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Chemical applications of topology and group theory. 23. A comparison of graph‐theoretical and extended Hückel methods for study of bonding in octahedral and icosahedral boranes

The graph‐theory derived model for the bonding topology in the globally delocalized polyhedral boranes B6H62− and B12H122− is evaluated by comparison of the energies of the core molecular orbitals with those obtained by the 1962 LCAO‐MO extended Hückel calculations of Hoffmann and Lipscomb. Of particular interest is how well the complete graphs K6 and K12 used in the graph‐theory derived model approximate the bonding topologies of the unique internal orbitals (radial orbitals) of the octahedron and icosahedron, respectively. In the case of the B6H62− octahedron the single positive eigenvalue of the K6 graph corresponds to the results from the extended Hückel calculations. In addition, the parameters from the latter calculations indicate a ratio of 0.625 for the overlap of the unique internal orbitals of the trans atom pairs relative to cis atom pairs as compared with unity implied by an unweighted K6 graph. In the case of the B12H122− icosahedron the graph‐theory derived model is far less satisfactory since the single positive eigenvalue of the K12 graph disagrees with the four bonding core molecular orbitals (an A1g and three triply degenerate T1u molecular orbitals) found by the extended Hückel calculations after removing the effect of the mixing of core and surface bonding orbitals corresponding to the same irreducible representations. However, this core‐surface orbital mixing raises the energy of the triply degenerate T1u core molecular orbitals to antibonding levels so that the graph‐theory derived model fortuitiously gives correct skeletal electron counts for the regular icosahedron despite this fundamental error.