A spherical expansion is commonly used as the input for machine learning models of molecules and materials, but can, in principle, be used for other tasks related to learning of labelled point clouds. In the language of atomistic ML, a spherical expansion computes a fixed-size representation of the arrangement of atoms $j$ in a neighbourhood around a central atom $i$, based on the vectors connecting $i$ and $j$, $\vec R_{ij}$, and the chemical species labels $Z_j$ (typically the atomic number). It works by expanding the different ingredients as follows:
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