Kekulé structures of polyphenes

A group-theoretical treatment of theKekulé structures of polyphenes is performed. General expressions in terms of the number of benzene rings are given for the symmetrical structure ( _Kek ) and the characters (Kek) of the representation on the basis of theKekulé structures.

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Kekulé Strukturen von Polyphenen

Zusammenfassung Es wurde eine gruppentheoretische Behandlung derKekulé-Strukturen von Polyphenen durchgeführt. Es werden — in Abhängigkeit von der Anzahl der Benzoleinheiten — für die symmetrischen Strukturen ( _Kek ) und die Charaktere (Kek) der Darstellung auf Basis derKekulé-Strukturen allgemeine Ausdrücke angegeben.

     

Kekulé structures as graph generators

Kekulé valence-bond structures of catacondensed conjugated hydrocarbons with no, one, two and three branched cycles (which may be 4-, 6- and/or 8-membered) are used to generate highly regular vertex-transitive graphs through the application of an equivalence relation to the sextet of -electrons in theterminal rings of the hydrocarbon. The partitioning of a given set of Kekulé structures allows the study of certain novel combinatorial aspects of Kekulé counts. The graph- generating character reported here is closely related to the recent work of Randi, Woodworth, Kleiner and Hosoya.     

Heats of atomization of some conjugated molecules containing nitrogen or oxygen by a novel semiempirical method

Heats of atomization for a range of conjugated molecules containing nitrogen or oxygen are calculated by a semiempirical method that combines some features of both the MO and VB theories. The π ground state of each conjugated molecule is represented as a linear combination of Kekulé structures. Unlike in the VB theory, each Kekulé structure is a determinant containing bond orbitals. Here experimental heats of atomization are reproduced approximately as well as by the more sophisticated SCF –MO approach. The use of this method is, however, much simpler since it amounts to a single diagonalization of a matrix of the order equal to the number of Kekulé structures only.     

Note on the application of the reduced graph model in conjunction with search trees to the enumeration ofKekulé structures

The reduced graph model, when used in conjunction with the search trees method, provides a novel combinatorial procedure for the enumeration and generation ofKekulé structures. The procedure is suited for large benzenoid hydrocarbons consisting of cata- and thin peri-condensed parts.

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Zur Anwendung des Reduced Graph Model im Zusammenhang mit Search Trees zur Ermittlung der Anzahl möglicherKekulé-Strukturen

Zusammenfassung Das Modell erlaubt mit der im Titel genannten Kombination eine neuartige Methode zur Ermittlung und Generierung vonKekulé-Strukturen. Das Verfahren ist für große benzoide Kohlenwasserstoffe geeignet, die aus cata- und (dünnen) peri-kondensierten Teilstrukturen bestehen.

     

Examination of the statistical method in the conjugated circuit theory

Summary The recently proposed statistical method for the calculation of resonance energies in the conjugated circuit theory is tested on the example of polyacenes. It is found that for the estimation of the resonance energy with a 10% average error, the method requires a sample consisting of about 20 (randomly chosen) Kekulé structures. Further increase of the sample size gives no significant gain in the accuracy of the method.

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Überprüfung der statistischen Methode in der Konjugierten-Ring-Theorie

Zusammenfassung Die unlängst vorgeschlagene statistische Methode in der Konjugierten-Ring-Theorie (conjugated circuit theory) wird am Beispiel der Polyacene getestet. Es wurde festgestellt, daß für die Abschätzung der Resonanzenergien bei einem mittleren Fehler von 10% für diese Methode ungefähr 20 (willkürlich gewählte) Kekulé-Strukturen nötig sind. Weitere Erhöhung dieser Anzahl bringt keine signifikante Verbesserung der Genauigkeit der Methode.

     

The existence of Kekulé structures in helicenes and enumeration of concealed non-Kekuléan helicenes withh ≤ 13

Helicenes are the simply connected helicenic (geometrically non-planar) polyhexes. In this paper, we give some necessary and sufficient conditions for a helicene to have Kekulé structures. For a helicene withh 14, the necessary and sufficient conditions are simpler. By using the conditions, we give a construction method for all the concealed non-Kekuléan helicenes withh 13, and rigorously prove that there are exactly one, seventeen, and two hundred and sixty-nine concealed non-Kekuléan helicenes withh = 11, 12, 13, respectively.Project supported by NSFC.     

The number ofKekulé structures of hexagon-shaped benzenoids and members of other related classes

Some results from the enumeration ofKekulé structures are reviewed; they pertain to parallelogram-shaped, bent strips, chevrons and symmetrical hexagon-shaped benzenoids. The existing formulas are extended to the class of asymmetrical hexagons. Applications of the new formula reproduces a number of known results for three-tier and four-tier strips. In the latter case also some new formulas are achieved.

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Die Anzahl derKekulé-Strukturen von Benzenoiden mit sechseckigem Umri und Mitgliedern anderer verwandter Klassen

Zusammenfassung Es wird ein Überblick über die Berechnung der Anzahl möglicherKekulé-Strukturen von benzenoiden Verbindungen unterschiedlicher Formenklassen gegeben. Dabei werden die existierenden Formeln für die Klasse asymmetrischer Sechseckformen ausgeweitet und die neue Formel auch an bekannten Ergebnissen erprobt.

     

Enumeration ofKekulé structures:Étagères and related benzenoid classes

Formulas are developed for the number ofKekulé structures of some benzenoid classes, which are interpreted in terms of annelations of a multiple linear chain (parallelogram). One-sided and two-sided annelations are considered, the latter category leading to the definition of a benzenoid class referred to asétagères.

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Ermittlung der Anzahl vonKekulé-Strukturen: Étagère und verwandte benzenoide Klassen

Zusammenfassung Es werden Formeln für die Ermittlung der Anzahl möglicherKekulé-Strukturen für einige benzenoiden Klassen angegeben, wobei diese durch annelierte, mehrfache, lineare Ketten (Parallelogramme) charakterisiert werden. Dabei wird einseitige und zweiseitige Annelierung berücksichtigt, für die letztgenannte Gruppe wird der Ausdruck Étagère geprägt.

     

Characterization of distribution of pi-electrons amongst benzenoid rings for Randić’s “algebraic” Kekulé structures

For benzenoid hydrocarbons the distribution of pi-electrons amongst rings is characterized in the context of Randis mode of assignment attending to the different Kekulé structures. In particular the mean and mean deviation from the mean are considered, and the benzenoids which achieve maximum deviation are identified.     

Topological hamiltonian spectra of polycyclic benzenoid hydrocarbons

A general theory which points out the relations between Hückel-electron energy, the number of Kekulé structures and the HOMO-LUMO separation is presented. Some normalized topological invariants are derived from the concept of the spectral density function. A reasonably simple (three parameters) model spectral density function leads to universal relations between topological invariants that, although valid for any alternant molecule, were tested numerically for polycyclic benzenoid hydrocarbons. Some general conclusions concerning a distribution of the adjacency matrix eigenvalues are drawn.     

On the Distribution of π-Electrons into Rings of Conjugated Hydrocarbons Containing a Linear Polyacene Fragment

Summary. A method for assessing the -electron contents (EC) of rings of benzenoid hydrocarbons, based on the examination of their Kekulé structures, was recently put forward by Balaban and Randi. We now show that all hexagons belonging to a linear polyacene fragment of a conjugated hydrocarbon (not necessarily benzenoid) have mutually equal EC-values.     

On the Distribution of π-Electrons into Rings of Conjugated Hydrocarbons Containing a Linear Polyacene Fragment

A method for assessing the -electron contents (EC) of rings of benzenoid hydrocarbons, based on the examination of their Kekulé structures, was recently put forward by Balaban and Randi. We now show that all hexagons belonging to a linear polyacene fragment of a conjugated hydrocarbon (not necessarily benzenoid) have mutually equal EC-values.     

Cyclic Conjugation in Benzo-Annelated Perylenes. How Empty is the “Empty” Ring?

Summary. Cyclic conjugation in benzo-annelated perylenes is studied by means of the energy-effects of their six-membered rings. Classical theoretical approaches (based on Kekulé structures, Clar formulae, or conjugated circuits) predict that the central ring in benzo-annelated perylenes is empty and thus negligibly contributes to cyclic conjugation. Our calculations show that this is true only to a limited degree. In particular, rings angularly annelated relative to the central ring significantly increase the extent of its cyclic conjugation.     

Analysis of the π-electronic structure of infinitely large networks

Summary Applying the Coulson and Longuet-Higgins integral method to polycyclic aromatics, the analytical solutions of bond orders, -electronic energy and benzene character for the infinitely large cyclic polyacene and polyphenanthrene with various modes of bond alternation are obtained in the HMO scheme. Most of the results are explicitly and newly expressed in terms of three kinds of the elliptic integrals. Judging from the magnitudes of bond orders and benzene character the most probable modes of the bond alternation for these two networks are discussed with their Kekulé structures. It was shown that if bond alternation is properly taken into consideration, HMO calculation can fairly well reproduce the results obtained by more sophisticated methods.     

Wavefunction comparisons for the valence-bond model for conjugated π-networks

Approximate ground-state wavefunctions for valence-bond (or Heisenberg) models are obtained both within Néel-state-based and within Kekulé-state-based resonance-theoretic approaches. Comparisons are made between these and other general approaches, with particular emphasis on organic -network systems. Attention is drawn to the manner in which the quality of the different approximation schemes changes with variations in structural characteristics of the system. It is suggested that resonance-theoretic ideas are most appropriate for (aromatic benzenoid) systems with low coordination number, whereas Néel-state based ideas are most appropriate for (3-dimensional) structures with higher coordination number (and little frustration).Work supported by The Robert A. Welch Foundation of Houston, Texas     

Theorems for carbon cages

New theorems are established for cages (or polyhedra) with trivalent vertices. One theorem says that all such cages have at least three Kekulé structures (or perfect matchings). Thence, resonance generally appears as a possibility. Another theorem says that for every even vertex count >70 there is at least one cage of a preferable subclass, while for vertex count <70 the sole preferable cage is that of the truncated icosahedron. Thence, the unique role of the buckminsterfullerene structure for C₆₀ is mathematically indicated.[/p]Work supported by the Welch Foundation of Houston, Texas.     

Enumeration of Kekulé structures for some coronoid hydrocarbons: “Waffles”

Summary Primitive coronoids of hexagonal symmetry (D _6h orC _6h ) are referred to as waffles. Some theorems about unbranched catacondensed benzenoids are presented and used to derive a general combinatorial formula for the number of Kekulé structures (K) for waffles. The symmetry-adapted method of fragmentation is employed. SeveralK formulas for special classes of waffles are also reported.

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Die Anzahl von Kekulé-Strukturen für einige aromatische Coronoid-Kohlenwasserstoffe: Waffel-Strukturen

Zusammenfassung Einfache Coronoide von hexagonaler Symmetrie (D _6h oderC ₆ h) werden als Waffel bezeichnet. Einige Theoreme bezüglich unverzweigter catakondensierter Benzenoide werden angegeben und zur Ableitung einer generellen Formel für die Anzahl von Kekulé-Strukturen (K) für Waffel benutzt. Es wird die symmetrie-adaptierte Methode zur Fragmentierung angewendet. Außerdem werden einigeK-Formeln für spezielle Klassen von Waffeln angegeben.

     

A new theoretical approach to the empirical resonance energies of the aromatic hydrocarbons

In the framework of the Hückel MO approximation, the differences in total binding energy between a given molecule and the corresponding distorted Kekulé-type structure are calculated for a variety of benzenoid hydrocarbons. The total binding energy is assumed to be given by the sum of the -electron and -electron binding energies. It is shown that there is a good linear relationship between the calculated differences in total binding energy and the -electron delocalization energies (DE) as obtained by using the simple Hückel MO method. This provides a physical basis for the use of the -electron DE as a theoretical index to the empirical resonance energy (RE). Further, by examining the changes in -electron binding energy between a given molecule and the corresponding distorted Kekulé-type structure, it is concluded that in benzenoid hydrocarbons the main contributor to the RE is not the -electron DE but the compressional energy of bonds.     

Cyclic Conjugation in Benzo-Annelated Perylenes. How Empty is the “Empty” Ring?

Cyclic conjugation in benzo-annelated perylenes is studied by means of the energy-effects of their six-membered rings. Classical theoretical approaches (based on Kekulé structures, Clar formulae, or conjugated circuits) predict that the central ring in benzo-annelated perylenes is empty and thus negligibly contributes to cyclic conjugation. Our calculations show that this is true only to a limited degree. In particular, rings angularly annelated relative to the central ring significantly increase the extent of its cyclic conjugation.     

Partitioning of <Emphasis Type="Italic">π</Emphasis> -electrons in rings of polycyclic conjugated hydrocarbons: Part 6. Comparison with other methods for estimating the local aromaticity of rings in polycyclic benzenoids

By adopting the convention that shared double bonds in polycyclic conjugated hydrocarbons contribute with one -electron and unshared ones with two -electrons, a partition of -electrons in each ring (-electron content, EC) can be obtained by averaging over all Kekulé structures, which are assumed to have equal weights. This affords a simple measure of local aromaticity that is comparable with other such local aromaticity indices in polycyclic benzenoids.