On the anti-Kekulé number and anti-forcing number of cata-condensed benzenoids

Previously introduced concepts the anti-Kekulé number and anti-forcing number (Vukicevic et al., J. Math. chem., in press) are applied to cata-condensed benzenoids. It is shown that all cata-benzenoids have anti-Kekulé number either 2 or 3 and both classes are characterized. The explicit formula for anti-forcing number of chain (unbranched) cata-benzenoids is given. It is also shown that anti-forcing number of any cata-benzenoid goes up to h/2 where h is the number of hexagons in a cata-benzenoid.      

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.     

QSPR for physical properties of cata-condensed benzenoids using two simple dualist-based descriptors

Dualists of benzenoids are a special type of graphs that have as vertices the centers of hexagons and as edges the pairs of condensed rings. Catafusenes (cata-condensed benzenoids) have acyclic dualists. Codes of catafusenes with h benzenoid rings start from one end of a dualist and consist of digits 0 (indicating linear annelation, angle of 180 degrees between two adjacent edges) and 1 or 2 for kinks (120 degrees or 240 degrees angles) with the convention of choosing the lowest number formed from these h-2 digits. A shape parameter (n) for the dualist consists of the number of adjacent zeros in the code plus 1. It is shown in the present communication that many molecular properties of catafusenes depend mainly on n, whereas most bulk properties depend mainly on h. Monoparametric or biparametric correlations in terms of the simple integer parameters n and/or h are presented for molecular properties (energy of the p-band in electronic absorption spectra, ionization potentials, electron affinities, hardness and softness) and bulk properties (normal boiling point, chromatographic retention index, the logarithm of the water solubility, and lipophilicity).     

Contribution to the techniques of enumeration ofKekulé structures

Four auxiliary classes of benzenoids are introduced, and formulas are given for their number ofKekulé structures (K). An enumeration method forK of different important classes of benzenoids is illustrated by examples. The utilization of essentially disconnected benzenoids is a special feature of the method.

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Ein Beitrag zur Methodik der Bestimmung möglicherKekulé-Strukturen

Zusammenfassung Es werden Hilfsklassen von benzenoiden Aromaten eingeführt und Formeln zur zahlenmäßigen Auswertung möglicherKekulé-Strukturen angegeben (K). Die Auswertungsmethode fürK wird an verschiedenen wichtigen Klassen benzenoider Verbindungen exemplarisch gezeigt. Die Verwendbarkeit essentiell abgekoppelter aromatischer Bauelemente ist eine spezielle Eigenschaft dieser Methode.

     

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     

Enumeration of the isomers of phenylenes

Summary The [h]phenylene C_6h H_2h+4 isomers are enumerated up toh=12. The numbers are compared with old and new data for C _n H₅ isomers of benzenoids, fluoranthenoids and biphenylenoids.

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Anzahl möglicher Isomerer von Phenylenen

Zusammenfassung Die Anzahl der [h]Phenylen-Isomeren C_6h H_2h+4 wurde bish=12 ausgewertet. Die Zahlen wurden mit alten und neuen Daten für C _n H _s -Isomere von Benzenoiden, Fluoranthenoiden und Biphenyloiden verglichen.

     

Global Forcing Number of Benzenoid Graphs

A global forcing set in a simple connected graph G with a perfect matching is any subset S of E(G) such that the restriction of the characteristic function of perfect matchings of G on S is an injection. The number of edges in a global forcing set of the smallest cardinality is called the global forcing number of G. In this paper we prove several results concerning global forcing sets and numbers of benzenoid graphs. In particular, we prove that all catacondensed benzenoids and catafused coronoids with n hexagons have the global forcing number equal to n, and that for pericondensed benzenoids the global forcing number is always strictly smaller than the number of hexagons.     

A Semiempirical Study of Carbon Nanotubes with Finite Tubular Length and Various Tubular Diameters

A semiempirical PM3 quantum computational method has been used to generate the electronic and optimized geometrical structure of SWNT of zigzag and armchair types. We shed light on the electronic structures of SWNT with various diameters and lengths of the tube. Particularly, the calculated HOMO, LUMO and band‐gap of SWNT are not monotonic but exhibit a well‐defined oscillation, which depends on the tubular diameter and the tubular length. Calculated HOMO, LUMO and band‐gap of the zigzag SWNTs have oscillated with tubular diameter as they contain an odd or even number of benzenoids in the circular plane of the carbon nanotube. The zigzag SWNTs with an odd number of benzenoids have a higher band‐gap than those of SWNTs with an even number of benzenoids in the circular plane of the carbon nanotube. Calculated results also reveal that the tubular length in the zigzag SWNTs influences the band‐gaps very little. For the armchair SWNT, calculated HOMO, LUMO and band‐gap contained the oscillate depending on the number of carbon sections in the tubular length axis. Their repeat sections are 3n‐1, 3n and 3n+1. The armchair SWNT with 3n+1 sections has a high band‐gap while the SWNTs with 3n‐1 sections have a low band‐gap. The tubular diameters of armchair SWNT influence the HOMO, LUMO and band gap very little.     

A comparison between 1-factor count and resonant pattern count in plane non-bipartite graphs

The concept of resonant (or Clar) pattern is extended to a plane non-bipartite graph G in this paper: a set of disjoint interior faces of G is called a resonant pattern if such face boundaries are all M-conjugated cycles for some 1-factor (Kekulé structure or perfect matching) M of G. In particular, a resonant pattern of benzenoids and fullerenes coincides with a sextet pattern. By applying a novel approach, the principle of inclusion and exclusion in combinatorics, we show that for any plane graphs, 1-factor count is not less than the resonant pattern count, which generalize the corresponding results in benzenoid systems and plane bipartite graphs. Applications to fullerenes are also discussed.AMS Subject classification: 05C70, 05C90, 92E10     

Resonance structure counts in parallelogram-like benzenoids with holes

If a hexacyclic graph G represents a benzenoid, a perfect matching corresponds to a configuration of π-bonds. We present an algorithm for counting the number of configurations of π-bonds for parallelogram-like benzenoids with parallelogram-like holes by counting descending paths in a corresponding rectangular mesh with rectangular holes.     

The valence bond calculations for conjugated hydrocarbons having 24–28 π‐electrons

A novel algorithm is introduced for coding all Slater determinants in the covalent space with conserved S_Z, the z component of total spin S for a classical valence bond (VB) model. It effectively minimizes the search time and the storing space in the central memory of the computer. In cooperation with symmetry reductions based on molecular point group and spin inversion, the VB calculations have been extended to benzenoid hydrocarbons of up to 28 π‐electrons that have 4×10⁷ configurations. The low‐lying states of benzenoids with 24, 26, and 28 π‐electrons have been obtained for 62 species. To rationalize the aromaticity of benzenoids in a VB scheme, the resonance energy per hexagon (REPH) is defined. A linear correlation between the REPH and the energy gap of the ground (singlet) state and the first excited (triplet) state for 89 benzenoids is established. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 856–869, 2000     

A periodic table for all-benzenoid hydrocarbons and enumerations of some polyrex isomers

A polyhex (benzenoid or coronoid) isomer is characterized by the formula C _n H _s , which pertains to the corresponding hydrocarbon. The periodic table for benzenoid hydrocarbons is revisited, and the construction of it is simplified. All-benzenoids are treated with respect to their isomers. A periodic table for all-benzenoid hydrocarbons is proposed, and an easy construction of it is described. It involves the process of super-circumscribing, a new concept. Numerical data are given for a number of isomer classes of benzenoids, all-benzenoids and single coronoids. These data represent a substantial amount of supplements to the previously known numbers of different polyhex isomers.     

Free volume,glass transition,and degree of branching in metallocene‐based propylene/α‐olefin copolymers: Positron lifetime,density, and differential scanning calorimetric studies

Positron annihilation lifetime spectroscopy (PALS), density, and differential scanning calorimetric (DSC) measurements were used to study systematically the variation of the glass‐transition temperature (T_g) and the size v and number density N_h of local free volumes in n‐alkyl‐branched polypropylenes. The samples were metallocene‐catalyzed propylene copolymers with different α‐olefins (from C₄ to C₁₆) and a different α‐olefin content (between 0 and 20 mol %). From the total specific volume and crystallinity the specific volume of the amorphous phase V_a was estimated and used to calculate the fractional free (hole) volume h and value of N_h. The variations of T_g, v, V_a, h, and N_h were related to the degree (number and length) of branching. T_g decreases and v increases linearly with the number and length of n‐alkyl branches. This behavior was attributed to an increased segmental mobility caused by branching. Both values, T_g and v, follow linear master curves as a function of the degree of branching (DB) if this is defined as the number of all side‐chain carbons with respect to a total of 1000 (main‐chain and side‐chain) carbons. Only propylene/1‐butene copolymers deviated from these relations. A linear relation between v and T_g was also found. The number density of holes was estimated to be N_h = 0.49(±0.07) nm^?3 and N_h′ = 0.58(±0.11) × 10²¹ g^?1, respectively. It shows a slight variation with the DB, which is also seen in the behavior of the specific volume V_a. This variation was explained by the appearance of sterical hindrances resulting from short‐chain branches that may prevent an efficient packing of the chains. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 434–453, 2002; DOI 10.1002/polb.10108     

Perfect Matchings in Bipartite Lattice Animals: Lower Bounds and Realizability

In the first part of this paper we establish sharp lower bounds on the number of perfect matchings in benzenoid graphs and polyominoes. The results are then used to determine which integers can appear as the number of perfect matchings of infinitely many benzenoids and/or polyominoes. Finally, we consider the problem of concealed non-Kekuléan polyominoes. It is shown that the smallest such polyomino has 15 squares, and that such polyominoes on n squares exist for all n ≥ 15.     

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

An array of integers p_ij 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 (R_i sequence) which after division by K affords the r_i sequence; this is finally converted into the signature of the benzenoid s_i 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     

Numerical calculations of the viscoelastic properties of solutions of branched polymers based on the Zimm-Kilb theory

Numerical calculations were performed for the viscoelastic properties of dilute solutions of branched star polymers with equal branch lengths as formulated in terms of a bead-spring model by Zimm and Kilb without using the integrodifferential equation approximation method to calculate the eigenvalues. The complex modulus and complex viscosity were calculated as functions of frequency for various combinations of the number of branches f (4, 8, and 13), the number of beads in one branch N_b (= N/f; 20 to 100, where N + 1 is the total number of beads, N the number of springs in the molecule) and the reduced hydrodynamic interaction parameter h^* (= h/N^1/2 0.05 to 0.3, where h is the hydrodynamic interaction parameter of Zimm and Kilb). The frequency dependence of the complex modulus in the low-frequency range depends mainly on h^* and not on N_b if N_b is large enough, and it is very close to that calculated from the eigenvalues for h→∞ obtained by Zimm and Kilb, if h^* is about 0.25. As h^* decreases from 0.25, the frequency dependence gradually approaches that of the free-draining cash (h→0). Calculations may be carried out for h^* values somewhat larger than 0.25 and result in a frequency dependence that is not intermediate to the h → 0 and h → ∞ cases as evaluated by Zimm and Kilb. The physical meaning of such “super-non-free-draining” values of h^* is uncertain, however. The intrinsic viscosity ratio g′ = [η]_f/[η]_lin is an increasing function of h^* and changes very slowly with N. For h^* = 0.25, g′ is close to the non-free-draining limit for any value of N.     

关于正规苯烃和正规晕苯烃的拓扑性质的研究

本文较系统地研究了正规苯烃和正规晕苯烃的拓扑性质,给出了Cyvin和Gutman于1986年提出的两个猜想的证明。这些拓扑性质的给出和猜想的证明对正规苯烃和正规晕苯烃的判定、产生、分类、计数以及合成和理论研究都有一定的意义。     

Developmental variation in floral volatiles composition of a fragrant orchid Zygopetalum maculatum (Kunth) Garay

Emitted scent volatile profile of an orchid species Zygopetalum maculatum was studied using dynamic headspace sampling technique with four different adsorbent matrices, namely Porapak Type Q polymer (mesh size: 80/100), Tenax (mesh size: 60/80), activated charcoal and graphite. In addition, developmental variations in scent emissions and endogenous volatile levels were also investigated. Gas chromatography-mass spectrometry analysis revealed the presence of 21 volatile compounds in the headspace, which was predominantly enriched with benzenoid compounds. Among these benzenoids, o-diethylbenzene and p-diethylbenzene were the major compounds followed by benzyl acetate and methyl salicylate. Among the phenylpropanoid compounds, 2-phenylethyl acetate was the major volatile. However, as compared to benzenoids, the quantity was much lesser, indicating the inclination of phenylalanine flux towards benzenoid pathway. The outcome of this study has the implications in enhancing fragrance and vase life of orchids of the Sikkim Himalaya region and thus may further help to meet the growing market demand.     

Crystallographic Evidence for Direct Metal–Metal Bonding in a Stable Open‐Shell La2@Ih‐C80 Derivative

Endohedral metallofullerenes (EMFs) have novel structures and properties that are closely associated with the internal metallic species. Benzyl radical additions have been previously shown to form closed‐shell adducts by attaching an odd number of addends to open‐shell EMFs (such as Sc₃C₂@I_h‐C₈₀) whereas an even number of groups are added to closed‐shell EMFs (for example Sc₃N@I_h‐C₈₀). Herein we report that benzyl radical addition to the closed‐shell La₂@I_h‐C₈₀ forms a stable, open‐shell monoadduct instead of the anticipated closed‐shell bisadduct. Single‐crystal X‐ray diffraction results show the formation of a stable radical species. In this species, the La?La distance is comparable to the theoretical value of a La?La covalent bond and is shorter than reported values for other La₂@I_h‐C₈₀ derivatives, providing unambiguous evidence for the formation of direct La?La bond.     

Hydrogen-Bond Structure and Low-Frequency Dynamics of Electrolyte Solutions: Hydration Numbers from ab Initio Water Reorientation Dynamics and Dielectric Relaxation Spectroscopy

We present an atomistic simulation scheme for the determination of the hydration number (h) of aqueous electrolyte solutions based on the calculation of the water dipole reorientation dynamics. In this methodology, the time evolution of an aqueous electrolyte solution generated from ab initio molecular dynamics simulations is used to compute the reorientation time of different water subpopulations. The value of h is determined by considering whether the reorientation time of the water subpopulations is retarded with respect to bulk-like behavior. The application of this computational protocol to magnesium chloride (MgCl₂) solutions at different concentrations (0.6–2.8 mol kg⁻¹) gives h values in excellent agreement with experimental hydration numbers obtained using GHz-to-THz dielectric relaxation spectroscopy. This methodology is attractive because it is based on a well-defined criterion for the definition of hydration number and provides a link with the molecular-level processes responsible for affecting bulk solution behavior. Analysis of the ab initio molecular dynamics trajectories using radial distribution functions, hydrogen bonding statistics, vibrational density of states, water-water hydrogen bonding lifetimes, and water dipole reorientation reveals that MgCl₂ has a considerable influence on the hydrogen bond network compared with bulk water. These effects have been assigned to the specific strong Mg-water interaction rather than the Cl-water interaction.