Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons

Alphonse Emadak,Robert Martin Nemba,Jules Cesar Epee Ndongue,Leonel Tsassi Djoumessi

doi:10.11648/j.sjc.20190703.11

Abstract

A mathematical process of enumeration of Kekule structures of any polybenzenoid monoradical having only linear assemblies of benzenoid hydrocarbons is given by using a method which consisted of the excision of the radical center in order to obtain a conjugated hydrocarbon having a 12-annulene core ring appended to one or more polybenzenoid fragments, the partition of the resulting conjugated hydrocarbon into smaller independent resonant circuits. Polyhex monoradical skeletons are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoid fragments. Mathematical formulas are established for three cases of assemblies, one edge linear assemblies, two edges linear assemblies with two linear polybenzenoid fragments located at right angles θ = 60°, 120° or 180° and three edges linear assemblies with three linear polybenzenoid fragments located at right angles θ= 60°, 120° or 180°. Results show that the number of Kekule structures of any polybenzenoid monoradical having only linear assemblies of benzenoid hydrocarbons increases with the number of fused benzenoids according to results compiled in the table for h equal 1, 2, 3 and 4. Difficulty to represent graphs when the number of structures increases with the number of benzenoids cannot be neglected.

Highlights

Polycyclic aromatic hydrocarbons are the most abundant class of molecules in the known universe, occurring in meteorites, carbon stars in interstellar space and as pyrolytic products in the combustion engine
Polyhex monoradical skeletons examined in this work are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoïd fragments
If the previous conditions are satisfied during the dissection of the parent conjugated circuit one may obtain the solution of the problem of the enumeration of Kekule structures of any monoradical G0 by applying the following

Summary

Introduction

Polycyclic aromatic hydrocarbons are the most abundant class of molecules in the known universe, occurring in meteorites, carbon stars in interstellar space and as pyrolytic products in the combustion engine. The discovery and introduction of the concept of resonance embedding in this chemical compounds family has generated the problem of enumerating Kekule structures of polyhex molecules having several conjugated double bonds. A whole book by Cyvin and Gutman [14] is devoted to that topic It is discussed at length in several surveys of two recent volumes on Advances in the theory of benzenoid hydrocarbons [16, 17] and in numerous papers cited there. We are interested in determining the number of Kekule structures of any polybenzenoïd monoradical having only symmetrical linear assemblies of benzenoid hydrocarbons. Beyond the direct formulas of computation established, this work provides another insight on the problem of enumeration of Kekule structures of polyhexes monoradicals. Alphonse Emadak et al.: Enumeration of Kekule Structures of Polyhex Monoradical Having. Linear Assemblies of Benzenoid Hydrocarbons which still to be documented comparatively to others category of polyhexes such as fusene or perifusene

Generation of Polyhex Monoradical Skeletons

Mathematical Formulation

Application, Results and Graphical Representations

Conclusion