Defining rules of aromaticity: a unified approach to the Huckel, Clar and Randic 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 pi-electron-containing conjugated hydrocarbons, this matrix may be modified in order to represent one of the possible Kekule structures by assigning entry 1 to double bonds and entry 0 to single bonds, leading to the Kekule matrix K which can be obtained from the A matrix by subtracting 1 from elements a(pq) that represent single bonds in the Kekule structure. The A and K matrices are the boundary cases of a general matrix A(epsilon), named perturbation matrix, in which from elements apq that represent single bonds is subtracted a value epsilon is an element of < 0,1 > representing the magnitude of the perturbation. The determinant of the A(epsilon) matrix is unambiguously represented by an appropriate polynomial that, in turn, can be written in a form containing terms +/-(1 epsilon)(N/2) that identify types of pi-electron conjugated cycles (N is the corresponding number of pi-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 Huckel rule works, how the Randic 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).