Chemical applications of topology and group theory. 12. Post-transition element clusters with particular emphasis on nine vertex

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The previously reported graph theoretical model of aromaticity in two and three dimensions is used to treat homoatomic post-transition element ions containing germanium, tin, lead, antimony, bismuth, selenium, and tellurium. Localized bonding models are sufficient to treat three and seven vertex systems as well as 12 skeletal electron trigonal bipyramidal five vertex systems such as Sn 2− 5, Pb 2− 5, and Bi 3+ 5. The squares Bi 2− 4, Se 2+ 4, and Te 2+ 4 are two-dimensional aromatic systems completely analogous to C 4H 2− 4. Most interesting are the nine vertex systems which can be classified into the following three types: (1) The 20 skeletal electron Ge 2− 9 system which adopts the tricapped trigonal prismatic configuration expected for a closo 2n + 2 skeletal electron system; (2) The 22 skeletal electron Ge 4− 9, Sn 4− 9, and Pb 4− 9 systems which adopt the capped square antiprismatic configuration with one non-triangular face expected for a nido 2n + 4 skeletal electron system; (3) The 22 skeletal electron Bi 5+ 9 system which is ‘anomalous’ since it adopts the closo deltahedral tricapped trigonal prismatic configuration rather than the nido capped square antiprismatic configuration expected for a 2n + 4 skeletal electron system. However, the tricapped trigonal prism in the 22 electron Bi 5+ 9 system is more ‘elongated’ than the tricapped trigonal prisms in 20 skeletal electron nine vertex clusters such as B 9H 2− 9, B 7H 7C 2(CH 3) 2, and Ge 2− 9. Detailed graph theoretical calculations show that such elongation of the tricapped trigonal prism can lead to incomplete overlap of the nine unique internal orbitals which generates the two core bonding orbitals required to accommodate the four core bonding electrons in a 2n + 4 skeletal electron system. These calculations also provide an illustration of a graph splitting algorithm for the symmetry factoring of graph characteristic polynomials.