Chemical Applications of Topology and Group Theory. 33. Symmetry-Forbidden Coordination Polyhedra for Spherical Atomic Orbital Manifolds1

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The four chemically significant spherical manifolds of atomic orbitals are the sp3, sd5, sp3d5, and sd5f 7 manifolds of 4, 6, 9, and 13 orbitals found in the chemistry of the main-group elements, the early transition metal homoleptic hydrides and alkyls, most other compounds of the d-block transition metals, and the actinides, respectively. Coordination geometries with an inversion center (e.g., the octahedron) or a unique reflection plane passing through no vertices (e.g., the trigonal prism) are symmetry forbidden for the sd5 manifold thereby accounting for some unusual experimental and computed geometries for six-coordinate early transition metal hydrides and alkyls. The maximum coordination numbers for polyhedra with inversion centers for the nine-orbital sp3d5 and 13-orbital sd5f 7 manifolds are 6 and 12 corresponding to the regular octahedron and regular icosahedron, respectively, for the most symmetrical manifestations of these coordination numbers.