Retro-leapfrog and related retro map operations

Operations on maps are well-known theoretical tools for transforming a given polyhedral tessellation. Several theoretical investigations of fullerenes, such as their pi-electronic structure and stability, need information on the original map which was transformed into a larger molecular structure. In this respect, retro-operations, particularly those of the most used leapfrog, chamfering, and capra operations, appear particularly useful in searching the associate graphs of fullerenes. A series of analyzed cages proved to be leapfrog transforms of smaller cages. This information was useful in understanding their closed pi-electronic structure and related properties including the local aromaticity. An index based on the optimized geometries enabled the evaluation of aromaticity of their various substructures. Pictorial images of the pi-electron distribution as the main Kekulé valence structures have been performed by the aid of the JSCHEM software package.     

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.     

Applying the conjugated circuits method to Clar structures of [n]phenylenes for determining resonance energies

We consider the aromaticity of biphenylene and structurally related linear or angular [n]phenylenes for which the direct application of the model of conjugated circuits does not offer valid expressions for resonance energy and aromaticity. We located the cause of this problem as being due to Kekulé valence structures in which neighboring benzenoid rings are connected by two CC double bonds. By restricting the selection of Kekulé valence structures to those that contribute to Clar structures of such systems, we were able to show that linear and angular [n]phenylenes have approximately similar resonance energies, with angular [n]phenylenes being slightly more stable due to second order contributions arising from disjoint conjugated circuits. Expressions for resonance energies of [n]phenylenes up to n = 8 are listed and recursion expressions for higher n values are outlined.     

Numerical Kekulé structures of fullerenes and partitioning of pi-electrons to pentagonal and hexagonal rings

We calculated the partitioning of pi-electrons within individual pentagonal and hexagonal rings of fullerenes for a collection of fullerenes from C20 to C72 by constructing their Kekulé valence structures and averaging the pi-electron content of individual rings over all Kekulé valence structures. The resulting information is collected in Table 2, which when combined with the Schlegel diagram of fullerenes (illustrated in Figure 7) uniquely characterizes each of the 19 fullerenes considered. The results are interpreted as the basic information on the distributions (variation) of the local (ring) pi-electron density.     

Nanotube junctions and the genus of multi-tori

Carbon nanotube junctions can be modeled by fullerene spanning or by using some operations on map. They can self-assemble into more complex structures, such as finite or infinite high genera multi-tori. Four junctions of tetrahedral and octahedral symmetry, covered by patches consisting only of hexagons, were designed. Their stability is discussed in terms of total energy, evaluated at Hartree-Fock (HF) level of theory, HOMO-LUMO gap, strain energy, HOMA index of aromaticity and the Kekulé structure count. Vibrational spectra of these junctions are given as well. A new multi-toroidal structure, of octahedral symmetry, is presented for the first time. The study on topology of the multi-tori herein designed revealed the relation of these structures with the genus of their embedding surface.     

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.     

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.     

Molecules-in-molecule estimation of the extent of localization of Kekuléan substructures in polycyclic aromatic hydrocarbons

This article first revises graph-theoretical (local aromaticity and overall molecular) indices, introduced by M. Randi? in 1975, for benzenoid hydrocarbons and somewhat improves them for computer enumeration. This goes beyond total Kekulé structure enumeration, yielding an index calculation useful for the quantitative estimation of localization of different Kekuléan substructures (including ethylene-, benzene-, annulene-, and radialene-units). This may be viewed as a "molecules-in molecule" approach to polycyclic aromatic hydrocarbons within the context of graph theoretical partitioning.     

An alternative strategy for count and storage of Kekulé and longer range resonance valence bond structures

In this paper, we suggest a simple representation for notation of Kekulé valence bond (KVB) structures and longer range resonance valence bond (RVB) structures, which is called "the adjacency bonding array". In this representation, only an N component one-dimensional array is needed for inscribing each KVB or longer range RVB structure for an N-carbon system. Based on the adjacency bonding arrays, we develop very efficient algorithms for the systematic search of KVB and RVB structures as well as evaluation of the basis set overlap and Hamiltonian matrices.     

Aromaticity of carbon nanotubes

Carbon nanotubes are composed of cylindrical graphite sheets. Both nanotubes and graphite sheets are benzenoid derivatives composed of sp2 carbon atoms arranged in a hexagonal pattern. Therefore both systems are aromatic. The extent of the aromatic character of a molecule G (here benzenoids) can be explained in terms of the number of possible Kekulé structures in G. In this work the Kekulé structures in carbon nanotubes and the corresponding, rectangular, graphite-sheets the tubes might originate from, were enumerated. It was shown that (2,2), (3,3), and (4,4) carbon nanotubes are more aromatic than the corresponding, rectangular, planar structures. This explains why it might be more difficult to saturate nanotubes by addition reactions than the respective, "narrow", graphite sheets.     

Resonance and aromaticity: an ab initio valence bond approach

Resonance energy is one of the criteria to measure aromaticity. The effect of the use of different orbital models is investigated in the calculated resonance energies of cyclic conjugated hydrocarbons within the framework of the ab initio Valence Bond Self-Consistent Field (VBSCF) method. The VB wave function for each system was constructed using a linear combination of the VB structures (spin functions), which closely resemble the Kekulé valence structures, and two types of orbitals, that is, strictly atomic (local) and delocalized atomic (delocal) p-orbitals, were used to describe the π-system. It is found that the Pauling-Wheland's resonance energy with nonorthogonal structures decreases, while the same with orthogonalized structures and the total mean resonance energy (the sum of the weighted off-diagonal contributions in the Hamiltonian matrix of orthogonalized structures) increase when delocal orbitals are used as compared to local p-orbitals. Analysis of the interactions between the different structures of a system shows that the resonance in the 6π electrons conjugated circuits have the largest contributions to the resonance energy. The VBSCF calculations also show that the extra stability of phenanthrene, a kinked benzenoid, as compared to its linear counterpart, anthracene, is a consequence of the resonance in the π-system rather than the H-H interaction in the bay region as suggested previously. Finally, the empirical parameters for the resonance interactions between different 4n+2 or 4n π electrons conjugated circuits, used in Randi?'s conjugated circuits theory or Herdon's semi-emprical VB approach, are quantified. These parameters have to be scaled by the structure coefficients (weights) of the contributing structures.     

Defining rules of aromaticity: a unified approach to the Hückel, Clar and Randić concepts

The molecular structure of any system may be unambiguously described by its adjacency matrix, A, in which bonds are assigned entry a(ij) = 1 and non-bonded pairs of atoms entry a(ij) = 0. For π-electron-containing conjugated hydrocarbons, this matrix may be modified in order to represent one of the possible Kekulé structures by assigning entry 1 to double bonds and entry 0 to single bonds, leading to the Kekulé matrix K which can be obtained from the A matrix by subtracting 1 from elements a(pq) that represent single bonds in the Kekulé structure. The A and K matrices are the boundary cases of a general matrix A(ε), named perturbation matrix, in which from elements a(pq) that represent single bonds is subtracted a value ε∈<0,1> representing the magnitude of the perturbation. The determinant of the A(ε) matrix is unambiguously represented by an appropriate polynomial that, in turn, can be written in a form containing terms ±(1-ε)(N/2) that identify types of π-electron conjugated cycles (N is the corresponding number of π-electrons). If the sign before the term is (+), then the contribution is stabilizing, but if it is (-) the contribution is destabilizing. The approach shows why and how the Hückel rule works, how the Randi? conjugated circuits result from the analysis of canonical structures, and also how the Clar rule may be extended to include aromatic cycles larger than six-membered (aromatic sextet).     

Valence-bond determination of bond lengths of polycyclic aromatic hydrocarbons: comparisons with recent experimental and ab initio results

Pauling's valence-bond (VB) method for determining bond lengths is compared to ten recent literature experimental and theoretical results and is shown to give comparable results. His method only requires computation of the number of Kekulé (K) and Dewar structures (DS) of conjugated hydrocarbons. Both K and DS are obtained from the last two coefficients of the matching polynomial which is also used to obtain topological resonance energy (TRE). A molecular fragmentation method is given for determining DS of essentially disconnected polycyclic aromatic hydrocarbons (PAHs). Both Kekuléan alternant and nonalternant PAHs, including essentially disconnected and non-Kekuléan systems, have bond lengths that are easily determined by this method.     

Aromaticity of corazulenic fullerenes

A novel class of non-classical fullerenes, having pentagon–heptagon pairs, as in azulene, is modeled. The various coverings, sometimes alternating azulenic and benzenic units, are designed by some new sequences of map operations or generalized operations. The hypothetical azulenic fullerenes are characterized by PM3 semiempirical data and POAV1 strain energy SE. Their aromaticity is discussed in the light of several criteria. The HOMA index of aromaticity enabled evaluation of global and local aromaticity of the designed fullerenes.     

Edge effects, electronic arrangement, and aromaticity patterns on finite-length carbon nanotubes

Previous investigations have revealed that even long carbon nanotubes (CNTs) retain bond patterns that are characterized by the localization of Clar rings. Even for CNTs with 10 nm length, an alternated, oscillating structure of Clar and Kekulé patterning was also found, indicating that these arrangements may possibly persist for even longer nanotubes, given that they are finite. In the present work, we perform a detailed and comprehensive theoretical study of this phenomenon, in order to find the causes that give rise to these patterns. A complete set of CNTs with different chiralities, diameters (up to 2 nm), lengths (up to 10 nm) and endings (capped, uncapped, and tailored endings) was considered for such purposes. The results indicate that the Clar patterning appears not only on armchair CNTs, but also on those with chiral angle values close to 30°, and this results in a stabilization of the structure, when compared with the uniform, zigzag CNTs. This stabilizing effect points to the causes that underlie the three Nakamura CNT types, resulting as the superposition of structures with a maximal number of Clar rings. Although there is a strict dependence on the border shape, the main cause of the bond patterning in long tubes is to be found in the intrinsic wrapping of each CNT, because the type and number of oscillations present in the longest structures do not depend on the particular length. Nevertheless, the three Nakamura types of armchair tubes appear to subsist beyond the appearance of oscillations, because each of these sets evolves in a different manner, and energy properties that link them together. Apart from the geometry, Clar patterning was investigated through NICS (Nucleus Independent Chemical Shifts) measures, which reveal a connection between the Clar rings and a local concentration of aromaticity.     

Aromaticity in transition metal oxide structures.

The concept of aromaticity is useful for understanding the properties of some polyoxometalates containing transition metals such as vanadium, molybdenum, and tungsten having structures based on metal macropolygons and macropolyhedra with M-O-M edges. Thus, the aromatic macrocuboctahedral Keggin ions readily undergo one-electron reductions to highly colored mixed-valence "blues" (e.g., molybdenum blue), whereas the macroicosahedral Silverton ions, M(IV)Mo12O42(8-) (M(IV) = Ce, Th, U), which, like cyclohexane, do not have vertex valence orbitals available for delocalization, do not undergo analogous reduction reactions. A macrohexagon of d1 vanadium(IV) atoms as V-O-V units has been imbedded into an electronically inactive borate matrix in the ion [V6B20O50H8](8-). The small beta unit for the V-O-V interactions in this V6 macrohexagon leads to an unprecedented example of high spin aromaticity with a paramagnetism corresponding to four unpaired electrons per V6 unit in contrast to benzene, which is diamagnetic and hence exhibits low spin aromaticity. The M-O-M interactions in these aromatic metal oxides are closely related to the Cu-O-Cu interactions in the high critical temperature superconducting copper oxides which are essential to the electron transport in these systems.     

Frequency upshift in BO2 and CO2+ upon electronic excitation: a twin-state model rationalization

The twin-state model, previously shown to provide a simple physical rationalization for the frequency upshift (exaltation) of the Kekulé mode in benzene upon S(0) to S(1) electronic excitation, is extended to the case of the BO(2) radical and the CO(2)(+) radical cation. In the case of BO(2)/CO(2)(+), the ground and excited states are degenerate, yet the model applies to the degenerate two-state system as well. In contrast with a pseudo-Jahn-Teller model, the twin-state one can predict which frequency is exalted and also which pair of electronic states are coupled, thus explaining the specificity of the phenomenon. The frequency exaltation is a spectroscopic manifestation of the resonance between the pair of VB structures describing twin states. In analogy with the case of benzene, it is predicted that the ν(3) asymmetric stretch fundamental will be a dominant peak in the two-photon absorption spectrum of BO(2) and CO(2)(+).     

Ab initio study of cyclobutadiene in excited states: optimized geometries,electronic transitions and aromaticities

An Ab initio SCF-CI study of planar cyclobutadiene (CB) in ground and excited states has been carried out. The equilibrium geometries of some valence and Rydberg states have been calculated, as well as the energies of the vertical (absorption and emission) and non-vertical transitions. Using the optimized geometries, it is discussed how the aromaticity changes upon excitation of CB to the lowest-lying singlet and triplet states. The following conclusion is made: upon excitation to the fluorescent (S₁) or phosphorscent (T₁), states, the aromaticity of the anti-Hückel system cyclobutadiene increases significantly, whereas that of the Hückel system benzene descreases.