Surface Harmonics of a Non-Crystallographic Symmetry Group D

The importance of group theory in its applications to a variety of problems in physics and chemistry is well known. Symmetry plays an important role in the study of properties of crystals and molecules. Given a particular matrix representation of a symmetry group, it is often necessary to obtain expansions in spherical harmonics that belong to the given irreducible representation of that symmetry group. For instance, when the cellular method is applied to investigate the band structure of a metal, expansions are required that produce the symmetry of the group of the k-vector and necessity of such expansions appeared when hybrid orbitals are obtained for a central atom in a molecule of given symmetry. The author considered the group D4d, all its irreducible matrix representations were presented explicitly since they are not available in literature so far. The surface harmonics corresponding to all the irreps were obtained using those matrix representations.