From polygons to polyhedra through intermediate structures. A shape measures study of six-atom inorganic rings and clusters

Among the wealth of well-established molecular structures, inorganic rings and clusters present an overwhelming variety of geometries that chemists try to describe with a limited assortment of regular polygons and polyhedra. In the case of six-atom structures we usually employ the hexagon, the pentagonal pyramid, the trigonal prism and the octahedron. More often than not, however, real world structures deviate from those ideal geometries, and we try to cope with non-ideality by adding adjectives such as distorted, twisted, puckered or flattened, additionally nuanced by adverbs such as slightly, significantly or severely. This contribution presents a systematic structural perspective of six-atom groups in molecules by means of a continuous shape measures (CShM) analysis. The shape of a group of N points is defined by all the sets of 3 N Cartesian coordinates that can be generated by rigid translation, rotation, or isotropic scale change. Among all possible arrangements of N points in space, we select as reference shapes the corresponding regular N-vertex polygons and polyhedra, together with univocally defined combinations thereof (e.g., two coplanar or perpendicular edge-sharing squares). The present CShM study allows us to classify most of the structures not only by their closeness to a particular regular shape, but also by quantifying their position along minimal distortion interconversion pathways between two regular shapes.