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

Fast evaluation of spherical harmonics with sphericart