Graph theory in the study of metal cluster bonding topology: Applications to metal clusters having fused polyhedra

Abstract

The energy levels in a delocalized two- or three-dimensional chemical structure are related to the eigenvalues of the graph representing the corresponding bonding topology. Such relatively crude but computationally undemanding graph theory-derived models provide a clear demonstration of the close relationship between two-dimensional aromatic systems such as benzene and three-dimensional aromatic systems such as deltahedral boranes, carboranes, and metal clusters. The basic building blocks for the three-dimensional aromatic systems are deltahedra, having no degree 3 vertices. Delocalized bonding in such systems having v vertices requires two electrons for a multicenter core bond as well as 2v electrons for pairwise surface bonding. A problem of particular interest is how metal cluster polyhedra can fuse together, leading ultimately to the infinite structures of the bulk metals. As a model for such processes the fusion of rhodium carbonyl octahedra is examined using graph theory-derived methods. These lead to reasonable electron-precise models for the bonding topologies in the “biphenyl analogue” [Rh12(CO)30]2−, the “naphthalene analogue” [Rh9(CO)19]3−, the “anthracene analogue” H2Rh12(CO)25, and the “perinaphthene analogue” [Rh11(CO)23]3−. Similar models can also be developed for clusters based on centered larger rhodium polyhedra as exemplified by the centered cuboctahedral clusters of the type [Rh13(CO)24H5–q]q− (q = 2, 3, 4) representing a fragment of the hexagonal close-packed metal structure.