deep neural networks. CodeOcean (2022).

Download references


We gratefully acknowledge financial support from the following grants: Austrian Science Fund FWF-I-3403 (L.G.), FWF-M-2528 (R.R.) and WWTF-ICT19-041 (L.G.). The computational results have been achieved using the Vienna Scientific Cluster (VSC). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

Author information

Authors and Affiliations



P.G., P.M. and R.R. conceived the project. M.S., L.G. and R.R. developed the detailed method. M.S. and L.G. wrote the Python code with contributions from R.R. The numerical experiments were designed and performed by M.S., L.G. and P.M. with support from R.R. R.R., M.S. and L.G. wrote the manuscript with input from P.G. and P.M. P.G. supervised the project. R.R. and P.G. obtained funding.

Corresponding author

Correspondence to Rafael Reisenhofer.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Computational Science thanks Huan Tran, Linfeng Zhang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available. Primary Handling Editor: Jie Pan, in collaboration with the Nature Computational Science team.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

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

Peer Review File

Supplementary Data 1

Source data for Supplementary Fig. 1.

Supplementary Data 2

Source data for Supplementary Fig. 2.

Supplementary Data 3

Source data for Supplementary Fig. 3.

Source data

Source Data Fig. 1

Statistical source data for Fig. 1b.

Source Data Fig. 2

Statistical source data for Fig. 2.

Source Data Fig. 3

Statistical source data for Fig. 3.

Source Data Fig. 4

Statistical source data for Fig. 4.

Source Data Fig. 5

Statistical source data for Fig. 5.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Scherbela, M., Reisenhofer, R., Gerard, L. et al. Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks. Nat Comput Sci 2, 331–341 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing