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.

  • Article
  • Published:

Accurate transition state generation with an object-aware equivariant elementary reaction diffusion model

A preprint version of the article is available at arXiv.


Transition state search is key in chemistry for elucidating reaction mechanisms and exploring reaction networks. The search for accurate 3D transition state structures, however, requires numerous computationally intensive quantum chemistry calculations due to the complexity of potential energy surfaces. Here we developed an object-aware SE(3) equivariant diffusion model that satisfies all physical symmetries and constraints for generating sets of structures—reactant, transition state and product—in an elementary reaction. Provided reactant and product, this model generates a transition state structure in seconds instead of hours, which is typically required when performing quantum-chemistry-based optimizations. The generated transition state structures achieve a median of 0.08 Å root mean square deviation compared to the true transition state. With a confidence scoring model for uncertainty quantification, we approach an accuracy required for reaction barrier estimation (2.6 kcal mol–1) by only performing quantum chemistry-based optimizations on 14% of the most challenging reactions. We envision usefulness for our approach in constructing large reaction networks with unknown mechanisms.

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

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Overview of equivariant diffusion models (EDMs) for generative molecular system sampling.
Fig. 2: Object-aware SE(3) equivariance and its implementation based on SE(3)-equivariant GNNs.
Fig. 3: Analysis of samples generated by OA-ReactDiff on select elementary reactions.
Fig. 4: Evaluation of structural similarities for TS structures generated by OA-ReactDiff and true TS structures.
Fig. 5: Energetic performance for OA-ReactDiff plus recommender TS structures.

Similar content being viewed by others

Data availability

The Transition1x dataset55 used in this work can be found at GitLab,, with Source data are provided with this paper.

Code availability

Codebase for OA-ReactDiff is available as an open-source repository on GitHub for contiguous development, A stable version of the code56 used in this work is available at Zenodo,


  1. Dewyer, A. L., Argüelles, A. J. & Zimmerman, P. M. Methods for exploring reaction space in molecular systems. WIREs Comput. Mol. Sci. 8, e1354 (2018).

    Article  Google Scholar 

  2. Unsleber, J. P. & Reiher, M. The exploration of chemical reaction networks. Annu. Rev. Phys. Chem. 71, 121–142 (2020).

    Article  Google Scholar 

  3. Truhlar, D. G., Garrett, B. C. & Klippenstein, S. J. Current status of transition-state theory. J. Phys. Chem. 100, 12771–12800 (1996).

    Article  Google Scholar 

  4. Mardirossian, N. & Head-Gordon, M. Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals. Mol. Phys. 115, 2315–2372 (2017).

    Article  Google Scholar 

  5. Durant, J. L. Evaluation of transition state properties by density functional theory. Chem. Phys. Lett. 256, 595–602 (1996).

    Article  Google Scholar 

  6. Simm, G. N., Vaucher, A. C. & Reiher, M. Exploration of reaction pathways and chemical transformation networks. J. Phys. Chem. A 123, 385–399 (2019).

    Article  Google Scholar 

  7. Wang, L.-P. et al. Discovering chemistry with an ab initio nanoreactor. Nat. Chem. 6, 1044–1048 (2014).

    Article  Google Scholar 

  8. Pieri, E. et al. The non-adiabatic nanoreactor: towards the automated discovery of photochemistry. Chem. Sci. 12, 7294–7307 (2021).

    Article  Google Scholar 

  9. Zeng, J., Cao, L., Xu, M., Zhu, T. & Zhang, J. Z. H. Complex reaction processes in combustion unraveled by neural network-based molecular dynamics simulation. Nat. Commun. 11, 5713 (2020).

    Article  Google Scholar 

  10. Van de Vijver, R. & Zádor, J. Kinbot: automated stationary point search on potential energy surfaces. Comput. Phys. Commun. 248, 106947 (2020).

    Article  Google Scholar 

  11. von Lilienfeld, O. A., Müller, K.-R. & Tkatchenko, A. Exploring chemical compound space with quantum-based machine learning. Nat. Rev. Chem. 4, 347–358 (2020).

    Article  Google Scholar 

  12. Margraf, J. T., Jung, H., Scheurer, C. & Reuter, K. Exploring catalytic reaction networks with machine learning. Nat. Catal. 6, 112–121 (2023).

    Article  Google Scholar 

  13. Sheppard, D., Terrell, R. & Henkelman, G. Optimization methods for finding minimum energy paths. J. Chem. Phys. 128, 134106 (2008).

    Article  Google Scholar 

  14. Schreiner, M., Bhowmik, A., Vegge, T., Busk, J. & Winther, O. Transition1x - a dataset for building generalizable reactive machine learning potentials. Sci. Data 9, 779 (2022).

    Article  Google Scholar 

  15. Zhao, Q. & Savoie, B. M. Simultaneously improving reaction coverage and computational cost in automated reaction prediction tasks. Nat. Comput. Sci. 1, 479–490 (2021).

    Article  Google Scholar 

  16. Lemm, D., von Rudorff, G. F. & von Lilienfeld, O. A. Machine learning based energy-free structure predictions of molecules, transition states, and solids. Nat. Commun. 12, 4468 (2021).

    Article  Google Scholar 

  17. Zhang, J. et al. Deep reinforcement learning of transition states. Phys. Chem. Chem. Phys. 23, 6888–6895 (2021).

    Article  Google Scholar 

  18. Pattanaik, L., Ingraham, J. B., Grambow, C. A. & Green, W. H. Generating transition states of isomerization reactions with deep learning. Phys. Chem. Chem. Phys. 22, 23618–23626 (2020).

    Article  Google Scholar 

  19. Makoś, M. Z., Verma, N., Larson, E. C., Freindorf, M. & Kraka, E. Generative adversarial networks for transition state geometry prediction. J. Chem. Phys. 155, 024116 (2021).

    Article  Google Scholar 

  20. Choi, S. Prediction of transition state structures of gas-phase chemical reactions via machine learning. Nat. Commun. 14, 1168 (2023).

    Article  Google Scholar 

  21. Schreiner, M., Bhowmik, A., Vegge, T., Jørgensen, P. B. & Winther, O. NeuralNEB—neural networks can find reaction paths fast. Mach. Learn. Sci. Technol. 3, 045022 (2022).

    Article  Google Scholar 

  22. Ho, J., Jain, A. & Abbeel, P. in (eds Larochelle, H. et al.) Advances in Neural Information Processing Systems Vol. 33, 6840–6851 (Curran Associates, 2020).

  23. Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N. & Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning 2256–2265 (2015).

  24. Song, Y. et al. Score-based generative modeling through stochastic differential equations. In Int. Conference on Learning Representations (2020).

  25. Hoogeboom, E., Satorras, V. G., Vignac, C. & Welling, M. Equivariant diffusion for molecule generation in 3D. In Proc. 39th International Conference on Machine Learning 8867–8887 (2022).

  26. Corso, G., Stärk, H., Jing, B., Barzilay, R. & Jaakkola, T. DiffDock: Diffusion steps, twists, and turns for molecular docking. In Int. Conference on Learning Representations (2023).

  27. Schneuing, A. et al. Structure-based drug design with equivariant diffusion models. Preprint at (2022).

  28. Thomas, N. et al. Tensor field networks: rotation- and translation-equivariant neural networks for 3D point clouds. Preprint at (2018).

  29. Satorras, V. G., Hoogeboom, E. & Welling, M. E(n) equivariant graph neural networks. In Proc. 38th International Conference on Machine Learning 9323–9332 (2021).

  30. Batzner, S. et al. E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nat. Commun. 13, 2453 (2022).

    Article  Google Scholar 

  31. Zhou, H.-C., Long, J. R. & Yaghi, O. M. Introduction to metal-organic frameworks. Chem. Rev. 112, 673–674 (2012).

    Article  Google Scholar 

  32. Du, W. et al. A new perspective on building efficient and expressive 3D equivariant graph neural networks. Preprint at (2023).

  33. Lugmayr, A. et al. Repaint: inpainting using denoising diffusion probabilistic models. In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2022);

  34. Henkelman, G., Uberuaga, B. P. & Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Phys. Chem. 113, 9901–9904 (2000).

    Article  Google Scholar 

  35. Chai, J.-D. & Head-Gordon, M. Systematic optimization of long-range corrected hybrid density functionals. J. Phys. Chem. 128, 084106 (2008).

    Article  Google Scholar 

  36. Ditchfield, R., Hehre, W. J. & Pople, J. A. Self-consistent molecular-orbital methods. ix. an extended gaussian-type basis for molecular-orbital studies of organic molecules. J. Phys. Chem. 54, 724–728 (1971).

    Article  Google Scholar 

  37. Grambow, C. A., Pattanaik, L. & Green, W. H. Reactants, products, and transition states of elementary chemical reactions based on quantum chemistry. Sci. Data 7, 137 (2020).

    Article  Google Scholar 

  38. Grambow, C. A., Pattanaik, L. & Green, W. H. Deep learning of activation energies. J. Phys. Chem. Lett. 11, 2992–2997 (2020).

    Article  Google Scholar 

  39. Ruddigkeit, L., van Deursen, R., Blum, L. C. & Reymond, J.-L. Enumeration of 166 billion organic small molecules in the chemical universe database GDB-17. J. Chem. Inf. Model. 52, 2864–2875 (2012).

    Article  Google Scholar 

  40. Jumper, J. et al. Highly accurate protein structure prediction with alphafold. Nature 596, 583–589 (2021).

    Article  Google Scholar 

  41. Duan, C., Nandy, A., Meyer, R., Arunachalam, N. & Kulik, H. J. A transferable recommender approach for selecting the best density functional approximations in chemical discovery. Nat. Comput. Sci. 3, 38–47 (2023).

    Article  Google Scholar 

  42. Seifert, G. & Joswig, J.-O. Density-functional tight binding—an approximate density-functional theory method. WIREs Comput. Mol. Sci. 2, 456–465 (2012).

    Article  Google Scholar 

  43. Liu, W.-G. & Goddard, W. A. I. First-principles study of the role of interconversion between NO2, N2O4, cis-ONO-NO2, and trans-ONO-NO2 in chemical processes. J. Am. Chem. Soc. 134, 12970–12978 (2012).

    Article  Google Scholar 

  44. Duan, C., Chu, D. B. K., Nandy, A. & Kulik, H. J. Detection of multi-reference character imbalances enables a transfer learning approach for virtual high throughput screening with coupled cluster accuracy at dft cost. Chem. Sci. 13, 4962–4971 (2022).

    Article  Google Scholar 

  45. Lipman, Y., Chen, R. T. Q., Ben-Hamu, H., Nickel, M. & Le, M. Flow matching for generative modeling. In 11th International Conference on Learning Representations (2023).

  46. Liu, G.-H. et al. I2SB: Image-to-image schrödinger bridge. Preprint at (2023).

  47. Kim, S., Woo, J. & Kim, W. Y. Diffusion-based generative AI for exploring transition states from 2D molecular graphs. Preprint at (2023).

  48. Zhao, Q. et al. Comprehensive exploration of graphically defined reaction spaces. Sci. Data 10, 145 (2023).

    Article  Google Scholar 

  49. Serre, J.-P. et al. Linear Representations of Finite Groups (Springer, 1977).

  50. Bronstein, M. M., Bruna, J., Cohen, T. & Veličković, P. Geometric deep learning: grids, groups, graphs, geodesics, and gauges. Preprint at (2021).

  51. Köhler, J., Klein, L. & Noé, F. Equivariant flows: exact likelihood generative learning for symmetric densities. In International Conference on Machine Learning 5361–5370 (2020).

  52. Nichol, A. Q. & Dhariwal, P. Improved denoising diffusion probabilistic models. In Proc. 38th International Conference on Machine Learning 8162–8171 (2021).

  53. Du, W. et al. SE (3) equivariant graph neural networks with complete local frames. In International Conference on Machine Learning 5583–5608 (2022).

  54. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O. & Dahl, G. E. Neural message passing for quantum chemistry. In International Conference on Machine Learning 1263–1272 (2017).

  55. Transition1x data release. figshare (2023).

  56. OA-ReactDiff stable code release. Zenodo (2023).

Download references


This work was supported by the US Office of Naval Research under grant no. N00014-20-1-2150 (C.D. and H.J.K.) and National Science Foundation grant CBET-1846426 (H.J. and H.J.K.). C.D. thanks the Molecular Sciences Software Institute for the fellowship support under NSF grant OAC-1547580. C.D. thanks Q. Zhao and M. Monkey for discussions about elementary reactions. C.D. thanks A. Nandy and W. Du for discussions about equivariant graph neural networks. C.D. and Y.D. thank G.-H. Liu and T. Chen for discussions about diffusion model and Schrödinger bridge. C.D. and H.J. thank Y. Zhao for his help on preparing a demo Jupyter notebook for this work. The authors thank S. Choi and M. Schreiner for communications and providing their raw data that makes the comparison in Table 1 possible.

Author information

Authors and Affiliations



C.D. was responsible for conceptualization, methodology, software, validation, investigation, data curation, writing of the original draft, review, editing and visualization. Y.D. was responsible for methodology, software, writing of original draft, review and editing. H.J. was responsible for data curation, review and editing. H.J.K. was responsible for writing of the original draft, review and editing.

Corresponding author

Correspondence to Chenru Duan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Computational Science thanks Sunghwan Choi, Hyunwook Jung and Matteo Maestri for their contribution to the peer review of this work. Primary Handling Editor: Kaitlin McCardle, in collaboration with the Nature Computational Science team. Peer reviewer reports are available.

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 Tables 1–4, Figs. 1–10 and Text 1.

Peer Review File

Source data

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

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duan, C., Du, Y., Jia, H. et al. Accurate transition state generation with an object-aware equivariant elementary reaction diffusion model. Nat Comput Sci 3, 1045–1055 (2023).

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