Numerical determination of the Kekulé structure count of some symmetrical polycyclic aromatic hydrocarbons and their relationship with π-electronic energy: (A computational approach)

The perfect matching (Kekulé structure count) of certain polycyclic aromatic hydrocarbons (benzenoids, i.e., PAH6) having mirror plane symmetry is obtained through a computer program. A computer operation in the form of an initial approximation (P, Q) is selected such that it extracts the quadratic factors (QFs) like (X² - A_iX + B_i) and linear factors (LFs) like (X - a_i) from the characteristic polynomials (CPs) of the different components obtained from the mirror plane fragmentation technique following an energy scale. The most minimum energy factor is extracted first, and then the next higher factor is extracted. This process of gradual extraction of the energy factor concludes after the HOMO level of the fragment is extracted. These factors contain the positive Hückel eigenvalues which are responsible for the Kekulé structure count and the π-electron energy (E_π) of the benzenoid molecules. A_i, B_i, and a_i are used to correlate (E_π)_i and K_i of the fragments. A linear relationship between the total π-electronic energy of benzenoid hydrocarbons on the Kekulé structure count is established: E_π(total) = 2[Σ_i=1ⁿ(X_r)_i + Σ_i=1ⁿ[1/(X_r)_i]K_i + Σ_j=1ⁿK_j] where K_i and K_j are the parts of the total K obtained through the quadratic and linear operations, respectively, and X_r are the positive Hückel eigenvalues extracted by the quadratic operations. Further, n refers to all of the extracted factors, and r may be 1 or 2, depicting the first or the second eigenvalue extracted by the QF.