Vertex and face colorings of dodecahedron and its distortions for all irreducible representations: insights into dynamic chirality, pentagonal, Jahn–Teller and elongated distortions

Abstract

We present combinatorial cum group theoretical generating function methods for vertex and face colorings of a dodecahedron for all irreducible representations of the icosahedral group. We demonstrate the usefulness of the Mȍbius inversion method. We consider several types of distortions arising from the highly symmetric dodecahedron, in particular, to an elongated dodecahedron and a pyrithohedron (T_h). Elaborate combinatorial enumerations and tables of combinatorial numbers are explicitly constructed for all irreducible representations of both the distorted and the parent undistorted structures. It is shown that the combinatorial cum computational techniques provide new insights into the dynamic chirality arising from such distortions which include the pentagonal distortions, elongated distortions and so forth. We point out applications to the dynamic NMR and ESR spectroscopies as well to the dynamic stereochemistry of topological metamorphosis through a combination of combinatorics and group theory.