Defining rules of aromaticity: a unified approach to the Hückel, Clar and Randić concepts

The molecular structure of any system may be unambiguously described by its adjacency matrix, A, in which bonds are assigned entry a(ij) = 1 and non-bonded pairs of atoms entry a(ij) = 0. For π-electron-containing conjugated hydrocarbons, this matrix may be modified in order to represent one of the possible Kekulé structures by assigning entry 1 to double bonds and entry 0 to single bonds, leading to the Kekulé matrix K which can be obtained from the A matrix by subtracting 1 from elements a(pq) that represent single bonds in the Kekulé structure. The A and K matrices are the boundary cases of a general matrix A(ε), named perturbation matrix, in which from elements a(pq) that represent single bonds is subtracted a value ε∈<0,1> representing the magnitude of the perturbation. The determinant of the A(ε) matrix is unambiguously represented by an appropriate polynomial that, in turn, can be written in a form containing terms ±(1-ε)(N/2) that identify types of π-electron conjugated cycles (N is the corresponding number of π-electrons). If the sign before the term is (+), then the contribution is stabilizing, but if it is (-) the contribution is destabilizing. The approach shows why and how the Hückel rule works, how the Randić conjugated circuits result from the analysis of canonical structures, and also how the Clar rule may be extended to include aromatic cycles larger than six-membered (aromatic sextet).