As an aside, we note that atom-centered MLPs are fully invariant with respect to all permutations of like atoms. This was apparent even for the early and highly influential Tersoff potential. And there is no advantage in using a reduced symmetry. And to preview our new Many-Body PIP representation of molecules, clusters, etc described and illustrated below, we show that this representation is fully permutationally invaraiance with respect to all permutations of like atoms.