A note on the enumeration of Kekulé structures in a class of coronoids (original) (raw)
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Kekul� structure counts: General formulations for primitive coronoid hydrocarbons
Monatshefte f�r Chemie Chemical Monthly, 1991
The Kekul6 structure counts (K) for primitive coronoids are treated. The K formula which involves the trace of a matrix product is recalled and supplemented with new findings. In this way a kind of symmetry in the mathematical formulations is restored. Another general formulation for the K number is provided in terms of polynomials which, for a somewhat mysterious reason, are identified as the matching polynomials of cycles.
Journal of Chemical Information and Modeling, 1998
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 (X 2-A i X + B i) and linear factors (LFs) like (Xa 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[∑ i)1 n (X r) i + ∑ i)1 n [1/(X r) i ]K i + ∑ j)1 n K 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.
The McClelland number of conjugated hydrocarbons
2005
The McClelland number of a conjugated hydrocarbon is the integer k, satisfying the condition 2 -(1/2) k 2nm £ E < 2 -(1/2) k+1 2nm, where E is the HMO total p-electron energy, n the number of carbon atoms, and m the number of carbon-carbon bonds. If k = 3, then the respective conjugated system is said to be energy-regular. If k £ 2 and k ³ 4, then one speaks of energy-poor and energy-rich p-electron systems, respectively. We found that all polycyclic Kekuléan hydrocarbons, possessing condensed rings, are energy-regular, with only three exceptions: naphthalene, phenanthrene, and triphenylene (which are energy-rich). Energy-poor p-electron systems are some (but not all) non-Kekuléans, whereas many of the polycyclic Kekuléan hydrocarbons with non-condensed rings (polyphenyls, phenyl-substituted polyenes and similar) are energy-rich.
Coding and Ordering Kekul� Structures
J Chem Inf Model, 2004
The concept of numerical Kekulé structures is used for coding and ordering geometrical (standard) Kekulé structures of several classes of polycyclic conjugated molecules: catacondensed, pericondensed, and fully arenoid benzenoid hydrocarbons, thioarenoids, and [N]phenylenes. It is pointed out that the numerical Kekulé structures can be obtained for any class of polycyclic conjugated systems that possesses standard Kekulé structures. The reconstruction of standard Kekulé structures from the numerical ones is straightforward for catacondensed systems, but this is not so for pericondensed benzenoid hydrocarbons. In this latter case, one needs to use two codes to recover the geometrical Kekulé structures: the Wiswesser code for the benzenoid and the numerical code for its Kekulé structure. There is an additional problem with pericondensed benzenoid hydrocarbons; there appear numerical Kekulé structures that correspond to two (or more) geometrical Kekulé structures. However, this problem can also be resolved.
Monatshefte für Chemie - Chemical Monthly, 2006
Within classes of isomeric benzenoid hydrocarbons various Kekulé-and Clar-structurebased parameters (Kekulé structure count, Clar cover count, Herndon number, Zhang-Zhang polynomial) are all mutually correlated. This explains why both the total -electron energy (E), the Dewar resonance energy (DRE), and the topological resonance energy (TRE) are well correlated with all these parameters. Nevertheless, there exists an optimal value of the variable of the Zhang-Zhang polynomial for which it yields the best results. This optimal value is negative-valued for E, around zero for TRE, and positive-valued for DRE. A somewhat surprising result is that TRE and DRE considerably differ in their dependence on Kekulé-and Clar-structure-based parameters.
Chemical Combinatorics for Alkane-Isomer Enumeration and More
Journal of Chemical Information and Modeling, 1998
Standard combinatorial enumeration techniques for alkanes are considered with a view to the extension to a widened range of chemically interesting features. As one brief point it is noted that these standard techniques naturally associate to generational schemes and thence have nomenclatural interpretations, which may be made to achieve some similarity to the standard IUPAC nomenclature. Our primary focus is the illustration that such combinatorial techniques are sufficient to enable computation of several graph-theoretic structural invariants averaged over (different types of) isomer classes. Such averages (and associated isomer counts) are tabulated for structural isomers for up to N ) 40 carbons, where there are ∼10 14 isomers (though the computational methodology should rather readily extend to at least N ) 80 where there should be ∼10 28 isomers). The averages for invariants are utilized to estimate several physicochemical properties averaged over these same isomer classes. The properties currently so considered are heat of formation, index of refraction, and magnetic susceptibility. Further, various asymptotic results for counts, mean invariants, and mean properties are noted, so that the exact graph-theoretic data are extrapolated with high accuracy to arbitrarily large alkanes.
Benzenoids with maximum Kekule structure counts for given numbers of hexagons
Journal of Chemical Information and Modeling, 1993
Polycyclic aromatic hydrocarbons with the maximal numbers of Kekule structures for any number h of benzenoid rings are shown to be branched cata-condensed systems. Their structures present regularities with periodicities of three and four in terms of h. An analysis of their structures is presented, and the recurrences are highlighted. INTRODUCTION AND DEFINITIONS The number of Kekule structures (or the Kekule structure count) has served for a long time as an indicator for resonance energies of polycyclic aromatic hydrocarbons (PAH's), for computing Pauling bond orders, and for other properties. From the rich bibliography, one may note a few books,l4 book chapters,7-" and p a p e r~. I l-~~ Of course there are some qualifications in using Kekule structures: the more elaborate theoretical methods indicate, in agreement with experimental findings for enthalpies of formation, that triphenylene (1) and chrysene (2) have approximately equal stabilities, despite the higher Kekule structure count of the f~r m e r ;~~-~~ this is probably due to the slight nonplanarity of triphenylene.26 Alternative names for PAH's are polyhexes or benzenoids. The latter term, however, has sometimes been used in a restricted to denote only those PAH's which can be obtained from the graphite lattice by selecting a connected array of hexagons ("graphite-connected PAH'snZ7). Under such a restricted acceptance, helicenes with six or more hexagons would be excluded, yet they are stable and interesting compounds because of their nonplanar geometry and inherent chirality. [Nonplanar geometry is different from graphtheoretical nonplanarity, which is not relevant in the present context.] Hemdon and co-workers2628 have shown, moreover, that most PAH's are geometrically nonplanar. Trinajstic30 restricted further the term "benzenoid" to those PAH's whose Kekule structure counts are nonzero. In the present paper we shall use the term benzenoid as a synonym for "polyhex", including also geometrically nonplanar PAH's, Le. ignoring any restriction (actually, it so happens that all systems which will be presented here have nonzero Kekule structure counts). The number of hexagons in the benzenoid system will be denoted by h and the number of Kekule structures by K (or K with some subscript such as Kh). A simple means of computing the value of Kh for any given benzenoid structure is via its adjacency matrix and its determinant or via its characteristic polynomial. In graph theory, the Kekule structure count is called the number of perfect matchings of the Huckel graph. Benzenoids are of three types: (i) cata-condensed (catafusenes), in which nocarbon atom is common to three hexagons or whose dualist graph (inner dual) is acyclic [dualist graphs have vertices corresponding to the centers of benzenoid rings,