Clar Structures Research Articles

Clar's Sextet Rule Is a Consequence of the σ-Electron Framework

A number of condensed PAHs are examined to identify the underlying reasons governing empirical Clar's rule taking benzene as a limiting case. It is found that the so-called Clar's structures are the only minima on the MP2(fc) potential energy hypersurfaces, meaning that other conceivable valence isomers are nonexistent. The influence of the electron correlation energies to the stability of Clar's structures is substantial with predominating influence of the sigma-electrons. However, the contributions arising from the sigma- and pi-electron correlation energies are approximately the same, if Clar's structures are compared with some artificial pi-electron localized or graphite-like delocalized planar systems. Analysis of the Hartree-Fock (HF) energies provides a compelling evidence that the origin of stability of Clar's structures lies in a decrease of the positive T, V(ee) and V(nn) energy terms relative to some characteristic virtual "delocalized" or "localized" model geometries. Partitioning of the mixed and terms in the sigma- and pi-type contributions, by using the stockholder (SHR), equipartitioning (EQP) and standard pi (SPI) schemes, unequivocally shows that the driving force leading to Clar's structures are more favorable sigma-type interactions. All these conclusions hold for the archetypal benzene too, which could be considered as a limiting Clar system. Finally, the boundaries of Clar's hypothesis and some common misconceptions are briefly discussed. Perusal of the geometric parameters and pi-bond orders reveals that there are no benzene rings completely "vacant" or "fully occupied" by the pi-electrons, envisaged by Clar in his picture of condensed benzenoid compounds. Instead, there are six-membered rings with higher and lower total pi-electron density. The bond length anisotropy of the former rings is smaller. It is concluded that Clar's proposition is a useful rule of thumb providing qualitative information on the stability of the PAH systems, which in turn should not be overinterpreted.

Partitioning of π‐Electrons in Rings for Clar Structures of Benzenoid Hydrocarbons.

AbstractChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 200 leading journals. To access a ChemInform Abstract, please click on HTML or PDF.

Partitioning of π-Electrons in Rings for Clar Structures of Benzenoid Hydrocarbons

Resonance structures of polycyclic aromatic hydrocarbons can be associated with numerical formulas by assigning pi-electrons of C=C double bonds to individual benzenoid rings. Each C=C double bond in a resonance structure assigns two pi-electrons to a ring in a fused-benzenoid system if it is not shared by adjacent rings and one pi-electron when it is common to two rings, obtaining thus a "local" characterization of rings in polycyclic conjugated hydrocarbons. In the present contribution we extend this approach to the aromatic pi-sextet model of Clar, which offers an alternative description of benzenoid hydrocarbons. In this model local characteristics of individual benzenoid rings are based on partitioning of pi-electrons but only for those resonance structures (fewer in number) that contribute to Clar's formula of benzenoid hydrocarbons.

Assessment of Clar's aromatic π‐sextet rule by means of PDI, NICS and HOMA indicators of local aromaticity

AbstractThe local aromaticity of a series of benzenoid systems was determined through the use of structurally (HOMA) and magnetically (NICS) based measures, and also by using a new electronically based indicator of aromaticity, thepara‐delocalization index (PDI). The results were compared with the predictions of Clar's aromatic π‐sextet rule. It is found that, for all analyzed benzenoid hydrocarbons having a single Clar structure, local aromaticity orderings of the different six‐membered rings given by all descriptors of aromaticity tested are identical and in agreement with Clar's aromatic π‐sextet rule. For benzenoid species that are described by a superposition of Clar structures, HOMA and PDI indices provide local aromaticity values that are totally consistent with Clar's rule, whereas NICS results deviate somewhat from the estimations based on this model. Copyright © 2005 John Wiley & Sons, Ltd.

The Agreement between Clar Structures and Nucleus-Independent Chemical Shift Values in Pericondensed Benzenoid Polycyclic Aromatic Hydrocarbons: An Application of the Y-Rule

In the present study, we verify theoretically the performance of a previously introduced qualitative simple rule for aromaticity, the Y-rule, which determines the relative aromaticity of each hexagonal ring in pericondensed benzenoid polycyclic aromatic hydrocarbons (peri-PAHs). The Y-rule extends the sextet−double bond description of Clar's model for peri-PAHs by predicting, in an easy way, the most likely location and total number of aromatic sextets in the structure. The final π-electronic distribution thus obtained corresponds to the superposition of the most important Clar structures. The problem of the number and location of aromatic sextets in peri-PAHs involves the very concept of aromaticity. The π-electronic distribution obtained by the Clar sextet principle (or model) as determined by the Y-rule is compared with the π-electronic distribution obtained from nucleus-independent chemical shift (NICS) calculations for various series of benzenoid peri-PAHs. The NICS values were used to assess the relative aromaticities of the individual rings in the peri-PAH compounds. We found that the simple heuristic Y-rule predicts for all the calculated systems the same relative aromaticities given by the NICS calculations, and for the case of peri-systems with very high symmetry (like the compact peri-PAHs) it reinforces the picture derived from the NICS calculations. In these cases, the NICS results can be complex due to symmetry. The advantages of the qualitative Y-rule are that it is of particular importance for the case of large peri-PAHs, it takes only a few minutes to be applied following a very easy methodology, and it provides a quick answer about the aromaticity and location of the resonant sextets in peri-PAH compounds without having to carry out theoretical quantum chemistry calculations that take time and that can be costly depending on the size of the system.

Clar and sextet polynomials of buckminsterfullerene

A graph-theoretical definition of Herndon and Hosoya's Clar structure is given. The Fries Kekulé structure of buckminsterfullerene ( C 60) with I h symmetry implies that every independent set of hexagon-dual graph (dodecahedron) of C 60 corresponds to a sextet pattern; and every maximal independent set to a Clar structure. By proposing the maximal independent set polynomial of a graph and developing its various calculational methods the Clar polynomial of C 60, ξ( C 60, χ)=5 χ 8+280 χ 7+10 χ 6, is given, which says that C 60 possesses a total of 295 Clar structures, and thus corrects a corresponding result recently published. Furthermore, the sextet polynomial of C 60 is also produced.

Perfect Clar structures and 3-homogeneous simplicial complexes

In this paper we discuss the notion of a Clar structure on a 3-homogeneous simplicial complex, in particular on a triangulated surface. Such structures arise in theoretical chemistry, from where they also have received the name. We discuss various aspects of Clar structures, particularly existence of Clar structures on (closed compact) surfaces of general genus, and a possible variation of the definition using a splitting. We give an interpretation of Clar structures in terms of (mod 2) two step homology groups, which establishes the relationship of these structures with certain standard calculations in algebraic topology.

Clar sextet theory of buckminsterfullerene (C60)

An algorithm is described for the calculation of the number of Clar structures of C60 by considering various fragments which result from inscribing all possible numbers of Clar's circles in both central and peripheral shells of hexagons of the C60 Schlegel projection. There are six bonding schemes generating a total of 330 Clar structures of which only five are proper structures possessing eight circles each. They correspond to one Ih Kekulé structure which describes the ground-state wave function of C60. The Sachs graph associated with any of the proper Clar structures covers only 48 of (the 60) C atoms of C60, a finding which is consistent with its electron-deficient chemistry.

Topological Properties of Circumcoronenes

Abstract The regular-hexagon-shaped benzenoid hydrocarbons: B1 = benzene (C6H6), B2 = coronene (C24H12), B3 = circumcoronene (C54H18), B4 = circumcircumcoronene (C150H30), etc. possess unique topological properties. General expressions for the most important of such properties (number of fun-damental structural invariants, number of Kekule and Clar structures, number of aromatic sextets, Wiener and Szeged indices, spectral moments) are given, including a number of results that are communicated here for the first time. Cyclic conjugation in circumcoronenes is analyzed by means of its energy-effect, and found to agree with the predictions of Clar's aromatic sextet theory only in the case of B1 and B2.

On Construction of Clar Structures for Large Benzenoids

Abstract We present an algorithm for construction of Clar valence structures for large benzenoids. The algorithm starts with a single Clar structure and systematically generates the remaining structures by considering all possible “moves” for the π-sextets on the “board” of the fused hexagons representing carbon skeleton of a benzenoid. We illustrate the approach on hexabenzocoronene, its higher homologue and few other cases. We end the presentation by discussing some limitations of the Clar model.

Fractal relations. Modelling fractals and analogy with Feigenbaum's period-doubling diagram

The equivalence relation, l, defined previously on Kekulé spaces of benzenoid hydrocarbons (S. El-Basil, J. Chem. Soc., Faraday Trans., 89 (1993) 909; J. Mol. Struct. (Theochem), 288 (1993) 67; J. Math Chem., 14 (1993) 305) is used to map Kekulé spaces of benzenoid systems onto various stages of deterministic fractals including the Cantor set, the Sierpinski triangle, the Sierpinski carpet, the Koch curve and the box fractal as well as certain stages of cellular automata. Furthermore, the Kekulé spaces of quasicrystalline benzenoids (defined by S. El-Basil in J. Chem. Soc., Faraday Trans., 89 (1993) 909) exhibit a period-doubling pattern which manifests itself through a full analogy with Feigenbaum's scaling theorem (M. Feigenbaum, J. Stat. Phys., 19 (1978) 25) including a bifurcation universality constant, a control parameter (of the logistic equation) and cycle-sizes (which can only be integral powers of two). It was found that in certain instances l is also a “ percolation” process where larger clustering (of the Kekulé space) occurs (D. Stauffer, Introduction to Percolation Theory, Taylor and Francis, London, 1985). Cases in which l percolates the Kekulé space correspond to benzenoid systems which are energetically more stable than those in which the resulting clusters are too small to allow percolation of the space by l. A fractal-like scale factor, s, is defined as the limit of the ratio of Clar counts to Kekulé counts in a given homologous series of benzenoid hydrocarbons as the molecule's length approaches infinity. This limit shows that with quasicrystalline benzenoids (branched and/or unbranched), the extent of data reduction using Clar structures is maximum (i.e. s = 0) and hence the use of such structures as a quantum-mechanical basis set in the method of Herndon and Hosoya (W.C. Herndon and H. Hosoya, Tetrahedron, 40 (1984) 3987) is most suited for this class of benzenoid systems. The linear acenes, however, lead to a value of s = 1 and hence, as the size of the molecule gets larger, the computational scheme of Herndon and Hosoya becomes progressively less efficient. This scale factor has certain arithmetical properties which are in harmony with the ultraviolet spectra of the corresponding benzenoid systems.

Graph theoretical analysis of large benzenoid hydrocarbons

We consider the problem of enumeration of so called Clar valence structures in large benzenoid systems. It is known that computationally the problem is NP-hard (i.e. the algorithmic complexity is non-polynomial). We outline an algorithm which is rather straightforward in applications. The approach is constructional, however, its implementation is simple, as one is not required to test the generated valence structures for duplicates, and the systematic application makes it impossible to overlook any of the possible Clar structures. The approach is illustrated on numerous polycyclic benzenoid systems, including some relatively large structures. These valence structures are of quantum-chemical importance and have been used to reproduce SCF resonance energies of benzenoid systems as described by W.C. Herndon and H. Hosoya, Tetrahedron, 40 (1984) 3987.

Combinatorial properties of some macrocyclic polyhex systems

Explicit formulae are given for the number of Clar structures, ζ (which represent maximal independent sets of vertices), of several classes of corona-condensed polyhex systems (primitive coronoids). There is a striking equivalence between expressions for ζ and K, the Kekulé counts for the classes reported by Bergan, Cyvin and Cyvin.

On the Relative Stability of Nonbenzenoid Alternant Hydrocarbons

Abstract We reinvestigated a number of nonbenzenoid alternant hydrocarbons. The concepts of parity and algebraic structure count have been reevaluated and apparent difficulties with extension of such concepts to large nonbenzenoids clarified. This work advocates the approach based on the use of Clar's structures and maximal valence structures instead of ASC approach for characterizing non-benzenoid alternant hydrocarbons and for predicting their relative stabilities.

ChemInform Abstract: Fibonacci Numbers in the Topological Theory of Benzenoid Hydrocarbons and Related Graphs

Fibonacci numbers are studied with respect to the topological theory of benzenoid hydrocarbons. These numbers are identified as the number of Kekule structures of nonbranched all-benzenoid hydrocarbons, the number of matchings of paths, the number of independent sets of vertices of paths, the number of nonattacking rooks of certain rook boards, as well as the number of Clar structures of certain benzenoid hydrocarbons. Fibonacci numbers were also identified as the number of conjugated circuits of certain benzenoid hydrocarbons and thus they were also related to the structure-resonance model. Maximal independent sets of caterpillar trees are also shown to be Fibonacci numbers.

Fibonacci numbers in the topological theory of benzenoid hydrocarbons and related graphs

Fibonacci numbers are studied with respect to the topological theory of benzenoid hydrocarbons. These numbers are identified as the number of Kekule structures of nonbranched all-benzenoid hydrocarbons, the number of matchings of paths, the number of independent sets of vertices of paths, the number of nonattacking rooks of certain rook boards, as well as the number of Clar structures of certain benzenoid hydrocarbons. Fibonacci numbers were also identified as the number of conjugated circuits of certain benzenoid hydrocarbons and thus they were also related to the structure-resonance model. Maximal independent sets of caterpillar trees are also shown to be Fibonacci numbers.

Binomial-combinatorial properties of clar structures

A Clar structure is defined to be a maximal indepenent set of vertices of the Clar graph of the corresponding benzenoid hydrocarbon. A special type of coloring, called a Clar coloring which is a specific type of 2-coloring is applied to Clar structures of two types of benzenoid hydrocarbons, viz., (a) nonbranched all-benzenoid systems, and (b) necklace-type hydrocarbons. It is found that the Clar counts of these two systems define what may be termed a ‘delayed’ Fibonacci sequence, in contrast to their Kekulé counts which form a fibonacci sequence. Several combinatorial properties of Clar counts are given (eqs. (3)–(30)). The results emphasize the advantage of using Clar structures as bases of the (recently proposed [13]) valence-bond Hamiltonian instead of Kekulé structures.

Combinatorial Clar sextet theory

Clar structures recently used as basis-set to compute resonance energies [9] are identified as maximal independent sets of benzenoid hydrocarbons “colored” in a special way. Binomial properties of such objects are induced for several catafusenes and perifusenes (Eqs. 2–31). Novel polynomials, called Clar polynomials, are given for perifusens in terms of units of catafusenes which allow display and enumeration of the populations of their Clar structures. The work is particularly pertinent to that of [8] and [9].

Parameterized valence bond calculations for benzenoid hydrocarbons using clar structure

SCF (Dewar-de Llano) calculations are used to parameterize a valence bond theory with a basis of Clar structures. A precise correlation of calculated resonance energies is found (corr. coeff. 0.9998). Graphtheoretic algorithms for carrying out the calculations are illustrated. It is suggested that this theoretical approach would be useful in considering structure and reactivity problems in very large polycyclic benzenoid aromatic hydrocarbons.