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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.

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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.


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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.

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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.

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The authors declare no competing interests.

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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.

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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).

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