Enumeration of Möbius type cyclic polyazulenoids

A Möbius type cyclic polyazulenoid is a kind of nonalternant conjugated hydrocarbon, consisting of a cyclic series of alternatingly fused five membered rings and seven membered rings, which forms a “Möbius belt of rings” or say, “Möbius belt of azulenes”, more precisely. As a sequel of the recent paper (Deng and Zhang in J Math Chem 54(2):416–427, 2016), which counts “belt type” cyclic polyazulenoid, this article studies the enumeration problem of Möbius type cyclic polyazulenoid. We obtain the exact counting formula for the number of Möbius type cyclic polyazulenoids with n azulene units, by a method based on coding technique and a generalization of ‘Burnside’ lemma. And we give the numerical results for n not larger than 20, which may contribute to the fully synthesis of Möbius type cyclic polyazulenoids.     

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.     

A möbius inversion of the ulam subgraphs conjecture

The generalized Eichinger matrices are defined asE = _j ⁿ ₁( _j S ^T S)^–1, where _j M denotes the matrixM withj th row and column deleted.S is the incidence matrix andM ^T is the transposed matrix. The conjectureS ^T SE = S _K ^TS _K , where SK is the incidence matrix of the complete graph, is proven for trees, simple cycles and complete graphs. The consequence of the conjecture isS _G ^T S _G (E _G -I) = S _G ^TS _G , whereG is the complementary graph ofG. It leads to graphs with imaginary arcs as the complements of graphs with multiple arcs.     

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.     

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.     

Optical Physical Mechanisms of Helicene Carbon Nanohoop with Möbius Topology

Optical and spectral properties of carbon nanohoop with Möbius topology is of great interest in nano-science and nano-technology. And it can be imagined that it has a lot of unexpected potential application prospects. However, theoretical calculations based on some figure-of-eight helicene carbon nanohoop with Möbius topology are still insufficient. Therefore, in this paper, we theoretically study the optical and spectral properties of figure-of-eight helicene carbon nanohoop with Möbius topology. Optical and spectral properties are analyzed with visualization method of transition density matrix and charge density difference, which reveal the unique characterization of carbon nanohoop with Möbius topology. Our results can not only deepen the understanding of the optical physical mechanisms of the nanorings with mobius carbons, but also provide deeper insight on optical properties and potential design on optical nanodevices.     

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.     

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.     

Double-twist Möbius aromaticity in a 4n+ 2 electron electrocyclic reaction

One transition state for the 10 electron (4n+ 2) electrocyclisation of Z,E,Z-decapentaene is predicted to have the properties of a double-twist M?bius strip, with delocalised bond lengths and the NICS(0) magnetic index typical of aromaticity; a 5,6-di-(t)Bu substituted decapentaene is predicted to react in this mode in preference to the competitive six-electron electrocyclisation.     

Algebraic Structure Count of Some Cyclic Hexagonal-Square Chains on the Möbius Strip

The concept of ASC (Algebraic structure count) is introduced into theoretical organic chemistry by Wilcox as the difference between the number of so-called “even” and “odd” Kekulé structures of a conjugated molecule. Precisely, algebraic structure count (ASC-value) of the bipartite graph G corresponding to the skeleton of a conjugated hydrocarbon is defined by where A is the adjacency matrix of G. The determination of algebraic structure count of (bipartite) cyclic hexagonal-square chains in the the class of plane such graphs is known. In this paper we expand these considerations on the non-plane class. An explicit combinatorial formula for ASC is deduced in the special case when all hexagonal fragments are isomorphic.     

Assembly and autochirogenesis of a chiral inorganic polythioanion Möbius strip via symmetry breaking

Using [Mo(2)S(2)O(2)(H(2)O)(6)](2+) and squarate dianion, we synthesized the thiometalate ring compounds [(Mo(2)S(2)O(2))(x)(OH)(y)(C(4)O(4))(z)(Mo(2)O(8))(o)(H(2)O)(p)](n-), where [x,y,z,o,p,n] = [7,14,2,0,2,4] for 1, [6,8,2,2,4,8] for 2, and [4,6,1,1,0,4] for both 3a and 3b, which are chiral and nonchiral isomers, respectively. Not only do the four thiometalate clusters show decreasing symmetry at the molecular level across the series, but the incorporation of the "addendum" {Mo(2)O(8)}(o) unit also allows the thiometalate ring to twist. The reaction initially yields the chiral molecule 3a with a twisted ring, which undergoes spontaneous resolution upon crystallization; the reaction mixture later yields the intrinsically nonchiral isomer 3b with a nontwisted ring. In addition, the compounds are able to promote the electrocatalytic evolution of hydrogen.     

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     

Dynamic processes in [16]annulene: Möbius bond-shifting routes to configuration change

Density functional and ab initio methods have been used to study the mechanisms for key dynamic processes of the experimentally known S4-symmetric [16]annulene (1a). Using BH&HLYP/6-311+G** and B3LYP/6-311+G**, we located two viable stepwise pathways with computed energy barriers (Ea = 8-10 kcal/mol) for conformational automerization of 1a, in agreement with experimental data. The transition states connecting these conformational minima have M?bius topology and serve as starting points for non-degenerate pi-bond shifting (configuration change) via M?bius aromatic transition states. The key transition state, TS1-2, that connects the two isomers of [16]annulene (CTCTCTCT, 1 --> CTCTTCTT, 2) has an energy, relative to the S4 isomer, that ranged from 6.9 kcal/mol (B3LYP/6-311+G**) to 16.7 kcal/mol (BH&HLYP/6-311+G**), bracketing the experimental barrier. At our best level of theory, CCSD(T)/cc-pVDZ(est), this barrier is 13.7 kcal/mol. Several other M?bius bond-shifting transition states, as well as M?bius topology conformational minima, were found with BH&HLYP energies within 22 kcal/mol of 1a, indicating that many possibilities exist for facile thermal configuration change in [16]annulene. This bond-shifting mechanism and the corresponding low barriers contrast sharply with those observed for cis/trans isomerization in acyclic polyenes, which occurs via singlet diradical transition states. All M?bius bond-shifting transition states located in [16]- and [12]annulene were found to have RHF --> UHF instabilities with the BH&HLYP method but not with B3LYP. This result appears to be an artifact of the BH&HLYP method. These findings support the idea that facile thermal configuration change in [4n]annulenes can be accounted for by mechanisms involving twist-coupled bond shifting.     

Möbius aromaticity in [12]annulene: cis-trans isomerization via twist-coupled bond shifting

Density functional and coupled cluster calculations show that facile thermal configuration change in [12]annulene occurs via a twist-coupled bond-shifting mechanism. The transition state for this process is highly aromatic with M?bius topology. At the CCSD(T)/cc-pVDZ//BH&HLYP/6-311+G** level, the isomerization of tri-trans-[12]annulene 1a (CTCTCT) to its di-trans isomer 2 (CCCTCT) via such a mechanism has a barrier of 18.0 kcal/mol, in good agreement with earlier experiments. Two other aromatic M?bius bond-shifting transition states were located that result in configuration change for other [12]annulene conformers. This mechanism contrasts sharply with diradical configuration change for acyclic polyenes and with planar bond-shifting mechanisms generally assumed for annulenes. This constitutes evidence that neutral M?bius aromatic annulenes play a role in the dynamic processes of neutral [4n]annulenes.     

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...