Helical symmetry

With regard to orientation, consider a repeating side-group originating at atom the first atom of the side-group being BL. For certain chain symmetries (helical, for instance) the bond vectors b(A. B )... 

In catalysts obtained from chiral stereorigid metallocenes of class III with C2 molecular symmetry (helical), such as racemic isomers of ansa-metallocenes, e.g. rac.-(IndCH2)2MtX2 (Table 3.1) and rac.-(ThindCH2)2MtX2, the two coordination positions available for the incoming monomer and the growing... 

Macromolecules exist in a variety of conformational forms that range from randomly coiled chains to more spatially ordered structures. Of particular interest are the polymers that adopt helical symmetry. Helical conformation is a result of an orderly repeated unit with internal rotational angles along the polymer backbone. In the crystalline state, polyoxyethylene exists in a helical conformation that contains seven chemical units (-CH2CH2-O-) and two turns in a backbone identity period of 19.3 A (7-9). 

Nevertheless, the spatial groups, although introduced as extensions of the point groups by considering the translations in steps inside the elementary cell, include elements of symmetry (helicals and glide planes) which, added to the symmetries of the basic point group, make that their intersection can not be reduced to a point any longer. 

The structurally similar L and M subunits are related by a pseudo-twofold symmetry axis through the core, between the helices of the four-helix bundle motif. The photosynthetic pigments are bound to these subunits, most of them to the transmembrane helices, and they are also related by the same twofold symmetry axis (Figure 12.15). The pigments are arranged so that they form two possible pathways for electron transfer across the membrane, one on each side of the symmetry axis. 

This symmetry is important in bringing the two chlorophyll molecules of the "special pair" into close contact, giving them their unique function in initiating electron transfer. They are bound in a hydrophobic pocket close to the symmetry axis between the D and E transmembrane a helices of both... 

This pair of chlorophyll molecules, which as we shall see accepts photons and thereby excites electrons, is close to the membrane surface on the periplasmic side. At the other side of the membrane the symmetry axis passes through the Fe atom. The remaining pigments are symmetrically arranged on each side of the symmetry axis (Figure 12.15). Two bacteriochlorophyll molecules, the accessory chlorophylls, make hydrophobic contacts with the special pair of chlorophylls on one side and with the pheophytin molecules on the other side. Both the accessory chlorophyll molecules and the pheophytin molecules are bound between transmembrane helices from both subunits in pockets lined by hydrophobic residues from the transmembrane helices (Figure 12.16). 

Like the photosynthetic reaction center and bacteriorhodopsin, the bacterial ion channel also has tilted transmembrane helices, two in each of the subunits of the homotetrameric molecule that has fourfold symmetry. These transmembrane helices line the central and inner parts of the channel but do not contribute to the remarkable 10,000-fold selectivity for K+ ions over Na+ ions. This crucial property of the channel is achieved through the narrow selectivity filter that is formed by loop regions from thefour subunits and lined by main-chain carbonyl oxygen atoms, to which dehydrated K ions bind. 

Before we can analyze the electronic structure of a nanotube in terms of its helical symmetry, we need to find an appropriate helical operator S>(h,ip), representing a screw operation with a translation h units along the cylinder axis in conjunction with a rotation if radians about this axis. We also wish to find the operator S that requires the minimum unit cell size (i.e., the smallest set of carbon atoms needed to generate the entire nanotube using S) to minimize the computational complexity of calculating the electronic structure. We can find this helical operator by first... 

Fig. 2. Depiction of conformal mapping of graphene lattice to [4,3] nanotube. B denotes [4,3] lattice vector that transforms to circumference of nanotube, and H transforms into the helical operator yielding the minimum unit cell size under helical symmetry. The numerals indicate the ordering of the helical steps necessary to obtain one-dimensional translation periodicity.

The rotational and helical symmetries of a nanotube defined by B can then be seen by using the corresponding helical and rotational symmetry operators and C/ to generate the nanotube[13,14]. This is done by first introducing a cylinder of radius... 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

Now, let us return to our discussion of carrying out an electronic structure calculation for a nanotube using helical symmetry. The one-electron wavefunc-tions can be constructed from a linear combination of Bloch functions linear combination of nuclear-centered functions Xj(r),... 

This cell is veiy different from that of Ooct-OPV3 and contains four symmetry-related molecules. Due to the shorter substituents, the density is much higher. Despite this, there is no 7r-stacking (sec Fig. 16-21). The successive re-systems within one, non-centrosymmctric molecule are arranged in a helical fashion because the dihedral angles all have the same sign. 

It is typical, for instance, of syndiotactic polystyrene (s-PS) [7-9] and syndiotactic poly- p-methylstyrene (s-PPMS) [10] to present crystalline forms with a transplant conformation of the chains (shown for s-PS in Fig. 1) as well as crystalline forms with sequences of dihedral angles of the kind TTG+G+ (or the equivalent G G TT), corresponding to a s(2/l)2 helical symmetry of the chains (shown for s-PS in Fig. 1). 

Clathrate structures have been recently obtained also for s-PS [8] and s-PPMS [10]. In particular for s-PS, the treatment of amorphous samples, as well as of crystalline samples in the a or in the y form, produces clathrate structures including helices having s (2/1)2 symmetries, which present similar diffraction patterns, independently of the considered solvent. The treatment of samples of s-PPMS with suitable solvents also produces clathrate structures including s(2/l)2 helices however, the large differences in the X-ray diffraction patterns suggest different modes of packing, depending on the included solvent. 

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