Polyhedral structures with an odd number of vertices: nine-atom clusters and supramolecular architectures

The geometries of metal clusters and supramolecular architectures that contain nine metal atoms are analyzed within the framework of continuous shape measures (CShM). The most common polyhedra in nine coordinate complexes, the capped square antiprism and the tricapped trigonal prism, are also found among these families of compounds, even if much more scarcely. In addition, a variety of new shapes, not found among coordination polyhedra, can be identified and their proximity to the ideal geometries quantified. These include a linear chain, two types of trigonal columns, the planar regular enneagon, two-dimensional hexagonal and square grids, fragments of a close-packed structure, the triangular cupola, the tridiminished icosahedron or different fragments of the icosahedron. Among the nine-atom boranes and related clusters of the groups 13 and 14 elements, those having between 18 and 20 framework electrons present the structure of the tricapped trigonal prism, the expected closo structure. However, clusters with 21 and 22 framework electrons present a variety of structures with geometries covering nearly all the path that takes one from the capped square antiprism (nido form) to the tricapped trigonal prism (closo form).