Benzenoid Molecules Research Articles

Benzenoid Molecules with Uniform Distribution of π-Electrons within Rings

In a recent work it was demonstrated that in linear hexagonal chains the distribution of π-electrons into rings (as computed by means of the Randic–Balaban method) is uniform, irrespective of the nature of the terminal fragments. We now establish that an analogous, yet somewhat more complex, uniformity in the π-electron distribution exists also in double linear hexagonal chains, as well as in some other benzenoid systems.

ALGORITHM FOR SIMULTANEOUS CALCULATION OF KEKULÉ AND CLAR STRUCTURE COUNTS, AND CLAR NUMBER OF BENZENOID MOLECULES

The Chinese mathematicians Heping Zhang and Fuji Zhang conceived a combinatorial polynomial associated with benzenoid molecules. This “Zhang–Zhang polynomial” contains information on the Kekulé structure count (K), Clar structure count (C), Clar number (Cl), Hosoya-Yamaguchi sextet polynomial σ(B, x), and several other important characteristics of the underlying benzenoid molecule. We show how one can easily calculate the Zhang-Zhang polynomial, and thus arrive at an algorithm for simultaneous computation of K, C, Cl, and σ(B, x). Some applications of this algorithm are pointed out.

Dependence of Dewar resonance energy of benzenoid molecules on Kekulé structure count

The dependence of the Dewar resonance energy (DRE) on the Kekul? structure count (K) was found to be significantly different from that earlier anticipated. Within classes of benzenoid isomers, the DRE increases either as K? for ? ? 0.3 or as (ln K) for ? ? 2. Both functional dependencies result in approximate expressions for DRE of nearly equal accuracy. Approximations of the form DRE ? a K + b and DRE ? a' ln K + b' are somewhat less accurate, but can still be used in usual practical applications of the Dewar resonance energy. .

Benzenoid isomers with greatest and smallest Kekulé structure counts

In families of benzenoid isomers, the species with the greatest and smallest value of the Kekul? structure count (K) possess, respectively, the greatest and smallest thermodynamic stability and, respectively, the smallest and greates reactivity. In this paper the structure of the isomers with extremal K-values for the benzenoid molecules A = A(a, b, c, d, e) (with the formula C4h+2H2h+4) and B1 = B1(a, b, c, d)& B2 = B2(a, b, c, d) (with formula C4hH2h+2) were determined. The system A is obtained by attaching linear polyacene fragments of length a, b, c, and d to a linear polyacene of length e + 2, so that h = a + b + c + d + e + 2, the number of hexagons of A, is constant. The systems B1 and B2 are obtained by attaching linear polyacene fragments of length a, b, c, and d to pyrene, in two different ways, so that h = a + b + c + d + 4, the number of hexagons of B1 and B2, is constant. The main conclusion is that K is maximal if the parameters a, b, c, d differ by at most one, and that K is minimal if b, c, d are equal to zero.

A concealed difference between the structure-dependence of Dewar and topological resonance energy

In the 1970s, two Dewar-type resonance energies were put forward, one by Hess and Schaad (denoted here by DRE), the other by Gutman et al. and, independently, by Aihara (denoted here by TRE). Both were believed to be equivalent and, indeed, a good linear correlation between them could be established. We now show that there are some significant differences between the structure-dependencies of DRE and TRE. In particular, in the case of benzenoid molecules, DRE and TRE are found to be linearly related to ln ζ(0) and ln ζ(1), respectively, where ζ( x) is the Zhang–Zhang polynomial. Whereas ζ(0) is just the Kekulé structure count, ζ(1) is a novel structure-descriptor overlooked in the previous studies of resonance energy.

Annelated perylenes: Benzenoid molecules violating the Kekulé-structure-based cyclic conjugation models

Several currently used models for assessing the extent of cyclic conjugation in benzenoid hydrocarbons, all based on Kekul?-type structural formulas predict that there is no cyclic conjugation in the central, "empty", ring of perylene and its annelated derivatives. In this paper it is shown that in some annelated perylenes the cyclic conjugation in the "empty" ring (measured by its energy-effect) may be unexpectedly high. Therefore, in the case of these annelated perylenes, the Kekul?- structure-based models fail. The cause for such an "anomalous" behavior of annelated perylenes is discussed.

Balancing the Pauling bond orders in Kekuléan benzenoid hydrocarbons

We consider a cutting of the molecular graph B of a Kekuléan benzenoid molecule into two disconnected subgraphs, S and the other, by deleting from B certain edges. It is required that both subgraphs remain Kekuléan. The edges involved in this cutting are classified as starred and unstarred. A starred edge is incident to a starred carbon site of the subgraph S, whereas an unstarred edge to an unstarred carbon site of S. The following regularity is established: for any above-described cutting of any Kekuléan benzenoid system, the sum of the Pauling bond orders of the starred edges is equal to that for the unstarred edges.

On the π-Electron Content of Bonds and Rings in Benzenoid Hydrocarbons

The Pauling bond order can be viewed as a measure of the π-electron content of the respective carbon-carbon bond. In benzenoid hydrocarbons its values lie between 0 (in the case of essential single bonds) and 1 (in the case of essential double bonds). If the benzenoid molecule does not possess essential single and double bonds, then the Pauling bond orders are greater than 0 and less than 1, but may assume values arbitrarily close to 0 and 1. The π-electron content of a ring is equal to the sum of the π-electron contents of the carbon-carbon bonds forming this ring. We show that in benzenoid hydrocarbons the π-electron content of any (six-membered) ring lies between 0 and 5.5. If the molecule does not possess essential single and double bonds, then the π-electron content of any ring is greater than 0 and less than 5.5, but may assume values arbitrarily close to 0 and 5.5.

Relation between Wiener-type topological indices of benzenoid molecules

The distance d( u, v| G) between the vertices u and v of a molecular graph G is the length of a shortest u, v-path. We consider a class of Wiener-type topological indices W λ ( G), defined as the sum of the terms d( u, v| G) λ over all pairs of vertices of G. Several special cases of W λ ( G), namely for λ=+1 (the original Wiener number) as well as for λ=−2,−1, +1/2, +2 and +3, were previously studied in the chemical literature, and found applications as molecular structure-descriptors. We establish a relation between W λ+1 and W λ , applicable for benzenoid molecules, phenylenes, chemical trees, and other types of molecular graphs.

Synthesis, structure, and aromaticity of a hoop-shaped cyclic benzenoid [10]cyclophenacene.

The first hoop-shaped cyclic benzenoid compounds, [10]cyclophenacene derivatives that contain 40 pi electrons, have been synthesized in three or four steps from [60]fullerene by rationally designed chemical modification. The compounds thus synthesized are chemically stable, yellow-colored, luminescent, and EPR-silent. X-ray crystallographic analysis provided high precision structural data sets. On the basis of these results and theoretical investigations, the new cyclic benzenoid molecules were proven to be aromatic.

Mean-field resonating-valence-bond theory for unpaired π-electrons in benzenoid carbon species

A qualitative resonance-theoretic view is presented for the description of a variety of conjugated π-network species identified with “subgraphs” (either finite or infinite) of the graphite network. Within the framework of this resonance theory, simple rules are described to provide qualitative information: On ground-state spin multiplicities; on patterns of ground-state spin density; and on exchange splittings to low-lying “spin-flipped” excited states. Beyond ordinary benzenoid molecules, illustrative applications are noted to a diversity of extended species, including: Differently structured edges on semi-infinite graphite; corner structures where edges along different directions meet; conjugated polymer-strip ends; and local defect vacancy structures in extended graphite. The variety of simple resonance-theoretic predictions are compared against a semiempirical unrestricted Hartree–Fock view of some quantitative tight-binding molecular-orbital-theoretic computations. Agreement in predictions from the resonance- and band-theoretic viewpoints is taken to engender reliability of the so coincident predictions. A traditional organic chemical resonance-theoretic view is thence conveniently reformulated and brought to bear on several extended nano-structured systems to reveal systematic patterns of π-electronic behavior.

Binary coding of Kekulé structures of catacondensed benzenoid hydrocarbons

An algorithm is described by means of which the Kekulé structures of a catacondensed benzenoid molecule (with h hexagons) are transformed into binary codes (of length h). By this, computer-aided manipulations with, and memory-storage of Kekulé structures are much facilitated. Any Kekulé structure can easily be recovered from its binary code.

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 − AiX + Bi) and linear factors (LFs) like (X − ai) 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 (Eπ) of the benzenoid molecules. Ai, Bi, and ai are used to correlate (Eπ)i and Ki of the fragments. A linear relationship between the total π-electronic energy of benzenoid hydrocarbons on the Kekulé structure count is established: Eπ(total) = 2[ (Xr)i + [1/(Xr)i]Ki + Kj], where Ki and Kj are the parts of the total K obtained through the quadratic and linear operations, respectively, and Xr 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.

Kinetics of phenol biotransformation

Phenol and its homologues are aromatics containing hydroxyl, methyl, amide and sulphonic groups attached to the benzenoid molecules. These molecules are, both, anthropogenic and xenobiotics. Phenols are environmental pollutants discharged through wastewaters from fossil fuel refining processes, phenol manufacturing plants, pharmaceutical and a variety of other industries. Phenols are toxic to several biochemical reactions. However, biological transformation of phenols to non-toxic entities exists in specialized microbes, having enzymes of aromatic catabolic pathways. Transformation of phenols is also inhibited by the presence of other toxic materials such as cyanides and sulphides that are present in the industrial wastes having phenols as the dominant carbonaceous material. Kinetics of the biotransformation of phenols using pure cultures isolated in the laboratory, mixture of a consortium of the pooled cultures and enriched activated sludge biomass in pure substrates, mixed substrates and actual phenolic waste has been estimated under varying input concentrations. Experimental results from these studies have been subjected to analysis by mathematical models using Monod's and Haldane's equations to test the validity of these models in interpreting the data on inhibitory substances under steady as well as unsteady state operations of wastewater treatment plants. Comparative evaluation of the kinetic parameters reveals that Monod's model can be employed for the estimation of the kinetic constant μ only while Haldane's model has to be used for the calculation of μ max and K 1.

Resonance in elemental benzenoids

The phenomenon of resonance amongst a set of different classical chemical structures entails at an elementary level the enumeration of these resonance structures, corresponding (in benzenoid molecules) to perfect matchings of the underlying molecular (π-network) graph. This enumeration is analytically performed here for the finite-sized elemental benzenoid graphs corresponding to hexagonal coverings on suitable closed surfaces, namely the torus and the Klein bottle. Relevance is also indicated for a class of benzenoid structures corresponding to cylinders with polyhex coverings on the sides (but not the “ends”).

Regiospecific synthesis of polysubstituted furans and their application in organic synthesis

Abstract

Are kekulene, coronene, and corannulene tetraanion superaromatic? Theoretical examination using hardness indices

AbstractThe superaromaticity concept is examined and developed, taking into account what is known about aromaticity. Three new hardness indices indices are defined, and shown to be excellent parameters for characterizing superaromaticity. High superaromaticity indicates significant global annulenoid conjugation relative to local benzenoid conjugation in circularly annealed benzenoid molecules. Kekulene, coronene and corannulene tetraanion all are predicted to be superaromatic. The prediction for the corannulene tetraanion strongly supports the “annulene‐within‐an‐annulene” structure derived from NMR data.

Symmetry extensions of Euler's theorem for polyhedral, toroidal and benzenoid molecules

Euler's theorem is extended to the permutation symmetries of objects associated with vertices, edges and faces of general polyhedra imbedded in the sphere, torus or a surface of higher genus. An equivalent general theorem is derived for benzenoid hydrocarbons. Further relations are found for deltahedral and three-coordinate polyhedra imbedded in the various surfaces. Application of the new theorems to vibrational analysis and molecular electronic structure is sketched, with particular reference to bonding in possible toroidal analogues of the fullerenes.

On Spanning Trees in Catacondensed Molecules

Abstract Among all catacondensed benzenoid molecules the unbranched systems possess the largest number of spanning trees. Some other relations for the spanning-tree count of catacondensed benzenoids are also reported. The results obtained continue to hold for catacondensed molecules composed of rings the size of which differ from six. The propositions reported do, however, depend for their validity on there being rings of only one size in the catacondensed system under study.

Generalization of Cyvin's formula

A formula, recently discovered by Cyvin, relates the numbers of Kekule structures of unbranched catacondensed benzenoid molecules. It is shown that Cyvin's formula is a special case of a much more general result. The determinants occurring in Cyvin's formula as well as in our generalized version thereof can be interpreted in terms of correlation functions for die choices of pairs of double or pairs of single bonds in the Kekule structures of the respective conjugated system.