Chiral molecules symmetry

A schematic diagram of the surface of a liquid of non-chiral (a) and chiral molecules (b) is shown in figure Bl.5.8. Case (a) corresponds to oom-synnnetry (isotropic with a mirror plane) and case (b) to oo-symmetry (isotropic). For the crj/ -synnnetry, the SH signal for the polarization configurations of s-m/s-out and p-m/s-out vanish. From table Bl.5.1. we find, however, that for the co-synnnetry, an extra independent nonlinear susceptibility element, is present for SHG. Because of this extra element, the SH signal for... 

As witli tlie nematic phase, a chiral version of tlie smectic C phase has been observed and is denoted SniC. In tliis phase, tlie director rotates around tlie cone generated by tlie tilt angle [9,32]. This phase is helielectric, i.e. tlie spontaneous polarization induced by dipolar ordering (transverse to tlie molecular long axis) rotates around a helix. However, if tlie helix is unwound by external forces such as surface interactions, or electric fields or by compensating tlie pitch in a mixture, so tliat it becomes infinite, tlie phase becomes ferroelectric. This is tlie basis of ferroelectric liquid crystal displays (section C2.2.4.4). If tliere is an alternation in polarization direction between layers tlie phase can be ferrielectric or antiferroelectric. A smectic A phase foniied by chiral molecules is sometimes denoted SiiiA, altliough, due to the untilted symmetry of tlie phase, it is not itself chiral. This notation is strictly incorrect because tlie asterisk should be used to indicate the chirality of tlie phase and not tliat of tlie constituent molecules. 

We have seen in Section 4.1.4 that = n and that S2 = i, so we can immediately exclude from chirality any molecule having a plane of symmetry or a centre of inversion. The condition that a chiral molecule may not have a plane of symmetry or a centre of inversion is sufficient in nearly all cases to decide whether a molecule is chiral. We have to go to a rather unusual molecule, such as the tetrafluorospiropentane, shown in Figure 4.8, to find a case where there is no a or i element of symmetry but there is a higher-fold S element. In this molecule the two three-membered carbon rings are mutually perpendicular, and the pairs of fluorine atoms on each end of the molecule are trans to each other. There is an 54 axis, as shown in Figure 4.8, but no a or i element, and therefore the molecule is not chiral. 

In Section 4.2.1 it will be pointed out that hydrogen peroxide (Figure 4.1 la) has only one symmetry element, a C2 axis, and is therefore a chiral molecule although the enantiomers have never been separated. The complex ion [Co(ethylenediamine)3], discussed in Section 4.2.4 and shown in Figure 4.11(f), is also chiral, having only a C3 axis and three C2 axes. 

Compounds in which one or more carbon atoms have four nonidentical substituents are the largest class of chiral molecules. Carbon atoms with four nonidentical ligands are referred to as asymmetric carbon atoms because the molecular environment at such a carbon atom possesses no element of symmetry. Asymmetric carbons are a specific example of a stereogenic center. A stereogenic center is any structural feature that gives rise to chirality in a molecule. 2-Butanol is an example of a chiral molecule and exists as two nonsuperimposable mirror images. Carbon-2 is a stereogenic center. 

C atoms are labelled a-e (see text), (b), (c) Line drawings of the two enantiomers of C76 viewed along the short C2 rotation axis and illustrating the chiral D2 symmetry of the molecule. 

Chiral (Section 9.2) Having handedness. Chiral molecules are those that do not have a plane of symmetry and are therefore not superimposable on their mirror image. A chiral molecule thus exists in two forms, one right-handed and one left-handed. The most common cause of chirality in a molecule is the presence of a carbon atom that is bonded to four different substituents. 

Here, ry is the separation between the molecules resolved along the helix axis and is the angle between an appropriate molecular axis in the two chiral molecules. For this system the C axis closest to the symmetry axes of the constituent Gay-Berne molecules is used. In the chiral nematic phase G2(r ) is periodic with a periodicity equal to half the pitch of the helix. For this system, like that with a point chiral centre, the pitch of the helix is approximately twice the dimensions of the simulation box. This clearly shows the influence of the periodic boundary conditions on the structure of the phase formed [74]. As we would expect simulations using the atropisomer with the opposite helicity simply reverses the sense of the helix. 

A common finding of computational PECD studies is that a relatively large partial wave expansion, typically running to niax > 15 is required. Chiral molecules necessarily are of very low, or no, symmetry, and hence are quite... 

When the mesogenic compounds are chiral (or when chiral molecules are added as dopants) chiral mesophases can be produced, characterized by helical ordering of the constituent molecules in the mesophase. The chiral nematic phase is also called cholesteric, taken from its first observation in a cholesteryl derivative more than one century ago. These chiral structures have reduced symmetry, which can lead to a variety of interesting physical properties such as thermocromism, ferroelectricity, and so on. 

Chiral molecules are characterised by symmetry elements of the first kind, for example, axes of rotation. 

In the case of MAP, the concept of chirality was used so as to prevent centrosymmetry a chiral molecule cannot be superimposed on its image by a mirror or center of symmetry so that a crystal made only of left or right-handed molecules can accomodate neither of these symmetry elements. This use of the chirality concept ensures exclusion of a centrosymmetric structure. However as we shall see in the following, the departure of the actual structure from centrosymmetry may be only weak, resulting in limited nonlinear efficiencies. A prerequisite to the introduction of a chiral substituent in a molecule is that its location should avoid interfering with the charge-transfer process. 

The earliest approach to explain tubule formation was developed by de Gen-nes.168 He pointed out that, in a bilayer membrane of chiral molecules in the Lp/ phase, symmetry allows the material to have a net electric dipole moment in the bilayer plane, like a chiral smectic-C liquid crystal.169 In other words, the material is ferroelectric, with a spontaneous electrostatic polarization P per unit area in the bilayer plane, perpendicular to the axis of molecular tilt. (Note that this argument depends on the chirality of the molecules, but it does not depend on the chiral elastic properties of the membrane. For that reason, we discuss it in this section, rather than with the chiral elastic models in the following sections.)... 

From the form of the polarization it is clear that in order to observe any nonlinear optical effect, the input beams must not be copropagating. Furthermore, nonlinear optical effects through the tensor y eee requires two different input frequencies (otherwise, the tensor components would vanish because of permutation symmetry in the last two indices, i.e., ytfl eee = Xijy ) For example, sum-frequency generation in isotropic solutions of chiral molecules through the tensor y1 1 1 has been experimentally observed, and the technique has been proposed as a new tool to study chiral molecules in solution.59,61 From an NLO applications point of view, however, this effect is probably not very useful because recent results suggest that the response is actually very low.62... 

It should be emphasized that the existence of chiral zeroes is different from the vanishing of a chirality function due to a lack of qualitative completeness. When a function is not qualitatively complete, it will vanish identically for a wide class of chiral molecules and/or mixtures. A chiral zero of the type mentioned here, however, depends on particular values of the parameters, and one can get away from the zero by varying the parameters slightly, without changing any symmetry property. 

Finally, reference must be made to the important and interesting chiral crystal structures. There are two classes of symmetry elements those, such as inversion centers and mirror planes, that can interrelate. enantiomeric chiral molecules, and those, like rotation axes, that cannot. If the space group of the crystal is one that has only symmetry elements of the latter type, then the structure is a chiral one and all the constituent molecules are homochiral the dissymmetry of the molecules may be difficult to detect but, in principle, it is present. In general, if one enantiomer of a chiral compound is crystallized, it must form a chiral structure. A racemic mixture may crystallize as a racemic compound, or it may spontaneously resolve to give separate crystals of each enantiomer. The chemical consequences of an achiral substance crystallizing in a homochiral molecular assembly are perhaps the most intriguing of the stereochemical aspects of solid-state chemistry. 

Lord Kelvin lla> recognized that the term asymmetry does not reflect the essential features, and he introduced the concept of chiralty. He defined a geometrical object as chiral, if it is not superimposable onto its mirror image by rigid motions (rotation and translation). Chirality requires the absence of symmetry elements of the second kind (a- and Sn-operations) lu>>. In the gaseous or liquid state an optically active compound has always chiral molecules, but the reverse is not necessarily true. 

For example, in the tetracoordinate configuration 17 with a Tchiral molecules of Ruch s class A the same is true. 

Chiral molecules have a nonsuperposable mirror image (see Fig. 11.1) and so possess intrinsic handedness in three-dimensional space. In a perfectly symmetric, chi-rally unbiased world, each handed version (enantiomer) must exist with an equal probability. The observed preference of one enantiomer over the other in biomolecules implies that this symmetry has been broken. The initial induction of a symmetry-breaking chiral excess is the sine qua non of eventual chiral dominance. Several mechanisms have been proposed to bring about chiral symmetry breaking. 

Because proteins are made up of L-amino acids, they exhibit chirality in their structures, lacking planes or points of symmetry. Proteins also can exhibit chirality in their interactions with other chiral molecules as well as prochiral centers in other molecules. This latter point is beautifully illustrated by fumarase s catalysis of the dehydration of L-malate, a molecule containing two seemingly equivalent hydrogen atoms ... 

The stereospecificity of vibrational optical activity, on the other hand, arises from the fact that the vibrations take place in a chiral framework and VOA intensities depend in large part on the extent and phasing of coupling or mixing of vibrational motion in different parts of the molecule. In addition, many local symmetry restrictions are lifted in chiral molecules, for example, the local de-... 

Crystallization and reactivity in two-dimensional (2D) and 3D crystals provide a simple route for mirror-symmetry breaking. Of particular importance are the processes of the self assembly of non-chiral molecules or a racemate that undergo fast racemization prior to crystallization, into a single crystal or small number of enantiomorphous crystals of the same handedness. Such spontaneous asymmetric transformation processes are particularly efficient in systems where the nucleation of the crystals is a slow event in comparison to the sequential step of crystal growth (Havinga, 1954 Penzien and Schmidt, 1969 Kirstein et al, 2000 Ribo et al 2001 Lauceri et al, 2002 De Feyter et al, 2001). The chiral crystals of quartz, which are composed from non-chiral Si02 molecules is an exemplary system that displays such phenomenon. 

---