How icosahedral are icosahedral clusters?

Platonic and Archimedean solids in discrete metal-containing clusters

Metal-containing clusters have attracted increasing attention over the past 2-3 decades. This intense interest can be attributed to the fact that these discrete metal aggregates, whose atomically precise structures are resolved by single-crystal X-ray diffraction (SCXRD), often possess intriguing geometrical features (high symmetry, aesthetically pleasing shapes and architectures) and fascinating physical properties, providing invaluable opportunities for the intersection of different disciplines including chemistry, physics, mathematical geometry and materials science. In this review, we attempt to reinterpret and connect these fascinating clusters from the perspective of Platonic and Archimedean solid characteristics, focusing on highly symmetrical and complex metal-containing (metal = Al, Ti, V, Mo, W, U, Mn, Fe, Co, Ni, Pd, Pt, Cu, Ag, Au, lanthanoids (Ln), and actinoids) high-nuclearity clusters, including metal-oxo/hydroxide/chalcogenide clusters and metal clusters (with metal-metal binding) protected by surface organic ligands, such as thiolate, phosphine, alkynyl, carbonyl and nitrogen/oxygen donor ligands. Furthermore, we present the symmetrical beauty of metal cluster structures and the geometrical similarity of different types of clusters and provide a large number of examples to show how to accurately describe the metal clusters from the perspective of highly symmetrical polyhedra. Finally, knowledge and further insights into the design and synthesis of unknown metal clusters are put forward by summarizing these "star" molecules.

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.