Chemical applications of topology and group Theory. XXIV: Chiralization of chemically significant polyhedra [1]

Permutation group-theoretical methods are used to study the chiralization of achiral polyhedral skeletons with v vertices by successive ligand replacement. Starting from the fully symmetrical ligand partition (), such chiralization processes may be characterized either by the minimum number of ligand replacement steps m, or the minimum number of different kinds of ligandsi, required to destroy all improper rotations. These parameters are trivially related to the lowest degree chiral ligand partition(s) as determined by the subduction of the skeleton point group G into the corresponding symmetric groupS by the procedure of Ruch and Schönhofer. Two different chiralization pathways with different values ofm andi are found for the octahedron, cube, hexagonal bipyramid, and icosahedron. Many less symmetrical chemically significant polyhedra have the degree 2 ligand partition (v - 2, 2) as the lowest degree chiral ligand partition and thus have only one chiralization pathway. Such polyhedra include the bicapped tetrahedron, trigonal prism, capped octahedron, bisdisphenoid, square antiprism, 4, 4, 4-tricapped trigonal prism, 4-capped square antiprism, 4,4-bicapped square antiprism, and the cuboctahedron.