# Fast evaluation of spherical harmonics with sphericart

arXiv: https://arxiv.org/abs/2302.08381 · HTML: https://mlsanity.com/paper/2302.08381/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 34 references that are in the map.
- First posted on arXiv 16 Feb 2023.

## Builds on (most cited first, 2 of 2)

- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [DScribe: Library of Descriptors for Machine Learning in Materials Science](https://mlsanity.com/paper/1904.08875/) (2019)

## Built on by (most cited first, 6 of 6)

- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networks](https://mlsanity.com/paper/2501.13986/) (2025)
- [Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators](https://mlsanity.com/paper/2503.04649/) (2025)
- [How unconstrained machine-learning models learn physical symmetries](https://mlsanity.com/paper/2603.24638/) (2026)
- [Gaussian Process Regression of Steering Vectors With Physics-Aware Deep Composite Kernels for Augmented Listening](https://mlsanity.com/paper/2509.02571/) (2025)
- [Extending Neural Operators: Robust Handling of Functions Beyond the Training Set](https://mlsanity.com/paper/2603.03621/) (2026)

## Abstract (arXiv)

Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from geology and atmospheric sciences to signal processing and computer graphics. More recently, they have become a key component of rotationally equivariant models in geometric machine learning, including applications to atomic-scale modeling of molecules and materials. We present an elegant and efficient algorithm for the evaluation of the real-valued spherical harmonics. Our construction features many of the desirable properties of existing schemes and allows to compute Cartesian derivatives in a numerically stable and computationally efficient manner. To facilitate usage, we implement this algorithm in sphericart, a fast C++ library which also provides C bindings, a Python API, and a PyTorch implementation that includes a GPU kernel.