Computing the Kekulé structure count for alternant hydrocarbons

Abstract

ABSTRACT:A fast computer algorithm brings computation of the permanents of sparse matrices, specifically, molecular adjacency matrices. Examples and results are presented, along with a discussion of the relationship of the permanent to the Kekulé structure count. A simple method is presented for determining the Kekulé structure count of alternant hydrocarbons. For these hydrocarbons, the square of the Kekulé structure count is equal to the permanent of the adjacency matrix. In addition, for alternant structures the adjacency matrix for N atoms can be written in such a way that only an N/2 ϫ N/2 matrix need be evaluated. The Kekulé structure count correlates with topological indices. The inclusion of the number of cycles improves the fit. When comparing with previous results, the variance decreases 74%. The calculated standard heat of formation correlates with the logarithm of the Kekulé structure count. This heat increments 349 kJ/mol each time the Kekulé structure count increases by one order of magnitude.

Aromaticity of Polycyclic Conjugated Hydrocarbons

“…There are several computer programs that calculate K . , When one is considering benzenoid hydrocarbons, one can take advantage of the fact that the absolute value of the constant term in the characteristic polynomial is the square of K . , For bipartite graphs (benzenoids systems and other polycyclic systems having only even-member rings), as G. G. Hall 476 pointed out, one can obtain K as the value of the determinant of one of the blocks of the adjacency matrix. Torrens constructed a fast computer algorithm for permanents of sparse matrices of alternant hydrocarbons, and such are the adjacency matrix of molecular graphs of benzenoid hydrocarbons.…”

An Alternative Strategy for Count and Storage of Kekulé and Longer Range Resonance Valence Bond Structures

“…For a long time, enumeration of Kekulé valence structures has been one of the main focuses in graph theoretical study of conjugated hydrocarbons. Aside from the systematic search method of Kekulé valence structures suggested by Pauling, people have also found analytical formulas for specific kinds of systems − or derived recursion relations between smaller fragments and larger systems. − There were also particular techniques developed to work on this problem, − which made it possible to enumerate Kekulé valence structures for systems with hundreds of atoms, such as nanotubes. Readers may refer to several review articles for the detailed description of the enumeration techniques. − …”

Table of Periodic Properties of Fullerenes Based on Structural Parameters

“…In previous works, the calculation of Kekulé structure count and the permanent of adjacency matrices was applied to fullerenes with different structural parameters involving the presence of contiguous pentagons . Principal component analysis (PCA) of the structural parameters was performed. − The aim of the present report is to analyze the interdependence between the structural parameters and to classify fullerenes.…”

“…This conclusion is in accordance with the experimental discovery that fullerenes in flames are formed by the addition reaction of two PAHs. 27 In previous works, the calculation of Kekule ´structure count and the permanent of adjacency matrices 28 was applied to fullerenes with different structural parameters involving the presence of contiguous pentagons. 29 Principal component analysis (PCA) of the structural parameters was performed.…”