A Cartesian-3j Framework for Machine Learning Interatomic Potentials

Zemin Xu, Chenyu Wu, Wenbo Xie, Peijun Hu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:142308-142323, 2026.

Abstract

Machine learning interatomic potentials (MLIPs) have brought substantial gains in the extrapolation capability in computational chemistry. However, most equivariant models are typically built with spherical tensors (STs), while Cartesian tensor formulations remain less developed despite their natural alignment with atomic coordinates and tensorial targets. In this work, we develop a Cartesian framework for irreducible Cartesian tensors (ICTs) by introduce the Cartesian-3j symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the Wigner-3j symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the e3nn library to support ICT product, and use this framework to build Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first controlled comparison where architectures are held fixed and only the tensor basis is changed. Our experiments show that irreducible Cartesian models can achieve accuracy comparable to spherical counterparts, but direct Cartesianization incurs unfavorable compute and memory scaling, motivating dedicated Cartesian architectural choices. Leveraging ICTs and our framework, we introduce TACE-v1-OAM-M and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-xu26br, title = {A Cartesian-3j Framework for Machine Learning Interatomic Potentials}, author = {Xu, Zemin and Wu, Chenyu and Xie, Wenbo and Hu, Peijun}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {142308--142323}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/xu26br/xu26br.pdf}, url = {https://proceedings.mlr.press/v306/xu26br.html}, abstract = {Machine learning interatomic potentials (MLIPs) have brought substantial gains in the extrapolation capability in computational chemistry. However, most equivariant models are typically built with spherical tensors (STs), while Cartesian tensor formulations remain less developed despite their natural alignment with atomic coordinates and tensorial targets. In this work, we develop a Cartesian framework for irreducible Cartesian tensors (ICTs) by introduce the Cartesian-3j symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the Wigner-3j symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the e3nn library to support ICT product, and use this framework to build Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first controlled comparison where architectures are held fixed and only the tensor basis is changed. Our experiments show that irreducible Cartesian models can achieve accuracy comparable to spherical counterparts, but direct Cartesianization incurs unfavorable compute and memory scaling, motivating dedicated Cartesian architectural choices. Leveraging ICTs and our framework, we introduce TACE-v1-OAM-M and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.} }
Endnote
%0 Conference Paper %T A Cartesian-3j Framework for Machine Learning Interatomic Potentials %A Zemin Xu %A Chenyu Wu %A Wenbo Xie %A Peijun Hu %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-xu26br %I PMLR %P 142308--142323 %U https://proceedings.mlr.press/v306/xu26br.html %V 306 %X Machine learning interatomic potentials (MLIPs) have brought substantial gains in the extrapolation capability in computational chemistry. However, most equivariant models are typically built with spherical tensors (STs), while Cartesian tensor formulations remain less developed despite their natural alignment with atomic coordinates and tensorial targets. In this work, we develop a Cartesian framework for irreducible Cartesian tensors (ICTs) by introduce the Cartesian-3j symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the Wigner-3j symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the e3nn library to support ICT product, and use this framework to build Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first controlled comparison where architectures are held fixed and only the tensor basis is changed. Our experiments show that irreducible Cartesian models can achieve accuracy comparable to spherical counterparts, but direct Cartesianization incurs unfavorable compute and memory scaling, motivating dedicated Cartesian architectural choices. Leveraging ICTs and our framework, we introduce TACE-v1-OAM-M and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.
APA
Xu, Z., Wu, C., Xie, W. & Hu, P.. (2026). A Cartesian-3j Framework for Machine Learning Interatomic Potentials. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:142308-142323 Available from https://proceedings.mlr.press/v306/xu26br.html.

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