Dipole-induced-dipole polarizabilities of symmetric clusters

Symmetry arguments are applied to the determination of closed-form solutions to the classical dipole-induced-dipole model of interaction polarizabilities for polyhedral clusters of cubic and icosahedral symmetry. Analytical conditions for the range of applicability of the model and onset of the 'polarization catastrophe' are given. These conditions are expressed as transformed versions of the hyperbolic curve of divergence that is already present in the classical case of an interacting pair. Clusters based on the cube and the icosahedron support unique patterns of induced dipoles, but other physically accessible distributions are found for the octahedron (equatorial dipoles antiparallel to a large central moment), tetrahedron and dodecahedron (central and mean cage dipole moments antiparallel)