Explicit expressions for totally symmetric spherical functions and symmetry-dependent properties of multipoles

Closed expressions for matrix elements ⟨lm'|A(G)|lm⟩, where |lm⟩ are spherical functions and A(G) is the average of all symmetry operators of point group G, are derived for all point groups (PGs) and then used to obtain linear combinations of spherical functions that are totally symmetric under all symmetry operations of G. In the derivation, we exploit the product structure of the groups. The obtained expressions are used to explore properties of multipoles of symmetric charge distributions. We produce complete lists of selection rules for multipoles Ql and their moments Qlm, as well as of numbers of independent moments in a multipole, for any l and m and for all PGs. Periodicities and other trends in these properties are revealed.