Mobius twist

It has been pointed out that a different array of atomic orbitals might be conceived of in large conjugated rings. The array, called a Mobius twist, results in there being one point in the ring at which the atomic orbitals would show a phase discontinuity. ... 

Fig. 10. Cyoloaddition of k +1)77 and (m + l)n polyenes. The energies of the highest occupied (HO) and lowest vacant (LV) orbitals are shown for an unfavorable thermal cyclization (upper left) and a favorable cyclization (upper right). Below, three possible modes of favorable orbital overlap between the two polyenes are shown from left to right, symmetric + symmetric or aa antisymmetric+antisymmetric or bb , Mobius twist of one polyene to attain maximum overlap. These have also been termed ayn, syn and anti, respectively. 

The first real Mobius systems ([16]annulenes) were only synthesized a few years ago [105], In these systems the authors achieved enough rigidity for the molecules so that they would not flip back to a Hiickel system. It was also determined that these Mobius-twisted annulenes are more aromatic in their character than the Hiickel-systems [106],... 

Although rotation and translation in the plane of the objects can never bring them into coincidence, rotation about an axis in the plane achieves the conversion. Such an inversion is also brought about by the Mobius twist, figure 8. 

Figure 13.13 Mobius [28]hexapor-phin(1.1.1.1.1.1). Current density vectors are plotted on top of the ACID isosurface. The ACID isosurface is not shown to provide a clearer picture of the electron flow within the Mobius twisted % system. For technical details see Figure 13.8. The delocalized system in the periphery includes 28 % electrons...

FIGURE 15.7 A Mobius strip. Such a strip is easily constructed by twisting a thin strip of paper 180° and fastening the ends together. 

Figure 1.5. View of the Mobius stabilized annulene. Crystallographic data from Ajami, 2003. The tetradehydrodianthracene and twisted structures are shown left and right of the figure, respectively. C and H are represented by black and light grey balls, respectively.

We also considered in Section 1.2 the possibility of preparing Mobius-band molecules from cycloacenes. In fact regular cycloacenes, that is molecules built from joining both ends of linear acenes without twisting, have never been obtained, despite several experimental attempts. They are interesting since they represent the basic cylindrical carbon units of zig-zag ( ,0) nanotubes. 

Wires or strips at the nanometre scale are surprisingly flexible (surprising if we insist on thinking in terms of our macroworld ) and can even be twisted in order to form Mobius strips, as recently achieved with NbSea single crystals (Tanda et al, 2002). 

Figure 6. The Mobius strip approach (three half-twists) to a trefoil knot.

A directed knot synthesis relying upon the Mobius strip principle was conceived by Schill and coworkers [41-50] who synthesized the doubly bridged tetraamino-p-benzoquinone 9. Connection of three such molecules by long chains should give the molecular ladder 10, precursor of the three half-twist-containing Mobius strip 11 (Figure 8). 

B was just about to tear a strip off the edge of his pad and demonstrate when A surprised him with Oh, I know what a Mobius strip is. It s made by putting half a twist in a long strip before making it into a loop. It has only one side and only one edge. I once saw that well-known line from the Pervigilium Veneris... 

Figure 21-15 Normal (Hiickel) and Mobius rings of 7r orbitals. To clarify the difference between the two rings, visualize a strip of black-red typewriter ribbon, the black representing the + phase, and the red the — orbital phase. Now join the ends together without, or with, one twist in the strip. At the joint there then will be no node (left) or one node (right).

Propadiene is represented in Figure 13-4 as if it were two isolated Huckel ring systems. This molecule also may be represented as a stable Mobius system of Att electrons. Draw an orbital diagram of 1,2-propadiene to indicate this relationship. If 1,2-propadiene twisted so that the hydrogens on the ends all were in the same plane, 57, would it be a Huckel or a Mobius polyene, or neither ... 

The design of this SAR device was oriented on the so-called Mobius band, which is a twisted structure (see Figure 1.123) [141]. Two horizontal fluid layers are separated in the center. Thereafter, both double layers are turned by 90° in the same direction so that two vertically laminated systems are achieved. Then, the layers are joined to give a four-lamellae system. 

The overlap of this twisted radical with a monoatomic radical to give a twisted polyene is positive. It is zero for an annulene (Figure 3.5). A Mobius An + 2 annulene is... 

A 180 ° twist would correspond to a Mobius strip which could form a single large ring a 540 ° twist would yield a trifold knot and 720 ° twist a double treaded catenane. 

When a cyclic polyene is large enough, it can exist in both cis- and iraws-forms. Our approach to polyene cyclization has tacitly assumed an all cis -n chain in the form of a band or ribbon that would slip smoothly on to the surface of a cylinder of appropriate diameter. Should the orbitals of the two polyenes in (36) have a mismatch in their orbital symmetries, a single twist in the tt band of one of them could remedy this (Fig. 10c). Cycloaddition would now be allowed and the reaction would proceed, provided other factors were favorable. Such cases of Mobius (Zimmerman, 1966), anti (Fukui and Fujimoto, 1966b) or axisymmetric (Lemal and McGregor, 1966), as opposed to Hiickel, syn, or sigma-symmetric ring closure are unknown (or, at least, rare). A Mobius form has, however, been proposed as the key intermediate in the photochemical transformations of benzene (Farenhorst, 1966) in (48) in place of the disrotatory cyclization proposed by van Tamelen (1965). 

Even though the stability of Mobius systems was predicted over 40 years ago [103], for a long time no such systems were synthesized. One possible reason for this is the expected strain in the twisted structure. Based on quantum chemical calculations, different groups suggested the stability of [4 ]annulenes [(CH) , with n = 3-5], but it was also shown that there are many possible isomers of this system that are close in energy and thus the Mobius system might easily flip back to the less strained Hiickel structures [104],... 

Fig. 9 The Mobius strip approach to catenane synthesis. The number of half-twists in the strip determine the topology of the product after it is cut down the middle. Two half-twists would afford a [2]catenane. Reproduced with permission from [54] (copyright 1985 Elsevier)...

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