General formalism for machine-learning models based on multipolar spherical harmonics

The formulation of descriptors of the local chemical environment, enabling the construction of machine-learning models, is usually obtained by studying the properties of the expansion coefficients of a neighborhood density. In this work, we show that all the transformation properties of the descriptors and their behavior under rotation, inversion, and complex conjugation are derived from the choice of the basis over which the density is expanded. Furthermore, crucially, they are independent from the explicit mathematical form of the neighborhood density. In particular, we show that all the descriptors investigated can be obtained by an expansion in multipolar spherical harmonics, which constitutes the core of this work, and which is introduced and analyzed in great detail. By exploiting the orthogonality and the transformation rules of the multipolar spherical harmonics, we show that several formulations are simplified, such as the one needed to obtain the $\ensuremath{\lambda}\text{\ensuremath{-}}\mathrm{SOAP}$ kernel and its properties. We close this work by applying our framework to several multibody descriptors available in literature, providing an in-depth analysis of their main properties, as made clear from the vantage viewpoint of a basis-centered approach.