Algebraic Method for Solving Multiple Degenerate Eigenvalues in [r]Triangulenes.

An algebraic procedure for overcoming the multiple degeneracy problem in eigenvalue (root) determination of the characteristic polynomial of 3-fold symmetrical molecular graphs is given. This leads to the tabulation of Hückel molecular orbital binding energy (Eπ) and eigenvalues (roots) for [2]triangulene to [9]trianguene for the first time. Triangulenes are the smallest possible condensed benzenoid polyradicals.

Assessment of the Performance of the Bond Resonance Energy, Circuit Resonance Energy, Magnetic Resonance Energy, Ring Currents, and Aromaticity of Anthracene, Biphenylene, Phenalenyl, and p-Terphenyl.

This paper contributes to the further understanding of bond resonance energy (BRE), circuit resonance energy (CRE), and ring currents (RC = I) as indices for the measure of local ring aromaticity and topological resonance energy (TRE) and magnetic resonance energy (MRE) as indices for the measure of global molecular aromaticity. Calculations of ring currents in the 3-ring polycyclic hydrocarbons of anthracene, biphenylene, phenalenyl, and p-terphenyl are demonstrated. This comparative study provides new molecular insights. A number of shortcut techniques are briefly illustrated. p-Terphenyl is a benzenoid total resonant sextet (TRS) and biphenylene is shown to be a nonbenzenoid TRS, i.e., both obey Clar's sextet principle. p-Terphenyl has only ring circuits, whereas anthracene, biphenylene, and phenalenyl have ring circuits and larger circuits engulfing the ring circuits. The perimeter circuits of anthracene, biphenylene, and phenalenyl engulf all other circuits, thus leading to the definition of MRE. By these several aromatic indices, the monocation, monoanion, and monoradical of phenalenyl are equally aromatic. The agreement in the prediction of local ring aromaticity between the three indices of m-BRE, t-BRE, and CRE is evaluated.

Bond Resonance Energy Verification of σ-Aromaticity in Cycloalkanes.

The Klein-Larson bonding model is best suited for describing σ-conjugation and σ-aromaticity of cycloalkanes. Methylene groups (-CH2-) in cycloalkanes are isoconjugates with ethylene groups (-CH═CH-) in annulenes, both contributing two electrons to cyclic conjugation and aromaticity. As in the case of π-aromatic stabilization, σ-aromatic stabilization must be associated with ring-current diamagnetism. Both nucleus-independent chemical shift (NICS) and bond resonance energy (BRE) values support this interpretation. Bond resonance energy has been used to determine local ring aromaticity in nonalternant hydrocarbons, alternant hydrocarbons, heterocyclic conjugated molecules, fullerenes, pure boron clusters, and porphyrins. This contribution represents the first application of bond resonance energy toward the evaluation of σ-aromaticity/σ-antiaromaticity of polycyclic alkanes.

Edge Effects in Benzenoid and Total Resonant Sextet Benzenoid Hydrocarbons and Clar's Sextet Principle.

Because the effect of a benzenoid perimeter edge on a sextet pattern is of paramount importance, trends in the degree of sextet aromaticity are determined. The spectacular structural isomorphism between the total set of benzenoid hydrocarbons and its strain-free total resonant sextet (TRS) subset as described herein is an unequivocal demonstration of the validity of the Clar sextet principle. Aromaticity and the Clar's sextet principle are intricately related which form the basis of this isomorphism. Hexagons on the edge of a polycyclic benzenoid hydrocarbon tend to have higher spin density, have a higher probability of sextet residence, be more reactive to substitution reactions, and have a perimeter topology described by four distinct parameters by means of a very simple relation. It is shown that because the location of Clar sextets in TRS benzenoids is totally fixed, the relative order of ring aromaticity of the peripheral benzene hexagons can be predicted from their topology. Using Bosanac and Gutman's energy effect ( ef) and Aihara's bond resonance energy, we show for the first time that the relative order of decreasing ring aromaticity for perimeter edge hexagons in any specific TRS benzenoid is quarto > trio > duo > solo > empty. TRS benzenoid hydrocarbons are those isomers in benzenoids having their number of formula carbons divisible by six and a maximum number of perimeter bay regions (ηo) given by η0(TRS) = (1/2) NH(sextet) - 3 + F. NH(sextet) is the number of formula hydrogens and F is the number of perimeter fjords (triplet of adjacent bay regions); for strain-free TRS benzenoids, F = 0.

Practical Tight-Binding Hückel-Like Modeling of Electronic Properties of Saturated Hydrocarbons: On σ-Aromaticity of Cyclopropane and Correlations of Alkane Ionization Energies and Enthalpies of Formation Corrected for Protobranching.

This testing of σ-aromaticity in cyclopropane and protobranching in alkanes via a Hückel-like model proposed by Klein and Larson is expedited by mirror-plane fragmentation per the method of McClelland originally developed for conjugated polyenes. Alkane carbon sp3 orbitals are represented by edge weighted tetrahedral stellate graphs connected to hydrogen orbitals.

Systematic Construction and Calculation of Electronic Properties of Fullerene Series Related by Rotational Symmetry: From Fullerenes to Bicapped Nanotubes.

The results herein demonstrate that the methods of circumscribing and the facile calculation of Hückel molecular orbital (HMO) eigenvalues by mirror-plane fragmentation have a broad application in the construction of carbon cluster series and the systematic study of trends in their electronic properties. In comparing open-ended nanotubes and their isomeric elongated fullerenes (bicapped nanotubes), we show that the former are more aromatic but the latter are more conjugated and that progressive elongation increases aromaticity and conjugation in both. Recursion equations that will allow one to obtain the eigenvalues to all 5-endcapped nanotubes are given.

A quantitative metric for conjugation in polyene hydrocarbons having a single classical structure.

A quantitative measure of the extent of conjugation is introduced. The number of Dewar resonance structures (DS) in a conjugated hydrocarbon quantitatively determines its extent of conjugation interaction. The greater the degree of conjugation the more stable is the conjugated hydrocarbon. The number of single bonds joining the exo-double bonds of a purely cross-conjugated hydrocarbon is equal to its number of Dewar resonance structures (DS). Polyene cross-conjugated hydrocarbons are minimally conjugated and possess only one classical structure with a succession of fixed single and exo-double bonds. There are three increasing levels of conjugation for systems with a single classical structure according to whether they are cross-conjugated, linear extended two-way conjugated, and ring-enhanced conjugated. Numerous analytical expressions have been derived. Our valence-bond (VB) approach involves studying the interrelationship of several well-known series of conjugated hydrocarbons having only a single classical structure, i.e., the number of Kekulé resonance structures is one (K = 1). This work provides a totally new perspective to the concept of conjugation.

Valence-bond determination of diradical character of polycyclic aromatic hydrocarbons: from acenes to rectangular benzenoids.

The number of Kekulé resonance structures in a conjugated hydrocarbon is a measure of its singlet ground state, and its number of Dewar resonance structures is a measure of the triplet contribution to its ground state. We present a 2-dimensional organized study of rectangular benzenoids having both armchair and zigzag edges that play an important role in the electronic properties of nanographene patches and strips. Analytical expressions for the number of Dewar resonance structures (DS) for select series of rectangular benzenoid hydrocarbons are presented for the first time. An index of biradical character present (BRC) in polycyclic aromatic hydrocarbons is defined and illustrated. Biradical character is strongly influenced by zigzag edges and the presence of a diradical excised internal structure in polycyclic aromatic hydrocarbons.

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.

Bile acid-based cage compounds with lipophilic outer shells and inner cavities.

The uniquely functionalized steroid-based cyclodimers 4, cis-5, and trans-5 have been synthesized and fully characterized.The cyclodimer 5, with a cis-trans ratio of 3:1, is obtained by coupling the terminal alkenes of two 4-pentenoate groups on a cyclodimer 4 via Grubbs' intramolecular ring-closing metathesis. The crystal structure shows cis-5 to be a cagelike cyclic oligomer bridged by the flexible oct-4-enedioate link.

The polyhex/polypent topological paradigm: regularities in the isomer numbers and topological properties of select subclasses of benzenoid hydrocarbons and related systems.

The informatics on benzenoid hydrocarbons and related systems is reviewed. This informatics involves a collection of algorithms used to discover, process, and analyze patterns in structure/formula data sets. The unique organizational structure that emerges will be useful to chemists with an interest in polycyclic aromatic compounds, environmental chemists, material scientists, and other scientists wishing for a greater understanding of the formation of carbonaceous materials. This tutorial review will show that isomer enumeration can lead to significant chemical insights. The isomer number regularities will be shown to have a structural origin referred to as the polyhex/polypent topological paradigm.

13C nuclear magnetic resonance data of lanosterol derivatives--profiling the steric topology of the steroid skeleton via substituent effects on its 13C NMR.

The (13)C NMR spectra of over 24 tetracyclic triterpenoid derivatives have been structurally analyzed. The (13)C NMR chemical shifts allow one to probe the steric topology of the rigid steroid skeleton and inductive effects of its substituents. Use of deuterium labeling in chemical shift assignment and B-ring aromatic terpenoids are also featured.

Structure and electronic characteristics of coronoid polycyclic aromatic hydrocarbons as potential models of graphite layers with hole defects.

While the literature on coronoid hydrocarbons is comprehensively surveyed, new enumeration results on select coronoid classes are presented. The first examples of Kekulean polycyclic (even-carbon) alternant hydrocarbons with doubly degenerate zero eigenvalues at the HMO level are identified. The parallelism between a molecule removed and the antimolecule hole left in formation of a graphite vacancy defect is elaborated. It is argued that the formation of a single carbon atom vacancy hole defect in graphite should be facile because it would entail a minimum electronic reorganization. These results are supported by a substantial amount of eigenvalue calculations. This work provides a unique overview for the class of coronoid hydrocarbons.

On the spectacular structural isomorphism between CnHs monoradical and Cn+sHs+3 diradical benzenoid hydrocarbons: from reactive intermediates to vacancy (Hole) defects in graphite.

The formula/structure informatics of monoradical and diradical benzenoid hydrocarbons that are potential reactive intermediates is studied. Some new enumeration and structural results with analytical expressions are presented. The topological paradigm and one-to-one correspondence between the monoradical and diradical constant-isomer series is demonstrated. Constant-isomer benzenoid monoradicals of the formula CnHs have a one-to-one correspondence in isomer number and topology to constant-isomer diradicals of the formula Cn+sHs+3. Some electronic properties of benzenoid radicals are delineated. Excising out a monoradical or diradical benzenoid carbon molecule from a perfect hexagonal graphite layer leaves a matching monoradical or diradical vacancy hole defect called an antimolecule; this observation can be generalized to include excising out all nondisjoint and obvious benzenoid polyradicals from a perfect (Kekuléan) hexagonal graphite layer. It is shown that the characteristics of graphite vacancies (antimolecules) can be deduced from knowledge about the carbon molecules removed in their formation.

Mathematics of Periodic Tables for Benzenoid Hydrocarbons.

The upper and lower bounds for invariants of polyhex systems based on the Harary and Harborth inequalities are studied. It is shown that these invariants are uniquely correlated by the Periodic Table for Benzenoid Hydrocarbons. A modified periodic table for total resonant sextet (TRS) benzenoids based on the invariants of Ds and r(empty) is presented; Ds is the number of disconnections among the empty rings for fused TRS benzenoid hydrocarbons. This work represents a contribution toward deciphering the topological information content of benzenoid formulas.

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.

What do we know about C28H14 and C30H14 benzenoid hydrocarbons and their evolution to related polymer strips?

While critically reviewing the current status of what is known about C28H14 and C30H14 benzenoid isomers, which are ubiquitous pyrolytic constituents, some new insights will be presented. Representative isomers belonging to these benzenoid hydrocarbons are at the crossroads to homologous series that extend to infinite polymer strips that have been the object of numerous molecular modeling studies. The goal of these studies is to understand their potential electronic properties for material science applications. Along the way, some prior numerical results are supplemented.

Perimeter topology of benzenoid polycyclic hydrocarbons.

The equivalence of the perimeter topological equations derived by Dias and the 13 possible modes of hexagon adjacency in fused benzenoid systems subsequently presented by Cyvin, Gutman, and collaborators is shown. The aufbau principle for generation of all fused total resonant sextet (TRS) benzenoid hydrocarbons is proved. The precise distinction and topological properties between fused, nonfused, strain-free, strained, helical, and nonhelical benzenoid hydrocarbons are carefully delineated. The TRS benzenoid isomers have the maximum number of bay regions, and expressions for the number of bay regions that any given TRS benzenoid will have are given.

The most stable class of benzenoid hydrocarbons-new topological correlations of strain-free total resonant sextet benzenoids.

While briefly reviewing how the concepts of strictly pericondensed, strain-free, Clar's aromatic sextet, and symmetry are interconnected in the topological correspondence between strictly pericondensed and total resonant sextet (TRS) benzenoid hydrocarbons, new structural correlations in isomer numbers, symmetry distributions, and empty rings between various strain-free TRS benzenoids made up of only fused hexagonal rings are presented. These correlations are made evident by the Formula Periodic Table for Total Resonant Sextet Benzenoid Hydrocarbons [Table PAH6(sextet)] and application of the leapfrog operation. A new perimeter topology relationship for fused strain-free TRS benzenoids is derived. This work represents a contribution toward understanding formula/structure relationships of benzenoid hydrocarbons.