Chemical applications of topology and group theory

Abstract

Polyhedral isomerizations of the typeP ₁→P ₂→P ₃ are degenerate ifP ₁ is combinatorically equivalent toP ₃ and planar ifP ₂ is a planar polygon. This paper systematizes degenerate non-planar isomerizations of 5- and 6-vertex polyhedra by using their Gale diagrams which are 1- and 2-dimensional, respectively. Using this method, it is trivial to show that all degenerate non-planar isomerizations of 5-vertex polyhedra can be formulated as sequences of Berry pseudorotation processes, i.e. the prototypical diamond-squarediamond (dsd) process. The Gale diagrams of the 7 combinatorically distinct 6-vertex polyhedra consist necessarily of points on the circumference of the unit circle as well as the center in the case of the pentagonal pyramid. Study of allowed motions of these points along the circumference of the unit circle in these Gale diagrams reveal 6 different types of single or multiple parallel dsd processes or closely related dsd′ or sds processes connecting these 7 combinatorically distinct 6-vertex polyhedra. In addition, a study of allowed motions of the points on the circumference of the Gale diagrams of the 6-vertex polyhedra through the center reveal 2 additional degenerate nonplanar isomerization processes of 6-vertex polyhedra which involve pentagonal pyramid intermediates.