This amplitude is invariant both to translations and permutations of atoms α, but it is not rotationally invariant. By integrating over all relative rotations R tensor products between this atom density evaluated at different points, one can obtain a representation of the atomic environment that is spherical invariant, corresponding to n-body correlations between the atoms. In particular, the second-order (three-body) invariant can be easily computed, on a radial basis, in terms of the S3 spherical decomposition 4, and corresponds to the SOAP power spectrum