Peri-condensed Benzenoid Research Articles

Minimum Number of Kekulé Structures of Certain Benzenoid Graphs and -Polyacenic Nanotubes

In a molecular graph, a Kekulé structure and perfect matching resembles the same. Thus, the number of Kekulé structure(chemically) is equal to the number of perfect matching (mathematically) required to cover the edge set or covalent bonds of a molecular graph entirely. The number of possible Kekulé structures in a molecule (benzenoids) is used to describe the extent of its aromatic property and plays an significant role in the analysis of resonance energy and stability of certain chemical compounds. In this article, the minimum number of Kekulé structures (K) that covers all the edges of a class of peri-condensed benzenoid graphs possessing three rows of hexagon of various length and -polyacenic nanotubes were discussed and summarized.

SOMBOR INDEX OF LINE AND TOTAL GRAPHS AND PERICONDENSED BENZENOID HYDROCARBONS

Gutman proposed a new alternative interpretation of vertex-degree-based topological index, called Sombor index. It is defined via the term . In this paper, we determine the explicit expressions of Sombor index for line and total graphs and several pericondensed benzenoid hydrocarbons.

Zhang-Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John-Sachs Theorem.

Multiple zigzag chains of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains . The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains , i.e., derivatives of with a single attached polyacene chain of length k. All presented formulas are accompanied by formal proofs. The developed theoretical machinery is applied for predicting aromaticity distribution patterns in large and infinite multiple zigzag chains and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains .

Zhang–Zhang Polynomials of Ribbons

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids Rbn1,n2,m1,m2, usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of Rbn1,n2,m1,m2: the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes Ok,m,n and oblate rectangles Orm,n.

Sd-Index of Another Case of Pericondensed Benzenoid Graphs G(m, n, k)

Sd-Index of Another Case of Pericondensed Benzenoid Graphs G(m, n, k)

Growth reactions of polycyclic aromatic hydrocarbons during the supercritical pyrolysis of n-decane, as determined from doping experiments with 1-methylphenanthrene and fluorene

To better understand mechanisms for the growth of polycyclic aromatic hydrocarbons (PAH) under conditions relevant to the pre-combustion environments of future-high-speed-aircraft fuels, we have performed supercritical pyrolysis experiments with the model fuel n-decane, to which 1-methylphenanthrene and fluorene, two major three-ring PAH products of n-decane pyrolysis, have each been added as a dopant, at concentrations of 0.73mg/g n-decane and 2.3mg/g n-decane, respectively. The experiments are conducted in an isothermal, silica-lined stainless-steel flow reactor at 570°C, 94.6atm, and 133s, conditions of rapid PAH growth. Analyses of the PAH products by high-pressure liquid chromatography with diode-array ultraviolet–visible absorption and mass-spectrometric detection have exposed the important roles of four resonance-stabilized aromatic radicals—9-fluorenyl, 1-phenanthrylmethyl, 9-phenanthrylmethyl, and benz[de]anthracenyl—and revealed the channels through which these radicals react with the principal aliphatic (methyl, ethylene, propene, and 1-butene) and aromatic (benzyl, the two naphthylmethyls, and phenalenyl) reactants in the supercritical n-decane pyrolysis environment to effect PAH growth. 9-Fluorenyl's reactions with these principal reactants—in some cases involving “ring expansion” of fluorene's five-membered ring—result in the formation of phenanthrene, 9-methylphenanthrene, 9-ethylphenanthrene, fluoranthene, and eleven fluoranthene benzologues of five to seven rings. 1-phenanthrylmethyl, stemming from the 1-methylphenanthrene dopant, and 9-phenanthrylmethyl, stemming from the fluorene dopant's production of 9-methylphenanthrene and 9-ethylphenanthrene, react with n-decane's principal reactant species to form chiefly four- to six-ring cata-condensed benzenoid PAH. Their reaction with ethylene leads to formation of the four-ring phenalenyl-type radical benz[de]anthracenyl, which reacts with the principal alkenes and arylmethyl radicals in the system to produce several five- to seven-ring peri-condensed benzenoid PAH. Since the three-ring structures of 1-methylphenanthrene and fluorene are embedded in the structures of many of n-decane's higher-ring-number PAH products, we would expect the PAH-growth pathways outlined here to have widespread applicability to larger PAH in the reaction environment as well.

Variants of the Szeged index in certain chemical nanosheets

A topological index is a numeric quantity of a molecule that is mathematically derived in an unambiguous way from the structural graph of a molecule. In this paper, we compute the edge Szeged index, edge–vertex Szeged index, and total Szeged index of nanosheets such as type I and type II C4C8(S) nanosheets, peri-condensed benzenoid nanosheet, and H-naphtalenic nanosheet. In addition, we correct the computational errors occurring on the values of the Wiener and vertex Szeged indices of these nanosheets in the literature.

Analytical expressions for Wiener indices of n-circumscribed peri-condensed benzenoid graphs

We have employed cut axes and vertex partitioning (I-partition) methods to obtain exact analytical expressions for Wiener indices of n-circumscribed peri-condensed benezenoid graphs as a function of n, the order of circumscribing. Such expressions have not been obtained before for nth order circumscribing of peri-condensed benzenoid graphs with the exception of n-Circum-coronene which becomes a honeycomb lattice as n goes to infinity. The Wiener indices of these n-circum peri-condensed benzenoids are found to be polynomials of degree 5. Such expressions can be critical in topological characterization of peri-condensed benzenoid sheets of any order of circumscribing. A number of examples are provided ranging from Circum-polyacenes, Circum-ovalenes, Circum-pyrenes to more complex Circum-peri-condensed graphs of wide ranging complexities.

A New Technique to Compute Padmakar-Ivan Index and Szeged Index of Pericondensed Benzenoid Graphs

A New Technique to Compute Padmakar-Ivan Index and Szeged Index of Pericondensed Benzenoid Graphs

Aromaticity in pericondensed cyclopenta-fused polycyclic aromatic hydrocarbons determined by density functional theory nucleus-independent chemical shifts and the Y-rule — Implications in oil asphaltene stability

The characterization of the stability of the fused aromatic region (FAR) in oil asphaltenes in terms of kinetic and thermodynamic stability is primary. Such an understanding is important if we are to get the optimal use from the heavy fraction of any crude oil. The FAR region is composed of pericondensed cyclopenta-fused polycyclic aromatic hydrocarbon compounds (CPPAHs) with N, S, and O heteroatoms. The Clar model, which states that the most important representation of a PAH is one having the maximum number of disjoint π-sextets, depicted by inscribed circles, and a minimum number of fixed double bonds, captures the essence of the kinetic and thermodynamic stability arguments. This model is readily employed for complex aromatics of the sort to be considered for asphaltenes. In the present research we prove that the aromaticity of CPPAHs can be assessed by using the qualitative easy-to-apply Y-rule. In the literature, it is proven that the Y-rule is applicable to elucidate the aromaticity of benzenoid PAHs and it has been validated for pericondensed benzenoid PAHs but not for pericondensed CPPAHs. Here, we verify that it is applicable for CPPAHs. The applicability of the Y-rule has been theoretically proven by comparing the π-electronic distribution obtained with it with the one obtained from nucleus-independent chemical shift (NICS) calculations at the density functional theory (DFT) level. The importance of doing this is that due to the polydispersity in the composition of the oil asphaltenes, and to understand their aromatic core structure, it is necessary to be able to asses the aromaticity of many cyclopenta-fused PAHs (possibly more than 500), of different sizes (up to 15 rings between hexagons and pentagons), and different spatial rearrangements in a quick but realistic and effective way. To try to do this with NICS will be very time consuming and computationally expensive, especially in the case of big systems.

Stability order of isomeric benzenoid hydrocarbons and Kekulé structure count

The commonly accepted opinion that the thermodynamic stability of isomeric benzenoid hydrocarbons (assessed by their total ?-electron energy and various resonance energies) increases with increasing number of Kekul? structures is shown to be violated in numerous cases. The smallest examples of such anomalous behavior are two hexacyclic pericondensed benzenoids of formula C24H14 and several pairs of heptacyclic catacondensed benzenoids of formula C30H18.

A New Yardstick for Benzenoid Polycyclic Aromatic Hydrocarbons

The newly introduced signature of benzenoids (a sequence of six real numbers Si with i = 6-1) shows the composition of the pi-electron partition by indicating the number of times all rings of the benzenoid are assigned 6, 5, 4, 3, 2, or 1 pi-electrons. It allows the introduction of a new ordering criterion for such polycyclic aromatic hydrocarbons by summing some of the terms in the signature. There is an almost perfect linear correlation between sums S6 + S5 and S4 + S3 for isomeric cata- or peri-fused benzenoids, so that one can sort such isomers according to ascending s 6 + S5 or to descending S4 + S3 sums (the resulting ordering does not differ much and agrees with that based on increasing numbers of Clar sextets and of Kekule structures). Branched cata-condensed benzenoids have higher S6 + S5 sums than isomeric nonbranched systems. For nonisomeric peri-condensed benzenoids, both sums increase with increasing numbers of benzenoid rings and decrease with the number of internal carbon atoms. Other partial sums that have been explored are S6 + S5 + S3 And S6 + S2 + S1, and the last one appears to be more generally applicable as a parameter for the complexity of benzenoids and for ordering isomeric benzenoids.

π-Electron Partitions, Signatures, and Clar Structures of Selected Benzenoid Hydrocarbons

It is shown for a representative set of isomeric benzenoids that pi-electron partitions and signatures can serve for characterizing and ordering benzenoids. Benzenoid signatures (sequences s6 through s1 where the subscripts correspond to numbers of pi electrons in all rings) are obtained by examining the numbers of assigned pi electrons ranging from 6 to 1 for each ring in all resonance structures. The pi-electron partitions and signatures of all 33 non-isoarithmic peri-condensed benzenoid hydrocarbons with eight rings and four contiguous internal carbon atoms allow an ordering of these benzenoids that agrees fairly well with increasing numbers of Kekulé valence structures and Clar sextets. Interestingly, an excellent correlation (R2 > 0.99) is observed between s6 + s5 + s2 + s1 and s4 + s3, and an explanation for this observation is provided: the number P of pi electrons is divided unequally between two components: s34 = s4 + s3 and s1256 = s6 + s5 + s2 + s1 so that s1256 or the quotient s1256/s34 = Q can serve as a new metric for perfect matchings of polyhexes and a criterion for ordering and for evaluating the complexity of isomeric benzenoids quantitatively.

Ring signatures for benzenoids with up to seven rings, Part 2: Pericondensed systems

AbstractAn array of integers pij indicate in how many of the K resonance structures one finds 6 > i > 0 π‐electrons in the corresponding ring (which is labeled with capital letters A through a subsequent letter up to G). The index j for the rows indicates the ring type, denoted by capital letters (that is, j = 1 stands for A, j = 2 stands for B, etc.). Row sums lead to π‐electron ring partitions after division by the number K of resonance structures. Column sums lead to a sequence of seven integers (Ri sequence) which after division by K affords the ri sequence; this is finally converted into the signature of the benzenoid si sequence). The analysis of signatures for pericondensed benzenoids with h = 4 through 7 benzenoid rings has revealed interesting regularities and correlations with structure, and has also allowed a confirmation of Clar's theory by investigating five isomeric dibenzopyrenes. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Coding and Ordering Kekule Structures.

The concept of numerical Kekule structures is used for coding and ordering geometrical (standard) Kekule 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 Kekule structures can be obtained for any class of polycyclic conjugated systems that possesses standard Kekule structures. The reconstruction of standard Kekule 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 Kekule structures: the Wiswesser code for the benzenoid and the numerical code for its Kekule structure. There is an additional problem with pericondensed benzenoid hydrocarbons; there appear numerical Kekule structures that correspond to two (or more) geometrical Kekule structures. However, this problem can also be resolved.

Coding and ordering Kekulé structures.

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.

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.

Recognizing essentially disconnected benzenoids with fixed double bonds

A Kekulean benzenoid system is one with Kekule structures. A fixed double (single) bond of a Kekulean benzenoid systemH is an edge belonging to all (none) of the Kekule structures ofH. Essentially disconnected systems are Kekulean pericondensed benzenoid systems with some fixed double or single bonds. In this paper a necessary and sufficient condition for a Kekulean benzenoid system to be essentially disconnected benzenoid system with fixed double bonds is given and rigorously proved.

A periodic table for polycyclic aromatic hydrocarbons: Part X. On the characteristic polynomial and other structural invariants [1

A method for calculation of the HMO characteristic polynomial for many small molecular graphs by inspection is reviewed. Inspection methods for ascertaining the presence of HMO eigenvalues of ± 1 in molecular graphs are presented. The characteristic polynomial for many macromolecular, strictly peri-condensed benzenoid hydrocarbons are tabulated for the first time. Topological information transmitted by the coefficients of the characteristic polynomial are detailed. Relationships for quickly determining the number of Kekulé structures, the structure count, and pi electron energy are also reviewed. A nearly universal expression for the sixth coefficient of the characteristic polynomial is presented.

A periodic table for polycyclic aromatic hydrocarbons: Part IX. Isomer enumeration and properties of radical strictly peri-condensed polycyclic aromatic hydrocarbons

Isomer enumeration of strictly peri-condensed benzenoid (PAH6) radicals is simplified by enumerating their excised internal structures. The number of many radical benzenoid isomers are enumerated for the first time. Structural properties of this important class of high-spin hydrocarbons has been detailed. It is shown that for a given number of formula carbon atoms, the strictly peri-condensed benzenoid hydrocarbon structures are the most stable, and the possible pyrolytic formation of strictly peri-condensed benzenoid hydrocarbons via condensation of other benzenoid hydrocarbons can only occur with the evolution of hydrogen. If the excised internal structure of a strictly peri-condensed PAH6 is not a cross conjugated diradical species and it has a HMO eigenvalue of zero, then its corresponding strictly peri-condensed PAH6 structure will also have an eigenvalue of zero.