Singlet biradical character of phenalenyl-based Kekulé hydrocarbon with naphthoquinoid structure

[reaction: see text] A new Kekulé polycyclic hydrocarbon with a singlet biradical index of 50% was synthesized. The singlet biradical character was assessed with UV and 1H-NMR spectroscopy, cyclic voltammetry, SQUID magnetic susceptibility measurement, and quantum chemical calculations.     

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.     

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.     

Bis-periazulene: a simple Kekulé biradical with a triplet ground state

B3LYP/6-31G* calculations on bis-periazulene (cyclohepta[def]-fluorene) predict a triplet ground state for this molecule. The singlet has an aromatic 14π-electron periphery but is 2 kcal/mol higher in energy. The results agree with earlier predictions by Heilbronner. Received: 19 August 1998 / Accepted: 6 October 1998 / Published online: 23 February 1999     

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.     

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.     

Spectroscopy and reactivity of Kekulé hydrocarbons with very small singlet-triplet gaps.

Two Kekulé hydrocarbons, 2,2-dimethyl-2H-benzo[cd]fluoranthene (1) and its benzannellated analogue 2,2-dimethyl-2H-dibenzo[cd,k]fluoranthene (2), were generated photochemically from two different photoprecursors each and investigated spectroscopically in cryogenic matrices by UV-vis, fluorescence, and EPR and in solution using ns flash photolysis and chemical trapping experiments. Hydrocarbon 1 is a ground-state singlet species, whereas compound 2 has a triplet ground state, the first such neutral Kekulé hydrocarbon. This difference, which is supported by density functional calculations, has profound influence on the spectroscopy and reactivity of the two compounds. Using the results of the spectroscopic measurements, trapping experiments, and density functional calculations, the singlet-triplet gap for 1 is estimated to be 2.3-2.8 kcal mol(-1), with the singlet the ground state, and 0.8-1.3 kcal mol(-1) for 2, in favor of the triplet.     

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.     

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.     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

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.     

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.     

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.     

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.     

Kekulé structure counts for some classes of all-benzenoid and all-coronoid hydrocarbons

Summary A general formula for the Kekulé structure count (K) is deduced for the class of catacondensed all-benzenoids with unbranched backbones. The formula is extended to thin pericondensed all-benzenoids, where allowance is made for pyrene units. In this treatment the fragmentation matrices are employed. A generalization of these matrices is furnished. Next some generalK formulas for classes of catacondensed and thin pericondensed all-coronoids are deduced. Again the fragmentation matrices are employed, but the problem is also studied in terms of certain polynomials.Dedicated to the memory of Professor Oskar E. Polansky, who died in January 1989. He was the one who coined the term all-benzenoid.     

Generation of Kekulé valence structures and the corresponding valence bond wave function

A new scheme, called "list of nonredundant bonds", is presented to record the number of bonds and their positions for the atoms involved in Kekulé valence structures of (poly)cyclic conjugated systems. Based on this scheme, a recursive algorithm for generating Kekulé valence structures has been developed and implemented. The method is general and applicable for all kinds of (poly)cyclic conjugated systems including fullerenes. The application of the algorithm in generating Valence Bond (VB) wave functions, in terms of Kekulé valence structures, is discussed and illustrated in actual VB calculations. Two types of VBSCF calculations, one involving Kekulé valence structures only and the second one involving all covalent VB structures, were performed for benzene, pentalene, benzocyclobutadiene, and naphthalene. Both strictly local and delocalised p-orbitals were used in these calculations. Our results show that, when the orbitals are restricted to their own atoms, other VB structures (Dewar structures) also have a significant contribution in the VB wave function. When removing this restriction, the other VB structures (Dewar and also the ionic structures) are accommodated in the Kekulé valence structures, automatically. Therefore, at VBSCF delocal level, the ground states of these systems can be described almost quantitatively by considering Kekulé valence structures only at a considerable saving of time.     

Effect on ring current of the Kekulé vibration in aromatic and antiaromatic rings

Derivative current-density maps are used to follow the changes in ring-current (and hence, on the magnetic criterion, the changes in aromaticity) with the Kekulé vibrations of the prototypical aromatic, antiaromatic, and nonaromatic systems of benzene, cyclooctatetraene (COT), and borazine. Maps are computed at the ipsocentric CHF/6-31G**//RHF/6-31G** level. The first-derivative map for benzene shows a growing-in of localized bond currents, and the second-derivative map shows a pure, paratropic "antiring-current", leading to the conclusion that vibrational motion along the Kekulé mode will reduce the net aromaticity of benzene, on average. For planar-constrained D(4h) COT, the Kekulé mode (positive for reduction of bond-length alternation) increases paratropicity at both first and second order, indicating an average increase in antiaromaticity with zero-point motion along this mode. On the ring-current criterion, breathing expansions of benzene and D(4h) COT reduce aromaticity and increase antiaromaticity, respectively.     

Aromaticity and π-bond covalency: prominent intermolecular covalent bonding interaction of a Kekulé hydrocarbon with very significant singlet biradical character

An anthracene-linked bisphenalenyl Kekulé molecule with very significant singlet biradical character has shown a prominent covalent bonding interaction between molecules in a molecular aggregate. High aromatic stabilization energy in the anthracene linker is responsible for the significant singlet biradical character.