Hückel and Möbius Aromaticity in Charged Sigma Complexes

In sigma complexes, intermediates in nucleophilic and electrophilic aromatic substitution and other reactions, delocalization in the aromatic ring is formally disrupted. Unexpectedly, computational evidence is presented that favorable processes contain aromatic sigma complexes. Tetracoordinated carbon therein surprisingly employs orbitals that are more similar to sp² than to sp³ hybrids in sigma bonds with adjacent ring atoms. Both leaving groups and nucleo- or electrophiles may donate electrons to the π-system depending on the availability of p-type orbitals to fulfill Hückel (4N+ 2) or Möbius (4N) rules of aromaticity in analogy to conjugated transition-metal metallacycles.     

Mbius环与Hückel环

Mbius环是以1863年提出“Mbius带”的德国天文学家 A.F.Mbius的名字命名的,它有好些引人入胜的拓补结构特性。Huckel环则是以本世纪三十年代对有机化学做     

Möbius strip versus linear and cyclic polyacenes: a Hückel and semiempirical investigation

The electronic structure of finite and infinite linear, cyclic and M?bius strip polyacenes has been investigated by adopting Hückel and semiempirical schemes. Using the Hückel approach, it turns out that the M?bius belting process modifies the highest occupied molecular orbital (HOMO)–lowest unoccupied molecular orbital (LUMO) gap in such a way its evolution with chain length is similar to the linear polyacenes rather than their cyclic analogs. These results are corroborated at the Austin model 1 (AM1) level, where the geometry relaxation effects are taken into account. The optimized AM1 structures show that the M?bius defect is localized and extends over a third of the ring. With respect to the Hückel approach, accounting for geometry distortion at the AM1 levels results in an increase in the HOMO–LUMO gap of the M?bius strip relative to the linear and cyclic finite-size structures. On the other hand, when including electron-hole correlation at the configuration interaction singles/Zerner's intermediate neglect of differential overlap level the behavior with system size of the first excitation energy of cyclic and M?bius polyacenes differs from their linear analogs and leads to smaller singlet excitation energies. Received: 20 September 2000 / Accepted: 21 September 2000 / Published online: 28 February 2001     

Möbius bis and tris-spiroaromatic systems

We propose that a general class of bis and tris-spiro 7-membered ring systems with a common atom X, of which there are a number of examples characterised crystallographically, can in fact be considered as spiroaromatic molecules in which each ring exhibits some degree of M?bius 4n pi-electron aromaticity. The aromaticity is probed as a function of the spiro-atom using ab initio calculations of the NICS(0) values, which indicate that the M?bius-aromaticity increases as the spiro-atom is changed e.g. from Al to P, and from e.g. P to As.     

Interplay of Hückel and Möbius Aromaticity in Metal-Metal Quintuple Bonded Complexes of Cr,Mo, and W with Amidinate Ligand: Ab initio DFT and Multireference Analysis**

The aromaticity of metal-metal quintuple bonded complexes of the type M₂L₂ (M=Cr, Mo, and W; L=amidinate) are studied employing gauge including magnetically induced ring current (GIMIC) analysis and electron density of delocalized bonds (EDDB). It is found that the complexes possess two types of aromaticity: i) Hückel aromaticity through delocalization of ligand π electrons with metal-metal δ-bond-forming 6 conjugated electrons (4π and 2δ) ring; ii) Craig-Möbius aromaticity through delocalization of π electrons of both the ligands with metal d-orbitals in Craig type orientation forming 10π electrons ring with a double twist. Extended transition state natural orbital chemical valence (ETS-NOCV) and canonical molecular orbital natural chemical shielding (CMO-NCS) analysis confirm the Craig-Möbius type arrangement of the orbitals. Furthermore, the unprecedented Hückel and Möbius type aromaticity is confirmed from the plot of the current pathways using 3D line integral convolution (3D-LIC) plots. The metal-metal bond order also increases down the group as justified from the complete active space self-consistent field (CASSCF) analysis. Due to an increase in the π and δ electron conjugation, both the Hückel and Möbius aromaticity increase down the group.     

Möbius function and characteristic monomials for combinatorial enumeration

After the definitions of amplified representations and number-theoretical vectors, the markaracter table of a cyclic subgroup is converted into the corresponding Q-conjugacy character table. The conversion is shown to necessitate an interconversion matrix that contains M?bius functions as elements. Since the interconversion matrix gives characteristic monomials for cyclic groups, all the powers appearing in each of the characteristic monomials are shown to be integers. Characteristic monomials for finite groups are then built up by starting from those of cyclic groups. This procedure clarifies the fact that all the powers appearing in each characteristic monomial for finite groups are integers. The relationship between characteristic monomial tables and unit-subduced-cycle-index tables is discussed with respect to their application to isomer enumeration. Received: 15 July 1998 / Accepted: 16 December 1998 / Published online: 16 March 1999     

Synthesis and properties of the first Möbius annulenes

Heilbronner in 1964 predicted that annulenes with ".. a planar perimeter of N=4r AO's, which would yield an open shell configuration when occupied by 4r electrons, can be twisted into a closed shell M?bius strip perimeter without loss in pi electron energy". We have been able to synthesize the first [4n]annulene with such a M?bius topology and now present further M?bius isomers and the details of their preparation as stable compounds. To address the question whether the twist in the pi system has an effect on the properties we systematically investigate energy, geometry and magnetic parameters of a large number isomers of [16]annulenes. The M?bius twisted annulenes are consistently more aromatic than the non-twisted isomers. This is true for the parent as well as our benzoannelated systems. Our results are in contrast to those published recently by C. Castro, W. L. Karney, P. von R. Schleyer et al.     

Viability of möbius topologies in [26]- and [28]hexaphyrins

Recently, hexaphyrins have emerged as a promising class of π-conjugated molecules that display a range of interesting electronic, optical, and conformational properties, including the formation of stable M?bius aromatic systems. Besides the M?bius topology, hexaphyrins can adopt a variety of conformations with Hückel and twisted Hückel topologies, which can be interconverted under certain conditions. To determine the optimum conditions for viable M?bius topologies, the conformational preferences of [26]- and [28]hexaphyrins and the dynamic interconversion between the M?bius and Hückel topologies were investigated by density functional calculations. In the absence of meso?substituents, [26]hexaphyrin prefers a planar dumbbell conformation, strongly aromatic and relatively strain free. The M?bius topology is highly improbable: the most stable tautomer is 33?kcal?mol(-1) higher in energy than the global minimum. On the other hand, the M?bius conformer of [28]hexaphyrin is only 6.5?kcal?mol(-1) higher in energy than the most stable dumbbell conformation. This marked difference is due to aromatic stabilization in the M?bius 4n electron macrocycle as opposed to antiaromatic destabilization in the 4n+2 electron system, as revealed by several energetic, magnetic, structural, and reactivity indices of aromaticity. For [28]hexaphyrins, the computed activation barrier for interconversion between the M?bius aromatic and Hückel antiaromatic conformers ranges from 7.2 to 10.2?kcal?mol(-1) , in very good agreement with the available experimental data. The conformation of the hexaphyrin macrocycle is strongly dependent on oxidation state and solvent, and this feature creates a promising platform for the development of molecular switches.     

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.     

Toward Möbius and Tubular Cyclopolyarene Nanorings via Arylbutadiyne Macrocycles

By harnessing a highly efficient metal-catalyzed tandem cycloaddition reaction as the key benzannulation step, a series of cyclopolyarene nanorings of varied sizes are obtained from poly(arylene-butadiynylene) macrocyclic precursors, which can be synthesized relatively conveniently. Interestingly, due to the nonparallel bond connectivity of the repeat unit, unique Möbius topology is manifested by the cyclopolyarene nanorings composed of an odd number of repeat units, whereas cylindrical tubular structures with radial conjugation are formed with those consisting of an even number of repeat units.     

Graphene Nanobelts,Möbius and Higher Order Möbius Strips Formed from Graphene. A Combined CASSCF/DFT Study

Russian Journal of Physical Chemistry A - Twisted graphene nanoribbons with linking numbers (Lk) of 2 and 3 have been studied at hybrid DFT and CASSCF levels. CASSCF calculations demonstrated...     

Aromaticity and Möbius antiaromaticity in monocyclic [11]annulenium cations

Monocyclic [11]annulenium cations, which are experimentally unknown, have been studied primarily via DFT methods but also with some CCSD(T) validation. We have located six minima: two doubly trans (26, 27), one triply trans (28), one singly trans (29), one quintuply trans (trannulene-type, 33), and one all-cis (31). The first three are aromatic, 33 is modestly aromatic, 29 is nonaromatic, and the last is a M?bius antiaromatic species. We also investigated the fusion of various numbers of three-membered rings (3MRs) to the central 11-membered ring (11MR). We found several planar, all-cis-[11]annulenium ion derivatives as well as another M?bius antiaromatic species (52b); for comparison, we also found planar, antiaromatic all-cis-[12]annulene (60) and [15]annulenium cation (61) derivatives. The (anti)aromatic characterization of these compounds is based mainly on calculated magnetic data for the ground singlet and vertical triplet states, although aromatic stabilization energies (ASE) are also considered. Data for optimized triplets, several of which are M?bius aromatic systems (31t, 52t, 63t, 64t), are also included. Several of these cations are reasonable synthetic targets.     

Möbius and twisted graphene nanoribbons: stability, geometry, and electronic properties

Results of classical force field geometry optimizations for twisted graphene nanoribbons with a number of twists N(t) varying from 0 to 7 (the case N(t)=1 corresponds to a half-twist M?bius nanoribbon) are presented in this work. Their structural stability was investigated using the Brenner reactive force field. The best classical molecular geometries were used as input for semiempirical calculations, from which the electronic properties (energy levels, HOMO, LUMO orbitals) were computed for each structure. CI wavefunctions were also calculated in the complete active space framework taking into account eigenstates from HOMO-4 to LUMO+4, as well as the oscillator strengths corresponding to the first optical transitions in the UV-VIS range. The lowest energy molecules were found less symmetric than initial configurations, and the HOMO-LUMO energy gaps are larger than the value found for the nanographene used to build them due to electronic localization effects created by the twisting. A high number of twists leads to a sharp increase of the HOMO-->LUMO transition energy. We suggest that some twisted nanoribbons could form crystals stabilized by dipolar interactions.     

Lemniscular hexaphyrins as examples of aromatic and antiaromatic double-twist Möbius molecules

Two reported [26] and [28]hexaphyrins are analyzed via measured and computed geometries and NMR-shieldings as examples of respectively 4n + 2 pi-electron aromatic and 4n pi-electron antiaromatic double-twist M?bius ring systems, adopting a lemniscular/figure-eight topology with linking number LK = 2pi. Values of local twist (TW) and nonlocal writhe (WR) derived from the relation Lk = Tw + WR appear relatively insensitive to the aromatic/antiaromatic character. The [26]hexaphyrin may adopt differing solution and solid-state conformations.