The spherical harmonics with the symmetry of the icosahedral group

Abstract

For a long time there has been some interest in obtaining spherical harmonics with the symmetry of the regular polyhedrons, particularly for electrostatic problema involving polyhedral conductors. Work on the icosahedral group has been done, among others, by Meyer (7), Laporte (6), Hodgkinson (3) and Poole (8), apart from the classic work of Klein (4). In the several approaches of these authors, only spherical harmonics for the totally symmetric representation were obtained, the most complete table being that of Laporte who obtained the spherical harmonics up tol= 21. New interest in the icosahedral group has arisen in connexion with the structureof some proteins (5), and we obtain here, by the recently developed methodof Altmann(1), expansions in spherical harmonics for ali the representations of this group. This has been done up to and includingl= 14. (For the totally symmetric representation we have also includedl= 15.) In what follows we shall use Altmann's formulae and notation.