Chiral Spaces

PHARMACEUTICALS, CHIRAL. Stereoisomers are compounds which have the same molecular formula bnl differ in the arrangement of their atoms in space. Chiral compounds are compounds which... 

Three-dimensional chirality can likewise be resolved in four dimensions by transplantation in non-orientable projective space. This three-dimensional analog requires a four-dimensional twist that closes the three-dimensional universe onto itself and turns left-handed matter into right-handed antimatter. Considered as a single universe in three-dimensional space, chirality is preserved throughout. However, the interface created by the curvature separates regions of space with opposite chiralities. The interface cannot be crossed in three-dimensional motion, but allows interaction between entities near the interface to give rise to the quantum effects. 

An n-dimensional object is called chiral if it is nonsuperimposable on its mirror image by any rotation in the n-dimensional space. Chirality is the property of chiral objects. 

Chirality in chemistry is associated with 3D molecular forms. It is a consequence of the asymmetrical substitution of tetrahedral carbons. However, at the beginning of this century Kelvin had defined chirality for objects in/i-dimensional spaces. Chiral... 

The R, S convention is a scheme which has largely superseded the D, i. system to denote configuration about a chiral centre in a molecule. The convention allows unequivocal designation of the absolute configuration in a description of the positions in space of ligands attached to a chiral centre, in relation to an agreed standard of chirality like a right-hand helix. 

If compounds have the same topology (constitution) but different topography (geometry), they are called stereoisomers. The configuration expresses the different positions of atoms around stereocenters, stereoaxes, and stereoplanes in 3D space, e.g., chiral structures (enantiomers, diastereomers, atropisomers, helicenes, etc.), or cisftrans (Z/E) configuration. If it is possible to interconvert stereoisomers by a rotation around a C-C single bond, they are called conformers. 

Clearly, the next step is the handling of a molecule as a real object with a spatial extension in 3D space. Quite often this is also a mandatory step, because in most cases the 3D structure of a molecule is closely related to a large variety of physical, chemical, and biological properties. In addition, the fundamental importance of an unambiguous definition of stereochemistry becomes obvious, if the 3D structure of a molecule needs to be derived from its chemical graph. The moleofles of stereoisomeric compounds differ in their spatial features and often exhibit quite different properties. Therefore, stereochemical information should always be taken into ac-count if chiral atom centers are present in a chemical structure. 

The value of embodies the conformation-independent 3D arrangement of the atoms of the ligands of a chirality center in distance space and thus cannot distinguish between enantiomers. This distinction is introduced by the descriptor S , , . 

The experimental facts that led van t Hoff and Le Bel to propose that molecules having the same constitution could differ m the arrangement of their atoms m space concerned the physical property of optical activity Optical activity is the ability of a chiral sub stance to rotate the plane of plane polarized light and is measured using an instrument called a polarimeter (Figure 7 5)... 

Optically pure (Section 7 4) Descnbing a chiral substance in which only a single enantiomer is present Orbital (Section 1 1) Strictly speaking a wave function i i It is convenient however to think of an orbital in terms of the probability i i of finding an electron at some point relative to the nucleus as the volume inside the boundary surface of an atom or the region in space where the probability of finding an electron is high... 

As a result of having two chiral centers, four stereoisomers of ascorbic acid are possible (Table 1) (Fig. 2). Besides L-ascorbic acid (Activity = 1), only D-araboascorbic acid (erythorbic acid (9)) shows vitamin C activity (Activity = 0.025-0.05). The L-ascorbic acid stmcture (1) in solution and the soHd state are almost identical. Ascorbic acid crystallizes in the space group P2 with four molecules in the unit cell. The crystal data are summarized in Table 2. 

Because a hexose contains four chiral carbon atoms, there are 2 = 16 different possible arrangements of the hydroxyl groups in space, ie, there are 16 different stereoisomers. The stmctures of half of these, the eight D isomers, are shown in Figure 1. Only three of these 16 stereoisomers are commonly found in nature D-glucose [50-99-7] D-galactose [59-23-4] and D-mannose [3458-28-4]. 

Figure 2.24, Determination of the enantiomeric excess of 1-phenylethanol [30, 0.1 mmol in 0.3 ml CDCI3, 25 °C] by addition of the chiral praseodymium chelate 29b (0.1 mmol), (a, b) H NMR spectra (400 MHz), (a) without and (b) with the shift reagent 29b. (c, d) C NMR spectra (100 MHz), (c) without and (d) with the shift reagent 29b. In the C NMR spectrum (d) only the C-a atoms of enantiomers 30R and 30S are resolved. The H and C signals of the phenyl residues are not shifted these are not shown for reasons of space. The evaluation of the integrals gives 73 % R and 27 % S, i.e. an enantiomeric excess (ee) of 46 %...

The symmetry groups for the chiral tubules are Abelian groups. The corresponding space groups are non-symmorphic and the basic symmetry operations... 

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

The diffraction patterns due to different isochiral clusters are superimposed and well separated in a polychiral MWCNT diffraction pattern, suggesting that interference between waves scattered by tubes with different chiral angles can be neglected. It is therefore meaningful to discuss only isochiral clusters of tubes. Such clusters are only compatible with a constant intercylinder spacing c/2 for pairs of Hamada indices satisfying the condition = L +M +LM - (nc/a). Approximate solutions are for instance (8, 1) and (5, 5) [16,17]. 

Fig. 11. Simulated diffraction space of a chiral (40, 5) SWCNT. (a) Normal incidence diffraction pattern with 2mm symmetry (b),(c),(d) and (e) four sections of diffraction space at the levels indicated by arrows. Note the absence of azimuthal dependence of the intensity. The radii of the dark circles are given by the zeros of the sums of Bessel functions [17].

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

The minor images of bromochlorofluoromethane have the sane constitution. That is, the atoms are connected in the sane order. But they differ in the anangement of then-atoms in space they are stereoisomers. Stereoisomers that are related as an object and its nonsuperimposable minor image are classified as enantiomers. The word enantiomer describes a paiticulai- relationship between two objects. One cannot look at a single molecule in isolation and ask if it is an enantiomer any more than one can look at an individual human being and ask, Is that person a cousin Fuithennore, just as an object has one, and only one, minor image, a chiral molecule can have one, and only one, enantiomer. 

The type of problem just mentioned is of great importance in the enumeration of chemical isomers when the situation of a molecule in three-dimensional space is heeded, and leads to the consideration of "chirality". 

Structural isomers have identical molecular formulas, but their atoms are linked to different neighbors. Geometrical isomers have the same molecular and structural formulas but different arrangements in space. Molecules with four different groups attached to a single carbon atom are chiral they are optical isomers. 

A further complication is that an icosahedron sharing four faces loses its symmetry operations of the second sort and becomes chiral, D or L. The eclipsed configuration described above requires that D and l alternate. Hence, in a ring of five icosahedra one DD or LL bond must occur, with a strain in the ring of 40°, which is a rotation of 4 per shared face in each icosahedron. The space-group symmetry is thus changed from that of diamond to a subgroup. 

The compounds crystallise in noncentrosymmetric space groups namely PI, P2i, C2, and P2i2i2i (but with priority of P2i) due to the chirality of the molecules. Most of the compounds have a tilted layer structure in the crystalline state. The tilt angle of the long molecular axes with respect to the layer normal in the crystal phase of the compounds is also presented in Table 18. Some compounds show larger tilt angles in the crystalline state than in the smectic phase. In the following only the crystal structures of some selected chiral liquid crystals will be discussed. 

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