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

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.

Towards universal property prediction in Cartesian space: TACE is all you need

Machine learning has revolutionized atomistic simulations and materials science, yet current approaches often depend on spherical-harmonic representations. Here we introduce the Tensor Atomic Cluster Expansion and Tensor Moment Potential, the first unified framework formulated entirely in Cartesian space for the systematic prediction of arbitrary structure-determined tensorial properties. TACE achieves this by decomposing atomic environments into a complete hierarchy of (irreducible) Cartesian tensors, ensuring symmetry-consistent representations that naturally encode invariance and equivariance constraints. Beyond geometry, TACE incorporates universal embeddings that flexibly integrate diverse attributes including basis sets, charges, magnetic moments and field perturbations. This allows explicit control over external invariants and equivariants in the prediction process. Long-range interactions are also accurately described through the Latent Ewald Summation module within the short-range approximation, providing a rigorous yet computationally efficient treatment of electrostatic interactions. We demonstrate that TACE attains accuracy, stability, and efficiency on par with or surpassing leading equivariant frameworks across finite molecules and extended materials, including in-domain and out-of-domain benchmarks, spectra, hessians, external-field response, charged systems, magnetic systems, multi-fidelity training, and heterogeneous catalytic systems. Crucially, TACE bridges scalar and tensorial modeling and establishes a Cartesian-space paradigm that unifies and extends beyond the design space of spherical-harmonic-based methods. This work lays the foundation for a new generation of universal atomistic machine learning models capable of systematically capturing the rich interplay of geometry, fields and material properties within a single coherent framework.