# Complete and Efficient Covariants for 3D Point Configurations with Application to Learning Molecular Quantum Properties

arXiv: https://arxiv.org/abs/2409.02730 · HTML: https://mlsanity.com/paper/2409.02730/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 71 references that are in the map.
- First posted on arXiv 4 Sep 2024.

## Builds on (most cited first, 5 of 5)

- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Forces are not Enough: Benchmark and Critical Evaluation for Machine Learning Force Fields with Molecular Simulations](https://mlsanity.com/paper/2210.07237/) (2022)
- [Completeness of Atomic Structure Representations](https://mlsanity.com/paper/2302.14770/) (2023)
- [From Peptides to Nanostructures: A Euclidean Transformer for Fast and Stable Machine Learned Force Fields](https://mlsanity.com/paper/2309.15126/) (2023)
- [E3x: E(3)-Equivariant Deep Learning Made Easy](https://mlsanity.com/paper/2401.07595/) (2024)

## Built on by (most cited first, 2 of 2)

- [The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products](https://mlsanity.com/paper/2506.13523/) (2025)
- [E3J: An Efficient and Open-Source Backend for Euclidean Equivariant Operations on GPU and TPU](https://mlsanity.com/paper/2609.35099/) (2026)

## Abstract (arXiv)

When modeling physical properties of molecules with machine learning, it is desirable to incorporate SO(3)-covariance. While such models based on low body order features are not complete, we formulate and prove general completeness properties for higher order methods, and show that 6k-5 of these features are enough for up to k atoms. We also find that the Clebsch--Gordan operations commonly used in these methods can be replaced by matrix multiplications without sacrificing completeness, lowering the scaling from O(l6) to O(l3) in the degree of the features. We apply this to quantum chemistry, but the proposed methods are generally applicable for problems involving 3D point configurations.