C2 symmetry axis

The n ==> n transition thus involves ground Ai) and exeited Ai) states whose direet produet (Ai x Ai) is of Ai symmetry. This transition thus requires that the eleetrie dipole operator possess a eomponent of Ai symmetry. A glanee at the C2v point group s eharaeter table shows that the moleeular z-axis is of A symmetry. Thus, if the light s eleetrie field has a non-zero eomponent along the C2 symmetry axis (the moleeule s z-axis), the n ==> 71 transition is predieted to be allowed. Light polarized along either of the moleeule s other two axes eannot induee this transition. 

The favorable stereochemistry and the displacement of the very good leaving group promoted the formation of a [3.2.0] bicyclic intermediate (see Scheme 2). Owing to the absence of a C2 symmetry axis two different [3.2.0] bicyclic sulfonium salt intermediates, 8a and 8b, can be generated by nucleophilic transannular attack of sulfur on the... 

Furthermore, Oda et al. pointed out that there are two topologically distinct types of chiral bilayers, as shown in Figure 5.46.165 Helical ribbons (helix A) have cylindrical curvature with an inner face and an outer face and are the precursors of tubules. These are, for example, the same structures that are observed in the diacetylenic lipid systems discussed in Section 4.1. By contrast, twisted ribbons (helix B) have Gaussian saddlelike curvature, with two equally curved faces and a C2 symmetry axis. They are similar to the aldonamide and peptide ribbons discussed in Sections 2 and 3, respectively. The twisted ribbons in the tartrate-gemini surfactant system were found to be stable in water for alkyl chains with 14-16 carbons. Only micelles form... 

Figure 5.46 Schematic representation of helical and twisted ribbons as discussed in Ref. 165. Top Platelet or flat ribbon. Helical ribbons (helix A), precursors of tubules, feature inner and outer faces. Twisted ribbons (helix B), formed by some gemini surfactant tartrate complexes, have equally curved faces and C2 symmetry axis. Bottom Consequences of cylindrical and saddlelike curvatures in multilayered structures. In stack of cylindrical sheets, contact area from one layer to next varies. This is not the case for saddlelike curvature, which is thus favored when the layers are coordinated. Reprinted with permission from Ref. 165. Copyright 1999 by Macmillan Magazines.

Molecular chirality, however, proved an extremely powerful tool in the quest for polar LCs. In 1974 Robert Meyer presented to participants of the 5th International Liquid Crystal Conference his now famous observation that a SmC phase composed of an enantiomerically enriched compound (a chiral SmC, denoted SmC ) could possess no reflection symmetry.1 This would leave only the C2 symmetry axis for a SmC a subgroup of C. The SmC phase is therefore necessarily polar, with the polar axis along the twofold rotation axis. 

Along this line, several new linear hosts and flexible cyclic hosts having a C2-symmetry axis (bisDPGP, bisTAGP, and bisTMGP Scheme 8) have been designed and synthesized based on the structural feature of the highly selective MeFruNys." However, these tailor-made flexible hosts prove less effective than MeFruNys as chiral selector (Table 15). 

Later on, Kagan [7] reported an important result with DIOP bisdentate ligand, the first ligand with a C2-symmetry axis, in which the stereoselectivity is improved by restricting the mobility around the metal atom and Morrison [8] reported the interesting neomenthyl diphenylphosphine ligand, which is devoid of any chiral phosphorus atom. As stated by Kotha [lb] "Early lessons learned in asynunetric hydrogenation paved the way to some of the new asymmetric catalytic processes". 

In the first case, the equivalence of the two halves means that any line which is a perpendicular bisector of the molecular axis is a C2 symmetry axis there is an infinite number of such C2 axes. The equivalence of the two halves of the molecule also means that there is a plane of symmetry perpendicular to the molecular axis. Since there is an infinity of rotations about a unique, vertical axis, C, there is also an infinity of C2 axes perpendicular to C , and there is a horizontal plane of symmetry. The group is, very reasonably, designated D h. 

A third example of optically active thiophenes is provided by bithienyls. The best candidate for this object is 3,3 -bithienyl (166). Substitution at the 2- and 4-positions of both rings, if the substituents are bulky enough, is sufficient to prevent rotation about the common bond. If only a C2 symmetry axis is present a pair of enantiomers is generated. Numerous sets of enantiomers have been separated, some being shown in (167a-g). The enantiomeric separation is usually accomplished with a diacid derivative using alkaloid bases. 

Chiral bis-sulfoxides with a C2 symmetry axis can be readily prepared from the known (W)-methyl p-tolyl sulfoxide and commercially available methyl (S)-p-toluenesulfinate. Such chiral ligands are very attractive because of their easy synthesis and their ready availability in both enantiomers from inexpensive starting materials. Their complexes with Fel3 are shown to be good chiral catalysts for asymmetric Diels-Alder reactions [48] (Eq. 8A.26). 

A general feature of weso-phenyl substituted Co3 + corrolates is then that the planarity of the macrocyclic ligand is maintained in solution the pattern shown by the resonances due to the peripheral methyl groups in fact are indicative of the existence of a C2 symmetry axis the direct pyrrole-pyrrole bond typical of a planar corrole skeleton, confirmed also by the signals due to the meso-protons, if present. 

The device consists of a Ceo molecule fixed in the middle of two atomic scale gold leads, and the C2 symmetry axis of the molecule coincides with the horizontal axis (2-axis), see Fig. 1. Two distances between the electrodes are considered, 11.7 A and 13.7 A. They correspond to the minimum distance between an Au atom of the left (right) lead and a C atom of the Ceo at 2.3 A and 3.3 A, respectively. Each electrode consists of repeated unit cells with nine Au atoms in the (100) direction and extended to 2 = 00. We do not consider the Coulomb blockade effect because the C60 is strongly coupled to the electrodes at these small electrode separations. 

Structure of the molecule FC(0)000C(0)F showing (a) its C2 symmetry axis and (b) left-handed chiral conformation. 

In these equations, Hw(qa, q, 0) and jls(qa, qfo, 0) are the dipole moment operators at initial time belonging, respectively, to the irreducible representations B and A of the C2 symmetry group that transforms themselves, the first one, according to x and v, and the last one, according to x2 and y2 (allowed Raman transition), where x and y and the Cartesian coordinates that are perpendicular to the C2 symmetry axis. Here, we prefer the notations g and u in place of A and B of group theory. 

The crystallization of achiral molecules in chiral space groups, while rare and unpredictable, is well documented. Molecules with a C2 symmetry axis tend to crystallize in chiral structures, according to Jacques and coworkers, but despite impressive work on crystal engineering, predictions of a correlation between crystal symmetry and molecular structures are still hard to make [6]. 

Figure 1.26. Double Bingel addition to C70 leads to an achiral top) and two inherently chiral (center and bottom) addition patterns. Combination of each of the latter with chiral ester moieties affords two diastereoisomeric pairs of enantiomers. The enantiomers of each pair were prepared separately by addition of either (R,R) or (S, -configured malonates to C70, and all stereoisomers were isolated in pure state. The black dots mark intersections of the C2-symmetry axis with the [70]fullerene spheroid. Next to the three-dimensional representations, constitution and configuration of the addition patterns are shown schematically in a Newman type projection along the Cs-axis of C70. Of the two concentric five-membered rings, the inner one corresponds to the polar pentagon closest to the viewer, and the attached vertical line represents the bond C(l)-C(2) where the first addition occurred. The functionalized bonds at the distal pole depart radially from the outer pentagon.

Fig. 3.2. Profiles of the charge density along the C2 symmetry axis for the dissociation of the water molecule. In (a) p decreases monotonically from a maximum value at the oxygen nucleus. As Ro is increased, an abrupt change occurs in the form of the curve when a point of inflexion appears (b). This unstable critical point exists only for this single configuration X, of the nuclei and a further increase in Rq results in Its bifurcation to yield a maximum, corresponding to the formation of a bond critical point and a minimum, corresponding to the formation of a ring critical point a.s typified by (c). The system has entered the region of the ring structure.

Levy and coworkers97 have measured 13C spin-lattice relaxation times, 7), for 3- and 4-aminobiphenyls in a number of solvent systems, and of the corresponding ammonium ions in acidic and nonacidic media. The observed 7) values indicated that the molecular tumbling is anisotropic for these species. In addition, the known biphenyl geometry allowed indentification and semiquantitative evaluation of internal rotation-libration motion. The protonated amine function is motionally more restricted by solvent-solute and ion-pair interactions than the corresponding neutral amine. Thus, in the 3-biphenylammonium ion, the principal axis for molecular reorientation is aligned close to the C3—NHj-bond, whereas in the amine the principal axis lies closer to the biphenyl C2-symmetry axis. In both 3- and 4-aminobiphenyls, the unsubstituted phenyl rings are less restricted due to rapid phenyl rotation or libration. Table 14 presents 13C Tj-data for 4-aminobiphenyl 37 (NH2 on C4) and 4-biphenylammonium acetate 38 and trifluoroacetate 39. 

Many of the lower members of the four Rydberg series possess vibrational progressions. Following the notation of Wang et al. <77JCP(67)2614>, the Rydberg orbitals under C2l. symmetry where the Z axis is the C2 symmetry axis are ... 

A molecule that possesses a C2 symmetry axis may be rotated by 360°/2 (or 180°) to obtain a geometry equivalent to the starting geometry. In general, a C axis corresponds to rotation by 360°/n. Chiral molecules typically possess no elements of symmetry or only C axes. Molecules that possess a plane of symmetry are necessarily achiral. 

Because DIPHOS (or dppe) is an achiral ligand, the environment at the initial stages in the cycle is racemic, and no enantioselectivity is possible. The most common ligands used in asymmetric hydrogenation are, however, chiral diphosphines that usually possess a C2 symmetry axis Figure 12-1 shows just a few of the hundreds and hundreds of diphosphine and related ligands that have been reported in the literature. 

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