Cartesian-nj: Extending e3nn to Irreducible Cartesian Tensor Product and Contracion

Abstract

Equivariant atomistic machine learning models have brought substantial gains in both extrapolation capability and predictive accuracy. Depending on the basis of the space, two distinct types of irreducible representations are utilized. From architectures built upon spherical tensors (STs) to more recent formulations employing irreducible Cartesian tensors (ICTs), STs have remained dominant owing to their compactness, elegance, and theoretical completeness. Nevertheless, questions have persisted regarding whether ST constructions are the only viable design principle, motivating continued development of Cartesian networks. In this work, we introduce the Cartesian-3j and Cartesian-nj symbol, which serve as direct analogues of the Wigner-3j and Wigner-nj symbol defined for tensor coupling. These coefficients enable the combination of any two ICTs into a new ICT. Building on this foundation, we extend e3nn to support irreducible Cartesian tensor product, and we release the resulting Python package as cartnn. Within this framework, we implement Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first systematic comparison of Cartesian and spherical models to assess whether Cartesian formulations may offer advantages under specific conditions. Using TACE as a representative example, we further examine whether architectures constructed from irreducible Cartesian tensor product and contraction(ICTP and ICTC) are conceptually well-founded in Cartesian space and whether opportunities remain for improving their design.