The topology of coordination polyhedra and their rearrangements

The s, p, d and f atomic orbitals are classified by the numbers and orientations of their major lobes, which can be conveniently depicted as an orbital graph. Coordination polyhedra are formed from these s, p, d and f orbitals by adding or subtracting orbitals from the spherical four-orbital sp3 and nine-orbital sp3d5 manifolds. The five-coordinate trigonal bipyramid and square pyramid arise from adding the z2 and x2−y2 orbitals, respectively, to a spherical sp3 manifold. Six-coordinate polyhedra are formed by adding pairs of d orbitals to the spherical sp3 manifold with the pairs (x2−y2,z2), (xz,z2), (xy,x2−y2) and (xy,xz), giving the octahedron, bicapped tetrahedron, pentagonal pyramid and trigonal prism, respectively. Similarly, the eight-coordinate square antiprism and bisdisphenoid (D2d dodecahedron) arise from subtracting the z2 and x2−y2 orbitals, respectively, from a spherial nine-orbital sp3d5 manifold. The seven-coordinate pentagonal bipyramid, capped octahedron and capped trigonal prisms can arise from subtractions of various d orbital pairs from the spherical nine-orbital sp3d5 manifold. The stereochemical non-rigidity of many five-coordinate ML5 complexes and eight-coordinate ML8 complexes can arise from continuous transformations of the d orbital added to the sp3 manifold or subtracted from the sp3d5 manifold, respectively, from the shape of a z2 orbital to that of an z2−y2 orbital. The seven-vertex hexagonal bipyramid and eight-vertex cube, hexagonal bipyramid and D3h 3,3-bicapped trigonal prism cannot be formed using hybrids of only s, p and d orbitals, but require an additional f orbital, which is an xyz orbital with eight major lobes in case of the cube, an x(x2−3y2) orbital with six major lobes in the cases of the hexagonal pyramid and bipyramid with six co-planar atoms, and a z3 orbital in the case of the 3,3-bicapped trigonal prism. The highly symmetrical 12-coordinate icosahedron and cuboctahedron can arise by addition of a triply degenerate set of cubic fε orbitals, namely the [x(z2−y2),y(z2−x2),z(x2−y2)] set, to the spherical nine-orbital sp3d5 manifold.