mlsanityData to 7 Oct 2026

Cartesian atomic cluster expansion for machine learning interatomic potentials

arXiv 2402.07472 ↗First posted 12 Feb 2024physics.comp-ph cs.LG physics.chem-phcited by 31

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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.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). Builds on 24 of its 62 references that are in the map. First posted on arXiv 12 Feb 2024.

Builds on

24 of 62 references.

References · 24

  1. 2017Attention Is All You Needcited by 59,932
  2. 2021E(n) Equivariant Graph Neural Networkscited by 951
  3. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  4. 2017SchNet: A continuous-filter convolutional neural network for modeling quantum interactionscited by 569
  5. 2020Directional Message Passing for Molecular Graphscited by 556
  6. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
  7. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  8. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  9. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  10. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
  11. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  12. 2020Machine Learning Force Fieldscited by 204
12 more
  1. 2022e3nn: Euclidean Neural Networkscited by 171
  2. 2021Spherical Message Passing for 3D Graph Networkscited by 166
  3. 2022The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentialscited by 100
  4. 2022Forces are not Enough: Benchmark and Critical Evaluation for Machine Learning Force Fields with Molecular Simulationscited by 92
  5. 2017DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamicscited by 79
  6. 2023TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentialscited by 53
  7. 2023Smooth, exact rotational symmetrization for deep learning on point cloudscited by 39
  8. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
  9. 2022Tensor-reduced atomic density representationscited by 10
  10. 2023Completeness of Atomic Structure Representationscited by 7
  11. 2023Fast evaluation of spherical harmonics with sphericartcited by 6
  12. 2022A smooth basis for atomistic machine learningcited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 10; 2025: 13; 2026: 8.

31 papers.

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  1. 2025Learning Smooth and Expressive Interatomic Potentials for Physical Property Predictioncited by 79
  2. 2024The Importance of Being Scalable: Improving the Speed and Accuracy of Neural Network Interatomic Potentials Across Chemical Domainscited by 40
  3. 2024Latent Ewald summation for machine learning of long-range interactionscited by 26
  4. 2025High-performance training and inference for deep equivariant interatomic potentialscited by 17
  5. 2024Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passingcited by 13
  6. 2025A universal augmentation framework for long-range electrostatics in machine learning interatomic potentialscited by 9
  7. 2024Learning charges and long-range interactions from energies and forcescited by 7
  8. 2025MatLLMSearch: Crystal Structure Discovery with Evolution-Guided Large Language Modelscited by 7
  9. 2024Overcoming systematic softening in universal machine learning interatomic potentials by fine-tuningcited by 6
  10. 2024High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spacescited by 6
  11. 2025Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Spacecited by 5
  12. 2024On the design space between molecular mechanics and machine learning force fieldscited by 3
19 more
  1. 2024Interpolation and differentiation of alchemical degrees of freedom in machine learning interatomic potentialscited by 2
  2. 2025Machine-Learning Interatomic Potentials for Long-Range Systemscited by 2
  3. 2025Generalization of Long-Range Machine Learning Potentials in Complex Chemical Spacescited by 2
  4. 2025A Cartesian-3j Framework for Machine Learning Interatomic Potentialscited by 2
  5. 2026Polarizable atomic multipoles for learning long-range electrostaticscited by 2
  6. 2026Equivariant learning of a transferable three-dimensional classical density functionalcited by 2
  7. 2024Response Matching for generating materials and moleculescited by 1
  8. 2025Optimal Invariant Bases for Atomistic Machine Learningcited by 1
  9. 2025Cross-functional transferability in universal machine learning interatomic potentialscited by 1
  10. 2025Efficient Parallelization of Message Passing Neural Network Potentials for Large-scale Molecular Dynamicscited by 1
  11. 2026From Evaluation to Design: Using Potential Energy Surface Smoothness Metrics to Guide Machine Learning Interatomic Potential Architecturescited by 1
  12. 2026Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentialscited by 1
  13. 2024Pretraining Strategy for Neural Potentialscited by 0
  14. 2025SSLFusion: Scale & Space Aligned Latent Fusion Model for Multimodal 3D Object Detectioncited by 0
  15. 2025Long-range electrostatics for machine learning interatomic potentials is easier than we thoughtcited by 0
  16. 2026Efficient Equivariant High-Order Crystal Tensor Prediction via Cartesian Local-Environment Many-Body Couplingcited by 0
  17. 2026Universal and efficient graph neural networks with dynamic attention for machine learning interatomic potentialscited by 0
  18. 2026Distilling latent electrostatics from foundation machine learning interatomic potentialscited by 0
  19. 2026Rotational Equivariance in Machine Learning: A Comprehensive Tutorialcited by 0