Graph-theoretic clustering method

The interstitial electron rule can be applied more directly to pyramidal clusters than the graph-theoretical approximation since the latter breaks down by giving zero eigenvalues in Eq. (3) when applied to pyramids. The same ideas as those in the Jemmis-Schleyer method are needed to treat nido pyramids graph-theoretically. 

A brief review of graph-theoretical ideas is to be found in Mathematical methods in coordination chemistry topological and graph-theoretical ideas in the study of metal clusters and polyhedral isomerizations by R. B. King, Coord. Chem. Rev. (1993) 122, 91. 

In recent years the fundamental ideas of Huckel molecular orbital theory, the Huckel rule, and other aspects of aromaticity have been extended to polyhedral three-dimensional inorganic structures regarded as aromatic like the two-dimensional aromatic hydrocarbons. Such an extension of Huckel molecular orbital theory requires recognition of its topological foundations so that they can be applied to three-dimensional structures as well as two-dimensional structures. In this connection graph theoretical methods can be used to demonstrate the close analogy between the delocalized bonding in two-dimensional planar aromatic systems such as benzene and that in three-dimensional deltahedral boranes, and carboranes. Related ideas can be shown to be applicable for metal carbonyl clusters, bare post-transition metal clusters, and polyoxometallates. ... 

Abstract We have examined the predictive capability of density functional theory methods in calculations of electric polarizability and hyperpolarizability. We have focused on test cases belonging to three high-priority classes of molecular systems soft metal clusters, novel types of compounds, and weakly bonded molecules. The performance of theoretical methods over arbitrary collections of molecular properties can be analyzed and classified by the introduction of a new methodology based on graph theoretic considerations and pattern recognition techniques. 

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