Novel graph-theoretical approach to estimating the relative importance of individual Kekulé valence structures. I. Simple catacondensed systems

A novel graph-theoretical approach for ordering Kekulé valence structures of benzenoid hydrocarbons is presented. The approach involves the transformation of the Kekulé structures into the subspaces of their individual double bonds. The submolecules generated in this way [H. Joela, Theor. Chim. Acta 39 , 241 (1975)] are ordered according to suitable connectivity indices. The resulting orders parallel those predicted from the so called Kekulé indices [A. Graocvac, I. Gutman, M. Randi?, and N. Trinajst?, J. Am. Chem. Soc. 95 , 6267 (1973)]. A relation is thus illustrated between VB and MO theories. The method is new and allows the prediction of the relative stabilities of structures from purely combinatorial vent without resort to computer.     

Novel graph-theoretical approach to estimating the relative importance of individual Kekulé valence structures. II. Benzenoid systems having four to seven fused benzene rings: “Pseudodegenerate” states

Kekulé structures of 10 nonlinear acenes comprising 83 graphs are studied through the use of connectivities [M. Randi?, J. Am. Chem. Soc. 97 , 6609 (1975)] of their corresponding submolecules [H. Joela, Theor. Chim. Acta 39 , 241 (1975)]. In certain rare cases states were identified to have identical branching indices but different Kekulé indices [A. Graovac, I. Gutman, M. Randi?, and N. Trinajsti?, J. Am. Chem. Soc. 95 , 6267 (1973)]. Such states are termed pseudodegenerate states. A method is described to forecast and another to remedy such situations. The method emphasizes the relation between VB (resonance) and MO theories using graph-theoretical concepts.     

Ordering of Kekulé structures using nonadjacent numbers

Kekulé indices and conjugated circuits are computed for 36 Kekulé structures, together with two VB quantities associated with the corresponding factor graphs (previously called submolecules). These latter quantitites are nonadjacent numbers of Hosoya and the reciprocal of the connectivity indices of Randi?. It was found that the index of Hosoya successfully orders a set of Kekulé structures belonging to the same hydrocarbon in a parallel order as their Kekulé indices and branching indices. This substantiates the relation between VB and MO theories. A code is derived by summing contributions of nonadjacent numbers in all Kekulé stuctures of a hydrocarbon. The order of the resulting codes is found to be identical to the order of the molecular properties (resonance energies, π-energies, and eigenvalues) of the hydrocarbons.     

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.     

Computing the Kekulé structure count for alternant hydrocarbons

A fast computer algorithm brings computation of the permanents of sparse matrices, specifically, molecular adjacency matrices. Examples and results are presented, along with a discussion of the relationship of the permanent to the Kekulé structure count. A simple method is presented for determining the Kekulé structure count of alternant hydrocarbons. For these hydrocarbons, the square of the Kekulé structure count is equal to the permanent of the adjacency matrix. In addition, for alternant structures the adjacency matrix for N atoms can be written in such a way that only an N/2 × N/2 matrix need be evaluated. The Kekulé structure count correlates with topological indices. The inclusion of the number of cycles improves the fit. When comparing with previous results, the variance decreases 74%. The calculated standard heat of formation correlates with the logarithm of the Kekulé structure count. This heat increments 349 kJ/mol each time the Kekulé structure count increases by one order of magnitude. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

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.     

Ground-state properties of benzenoid hydrocarbons by simple bond orbital resonance theory approach

π-electron energies and bond orders of benzenoid hydrocarbons with up to five fused hexagons have been considered by the simple Bond Orbital Resonance Theory (BORT) approach. The corresponding ground states were determined according to four BORT models. In the first three models a diagonalisation of the Hückel-type Hamiltonian was performed in the bases of Kekulé, of Kekulé and mono-Claus and of Kekulé and Claus resonance structures, respectively. In the fourth model a simple BORT ansatz was used. According to this ansatz, the ground state is a linear combination of the positive Kekulé structures, all with equal coefficients. It was shown that π-electron energies and bond orders obtained by these models correlate much better with the PPP energies and bond orders than with the Hückel energies and bond orders. This indicates that a simple BORT approach is quite reliable in predicting the more sophisticated PPP results. Concerning the relative performance of the four BORT models, the best results were obtained with the BORT ansatz. The performance deteriorates with the expansion of the basis set. This is attributed to the fact that in these models the improvement of the basis set is not accompanied with the corresponding improvement of the Hamiltonian. Comparing the BORT-ansatz bond orders with the Pauling bond orders, it was shown that BORT-ansatz bond orders correlate much better with the PPP bond orders. This revised version was published online in July 2006 with corrections to the Cover Date.     

Partitioning of pi-electrons in rings of polycyclic conjugated hydrocarbons. 5. Nonalternant compounds

All possible nonalternant hydrocarbons with a total of two, three, or four 5-, 6-, and 7-membered rings have been examined for the partition of their pi-electrons by averaging over all Kekulé structures (considered to contribute equally to the electron distribution) the pi-electrons in each ring in accordance to the rules introduced earlier: for each double bond shared with another ring one pi-electron is taken into account, and for double bonds that are not shared two pi-electrons are added. The trends observed for the partitions are discussed, and a comprehensive bibliography is provided as Supporting Information for all such systems, including both experimental and theoretical published data.     

A combination of Clar number and Kekulé count as an indicator of relative stability of fullerene isomers of C<Subscript>60</Subscript>

Kekulé count is not as useful in predicting the thermodynamic stability of fullerenes as it is for benzenoid hydrocarbons. For example, the Kekulé count of the icosahedral C₆₀, the most stable fullerene molecule, is surpassed by its 20 fullerene isomers (Austin et al. in Chem Phys Lett 228:478–484, 1994). This article investigates the role of Clar number in predicting the stability of fullerenes from Clar’s ideas in benzenoids. We find that the experimentally characterized fullerenes attain the maximum Clar numbers among their fullerene isomers. Our computations show that among the 18 fullerene isomers of C₆₀ achieving the maximum Clar number (8), the icosahedral C₆₀ has the largest Kekulé count. Hence, for fullerene isomers of C₆₀, a combination of Clar number and Kekulé count predicts the most stable isomer.     

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.     

π‐electron currents in polycyclic conjugated hydrocarbons: Coronene and its isomers having five and seven member rings

We have outlined novel graph theoretical model for computing π‐electron currents in π‐electron polycyclic conjugated hydrocarbons. We start with Kekulé valence structures of a polycyclic conjugated hydrocarbon and their conjugated circuits. To each 4n+2 conjugated circuits we assign counter clockwise current i and to each 4n conjugated circuit we assign clockwise current i. By adding the contributions from all conjugated circuits in a single Kekulé valence structure one obtains π‐electron current pattern for the particular Kekulé valence structure. By adding the conjugated circuit currents in all Kekulé valence structure one obtains the pattern of π‐electron currents for considered molecule. We report here π‐electron current patters for coronene and 17 its isomers, which have been recently considered by Balaban et al., obtained by replacing one or more pairs of peripheral benzene rings with five and seven member rings. Our results are compared with their reported π‐electron current density patters computed by ab initio molecular orbital (MO) computations and satisfactory parallelism is found between two so disparate approaches. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

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.     

Radical Ions in the Pentalene Series. Part. I. 1,3,5-tri-t-butylpentalene

The ESR. spectra of the radical anion and cation of 1, 3, 5-tri-t-butylpentalene (II) have been reexamined under higher resolution. With the assistance of the ENDOR. technique, the coupling constants of all protons in II?. and II?. could be determined. The experimental data agree satisfactorily with the values predicted by the simple MO model which suggests that the π-spin distributions in II?. and II?. should not strongly differ from those in the corresponding radical ions of parent pentalene (I). As in the case of other non-alternant hydrocarbons, the proton coupling constants for II?. are very sensitive to experimental conditions, due to the association of the radical anion with its counterion. Spectra of II?. taken at low temperatures (down to 133 and 163 K for ESR. and ENDOR., respectively) have not revealed specific line-broadening which could arise from the bond shift between the two Kekulé-structures of pentalene.     

On the number of kekulé structures of unbranched cata-condensed benzenoid chains

It is known that an alternative algorithm to the Gordon-Davidson algorithm for counting the Kekulé valence structures of catacondensed non-branched benzenoid hydrocarbons is a reformulation of the original algorithm.     

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

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

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.     

Stability of Conjugated Carbon Nanocones

We use interlacing techniques to prove that carbon nanocones who have a Fries Kekulé structure have closed Hückel shells, and that this result can be extended to all conjugated cones where each edge belongs to a hexagonal face and the configuration of the non-hexagonal faces are consistent with a Fries Kekulé structure. Cones with Fries Kekulé structure or substructure are topical—not only from a valence bond theoretical point of view—since a previous ab initioanalysis favored cones where the pentagons at the tip are configured as in a Fries Kekulé structure. The question of interdependence will therefore be addressed.