Abstract
Capturing the structural changes that molecules undergo during chemical reactions in real space and time is a long-standing dream and an essential prerequisite for understanding and ultimately controlling femtochemistry. A key approach to tackle this challenging task is Coulomb explosion imaging, which has benefited decisively from recently emerging high-repetition-rate X-ray free-electron laser sources. With this technique, information on the molecular structure is inferred from the momentum distributions of the ions produced by the rapid Coulomb explosion of molecules. Retrieving molecular structures from these distributions poses a highly nonlinear inverse problem that remains unsolved for molecules consisting of more than a few atoms. Here, we address this challenge using a diffusion-based Transformer neural network. We show that the network reconstructs unknown molecular geometries from ion-momentum distributions with a mean absolute error below one Bohr radius, which is half the length of a typical chemical bond.
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Data availability
The datasets and model weights used in this study have been deposited in the Zenodo database under accession code 1579447053. The raw data recorded for the water, tetrafluoromethane, and ethanol experiments at the European XFEL are available at https://doi.org/10.22003/XFEL.EU-DATA-002150-00, https://doi.org/10.22003/XFEL.EU-DATA-002181-00, and https://doi.org/10.22003/XFEL.EU-DATA-002926-00, respectively. Source data are provided with this paper.
Code availability
The source code is publicly available at https://github.com/xli025/molexa54.
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Acknowledgements
We acknowledge the LCLS data team and the SLAC Shared Science Data Facility (S3DF) for providing the computing and data storage used in model development. We acknowledge the teams of the three European XFEL experiments (2150, 2181, and 2926) for sharing the associated data. X.L. would like to thank Patricia Vindel Zandbergen for her help related to the model testing and Philipp Schmidt for his support in experimental data processing. This work is supported by the Linac Coherent Light Source, SLAC National Accelerator Laboratory, which is funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. P.J.H. is supported by the U.S. DOE BES Chemical Sciences, Geosciences, and Biosciences Division under Contract No. DE-AC02-06CH11357. D.R. and A.R. are supported by grant no. DE-FG02-86ER13491 from the same funding agency, and also acknowledge dedicated support for ML/AI developments through the GRIPex program at Kansas State University. F.T. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project 509471550, Emmy Noether Program. T.J.A.W. was supported by the Atomic, Molecular, and Optical Sciences Program of the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, through Contract No. DE-AC0276SF00515.
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Conceptualization and methodology: X.L. with support from J.B.T., J.P.C. and P.J.H.; Dataset creation and curation: X.L. and P.J.H.; Generative model development: X.L., J.H., M.X. and S.E.; CEI experiments: X.L., T.J., R.B., M.M., M.N.P., D.R., A.R. and F.T.; CEI experiment data analysis: X.L. and T.J.; Original draft: X.L., T.J., R.B., J.H., M.X., D.R., F.T., T.J.A.W., S.E. and P.J.H.; Final draft: all authors.
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Nature Communications thanks Xinwen Ma, Henrik Stapelfeldt, Daniel Strasser, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
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Li, X., Jahnke, T., Boll, R. et al. Generative modeling enables molecular structure retrieval from Coulomb explosion imaging. Nat Commun (2026). https://doi.org/10.1038/s41467-026-70160-5
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DOI: https://doi.org/10.1038/s41467-026-70160-5
