Continuous symmetry measures. 5. The classical polyhedra

The continuous symmetry measures approach, designed to assess quantitatively the degree of any symmetry within any structure, is extended to the important class of the polyhedra. For this purpose, we developed a general methodology and a general computational tool, which identify the minimal distance of a given structure to a desired general shape with the same number of vertexes. Specifically, we employ this tool to evaluate quantitatively the degree of polyhedricity within distorted polyhedra, taking as examples the most central and abundant polyhedral structures in chemistry in general and in coordination chemistry in particular, namely the tetrahedron, the bipyramid, the octahedron, the cube, the icosahedron, and the dodecahedron. After describing the properties of the symmetry measurement tool, we show its application and versatility in a number of cases where the deviation from exact symmetry has been an issue, including z-axis Jahn-Teller type polyhedral distortions, tantalum hydride complexes, pentacoordinated zinc complexes, tetrahedral/octahedral Sn complexes, and icosahedrally distorted C-60-fullerene anions.