From Polygonal Aromatics to Deltahedral Boranes

The eigenvalues of the adjacency matrix are obtained from the following determi-nantal equation  

Note the general similarities between Eqs. (4) and (5). In equation (5) the energy matrix H and the overlap matrix S can be resolved into the identity matrix I and the adjacency matrix A as follows  

The energy levels of the Hiickel molecular orbitals [Eq. (5)] are thus related to the eigenvalues xk of the adjacency matrix A (equation 4) by the following equation  

These same principles can be extended to systems, such as H3+, with delocalized multicenter er-bonding derived solely from the radial s-orbital combinations. The descriptive term, in-plane aromaticity, [44] includes such cases (see Chapter 3.2.1 for further discussion). 

More advanced mathematical aspects of the graph-theoretical models for aromaticity are given in other references [36, 48, 49]. Some alternative methods, beyond the scope of this chapter, for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [51-53] and the topological solutions of non-linear field theory related to the Skyrmions of nuclear physics [54]. 

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