A Cartesian-3j and nj 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), and it remains unclear whether it is the only practical design principle, or whether irreducible Cartesian tensors (ICTs) can offer distinct advantages by operating directly in the Cartesian space that naturally aligned with atomistic coordinates and tensor targets. In this work, we introduce the Cartesian-3j and Cartesian-nj symbols, which serve as direct analogues of the Wigner-3j and Wigner-nj symbols defined for spherical tensor coupling. We further extend the e3nn library to support ICT products, 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. Leveraging the ICTs and Cartesian-based architecture, a universal interatomic potential is trained and demonstrated competitive performance on a widely used public benchmark for materials discovery against SOTA ST models.