# Cartesian atomic cluster expansion for machine learning interatomic potentials

arXiv: https://arxiv.org/abs/2402.07472 · HTML: https://mlsanity.com/paper/2402.07472/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 62 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Forces are not Enough: Benchmark and Critical Evaluation for Machine Learning Force Fields with Molecular Simulations](https://mlsanity.com/paper/2210.07237/) (2022)
- [DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics](https://mlsanity.com/paper/1712.03641/) (2017)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Smooth, exact rotational symmetrization for deep learning on point clouds](https://mlsanity.com/paper/2305.19302/) (2023)
- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [Tensor-reduced atomic density representations](https://mlsanity.com/paper/2210.01705/) (2022)
- [Completeness of Atomic Structure Representations](https://mlsanity.com/paper/2302.14770/) (2023)
- [Fast evaluation of spherical harmonics with sphericart](https://mlsanity.com/paper/2302.08381/) (2023)
- [A smooth basis for atomistic machine learning](https://mlsanity.com/paper/2209.01948/) (2022)

## Built on by (most cited first, 30 of 31)

- [Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction](https://mlsanity.com/paper/2502.12147/) (2025)
- [The Importance of Being Scalable: Improving the Speed and Accuracy of Neural Network Interatomic Potentials Across Chemical Domains](https://mlsanity.com/paper/2410.24169/) (2024)
- [Latent Ewald summation for machine learning of long-range interactions](https://mlsanity.com/paper/2408.15165/) (2024)
- [High-performance training and inference for deep equivariant interatomic potentials](https://mlsanity.com/paper/2504.16068/) (2025)
- [Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing](https://mlsanity.com/paper/2405.14253/) (2024)
- [A universal augmentation framework for long-range electrostatics in machine learning interatomic potentials](https://mlsanity.com/paper/2507.14302/) (2025)
- [Learning charges and long-range interactions from energies and forces](https://mlsanity.com/paper/2412.15455/) (2024)
- [MatLLMSearch: Crystal Structure Discovery with Evolution-Guided Large Language Models](https://mlsanity.com/paper/2502.20933/) (2025)
- [Overcoming systematic softening in universal machine learning interatomic potentials by fine-tuning](https://mlsanity.com/paper/2405.07105/) (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)
- [On the design space between molecular mechanics and machine learning force fields](https://mlsanity.com/paper/2409.01931/) (2024)
- [A Cartesian-3j Framework for Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2512.16882/) (2025)
- [Interpolation and differentiation of alchemical degrees of freedom in machine learning interatomic potentials](https://mlsanity.com/paper/2404.10746/) (2024)
- [Machine-Learning Interatomic Potentials for Long-Range Systems](https://mlsanity.com/paper/2502.04668/) (2025)
- [Generalization of Long-Range Machine Learning Potentials in Complex Chemical Spaces](https://mlsanity.com/paper/2512.10989/) (2025)
- [Polarizable atomic multipoles for learning long-range electrostatics](https://mlsanity.com/paper/2605.05746/) (2026)
- [Equivariant learning of a transferable three-dimensional classical density functional](https://mlsanity.com/paper/2608.13506/) (2026)
- [Response Matching for generating materials and molecules](https://mlsanity.com/paper/2405.09057/) (2024)
- [Optimal Invariant Bases for Atomistic Machine Learning](https://mlsanity.com/paper/2503.23515/) (2025)
- [Cross-functional transferability in universal machine learning interatomic potentials](https://mlsanity.com/paper/2504.05565/) (2025)
- [Efficient Parallelization of Message Passing Neural Network Potentials for Large-scale Molecular Dynamics](https://mlsanity.com/paper/2505.06711/) (2025)
- [Efficient Equivariant High-Order Crystal Tensor Prediction via Cartesian Local-Environment Many-Body Coupling](https://mlsanity.com/paper/2602.04323/) (2026)
- [From Evaluation to Design: Using Potential Energy Surface Smoothness Metrics to Guide Machine Learning Interatomic Potential Architectures](https://mlsanity.com/paper/2602.04861/) (2026)
- [Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials](https://mlsanity.com/paper/2606.29584/) (2026)
- [Pretraining Strategy for Neural Potentials](https://mlsanity.com/paper/2402.15921/) (2024)
- [SSLFusion: Scale & Space Aligned Latent Fusion Model for Multimodal 3D Object Detection](https://mlsanity.com/paper/2504.05170/) (2025)
- [Long-range electrostatics for machine learning interatomic potentials is easier than we thought](https://mlsanity.com/paper/2512.18029/) (2025)
- [Universal and efficient graph neural networks with dynamic attention for machine learning interatomic potentials](https://mlsanity.com/paper/2603.22810/) (2026)
- [Distilling latent electrostatics from foundation machine learning interatomic potentials](https://mlsanity.com/paper/2606.15001/) (2026)

## Abstract (arXiv)

Machine learning interatomic potentials are revolutionizing large-scale, accurate atomistic modelling in material science and chemistry. Many potentials use atomic cluster expansion or equivariant message passing frameworks. Such frameworks typically use spherical harmonics as angular basis functions, and then use Clebsch-Gordan contraction to maintain rotational symmetry, which may introduce redundancies in representations and computational overhead. We propose an alternative: a Cartesian-coordinates-based atomic density expansion. This approach provides a complete set of polynormially indepedent features of atomic environments while maintaining interaction body orders. Additionally, we integrate low-dimensional embeddings of various chemical elements and inter-atomic message passing. The resulting potential, named Cartesian Atomic Cluster Expansion (CACE), exhibits good accuracy, stability, and generalizability. We validate its performance in diverse systems, including bulk water, small molecules, and 25-element high-entropy alloys.