Kekule Valence Structures Research Articles

Mathematical chemistry illustrations: a personal view of less known results

We have selected to review half a dozen outstanding results of application of mathematics to chemistry, which despite their importance are not well and widely known to chemists. Most of them involve novel mathematical concept or part of mathematics not widely known in chemistry, such as concept of circuits with alternating C=C and C–C bonds within individual Kekule valence structures, known as conjugated circuits; paths and partial ordering leading to periodic tables of isomers; aromatic sextets in polycyclic benzenoid compounds; local aromaticity derived from ring bond orders based on Pauling CC bond orders; construction of orthogonal molecular descriptors producing stable regression equations; graphical and numerical representation of DNA and proteins; numerical characterization and comparative study of proteomics maps; and finally we will overview the exact solution to the protein alignment problem.

An extension of the Fries rule to non-benzenoid hydrocarbons having four-atomic rings

The study contains a deductive search for the principal monocyclic substructures determining different importances (weights) of separate Kekule valence structures of phenylenes and their congeners. Individual Kekule structures are modelled as continuous conjugated systems consisting of uniform double (C=C) bonds connected by uniform single (C–C) bonds, the latter being substantially weaker as compared to the former. The relevant total $$\pi$$ -electron energies are shown to offer an adequate criterion for ordering of the structures concerned according to their weights. These energies, in turn, are derived in the form of power series with respect to the small resonance parameter of C–C bonds. Analysis of expressions for separate members of this series shows that the cyclobutadienoid rings (if any) are the most important destabilizing contributors to the energy of the given structure, whereas the benzenoid rings and cycles like 3,4-dimethylene cyclobutene take the second place (their increments are stabilizing and destabilizing, respectively). Additivity and transferability of the above-enumerated contributions to energies of Kekule valence structures also are among the conclusions. These results provide us with an extension of the classical Fries rule to non-benzenoid hydrocarbons having four-atomic rings. Specific examples of the given class of compounds are considered in a detail, viz. biphenylene, [3]phenylene, as well as benzo- and naphthocyclobutenes. The relation of the approach applied to the theory of conjugated circuits also is discussed.

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 Kekule valence structures of these molecules.

Characteristic polynomials of alternant edge weighted linear chains with subsequent application to some linear poly (p-phenylene) graphs

The algorithms for the calculation of the characteristic polynomial coefficients for the alternant edge weighted graphs of linear chains have been developed. These algorithms have been utilized to calculate the characteristic polynomials of the linear poly (p-phenylene) and the methylene substituted linear poly (p-phenylene) compounds. These compounds are found to be important materials for electro-optical and electronic applications. The recurrence relation to obtain the sum of the CP coefficients \({\left({S_{\rm CP}^{N_r }}\right)}\) of any such poly (p-phenylene) in terms of the respective sums of its lower analogs has been derived. The number of Kekule valence structures (K) for such molecular graphs have been presented. Excellent linear correlations of logarithm of sum of CP coefficients \(({\rm log} \, {S_{\rm CP}^{N_r }})\) and logarithm of Kekule valence structures count (log K) with the ambient conditions density and bulk modulus of linear poly (p-phenylene) have also been found.

ChemInform Abstract: Maximum Valence Structures in Nonbenzenoid Polycyclic Hydrocarbons

A number of polycyclic nonbenzenoids obtained by fusion benzene and cyclobutadiene rings were examined using the maximum valence structure model. The model assumes that only Kekule valence structures associated with the largest weight contribute to the overall molecular description. The weights of individual valence structures were determined by the smallest Pauling bond order found for any of CC double bonds occurring in a particular structure. Similarities and differences between the maximum valence structure model and models using all Kekule valence structures are discussed.

On Kekulé structures count

Kekule valence structures offer qualitative description of conjugated cyclic hydrocarbons, suggesting localization of CC single and CC double bonds. Generation of all geometric Kekule structures for families of polyhex tori of a given tube cross-section, provided by the TORUS software, was performed by the new Kekule Count software, developed on a coloring algorithm. The sets of all non-isomorphic Kekule structures for the considered molecules, as the main result of this study, are presented.

Correlations between various ways of accounting for the distribution of π-electrons in benzenoids

The π-electrons in polycyclic benzenoid hydrocarbons can be counted in two different ways: (i) as partitions of π-electrons in each ring of the benzenoid (according to the conventions that all resonance structures contribute equally, that shared double bonds resonance structures contribute with one π-electron to each of the condensed rings, and that both π-electrons of an unshared double bond belong to the corresponding ring) and (ii) as benzenoid signature, i.e. a sequence of six real numbers counting for all resonance structures the numbers of times a ring is assigned according to the above conventions si π-electrons, where i takes integer values between 6 and 1. For bridging the gap between these two ways of accounting for the same number of π-electrons, various types of resonance structures (also called Kekule valence structures) are examined and mixed: Fries, Clar, anti-Fries, and quasi-Clar structures.

Algebraic Kekule Formulas for Benzenoid Hydrocarbons.

By assigning two π-electrons of CC double bonds in a Kekule valence structure to a benzene ring if not shared by adjacent rings and one π-electron if CC double bond is shared by two rings we arrived at numerical valence formulas for benzenoid hydrocarbons. We refer to numerical Kekule formulas as algebraic Kekule valence formulas to contrast them to the traditional geometrical Kekule valences formulas. The average over all numerical Kekule 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 π-electron sextets to indicate aromatic benzenoids by placing inscribed circles in adjacent ringswhich contradicts Clar's characterization of benzenoid hydrocarbons.

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

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

Generalized bond orders

AbstractWe introduce generalized bond orders defined in terms of weighted Kekule valence structures. The weights were determined by the contributions of linearly independent and minimal conjugated circuits in individual Kekule valence structure. When special values for the contributions of conjugated circuits of different size are assumed, one obtains quantities that show considerable similarity to the Pauling and the Clar's bond orders. Pauling bond orders are obtained when one assumes that all conjugated circuits make equal contribution to bond orders. © 1994 John Wiley & Sons, Inc.

Maximum valence structures in nonbenzenoid polycyclic hydrocarbons

A number of polycyclic nonbenzenoids obtained by fusion benzene and cyclobutadiene rings were examined using the maximum valence structure model. The model assumes that only Kekule valence structures associated with the largest weight contribute to the overall molecular description. The weights of individual valence structures were determined by the smallest Pauling bond order found for any of CC double bonds occurring in a particular structure. Similarities and differences between the maximum valence structure model and models using all Kekule valence structures are discussed.

Fully benzenoid systems revisited

Fully benzenoid hydrocarbons are benzenoid hydrocarbons in which benzene rings of a polycyclic benzenoid system are either assigned to isolated π sextets or are shown as “empty”. For a number of polycyclic benzenoid hydrocarbons we examined Kekule valence structures in order to characterize those that contribute to Clar's valence formulae depicting π sextets. We found that a particular weighting approach to Kekulé valence structures can result in a structural formula which in a number of benzenoids coincides with their Clar structure. The weights are determined by the smallest Pauling bond order that occurs for CC double bonds appearing in a Kekule valence structure. All valence structures which have maximal weights upon superposition produce a molecular structural formula which in some fully benzenoid systems coincides with Clar's representation but in others suggests a structure with fewer π sextets, suggesting, so to speak, less “perfect” fully benzenoid systems.

Conjugation and aromaticity of macrocyclic systems

Using a graph theoretical approach we have analyzed the aromatic stability of macrocylic systems having ten and fewer fused benzene rings. The approach requires construction and examination of Kekule valence structures of the macrocyclic systems considered. The Kekule valence structures for a molecule were systematically derived and the conjugated circuits in the individual Kekule formulas were enumerated subsequently. The counting results is the expression for molecular resonance energy, from which relative aromatic stabilities can be established. We report also on molecular local features, including ring benzene characters and Pauling bond orders.

On global and local properties of clar Pi-Electron sextets

The individual rings in benzenoid systems are studied via a new structural index, called Clar's ring character. The new index, which is based onselected Kekule valence structures, differs significantly in a number of cases from similar ring characterizations based onall Kekule valence structures. It is related to previously introduced quantities based on MO functions, but it can be obtained simply (i.e. without the need of a computer). We have illustrated the new ring index for several families of compounds, and report a number of regularities for a homologous series of molecules.

A statistical approach to resonance energies of large molecules

For large conjugated molecules, resonance energies of SCF MO quality are not available. Here we outline a method of determining molecular resonance energies by combining a graph theoretical approach to aromaticity with a statistical analysis of random Kekule valence structures. The approach involves construction of random Kekule valence forms and subsequent enumeration of conjugated circuits within each such structure.

On the stability of conjugated hydrocarbon ions

We consider the question of predicting the stability of hypothetical structures that are too large for ab initio calculations, too complex for simple MO calculations, and too few for reliable semiempirical SCFMO schemes. The problem is illustrated on a polycyclic conjugated cation having 21 carbon atoms. The analysis is general and applies to conjugated hydrocarbons and their ions for which it is not apparent whether they exhibit aromatic characteristics (and, if they do, to what extent). For the investigated polycyclic cation we find that to a large extent its aromatic components are cancelled by its antiaromatic components resulting in an essentially nonaromatic system. The analysis, which is based on close examination of various contributing conjugated circuits, allows one to examine the partial role of the individual Kekule valence structures as well as to foretell the nature of both the positively and the negatively charged system. We conclude that the particular cation should be stable but represents a challenge as it would be a cation with a rather small resonance energy. In contrast, the corresponding anion is relatively stable and, at least from a theoretical point of view, is less interesting.

On the role of kekule valence structures

Abstract