Chirality about an axis

Some structures are said to have chirality about an axis, which can be considered to be one of the rotational axes in a more symmetric structure that no longer exists because substituents have been moved due to "stretching" of... 

Stretching a structure with chirality abouta pointto produce structures with chirality about an axis. (Adapted from reference 55.)... 

Chirality about an axis is exemplified by allenes, such as the 2,3-penta-diene enantiomer 23. In 23 the methyl and hydrogen substituents on C2 lie in a plane (in the page) that is perpendicular to the plane containing the methyl and hydrogen substituents on C4. Here the axis of chirality is coincident with the C2—C3—C4 bond axis. The structure has a C2 symmetry element, so it is dissymmetric, not asymmetric. The C2 rotation axis is perpendicular to the axis of chirality, as illustrated in Figure 2.13. It must be emphasized that not all structures that are chiral about an axis have a C2 rotation axis. For example, the 2,3-hexadiene enantiomer 24 also has an axis of chirality coincident with the C2-C3-C4 bond axis, but it does not have a C2 rotation axis. The adamantane 25 and appropriately substituted spiro compounds, such as 26, are also chiral about an axis. ... 

The R or S designation of frans-cyclooctene (28, Figure 2.17) can most easily be determined by considering the structure to have chirality about an axis. Viewing 28 as indicated by the eye symbol reveals it to have the R configuration. Krow detailed the application of the rules for planar chirality to... 

The second issue concerns the anisotropy of the membrane. The models presented in this section all assume that the membrane has the symmetry of a chiral smectic-C liquid crystal, so that the only anisotropy in the membrane plane comes from the direction of the molecular tilt. With this assumption, the membrane has a twofold rotational symmetry about an axis in the membrane plane, perpendicular to the tilt direction. It is possible that a membrane... 

Axial Chirality. For a system with four groups arranged out of the plane in pairs about an axis, the system is asymmetric when the groups on each side of the axis are different. Such a system is referred to as an axial chiral system. This structure can be considered a variant of central chirality. Some axial chiral molecules are allenes, alkylidene cyclohexanes, spiranes, and biaryls (along with their respective isomorphs). For example, compound 7a (binaphthol), which belongs to the class of biaryl-type axial chiral compounds, is extensively used in asymmetric synthesis. Examples of axial chiral compounds are given in Figure 1-5. 

The interconversion of enantiomers can be viewed in general as requiring inversion at a particular atom or twisting about an axis of the molecule. Provided these processes are inhibited to some degree, the chirality of a molecule will be detectable so that any chiral species may be said to contain centres and/or axes and/or planes of chirality.115,116,117,118 The precise meanings and utility of these concepts are, however, a matter of some debate,115,116 129 and they have not been extensively applied to coordination compounds. 

Figure 4. A translating spinning cylinder. The polar vector in the rotation-translation (screw displacement) corresponds to the direction of translation and the axial vector to the direction of spin. Time reversal (7) does not change the sense of chirality of homomorphous systems (a) and (b) in terms of the helicity generated by the product of the two vectors, (a) and (b) are both right-handed. Space inversion (P) of (a) yields a left-handed system (c), the enantiomorph of (a). Time reversal of (a), followed by rotation of (b) by 180° (Rn) about an axis perpendicular to the cylindrical axis, yields (d), a homomorph of (a). Space inversion of (d) brings us back to (c).

Axial chirality arises from the disposition of groups about an axis, e.g., in an allene see above. 

A number of l,4-dihydrop)rridines (17-20), exhibiting axial chirality (chiralty stemming from the nonplanar arrangement of four groups about an axis), have been separated by small-scale HPLC methods. This is an impor-... 

When the molecules that form a liquid-crystalline phase are chiral, the structure of these mesophases can have an additional property. In the chiral nematic phase (N ) the director precesses about an axis perpendicular to the director and describes in this way a helix (Figure 2.7). The pitch of a chiral nematic phase is the distance along the helix over which the director rotates over 360°. The chiral nematic phase is sometimes... 

The parallel arrangement is formed when molecule b is related by a translation to molecule a, and then allowed rotational freedom about R. The antiparallel arrangement involves translation and a rotation by n of molecule b about an axis perpendicular to R, so that the electric dipole sense is reversed from that of a. In the first (parallel) arrangement 6a = 6b=y=0 and the interaction between molecules of the same chirality is... 

A rotation about an axis transforms a left hand into a left hand and a right hand into a right hand. It conserves the chirality. It is called an operation of the first type. The determinant of any matrix representing an operation of the first typeis AJ= +1. 

The fact that the three compounds we have just introduced (along with Feist s acid in the box on p. 319) were chiral might have surprised you, because at first glance they do look quite symmetrical. In fact, they do all have an element of symmetry, and it is only one which is compatible with chirality an axis of symmetry. If a molecule can be rotated through 180° about an axis to give exactly the same structure then it has twofold axial symmetry, or C2 symmetry. Compounds with an axis of symmetry will still be chiral, provided they lack either a plane or a centre of symmetry. 

If the molecules that form a hquid crystal phase are chiral (lack inversion symmetry), then chiral phases exist in place of certain non-chiral phases. In calamitic liqnid crystals, the nematic phase is replaced by the chiral nematic phase, in which the director rotates in helical fashion about an axis perpendicirlar to the director. Such a phase is illustrated in Figme 1.11. 

Chiral disc-like molecules also form chiral phases, the best example being the chiral nematic phase. From the standpoint of the director, the structure of this phase is identical to the chiral nematic phase in rod-like molecules. That is, the director adopts a heUcal structure by rotating about an axis perpendicular to the director. 

Consider the other diastereomer of pentane-2,4-diol, the chiral d,l isomer. This isomer has an axis of symmetry. If the molecule is rotated 180° about an axis in the plane of the paper passing through the central carbon, the molecule can be converted into itself ... 

In polarimetry and ORD, the sample is placed between the first polarizing element (the polarizer), which remains fixed, and the second element (the analyzer), which can be rotated about the axis of propagation. Maximum intensity of the transmitted light is observed when the principal axis of the polarizer and analyzer are colinear and exactly parallel. The intensity is zero when they are crossed that is, when the principal axes are orthogonal to each other. The most accurate way to determine the rotation angle a is to set the polarizer and analyzer in the crossed position using an achiral substrate and to measure the extent to which the analyzer has to be turned to restore the optical null position when the achiral sample is replaced by a chiral substrate. 

Although we have shown that 4 is chiral, and that it has no stereogenic centre, it is not without symmetry. This is best seen with the aid of a Newman projection of 4, which is shown in 5. A two-fold (C2) axis of symmetry exists as shown this passes through the central carbon, C(2), and bisects the right angle between the two chlorine atoms (and likewise the two hydrogen atoms). Rotation about this axis by 180° gives an identical molecule. Because of the axis of symmetry, 4 cannot be said to be asymmetric, and similar situations are found in certain other chiral molecules. This has had implications for stereochemical nomenclature. In particular, to avoid confusion the term asymmetric carbon is now little used even for an sp3 hybridized carbon that carries four... 

An axis about which a set of ligands is held so that it results in a spatial arrangement that is not superposable on its mirror image. For example with an allene abC=C=Ccd the chiral axis is defined by the C=C=C bonds and with an orf/zo-substituted biphenyl the atoms C-1, C-1, C-4, and C-4 lie on the chiral axis. 

Figure 13.21 presents the picture of the dielectric ellipsoid orientation within each unit cell that is at the nanometer scale. The weak molecular chirality results in additional weak twisting of all structures with characteristic pitch of about Po 0.1-1 pm. An example of a such twisted structure is shown in Fig. 13.22 it is an antiferroelectric double-layer cell describing two geared helices upon rotation about z-axis. The helices are shifted in phase by (p = 7t and have the same handedness. On the molecular scale, due to molecular chirality, the c-director turns from layer to layer by a small angle 5cp = 2nllPo, therefore, for / Inm,... 

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