Jemmis mno rules

Jemmis mno rules represent a unified rule for predicting and systematizing structures of compounds, usually clusters. The rules involve electron counting. They were formulated by Eluvathingal Devassy Jemmis to explain the structures of condensed polyhedral boranes such as B₂₀H₁₆, which are obtained by condensing polyhedral boranes, by sharing a triangular face, an edge, a single vertex or even four vertices. These rules are addition and extensions to Wade’s rules and polyhedral skeletal electron pair theory. Jemmis mno rule provides the relationship between polyhedral boranes, condensed polyhedral boranes and β-rhombohedral boron. This is similar to the relationship between benzene, condensed benzenoid aromatics and graphite, brought by Hückel’s 4n+2 rule, as well as that between tetracoordinate tetrahedral carbon compounds and diamond. The Jemmis mno rule reduces to Hückel rule when restricted to two dimensions and reduces to Wade’s rules when restricted to one polyhedron.

Electron counting rules are the tools to predict the preferred electron count for molecules. Octet rule, 18-electron rule and Hückel’s 4n+2 π-electron rule are proved to be very useful in predicting the stability of molecules. Wade’s rules were formulated to explain the electronic requirement of monopolyhedral borane clusters. Jemmis mno rule is an extension of Wade’s rule, generalized to include condensed polyhedral boranes as well. The first condensed polyhedral borane B₂₀H₁₆, is formed by the sharing of 4 vertices between two icosahedra. According to Wade’s n+1 rule for closo structures, B₂₀H₁₆ should have a charge of +2 (n+1 = 20+1 = 21 pairs required; 16 BH units provide 16 pairs; four shared boron atoms provide 6 pairs; thus 22 pairs are available). To account for the existence of B₂₀H₁₆ as a neutral species, and to understand the electronic requirement of condensed polyhedral clusters, a new variable, “m”, corresponding to the number of polyhedra (sub-clusters) was introduced. In Wade’s n+1 rule, “1” corresponds to the core bonding molecular orbital (BMO) and “n” corresponds to the number of vertices, which in turn is equal to the number of tangential surface BMOs. If m polyhedra condense to form a macropolyhedron, m core BMOs will be formed. Thus the SEP requirement of closo-condensed polyhedral clusters is m+n. Single-vertex sharing is a special case, where each sub-cluster needs to satisfy Wade’s rule separately. Let a and b be the number of vertices in the sub-clusters including the shared atom. The first cage requires a+1 and second cage requires b+1 skeletal electron pairs (SEPs). Thus a total of a+b+2 or a+b+m SEPs are required; but a+b = n+1, as the shared atom is counted twice. Thus the rule can be modified to m+n+1or generally, m+n+o, where “o” corresponds to the number of single-vertex sharing condensations. The rule can be made more general by introducing a variable, “p”, corresponding to the number of missing vertices, and “q”, the number of caps. Thus the generalized Jemmis rule can be stated as follows:
The SEP requirement of condensed polyhedral clusters is m+n+o+p-q, where, “m” is the number of sub-clusters, “n” is the number of vertices, “o” is the number of single-vertex shared condensations, “p” is the number of missing vertices and “q” is the number of caps.

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