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A Cartesian-3j Framework for Machine Learning Interatomic Potentials

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.

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