A Cartesian-3j Framework for Machine Learning Interatomic Potentials
Not on the map. Why?
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.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 21 of its 50 references that are in the map. First posted on arXiv 18 Dec 2025.
Builds on
21 of 50 references.
References · 21
- 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
- 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
- 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
- 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
- 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
- 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
- 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 168
- 2022The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentialscited by 100
- 2025Learning Smooth and Expressive Interatomic Potentials for Physical Property Predictioncited by 79
- 2023TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentialscited by 53
- 2024Open Materials 2024 (OMat24) Inorganic Materials Dataset and Modelscited by 50
- 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
9 more
- 2024Cartesian atomic cluster expansion for machine learning interatomic potentialscited by 31
- 2023Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Sciencecited by 25
- 2025An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networkscited by 19
- 2025High-performance training and inference for deep equivariant interatomic potentialscited by 17
- 2020Machine learning of solvent effects on molecular spectra and reactionscited by 14
- 2025The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Productscited by 14
- 2024Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passingcited by 13
- 2024High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spacescited by 6
- 2025Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Spacecited by 5
Built on by
Citing papers per year: 2026: 2.
2 papers.