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 exttt{Cartesian-3j} symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the exttt{Wigner-3j} symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the exttt{e3nn} library to support ICT product, and use this framework to build Cartesian counterparts of exttt{MACE}, exttt{NequIP}, and exttt{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 exttt{TACE-v1-OAM-M} and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.

Source: arXiv cs.LG | 2026-06-03