Constant-isomer benzenoid series and their topological characteristics

Summary New constant-isomer series are presented. Two classes of constant-isomer series based on the topological correspondence of their member benzenoids are identified. The isomer numbers for the constant-isomer series alternate between singlet and doublet occurrence. Constant-isomer series with the same isomer number possess a pairwise one-to-one topological correspondence between their benzenoid membership. A correspondence also exists between threefold monoradical and diradical benzenoids belonging to these constant-isomer series. These topological relationships represent a new paradigm that we ascribe as an edge effect of our periodic table for benzenoids.     

Bond length interrelations in benzenoid hydrocarbons and their heteroatom analogues

The geometries of some benzenoid hydrocarbons and their analogues with XY (XY = BB, BC, BN, CN, and NN) bonds have been determined at the B3LYP/6-311++G^∗∗ level of theory. It is shown that the lengths of peripheral XY bonds are strictly correlated with the lengths of their CC counterparts in native hydrocarbons. No correlation of this type is observed for the XY bonds located inside the rings.     

Total resonant sextet benzenoid hydrocarbon isomers and their molecular orbital and thermodynamic characteristics

The more stable strain-free total resonant sextet isomers of benzenoid hydrocarbons are predicted to be probable combustion pollutants. These benzenoid hydrocarbons have been enumerated, and their topological characteristics examined. Total resonant sextet benzenoid isomers make up less than 1% of all the benzenoid hydrocarbons having formulas with N_c divisible by 6. There is a thermodynamic preference for the larger and more highly condensed polycyclic aromatic hydrocarbon molecule. The Hückel molecular orbital (HMO) and thermodynamic parameters of these total resonant sextet benzenoid hydrocarbons have been computed and tabulated.     

On isospectral benzenoid graphs

The recently Proposed procedure [5] for the construction of isospectral benzenoid graphs has been examined in detail. Necessary and sufficient conditions for the construction of isospectral benzenoid graphs with isomorphicH-graphs are formulated. The inapplicability of the Procedure for the construction of isospectral benzenoid graphs with an even number of vertices has been proven.     

A formula periodic table for benzenoid hydrocarbons and the aufbau and excised internal structure concepts in benzenoid enumerations

The genesis of the aufbau and excised internal structure concepts is traced. These concepts were pivotal to the first published enumerations of the strictly peri-condensed, non-Kekuléan, and essentially strain-free total resonant sextet benzenoid groups. Aperiodic table set is defined as a partially ordered set forming a two-dimensional array which complies with thetriad principle where a central element has a metric property that is the arithmetic mean of two adjacent surrounding member elements.     

Enumeration and classification of benzenoid hydrocarbons

The enumeration of benzenoids is studied with the aid of different modifications of a computer program. Known values for the number of cata-condensed and of all benzenoids with h (number of hexagons) up to eight are reproduced. These benzenoids are classified into (a) unbranched of different symmetries and branched cata-condensed, and (b) normal, essentially disconnected and non-Kekuléan peri-condensed. Within the normal benzenoids a more detailed classification is performed: basic benzenoids, fused, or annelated. For h = 9 an almost complete classification along the same lines was achieved.     

Resonance in catacondensed benzenoid hydrocarbons

We consider resonance in cata-condensed benzenoids having six and seven fused benzene rings. The resonance relationship between the Kekule valence structures of the molecules is represented by the resonance graphs in which the vertices represent the Kekule valence structures, and the edges, the presence of the quantum chemical resonance integral involving permutation of three CC double bonds (within a single six-membered ring). The construction of the resonance graphs for large benzenoids is outlined and the properties of the derived resonance graphs are discussed. It is shown that the leading eigenvalue of the resonance graphs correlates with the resonance energy of benzenoids. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 585–600, 1997     

Normal components of benzenoid systems

Summary Consider a benzenoid system with fixed bonds and the subgraph obtained by deleting fixed double bonds together with their end vertices and fixed single bonds without their end vertices. It has often been observed for particular benzenoid systems, and conjectured (or stated) that, in general, such a subgraph has at least two components, and that each component is also a benzenoid system and is normal. But there are no rigorous proofs for that. The aim of this paper is to present mathematical proofs of those two facts. It is also shown that if a benzenoid system has a single hexagon as one of its normal components then it has at least three normal components.     

Topological properties of benzenoid systems

The sextet polynomial of non-branched cata-condensed benzenoid molecules is proved to be related (Eq. (3)) to the characteristic polynomial of a tree.     

On the ordering of benzenoid chains and cyclo-polyphenacenes with respect to their numbers of Clar aromatic sextets

Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the concept of sextet polynomial introduced by Hosoya and Yamaguchi [Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986)], we define a new ordering of benzenoid systems. For two isomeric benzenoid systems B₁ and B₂, we say B₁>B₂ if each coefficient of sextet polynomial of B₁ is no less than the corresponding coefficient of sextet polynomial of B₂. In this paper, we consider the ordering of the benzenoid chains. The maximal and second maximal benzenoid chains as well as the minimal, the second minimal up to the fourth minimal benzenoid chains are determined. Furthermore, under this ordering, we determine the maximal and second maximal cyclo-polyphenacenes as well as the minimal, the second minimal, and up to the seventh minimal cyclo-polyphenacenes.     

PI indices of pericondensed benzenoid graphs

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., , where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.     

On the number of benzenoid hydrocarbons

We present a new algorithm which allows a radical increase in the computer enumeration of benzenoids b(h) with h hexagons. We obtain b(h) up to h = 35. We prove that b(h) approximately const.kappa(h), prove the rigorous bounds 4.789 < or = kappa < or = 5.905, and estimate that kappa = 5.16193016(8). Finally, we provide strong numerical evidence that the generating function summation operator b(h)z(h) approximately A(z) log(1 - kappa z), estimate A(1/kappa) and predict the subleading asymptotic behavior. We also provide compelling arguments that the mean-square radius of gyration (h) of benzenoids of size h grows as h(2 nu), with nu = 0.64115(5).     

k-resonant benzenoid systems and k-cycle resonant graphs.

A benzenoid system (or hexagonal system) H is said to be k-resonant if, for 1 < or = t < or = k, any t disjoint hexagons of H are mutually resonant; that is, there is a Kekule structure (or perfect matching) K of H such that each of the k hexagons is an K-alternating hexagon. A connected graph G is said to be k-cycle resonant if, for 1 < or = t < or = k, any t disjoint cycles in G are mutually resonant. The concept of k-resonant benzenoid systems is closely related to Clar's aromatic sextet theory, and the concept of k-cycle resonant graphs is a natural generalization of k-resonant benzenoid systems. Some necessary and sufficient conditions for a benzenoid system (respectively a graph) to be k-resonant (respectively k-cycle resonant) have been established. In this paper, we will give a survey on investigations of k-resonant benzenoid systems and k-cycle resonant graphs.     

Comparability graphs and electronic spectra of condensed benzenoid hydrocarbons

A systematic topological approach to the search for regularities in molecular properties has been proposed on the basis of the so-called comparability graphs of isomeric classes of molecules. It is shown that the ordering of the isomeric benzenoid hydrocarbons in the comparability graphs coincides with that of the longest wavelenght in the p band of their electronic spectra.     

Conjugated circuits and resonance energies of benzenoid hydrocarbons

A concept of conjugated circuits contained in Kekulé forms of benzenoid hydrocarbons is considered. Circuits of size (4n + 2) in which CC single and double bonds formally alternate are enumerated and serve as the basis for the characterization of a given system. Resonance energies of benzenoid systems are given by contributions of conjugated circuits of different size. The scheme permits the expression of resonance energies of different molecules in terms of selected reference compounds, such as linear acenes. The approach is illustrated on a dozen benzenoid hydrocarbons and the calculated resonance energies are on average within 0.05 eV of the values obtained by SCF MO calculations.     

A high-spin and durable polyradical: poly(4-diphenylaminium-1,2-phenylenevinylene)

A purely organic, high-spin, and durable polyradical molecule was synthesized: It is based on the non-Kekulé- and non-disjoint design of a pi-conjugated poly(1,2-phenylenevinylene) backbone pendantly 4-substituted with multiple robust arylaminium radicals. 4-N,N-Bis(4-methoxy- and -tert-butylphenyl)amino-2-bromostyrene 5 were synthesized and polymerized with a palladium-phosphine catalyst to afford the head-to-tail-linked polyradical precursors (1). Oxidation of 1 with the nitrosonium ion solubilized with a crown ether gave the aminium polyradicals (1(+)()) which were durable (half-life > 1 month) at room temperature in air. A high-spin ground state with an average S = (4.5)/2 for 1a(+) was proved even at room temperature by magnetic susceptibility, magnetization, ESR, and NMR measurements.     

Additivity of resonance energy in benzenoid hydrocarbons

A simple method is described for the calculation of resonance energies (RE ) of linear acenes based on their number of Kekulé structures. The values obtained for the first five linear acenes are used to graph–theoretically calculate RES of a wide variety of benzenoid hydrocarbons. Excellent linear relationships are found between RES and each of A-II, graph-theoretical (GT ), Hess–Schaad (HS ), and Dewar resonance energies (SCF ). These relations apply to 42 hydrocarbons and lead to the following equations: A-II = 0.084RE + 0.080 (0.9999); GT = 0.072RE + 0. 135 (0.9832); HS = 0. 106RE + 0. 169 (0.9889); and SCF = 0.316RE + 0. 166 (0.9899). Correlation coefficients are shown in parentheses. A linear relation also exists between RES and the square roots of the wavelengths of the UV spectra of hydrocarbons of the linear acenes and phene series. Least-squares analysis of the data leads to the following equation: RE = 0.412(λ)^½ ?15.479, with a correlation coefficient equal to 0.9903, in which λ is the wavelength of the β band of the UV spectra of these hydrocarbons. The method predicts no resonance energies for both open chain polyenes and the radialenes.