Molecules-in-molecule estimation of the extent of localization of Kekuléan substructures in polycyclic aromatic hydrocarbons

This article first revises graph-theoretical (local aromaticity and overall molecular) indices, introduced by M. Randi? in 1975, for benzenoid hydrocarbons and somewhat improves them for computer enumeration. This goes beyond total Kekulé structure enumeration, yielding an index calculation useful for the quantitative estimation of localization of different Kekuléan substructures (including ethylene-, benzene-, annulene-, and radialene-units). This may be viewed as a "molecules-in molecule" approach to polycyclic aromatic hydrocarbons within the context of graph theoretical partitioning.     

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.     

Kekulé’s Oscillating D3h Cyclohexatriene Structure of Benzene

Kekulé postulated that neighbouring carbon atoms in benzene undergo incessant collision with each other, thereby leading to the interchange of double and single bonds, which amounts to an oscillation between two cyclohexatriene structures in dynamic equilibrium. It has been claimed that Kekulé arrived at a fully symmetric D_6h structure of benzene and that the oscillation hypothesis should not be attributed to him. However, Clausius’ collision theory, which was known at the time, implies that, when the absolute temperature approaches zero, the collision frequency will tend toward zero too, i.e. collisions will stop, and a static, D_3h cyclohexatriene obtains. The classical collision theory did not allow Kekulé to construct the desired D_6h structure as the energy minimum. The theory of harmonic oscillators would have allowed it, but that was not attempted at Kekulé’s time.     

Kekulé-based Valence Bond Model. II. Diels-Alder Reactivity of Polycyclic Aromatic Hydrocarbons

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔｍｏｌｅｃｕｌａｒｏｒｂｉｔａｌ (ＭＯ)ｔｈｅｏｒｙｈａｓｐｌａｙｅｄａｎｉｍｐｏｒｔａｎｔｒｏｌｅｉｎｕｎｄｅｒｓｔａｎｄｉｎｇｖａｒｉｏｕｓｃｈｅｍｉｃａｌｒｅａｃｔｉｏｎｓｏｆｐｏｌｙｃｙｃｌｉｃａｒｏｍａｔｉｃｈｙｄｒｏｃａｒｂｏｎｓ .1Ｅｓｐｅｃｉａｌｌｙ ,ｔｈｅＤｉｅｌｓ Ａｌｄｅｒｒｅａｃｔｉｏｎｓｏｆｍａｎｙｐｏｌｙｃｙｃｌｉｃｂｅｎｚｅｎｏｉｄｈｙｄｒｏｃａｒｂｏｎｓｗｉｔｈｍａｌｅｉｃａｎｈｙｄｒｉｄｅｈａｖｅｂｅｅｎｓ…     

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.     

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.     

Kekulé-based Valence Bond Model. I. The Ground-state Properties of Conjugated π-Systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｈａｓｂｅｅｎａｃｋｎｏｗｌｅｄｇｅｄｔｈａｔｂｏｔｈｔｈｅＨ櫣ｃｋｅｌｍｏｌｅｃｕｌａｒｏｒｂｉｔａｌ (ＨＭＯ)ａｎｄｖａｌｅｎｃｅｂｏｎｄ (ＶＢ)ｍｅｔｈ ｏｄｓａｒｅｖａｌｉｄｓｔａｒｔｉｎｇｐｏｉｎｔｓｆｏｒｄｅｓｃｒｉｂｉｎｇｔｈｅｃｈｅｍｉｃａｌｂｅｈａｖｉｏｒｓｏｆａｌｔｅｒｎａｎｔｃｏｎｊｕｇａｔｅｄｓｙｓｔｅｍｓｗｉｔｈｏｕｔｄｅｇｅｎｅｒ ａｔｅｇｒｏｕｎｄｓｔａｔｅｓ (ｆｏｒｅｘａｍｐｌｅ ,ｃｙｃｌｏｂｕｔａｄｉｅｎｅｉｓｅｘｃｌｕｄ ｅｄ) .1Ｒｅｃ…     

Algebraic Kekulé formulas for benzenoid hydrocarbons

By assigning two pi-electrons of CC double bonds in a Kekulé valence structure to a benzene ring if not shared by adjacent rings and one pi-electron if CC double bond is shared by two rings we arrived at numerical valence formulas for benzenoid hydrocarbons. We refer to numerical Kekulé formulas as algebraic Kekulé valence formulas to contrast them to the traditional geometrical Kekulé valences formulas. The average over all numerical Kekulé valence structures results in a single numerical structure when a benzenoid hydrocarbon molecule is considered. By ignoring numerical values the novel quantitative formula transforms into a qualitative one which can replace incorrectly used notation of pi-electron sextets to indicate aromatic benzenoids by placing inscribed circles in adjacent rings-which contradicts Clar's characterization of benzenoid hydrocarbons.     

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.     

Algebraic Kekulé structures of benzenoid hydrocarbons

An algebraic Kekulé structure of a benzenoid hydrocarbon is obtained from an ordinary Kekulé structure by inscribing into each hexagon the number of pi-electrons which (according to this Kekulé structure) belong to this hexagon. We show that in the case of catafusenes, there is a one-to-one correspondence between ordinary and algebraic Kekulé structures. On the other hand, in the case of perifusenes, one algebraic Kekulé structure may correspond to several ordinary Kekulé structures.     

Kekulé count in capped zigzag boron-nitride nanotubes

Hemi-B16N16 capped zigzag boron-nitride nanotube is introduced, and its Kekulé count is studied. With a bond-allocating and coding scheme, recurrence formulas are established as well as for the case of a hemi-B36N36 capped zigzag nanotube. Numerical results reveal that the Kekulé counts increase exponentially with respect to the number of layers in the nanotubes concerned.     

Unicyclic Graphs Possessing Kekulé Structures with Minimal Energy

Unicyclic graphs possessing Kekulé structures with minimal energy are considered. Let n and l be the numbers of vertices of graph and cycle C _l contained in the graph, respectively; r and j positive integers. It is mathematically verified that for and l = 2r + 1 or has the minimal energy in the graphs exclusive of , where is a graph obtained by attaching one pendant edge to each of any two adjacent vertices of C ₄ and then by attaching n/2 − 3 paths of length 2 to one of the two vertices; is a graph obtained by attaching one pendant edge and n/2 − 2 paths of length 2 to one vertex of C ₃. In addition, we claim that for has the minimal energy among all the graphs considered while for has the minimal energy.      

Nanotubes: number of Kekulé structures and aromaticity

Carbon nanotubes (CNTs) are composed of cylindrical graphite sheets consisting of sp(2) carbons. Due to their structure CNTs are considered to be aromatic systems. In this work the number of Kekulé structures (K) in "armchair" CNTs was estimated by using the transfer matrix technique. All Kekulé structures of the cyclic variants of naphthalene and benzo[c]phenanthrene have been generated and the basic patterns have been obtained. From this information the elements of the transfer matrix were derived. The results obtained indicate that K (and the resonance energy) is greater if tubulenes are extended in the vertical than in the horizontal direction. Tubulenes are therefore more stabile than cyclic strips. An illustration, obtained by using scanning probe microscope, has been attached to affirm the existence of thin CNTs.     

Kekulé structures of hexagonal chains—some unusual connections

It is known since 1977 that the number K of Kekulé structures of a hexagonal chain is equal to the topological Z-index of a pertinently constructed “caterpillar” tree. Based on this relation we now connect K with some of other, seemingly unrelated, concepts: continuants (from number theory) and matchings of the path–graph (further related to Fibonacci numbers). We also arrive at a tridiagonal determinantal expression for K.     

On the number of Kekulé structures in capped zigzag nanotubes

Two theoretical formulae for the number of Kekulé structures in general capped zigzag nanotubes are established: one of which is by using the techniques of the transfer matrices, the other involves the eigenvalues of the transfer matrix which reveals the asymptotic behaviour of this index. In effective, according to the symmetric aspect of the tubule, the order of the transfer matrix could be notably decreased. As an application, the closed expressions for four types are given out and the relevant numerical results for those of length up to 50 are listed.     

Kekulé count and algebraic structure count for unbranched alternant cata-fusenes

Algorithms for making Kekulé-structure and algebraic-structure enumerations for unbranched catacondensed hydrocarbons with even-membered rings are described. A pictorial presentation of the algorithms is obtained and is shown to reduce to an earlier well-known recursion (of Gordon and Davison) applicable to polyhex chains.     

The number of Kekulé structures of polyominos on the torus

Let G be a (molecule) graph. A perfect matching, or Kekulé structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekulé structures of a (molecule) graph is interest in chemistry, physics and mathematics. In this paper, we focus on some polyominos on the torus and obtain the explicit expressions on the number of the Kekulé structures of them.