Graph-Theory Derived Models for the Skeletal Chemical Bonding in Organometallic Metal Carbonyl Clusters

Ideas derived from topology and graph theory can be used to model the skeletal chemical bonding in metal carbonyl clusters as well as boranes and carboranes. In such molecules delocalized skeletal bonding occurs when there is a mismatch between the number of internal orbitals provided by a vertex atom and the number of polyhedral edges meeting at that vertex. Such global delocalization in deltahedral metal clusters may be regarded as a three-dimensional analogue of the aromaticity in planar polygonal molecules such as benzene. Elementary graph-theoretical considerations provide a basis for the 2n+2 skeletal electrons normally observed in n vertex delocalized deltahedral metal clusters lacking tetrahedral chambers. Electron-rich n-vertex metal clusters have more than 2n + 2 skeletal electrons and form polyhedra having at least one face with at least 4 edges. Electron-poor n-vertex metal clusters have less than 2n + 2 skeletal electrons and form deltahedra having at least one tetrahedral chamber. Ideas derived from group theory and graph theory can be used to analyze molecular orbital energy parameters for the most symmetrical deltahedral borane anions B_nH_n²⁻ (particularly n = 6 and 12) computed using either semiempirical extended Hückel methods or ab initio methods based on Gaussian orbitals. Some of the fundamental aspects of the graph-theory derived models for skeletal bonding in delocalized deltahedral metal clusters are closely related to the tensor surface harmonic theory of Stone.