# A Cartesian-3j Framework for Machine Learning Interatomic Potentials

arXiv: https://arxiv.org/abs/2512.16882 · HTML: https://mlsanity.com/paper/2512.16882/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 50 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction](https://mlsanity.com/paper/2502.12147/) (2025)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Open Materials 2024 (OMat24) Inorganic Materials Dataset and Models](https://mlsanity.com/paper/2410.12771/) (2024)
- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Science](https://mlsanity.com/paper/2305.14247/) (2023)
- [An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networks](https://mlsanity.com/paper/2501.13986/) (2025)
- [High-performance training and inference for deep equivariant interatomic potentials](https://mlsanity.com/paper/2504.16068/) (2025)
- [The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products](https://mlsanity.com/paper/2506.13523/) (2025)
- [Machine learning of solvent effects on molecular spectra and reactions](https://mlsanity.com/paper/2010.14942/) (2020)
- [Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing](https://mlsanity.com/paper/2405.14253/) (2024)
- [High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces](https://mlsanity.com/paper/2412.18263/) (2024)
- [Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Space](https://mlsanity.com/paper/2509.14961/) (2025)

## Built on by (most cited first, 3 of 3)

- [Edge Cluster Expansion with Radial Rotary Attention for Interatomic Potentials](https://mlsanity.com/paper/2607.10664/) (2026)
- [BranchIP: Learning Adaptive Equivariant Computation for Interatomic Potentials](https://mlsanity.com/paper/2610.02013/) (2026)
- [FlashCart: Fast Cartesian Tensor Products for Equivariant Interatomic Potentials](https://mlsanity.com/paper/2610.06409/) (2026)

## Abstract (arXiv)

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 textttCartesian-3j symbol and Cartesian Generalized Clebsch-Gordan Coefficients, which serve as direct analogues of the textttWigner-3j symbol and Generalized Clebsch-Gordan coefficients defined for ST coupling. We extend the texttte3nn library to support ICT product, and use this framework to build Cartesian counterparts of textttMACE, textttNequIP, and textttAllegro, 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 textttTACE-v1-OAM-M and demonstrate that it achieves competitive performance on Matbench Discovery compared to state-of-the-art ST models.