Violations of the Hückel (4n + 2) Rule

In polycyclic conjugated hydrocarbons the Hückel (4n + 2)-rule may be violated, so that certain (4n + 2)-membcred rings cause thermodynamic destabilization. The structural conditions for the occurrence of such an effect are analyze d. Eight examples (of which seven are new) of non-b enzenoid hydrocarbons are put forward, in which the Hückel rule is disobeyed.     

Double aromaticity in monocyclic carbon, boron, and borocarbon rings based on magnetic criteria

The double-aromatic character of selected monocyclic carbon, boron, and borocarbon rings is demonstrated by refined nucleus-independent chemical shift (NICS) analyses involving the contributions of individual canonical MOs and their out-of-plane NICS tensor component (CMO-NICS(zz)). The double aromaticity considered results from two mutually orthogonal Hückel p AO frameworks in a single molecule. The familiar pi orbitals are augmented by the in-plane delocalization of electrons occupying sets of radial (rad) p orbitals. Such double aromaticity is present in B(3) (-), C(6)H(3) (+), C(6) (4+), C(4)B(4) (4+), C(6), C(5)B(2), C(4)B(4), C(2)B(8), B(10) (2-), B(12), C(10), C(9)B(2), C(8)B(4), C(7)B(6), C(6)B(8), and C(14). Monocyclic C(8) and C(12) are doubly antiaromatic, as both the orthogonal pi and radial Hückel sets are paratropic. Planar C(7) and C(9) monocycles have mixed aromatic (pi) and antiaromatic (radial) systems.     

Quantitative approach to hückel rule the relations between the cycles of a molecular graph and the thermodynamic stability of a conjugated molecule

Contribution of the rings to the total π-electron energy (CE) is calculated for a number of polycyclic conjugated hydrocarbons. The Hückel 4m+2 rule is tested quantitatively in this way. It is shown that (4m)-membered rings always destabilize, while (4m + 2)-membered rings almost always stabilize the molecule. The magnitude of this effect is, however, subject to considerable variations. The main factors which determine the magnitude of CE are discussed. Examples of the alternant hydrocarbons are found for which the (4m + 2)-membered cycles have a destabilizing effect, and therefore violate the Hückel rule. The odd membered cycles are shown to have a negligibly small effect on the stability of the conjugated molecules.     

Retaining Hückel Aromaticity in the Triplet Excited State of Azobenzene

The implication of the potential concept of aromaticity in the relaxed lowest triplet state of azobenzene, an efficient molecular switch, using elementary aromaticity indices based on magnetic, electronic, and geometric criteria has been discussed. Azobenzene exhibits a major Hückel aromatic character retained in the diradical lowest relaxed triplet state (T₁) by virtue of a twisted geometry with partial delocalization of unpaired electrons in the perpendicular p-orbitals of two nitrogen atoms to the corresponding phenyl rings. The computational analysis has been expanded further to stilbene and N-diphenylmethanimine for an extensive understanding of the effect of closed-shell Hückel aromaticity in double-bond-linked phenyl rings. Our analysis concluded that stilbene has Hückel aromatic character in the relaxed T₁ state and N-diphenylmethanimine has a considerable Hückel aromaticity in the phenyl ring near the carbon atom while a paramount Baird aromaticity in the phenyl ring near the nitrogen atom of the C=N double bond. The results reveal the application of excited-state aromaticity as a general tool for the design of molecular switches.     

Directing power of cyclobutenoid annelations on the double bonds of planar cyclooctatetraenes

Ab initio and hybrid density functional quantum mechanical computations are applied to the structure and energetics of a series of two-atom-bridge annelated cyclooctatetraenes. The contribution of each annelation to the exo/endo relative energy is estimated. Key directing factors for a given type of annelation, such as strain, electronegativity, or cyclic electron count, can be sorted out by comparison of various bridge compositions. Overall, electron count and the essential components of the Clar/Robinson rule work well to predict the exo/endo preferences. Specifically, three 4-e(-) Hückel systems (CH-CH, NH-BH and NH-C(O)) display dominant exo forms whereas the three 4n + 2 Hückel counterparts (C(O)-C(O), BH-BH, and planar NH-NH) display a common preference for endo. These endo systems act like four independent four-membered "aromatic" rings linked by "single" bonds. An analysis based on the effective hybridization of carbon atoms in the annulene (Bent's rule) provides a rationale for subtle trends in their specific annulene geometry.     

Hückel theory applied to large linear and cyclic conjugated π-systems

By use of rotational symmetry (C_n-symmetry) a lower limit to the frontier orbital (HOMO-LUMO) gap in large molecules with linear and cyclic conjugated π-systems containing simple repetitive units has been calculated within the Hückel approxmation. The frontier orbitals are shown to be the same in series of cyclic oligomers and liner polymers containing the same repetitive units. The orbital gap is calculated from the repetitive units closed on themselves to give a ring of Hückel or, alternatively, Möbius topology depending on the number of conjugated π-electrons in a liner array between the ends of the repetitive unit. For 4n (4n+2) systems the small ring of Hückel (Möbius) topology will give the frontier orbitals.     

Planar Antiaromatic Core-Modified 24π Hexaphyrin(1.0.1.0.1.0) and 32π Octaphyrin(1.0.1.0.1.0.1.0) Bearing Alternate Hybrid Diheterole Units

The Lewis acid catalyzed self-condensation of hybrid diheterole (furan-pyrrole and thiophene-pyrrole) precursors has afforded novel Hückel antiaromatic 24π hexaphyrin(1.0.1.0.1.0) and 32π octaphyrin(1.0.1.0.1.0.1.0) structures without β-annulated bridges. Single-crystal X-ray diffraction analysis of the hybrid porphyrinoids ( S₃N₃-ox and O₄N₄-ox ) revealed a nearly planar conformation and the ¹H NMR spectra suggest the presence of paratropic ring currents. These antiaromatic macrocycles show characteristic optical features and underwent reversible two-electron reduction to Hückel aromatic 26π- and 34π-electron species, respectively, as is evident from the results of spectroscopic and theoretical studies (nucleus-independent chemical shift (NICS) and anisotropy of the current-induced density (ACID) calculations). The incorporation of hybrid diheteroles alternately into expanded porphyrin skeletons provides a novel approach to the fine-tuning of the electronic structures of planar antiaromatic macrocycles.     

Silicon‐Containing Formal 4π‐Electron Four‐Membered Ring Systems: Antiaromatic,Aromatic, or Nonaromatic?

Density functional theory calculations (B3LYP) have been carried out to investigate the 4π‐electron systems of 2,4‐disila‐1,3‐diphosphacyclobutadiene (compound 1 ) and the tetrasilacyclobutadiene dication (compound 2 ). The calculated nucleus‐independent chemical shift (NICS) values for these two compounds are negative, which indicates that the core rings of compounds 1 and 2 have a certain amount of aromaticity. However, deep electronic analysis reveals that neither of these two formal 4π‐electron four‐membered ring systems is aromatic. Compound 1 has very weak, almost negligible antiaromaticity, and the amidinate ligands attached to the Si atoms play an important role in stabilizing this conjugated 4π‐electron system. The monoanionic bidentate ligand interacts with the conjugated π system to cause π‐orbital splitting. This ligand‐induced π‐orbital splitting effect provides an opportunity to manipulate the gap between occupied and unoccupied π orbitals in conjugated systems. Conversely, compound 2 is nonaromatic because its core ring does not have a conjugated π ring system and does not fulfill the requirements of a Hückel system.     

Rotational Phenomena on Substituent Phenyl Rings of (η5-Cyclopentadienyl)(η4-Tetraphenylcyclobutadiene)Cobalt

We used extended Hückel calculations, variable-temperature, homonuclear long-range shift-correlated 2-D ¹H NMR and dynamic NOE measurements to investigate rotational phenomena of substituent phenyl rings on (η⁵-cyclopentadienyl)(η⁴tetraphenylcyclobutadiene)cobalt (1). Two closely related compounds, (η⁵-cyclopentadienyl)(η⁴-1,3-diphenylcyclobutadiene)cobalt (A) and (η⁵-cyclopentadienyl)(η⁴-1,2-diphenylcyclobutadiene)cobalt (A′), were prepared. Energy minima appeared at conformations of which the dihedral angles between phenyl and cyclobutadiene rings are about 30° for I and 0° for A according to extended Hückel calculations. In 1, ortho protons of phenyl rings belong to one set of multiplet in ¹H NMR; meta and para protons belong to the other. It was supported by a long-range coupling 2-D ¹H NMR and NOE experiments. A sharp line due to phenyl rings was observed in the low-temperature ¹H NMR spectrum of A, which indicates that the five protons are magnetically equivalent at that temperature.     

A Nonaromatic Thiophene‐Fused Heptalene and Its Aromatic Dianion

Heptalene, a nonaromatic, bicyclic 12 π‐electron system with a twisted structure, is of great interest with regard to its potential Hückel aromaticity in the two‐electron oxidized or reduced forms. The synthesis of thiophene‐fused heptalene 5 from the reductive transannular cyclization of bisdehydro[12]annulene 4 , and its solid‐state structure, which was confirmed by X‐ray crystallographic analysis, is presented. Chemical reduction of 5 readily generated the corresponding dianion, which was successfully isolated as [(K[2.2.2]cryptand)⁺]₂ 5 ^2?. The X‐ray crystallographic analysis of the dianion revealed a shallower saddle structure for the heptalene moiety and a lesser degree of bond alternation relative to 5 . ¹H NMR spectroscopy exposed the effect of a diamagnetic ring current on dianion 5 ^2?, which was corroborated by nucleus‐independent chemical shift (NICS) calculations. These results demonstrate that the heptalene dianion, containing 14 π‐electrons, does indeed exhibit pronounced degrees of Hückel aromaticity.     

Bond-alternating Hückel-Möbius and related,twisted linear,cyclic and helical systems—their molecular orbitals,energies and phase correlation upon dissociation

Cyclic group formalism and screw symmetry operation are used to clarify and generalize the definition of Hückel and Möbius systems. It is shown that the Möbius ring system has half-integral pseudo-angular momentum similar to that of spin space, and that applications of Möbius electronics to chemical reactions have been based on truncated single-circle Möbius rings which have unique beginning and end (Sect. 2). This concept is illustrated by application to the [1, 7] antarafacial hydrogen shift (Appendix A and Figs. A1–A3). Definition of a Hüickel versus Möbius ring system for in-plane and out-of-plane π, δ and φ orbitals as well as the appropriate relative angle of twists are given (Sect. 2 and Table 1). Using the concept of the compatibility of the twist (screw) angle and rotation around a ring, we also derive the proper phase coherence and energy correlation between a parent cyclic (Hückel or Möbius) molecule and its dissociated linear fragments (Sect. 4). The concept of parentage in diabatic fragmentation is discussed. Forfinite, open, helical chain molecules, an exact periodic boundary condition based on the compatibility of twist angle and number of turns in a helical ring parent molecule is applied to derive their analytic wave functions (Sect. 5 and Table 2). Forbond-alternating “linear” and cyclic Hückel and Möbius systems we also derive the explicit LCAO-MO wavefunctions, energies, their degeneracies and their exact corresponding quantum members for even and odd atom systems at highest bonding and lowest antibonding levels (Sect. 3, Figs. 1–3). The corresponding wavefunctions and energies for uniform-bond systems are given for comparison and for completeness (Sect. 3).     

[40]π Fused and Nonfused Core‐Modified Nonaphyrins: Syntheses and Structural Diversity

The synthesis of fused and nonfused core‐modified 40π nonaphyrins are reported. Spectroscopic and X‐ray structural studies reveal a twisted figure‐eight conformation in the freebase form that is nonaromatic. Structural changes occur, from figure‐eight to open extended conformation, upon protonation, thereby adopting 4nπ Hückel antiaromatic character, which is reflected in spectroscopic and theoretical studies. Such a structural change also induces ring inversions of specific heterocyclic rings by 180°.     

Simple prediction of cycloaddition orientation: II—2+2 thermal cycloadditions. Scope and limitations of the method

In the accompanying paper, it has been suggested that, in a cycloaddition, the first σ bond to be formed is that corresponding to the greatest overlap stabilization. This rule has now been tested, successfully on over one hundred 2+2 cycloadditions.Comparisons of our method with other theoretical treatments are made and the reasons for the discrepancies pointed out.The choice of parameters (for Hückel calculations) or molecular geometries (for CNDO/2 calculations) is discussed. No particular advantage of CNDO/2 calculations over Hückel calculations has been observed.     

ESI‐MS/MS of expanded porphyrins: a look into their structure and aromaticity

Electrospray mass spectrometry/mass spectrometry was used to investigate the gas‐phase properties of protonated expanded porphyrins, in order to correlate those with their structure and conformation. We have selected five expanded meso‐pentafluorophenyl porphyrins, respectively, a pair of oxidized/reduced fused pentaphyrins (22 and 24 π electrons), a pair of oxidized/reduced regular hexaphyrins (26 and 28 π electrons) and a regular doubly N‐fused hexaphyrin (28 π electrons). The gas‐phase behavior of the protonated species of oxidized and reduced expanded porphyrins is different. The oxidized species (aromatic Hückel systems) fragment more extensively, mainly by the loss of two HF molecules. The reduced species (Möbius aromatic or Möbius‐like aromatic systems) fragment less than their oxidized counterparts because of their increased flexibility. The protonated regular doubly fused hexaphyrin (non‐aromatic Hückel system) shows the least fragmentation even at higher collision energies. In general, cyclization through losses of HF molecules decreases from the aromatic Hückel systems to Möbius aromatic or Möbius‐like aromatic systems to non‐aromatic Hückel systems and is related to an increase in conformational distortion. Copyright © 2016 John Wiley & Sons, Ltd.     

Calculation of relative Raman intensities: II. Calculations using an extended Hückel valence basis set

Line intensities in Raman spectra are determined by means of a modified extended Hückel [1] method. Perturbation terms are developed which are inserted into the Hamiltonian used in the extended Hückel theory to yield the induced dipole moment as a function of the normal coordinate Q, from which the polarizability change on Q is obtained. The normal coordinate determination is carried out with a programme developed in this laboratory [2]. From a valence basis set within the extended Hückel theory, the Raman intensities of acetylene, formaldehyde, phosgene, thiophosgene, dichloromethane and tetrafluoroethylene calculated from the extended Hückel theory are compared with those obtained by CNDO and INDO calculations and with the measured data. It is found that the Raman intensities calculated from the extended Hückel method are similar to those obtained from the CNDO and INDO calculations.     

Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.     

Re-evaluation of the Thermodynamic Activity Quantities in Pure Aqueous Solutions of Chlorate, Perchlorate, and Bromate Salts of Lithium, Sodium or Potassium Ions at 298.15 K

The Hückel equation used to correlate the experimental activities of dilute alkali metal chlorate, perchlorate or bromate solutions up to a molality of about 1.5 mol⋅kg⁻¹ contains two electrolyte-dependent parameters: B {that is related closely to the ion-size parameter (a ^∗) in the Debye–Hückel equation} and b ₁ (this parameter is the coefficient of the linear term with respect to the molality, which is related to hydration numbers of the ions of the electrolyte). In more concentrated solutions up to 7 mol⋅kg⁻¹, an extended Hückel equation was used, it contains additionally a quadratic term with respect to the molality, and the coefficient of this term is the parameter b ₂. Parameters for the Hückel equations were evaluated from isopiestic data for LiClO₃, LiBrO₃, LiClO₄, NaClO₃, NaBrO₃, NaClO₄, KClO₃, and KBrO₃. In these estimations, the Hückel parameters determined recently for NaCl solutions were used. The resulting parameter values were tested with the vapor pressure and isopiestic data available in the literature for solutions of these salts. Most of these data were reproduced within experimental error by means of the Hückel or extended Hückel equations, at least up to a molality of 3.0 mol⋅kg⁻¹, for all salts considered. Reliable activity and osmotic coefficients for solutions of these salts can, therefore, be calculated by using the new Hückel equations, and they are tabulated here at rounded molalities. The activity and osmotic coefficients obtained from these equations were compared to the values reported in several previous tabulations.     

Theoretical model for the polarization molecular and Hückel treatment of PhosphoCyclopentadiene in an external electric field: Hirschfeld study

In this study we relate the Hückel method and molecular polarization of PhosphoCyclopentadiene (P-cyclopentadiene) with respect to the Cyclopentadienyl, and its consequent separation of charges of particular set of conjugated diene systems. The Hückel method and Molecular Polarization of P-cyclopentadiene is expressed as a function of the induced polarizability of Cyclopentadienylin an external electric field, presenting a technique to express the molecular polarizabilities and Hückel method of diene systems as a function of another in an external electric field, using local quantum similarity index (LQSI) based on the Hirschfeld partitioning in the framework of conceptual density functional theory, this index was introduced in the molecular polarization of cyclopentadienyl in an external electric field and in the secular determinant of the Hückel method applied to the Cyclopentadienylin order to express the molecular polarization and Hückel method as a function of P-cyclopentadiene using six local similarity index: Overlap, Overlap-Interaction, Coulomb, Coulomb-Interaction, Overlap-Euclidian distance and Coulomb-Euclidian distance.The topo-geometrical superposition approach (TGSA) was used as method of alignment, which allows us to obtain high results in the proposed LQSIs, this method to be a straightforward procedure to cope with the problem of relative orientation of the molecules when evaluating, developing a new technique that will allow us to study structural systems that differ in one atom in its structure, and proposing methodologies for future studies on a much broader range of systems in which the Hückel method and molecular polarization of two species that differ only in one atom in its structure can be approximated in this way.     

Debye–Hückel Contributions to the Gibbs Energy Interaction Parameters for Ternary Electrolyte + Non-Electrolyte + Water Systems

The Debye–Hückel and non-Debye–Hückel contributions to the Gibbs energy interaction parameters are investigated for electrolyte (E) + non-electrolyte (N) + water (W) systems. A method is proposed for calculating the interaction parameters, C _n ^DH,C _n ^N, and C _n ^T, which represent the Debye–Hückel, non-Debye–Hückel, and total contributions, respectively. Four ternary E + N + W systems are chosen and the interaction parameters are computed with different forms of the Debye–Hückel equation. Results show that: (1) the Gibbs energy interaction parameters between E and N can be divided into two parts: the Debye–Hückel contribution and the non-Debye–Hückel contribution, C _n ^T=C _n ^DH+C _n ^N; (2) the signs and magnitudes of the Debye–Hückel contribution to the interaction parameters, C _n ^DH, depend mainly on the change in the dielectric constant of the solvent due to the addition of the non-electrolyte into the solvent; and (3) when the addition of the non-electrolyte only affects slightly the dielectric constant of the solvent, C ₁^DH (indicating the Debye–Hückel contribution to the interaction parameter for E + N) has a very small value and consequently can be neglected. In general, C ₁^DH is large, even larger than C ₁^N. Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users.     

Extended Debye–Hückel Theory for Studying the Electrostatic Solvation Energy

The electrostatic part of the solvation energy has been studied by using extended Debye–Hückel (DH) theories. Specifically, our molecular Debye–Hückel theory [J. Chem. Phys. 2011 , 135, 104104] and its simplified version, an energy‐scaled Debye–Hückel theory, were applied to electrolytes with strong electrostatic coupling. Our theories provide a practical methodology for calculating the electrostatic solvation free energies, and the accuracy was verified for atomic and diatomic charged solutes.