Analytical expressions for the count of LM-conjugated circuits of benzenoid hydrocarbons

A recursive method for enumeration of linearly independent and minimal conjugated circuits of benzenoid hydrocarbons had previously been given which is valid for several classes of benzenoid hydrocarbons. In the present article, the properties and constructions of unique minimal conjugated circuits and pairs of minimal conjugated circuits of a ring s in a benzenoid hydrocarbon B are investigated. An analytical expression for the count of LM-conjugated circuits of B is given which is based on the counts of Kekulé structures of selected subgraphs of B. By using the method, the LMC expression of any benzenoid hydrocarbon can be obtained. © 1996 John Wiley & Sons, Inc.     

A revisit to molecular path codes: Ordering,comparability, and similarity of benzenoid,nonbenzenoid, and monoradical hydrocarbons

Molecular path codes are computed for 17 hydrocarbons, including benzenoid alternant, nonbenzenoid alternant, fully aromatic nonalternant, antiaromatic nonalternant, and monoradical hydrocarbons. Comparability conditions of Randi? and Wilkins [9–18] based on the theorems of Muirhead and Karamata are imposed on the partial sums of the codes. The distance/similarity matrix is constructed using the individual paths as projections in Euclidean space. Both resonance energies and total pi energies correlate with partial ordering predicted from both comparability and similarity schemes. The normalized codes succeed in discriminating combinatorial properties of the various topologies studied such as conjugated circuits.     

Benzenoid rings resonance energies and local aromaticity of benzenoid hydrocarbons

In this article, we consider partitioning of the analytical expression for resonance energy (RE) in smaller benzenoid hydrocarbons, to individual benzenoid rings of polycyclic molecules. The analytical expression for molecular RE, available since 1976, is given by the count of all linearly independent conjugated circuit in all Kekulé structures in a molecule. Analytical expression for local ring RE (RRE) is given by counting all linearly independent conjugated circuits involving single benzenoid ring in all Kekulé structures, which when added, gives the molecular RE. If for benzene ring the RRE is taken to be 1.000, rings in polycyclic benzenoid hydrocarbons have their ring RRE, which give the degree of their local aromaticity, smaller than 1.000. The difference to 1.000 is a measure of the similarity of a ring to benzene in this one-dimensional (1-D) representation of local aromaticities of benzenoid hydrocarbons. The plot of RRE against the distance of the same ring from benzene in the Local Aromaticity Map, in which benzenoid rings are characterized ring bond orders and average variations of adjacent CC bonds, shows linear correlation (with r = 0.91), reducing the local aromaticity in benzenoid hydrocarbons to 1-D molecular property. © 2018 Wiley Periodicals, Inc.     

Recursive Formulae for Enumeration of LM-Conjugated Circuits in Structurally Related Benzenoid Hydrocarbons

The linearly independent and minimal conjugated (LM-conjugated) circuits of benzenoid hydrocarbons play the central role in the conjugated circuit model. For a general case, the enumeration of LM-conjugated circuits may be tedious as it requires construction of all Kekule structures. In our previous work, a recursive method for enumeration of LM-conjugated circuits of benzenoid hydrocarbons was established. In this paper, we further extend the recursive formulae for enumerations of LM-conjugated circuits for both catacondensed benzenoid hydrocarbons and some families of structurally related pericondensed benzenoid hydrocarbons.     

Resonance and aromaticity: an ab initio valence bond approach

Resonance energy is one of the criteria to measure aromaticity. The effect of the use of different orbital models is investigated in the calculated resonance energies of cyclic conjugated hydrocarbons within the framework of the ab initio Valence Bond Self-Consistent Field (VBSCF) method. The VB wave function for each system was constructed using a linear combination of the VB structures (spin functions), which closely resemble the Kekulé valence structures, and two types of orbitals, that is, strictly atomic (local) and delocalized atomic (delocal) p-orbitals, were used to describe the π-system. It is found that the Pauling-Wheland's resonance energy with nonorthogonal structures decreases, while the same with orthogonalized structures and the total mean resonance energy (the sum of the weighted off-diagonal contributions in the Hamiltonian matrix of orthogonalized structures) increase when delocal orbitals are used as compared to local p-orbitals. Analysis of the interactions between the different structures of a system shows that the resonance in the 6π electrons conjugated circuits have the largest contributions to the resonance energy. The VBSCF calculations also show that the extra stability of phenanthrene, a kinked benzenoid, as compared to its linear counterpart, anthracene, is a consequence of the resonance in the π-system rather than the H-H interaction in the bay region as suggested previously. Finally, the empirical parameters for the resonance interactions between different 4n+2 or 4n π electrons conjugated circuits, used in Randi?'s conjugated circuits theory or Herdon's semi-emprical VB approach, are quantified. These parameters have to be scaled by the structure coefficients (weights) of the contributing structures.     

The conjugated-circuit model: The optimum parameters for benzenoid hydrocarbons

A search for the optimum set of parameters for the conjugated-circuit computations on benzenoid hydrocarbons in reported. The SCFπ-MO resonance energies (REs) of Dewar and de Llano were used as standards for the determination ofR _n (n= l,2,3) parameters, which correspond to 4n + 2 conjugated circuits. The following set of parameters:R ₁ = 0.827 eV.R ₂ = 0.317 andR ₃ = 0.111 eV produced the best agreement between the REs calculated by the conjugated-circuit model and the REs calculated using the SCF π-MO model.     

Elektronenstruktur und Stabilität kondensierter Kohlenwasserstoffe

The theory of significant electron structures is applied to benzenoid, nonbenzenoid and semibenzenoid condensed hydrocarbons. Stabilization energies are obtained in agreement with SCF MO resonance energies. The weights of the structures enabled to calculate benzene characters and other character indices.Clar's postulate of localized benzene-like regions is justified. The structural and energetic properties of semibenzenoid hydrocarbons are derived in a systematic manner.     

Resonance energy of very large benzenoid hydrocarbons

The resonance energy of conjugated benzenoid systems is expressed as contributions arising from independent conjugated circuits. The scheme has been applied to numerous very large conjugated systems. In many cases, it was possible to find regularities in the increments for the resonance energy within a family of benzenoid systems as the number of benzene rings is increased.     

Fibonacci numbers in the topological theory of benzenoid hydrocarbons and related graphs

Fibonacci numbers are studied with respect to the topological theory of benzenoid hydrocarbons. These numbers are identified as the number of Kekulé structures of nonbranched all-benzenoid hydrocarbons, the number of matchings of paths, the number of independent sets of vertices of paths, the number of nonattacking rooks of certain rook boards, as well as the number of Clar structures of certain benzenoid hydrocarbons. Fibonacci numbers were also identified as the number of conjugated circuits of certain benzenoid hydrocarbons and thus they were also related to the structure-resonance model. Maximal independent sets of caterpillar trees are also shown to be Fibonacci numbers.     

Applying the conjugated circuits method to Clar structures of [n]phenylenes for determining resonance energies

We consider the aromaticity of biphenylene and structurally related linear or angular [n]phenylenes for which the direct application of the model of conjugated circuits does not offer valid expressions for resonance energy and aromaticity. We located the cause of this problem as being due to Kekulé valence structures in which neighboring benzenoid rings are connected by two CC double bonds. By restricting the selection of Kekulé valence structures to those that contribute to Clar structures of such systems, we were able to show that linear and angular [n]phenylenes have approximately similar resonance energies, with angular [n]phenylenes being slightly more stable due to second order contributions arising from disjoint conjugated circuits. Expressions for resonance energies of [n]phenylenes up to n = 8 are listed and recursion expressions for higher n values are outlined.     

π-Electron currents in fixed π-sextet aromatic benzenoids

In view of different patterns of π-electron density currents in benzenoid aromatic compounds it is of interest to investigate the pattern of ring currents in various classes of compounds. Recently such a study using a graph theoretical approach to calculating CC bond currents was reported for fully benzenoid hydrocarbons, that is, benzenoid hydrocarbons which have either π-sextets rings or “empty” rings in the terminology of Clar. In this contribution we consider π-electron currents in benzenoid hydrocarbons which have π-electron sextets and C=C bonds fully fixed. Our approach assumes that currents arise from contributions of individual conjugated circuits within the set of Kekulé valence structures of these molecules.     

The use of 90°-1,3-butadiene as a reference structure for the evaluation of stabilization energies for benzene and other conjugated cyclic hydrocarbons

Reactions are described that employ 90°-1,3-butadiene as a reference structure for the evaluation of the stabilization energyof the benzenoid and other cyclic conjugated hydrocarbons. The unique benefits of this rotamer of butadiene as a reference molecule within the homodesmotic conceptual framework are discussed. Experimental stabilization energies are presented for a number of cyclic hydrocarbons.     

Resonance energies of large conjugated hydrocarbons by a statistical method

We developed a theoretical method for studying the aromatic stability of large molecules, molecules having a dozen and more fused benzene rings. Such molecules have so far often been outside the domain of theoretical studies. Combining the statistical approach and a particular graph theoretical analysis, it is possible to derive the expressions for molecular resonance energy for molecules of any size. The basis of the method is enumeration of conjugated circuits in random Kekulé valence structures. The method has been applied to evaluation of the resonance energies of conjugated hydrocarbons having about a dozen fused benzene rings. The approach consists of (1) construction of random Kekulé valence structures, (2) enumeration of conjugated circuits within the generated random valence structures, and (3) application of standard statistical analysis to a sufficiently large sample of structures. The construction of random valence forms is nontrivial, and some problems in generating random structures are discussed. The random Kekulé valence structures allow one not only to obtain the expression for molecular resonance energies (RE ) and numerical estimates for RE , but also they provide the basis for discussion of local molecular features, such as ring characterization and Pauling bond orders.     

π‐Electron currents in larger fully aromatic benzenoids

Recently, we have reported on calculation of π‐electron ring currents in several smaller fully benzenoid hydrocarbons having up to eight fused benzene rings and five Clar π‐aromatic sextets. In contrast to early HMO ring current calculations and more recent ab initio calculations of π‐electron density, our current calculations are based on a graph theoretical model in which contributions to ring currents comes from currents associated with individual conjugated circuits. In this contribution, we consider several larger fully benzenoid hydrocarbons having from 9 to 13 fused rings and from six or seven π‐aromatic sextets. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Efficient method for calculating the resonance energy expression of benzenoid hydrocarbons based on the eunumeration of conjugated circuits

To reduce the calculating time for the summations over linearly independent and minimal conjugated circuits of benzenoid hydrocarbons (BHs), an approximate method is proposed that counts only the numbers of the first four classes of conjugated circuits R1, R2, R3, and R4, respectively. By representation of BHs as custom-made "ring-block chains" and use of the techniques of Database and visual computing, an application software is realized that is much faster and more powerful than the old one based on an enumeration technique.     

计算共振能的一个新的经验方法

本文提出了一个很简单的计算共振能(RE)的方法。对苯系芳烃,可用下式计算: RE(cV)=0.215(0.89N_0+0.75N_1+0.36N_2)式中,N_0是顶点度为3的碳原子数,N_1是顶点度为2但被两个顶点度为2的原子所连结的碳原子数,N_2是顶点度为2但被一个顶点度为2和另一个顶点度为3的原子所连结的碳原子数。此或略加修正也可用于计算含四元环共轭烃、半苯型烃和奥系化合物等共轭烃的共振能。     

Determining the number of resonance structures in concealed non-Kekuléan benzenoid hydrocarbons

The number of resonance structures (SC) for previously published concealed non-Kekuléan benzenoid hydrocarbons is determined. Using a simple computer program, analytical expressions for determining SC for various classes of non-Kekuléan (free-radical) benzenoid hydrocarbons are derived, and some properties of concealed non-Kekuléan benzenoid hydrocarbons are studied.