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This work introduces a Cartesian tensor framework for machine learning interatomic potentials, offering a viable alternative to spherical tensor models with comparable accuracy. While direct conversion incurs higher computational costs, the study demonstrates that dedicated architectural choices can leverage this basis for competitive performance on materials discovery benchmarks. The extension of the e3nn library and the release of the TACE model provide a practical resource for researchers exploring tensor basis trade-offs in equivariant graph networks.
A Cartesian-3j Framework for Machine Learning Interatomic Potentials
arXiv:2512.16882v2 Announce Type: replace-cross 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 \texttt{Cartesian-3j} symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the \texttt{Wigner-3j} symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the \texttt{e3nn} library to support ICT product, and use this framework to build Cartesian counterparts of \texttt{MACE}, \texttt{NequIP}, and \texttt{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 \texttt{TACE-v1-OAM-M} and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.
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 \texttt{Cartesian-3j} symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the \texttt{Wigner-3j} symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the \texttt{e3nn} library to support ICT product, and use this framework to build Cartesian counterparts of \texttt{MACE}, \texttt{NequIP}, and \texttt{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 \texttt{TACE-v1-OAM-M} and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.
Chemical Physics (physics.chem-ph); Materials Science (cond-mat.mtrl-sci); Machine Learning (cs.LG)
arXiv:2512.16882 [physics.chem-ph]
(or arXiv:2512.16882v2 [physics.chem-ph] for this version)
https://doi.org/10.48550/arXiv.2512.16882 Focus to learn more arXiv-issued DOI via DataCite
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