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.     

Generalized bond orders

We 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.     

Flag graphs and their applications in mathematical chemistry for benzenoids

Flag graphs have been used in the past for describing maps on closed surfaces. In this paper we use them for the first time in mathematical chemistry for describing benzenoids and some other similar structures. Examples include catacondensed and pericondensed benzenoids. Several theorems are included. Symmetries of benzenoid systems, flag graphs, and symmetry type graphs are briefly discussed.     

Strain-free total resonant sextet benzenoids and their antisextet dualists and retro-leapfrogs

The relationships between inner dual, dualist, sextet dualist, and antisextet dualist graphs and excised internal structures are fully delineated. It is proven that the antisextet dualist graph uniquely characterizes all strain-free benzenoids. At the same time, it is shown that the excised internal structure and dualist graph of the retro-leapfrog of all strain-free total resonant sextet benzenoids correspond to their antisextet dualist and sextet dualist graphs, respectively.     

A new yardstick for benzenoid polycyclic aromatic hydrocarbons

The newly introduced signature of benzenoids (a sequence of six real numbers Si with i = 6-1) shows the composition of the pi-electron partition by indicating the number of times all rings of the benzenoid are assigned 6, 5, 4, 3, 2, or 1 pi-electrons. It allows the introduction of a new ordering criterion for such polycyclic aromatic hydrocarbons by summing some of the terms in the signature. There is an almost perfect linear correlation between sums S6 + S5 and S4 + S3 for isomeric cata- or peri-fused benzenoids, so that one can sort such isomers according to ascending s 6 + S5 or to descending S4 + S3 sums (the resulting ordering does not differ much and agrees with that based on increasing numbers of Clar sextets and of Kekule structures). Branched cata-condensed benzenoids have higher S6 + S5 sums than isomeric nonbranched systems. For nonisomeric peri-condensed benzenoids, both sums increase with increasing numbers of benzenoid rings and decrease with the number of internal carbon atoms. Other partial sums that have been explored are S6 + S5 + S3 And S6 + S2 + S1, and the last one appears to be more generally applicable as a parameter for the complexity of benzenoids and for ordering isomeric benzenoids.     

A combinatorial/geometric analysis of parallelogram-like benzenoids

We recently reported an algorithm to count Kekulé (resonance) structures for convex cyclofusenes using a combinatorial/geometric approach. Previously, we presented an algorithm for counting resonance structures for parallelogram-like benzenoids with holes by counting descending paths using rectangular meshes with holes. In this article, we employ a similar combinatorial/geometric approach to determine algorithms that will facilitate counting of the resonance structures in parallelogram-like benzenoids with no holes.     

An ab initio valence bond study on cyclopenta-fused naphthalenes and fluoranthenes

To probe the effect of external cyclopenta-fusion on a naphthalene core, ab initio valence bond (VB) calculations have been performed, using strictly atomic benzene p-orbitals and p-orbitals that are allowed to delocalize, on naphthalene (1), acenaphthylene (2), pyracylene (3), cyclopenta[b,c]acenaphthylene (4), fluoranthene (5), and cyclopenta[c,d]fluoranthene (6). For the related compounds 1-4 and 5,6 the total resonance energies (according to Pauling's definition) are similar. Partitioning of the total resonance energy in contributions from the possible 4n + 2 and 4n pi-electron conjugated circuits shows that only the 6pi-electron conjugated circuits (benzene-like) contribute to the resonance energy. The results show that cyclopenta-fusion does not extend the pi system in the ground state; the five-membered rings act as peri-substituents. As a consequence, the differences in (total) resonance energy do not coincide with the differences in thermodynamic stability. Notwithstanding, the relative energies of the Kekule structures can be estimated using Randic's conjugated circuits model.     

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.     

The valence bond calculations for conjugated hydrocarbons having 24–28 π‐electrons

A novel algorithm is introduced for coding all Slater determinants in the covalent space with conserved S_Z, the z component of total spin S for a classical valence bond (VB) model. It effectively minimizes the search time and the storing space in the central memory of the computer. In cooperation with symmetry reductions based on molecular point group and spin inversion, the VB calculations have been extended to benzenoid hydrocarbons of up to 28 π‐electrons that have 4×10⁷ configurations. The low‐lying states of benzenoids with 24, 26, and 28 π‐electrons have been obtained for 62 species. To rationalize the aromaticity of benzenoids in a VB scheme, the resonance energy per hexagon (REPH) is defined. A linear correlation between the REPH and the energy gap of the ground (singlet) state and the first excited (triplet) state for 89 benzenoids is established. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 856–869, 2000     

Qualitatively graph-theoretical study on stability and formation of fullerenes and nanotubes

Kekule structures of different carbon species have been determined. On the basis of Kekule structure and C-C bond counts as well as the surface curvature, stability of diverse carbon species, driving force for curling of graphite fragments and formation of fullerenes and nanotubes, have been discussed. Curling of graphite flat fragments, end-capping of nanotubes, and closure of curved structures are driven by a tremendous increase in Kekule structures as terminal carbon atoms couple their dangling bonds into C-C o bonds. The increasing tendency becomes particularly striking for large cages and nanotubes. Resonance among numerous Kekule structures will stabilize the curved structure and dominate formation of closed carbon species. For similar carbon cages with comparable Kekule structure counts in magnitude, the surface curvature of carbon cages, as a measure for the strain energy, also plays an important role in determining their most stable forms.     

Resonance structure counts in contorted and flat hexabenzocoronenes

In several articles we have reported algorithms to count resonance structures for convex cyclofusenes and parallelogram-like benzenoids with and without holes using a combinatorial/geometric approach. In this article, using the same approach we report algorithms that facilitate resonance structure counts in both contorted and flat hexabenzocoronene.     

Resonance energies of large conjugated hydrocarbons by a statistical method

We developed a theoretical method for studying the aromatic stability of large molecules, molecules having a dozen and more fused benzene rings. Such molecules have so far often been outside the domain of theoretical studies. Combining the statistical approach and a particular graph theoretical analysis, it is possible to derive the expressions for molecular resonance energy for molecules of any size. The basis of the method is enumeration of conjugated circuits in random Kekulé valence structures. The method has been applied to evaluation of the resonance energies of conjugated hydrocarbons having about a dozen fused benzene rings. The approach consists of (1) construction of random Kekulé valence structures, (2) enumeration of conjugated circuits within the generated random valence structures, and (3) application of standard statistical analysis to a sufficiently large sample of structures. The construction of random valence forms is nontrivial, and some problems in generating random structures are discussed. The random Kekulé valence structures allow one not only to obtain the expression for molecular resonance energies (RE ) and numerical estimates for RE , but also they provide the basis for discussion of local molecular features, such as ring characterization and Pauling bond orders.     

Fusenes and benzenoids with perfect matchings

In this paper, a method to constructively enumerate fusenes and benzenoids with perfect matchings is given. It is based on an algorithm for generating all fusenes and benzenoids and introduces new restrictions into the generation process that avoid the generation of structures without perfect matchings.      

A unified theory of the stability of benzenoid hydrocarbons

The spectral density operator technique is proved to be a convenient tool for derivation of approximate topological formulae for total pi-electron energy (E_pi) of benzenoid hydrocarbons (BH s). Developed mathematical formalism points out a common origin of three different measures for the stability of BH s, namely: resonance energy (RE), number of Kekule structures (K) and HOMO -LUMO separation (X_HL). In turn, a novel topological invariant corresponding to “normalized” RE is derived. Numerical calculations for a representative set of BH s demonstrate effectiveness of the present approach. Various approaches to an estimation of BH stabilities are discussed.     

k-resonant benzenoid systems and k-cycle resonant graphs.

A benzenoid system (or hexagonal system) H is said to be k-resonant if, for 1 < or = t < or = k, any t disjoint hexagons of H are mutually resonant; that is, there is a Kekule structure (or perfect matching) K of H such that each of the k hexagons is an K-alternating hexagon. A connected graph G is said to be k-cycle resonant if, for 1 < or = t < or = k, any t disjoint cycles in G are mutually resonant. The concept of k-resonant benzenoid systems is closely related to Clar's aromatic sextet theory, and the concept of k-cycle resonant graphs is a natural generalization of k-resonant benzenoid systems. Some necessary and sufficient conditions for a benzenoid system (respectively a graph) to be k-resonant (respectively k-cycle resonant) have been established. In this paper, we will give a survey on investigations of k-resonant benzenoid systems and k-cycle resonant graphs.     

Global Forcing Number of Benzenoid Graphs

A global forcing set in a simple connected graph G with a perfect matching is any subset S of E(G) such that the restriction of the characteristic function of perfect matchings of G on S is an injection. The number of edges in a global forcing set of the smallest cardinality is called the global forcing number of G. In this paper we prove several results concerning global forcing sets and numbers of benzenoid graphs. In particular, we prove that all catacondensed benzenoids and catafused coronoids with n hexagons have the global forcing number equal to n, and that for pericondensed benzenoids the global forcing number is always strictly smaller than the number of hexagons.     

The Clar covering polynomial of hexagonal systems Ⅱ.An application to resonance energy of condensed aromatic hydrocarbons

The Clar covering polynomial of hexagonal systems is a recently proposed1,2 concept which contains much more topological properties of condensed aromatic hydrocarbons, such as Kekule structure count, Clar number, first Herndon number, etc. It is shown that this polynomial can be used for calculating the resonance energy of condensed aromatic hydrocarbons with better accuracy.