Twofold axes

Orthorhombic (or rhombic) Three unequal axes mutually perpendicular Three mutually perpendicular twofold axes, or two planes intersecting in a twofold axis a b c a = /3 = 7 = 90°... 

A, B, C, and D. We have now twisted the icosahedron somewhat, compared with Figure 16.6, so that we view the structure along one of the twofold axes, the same view as in Figure 16.3c. 

In the T = 4 structure there are 240 subunits (4 x 60) in four different environments, A, B, C, and D, in the asymmetric unit. The A subunits interact around the fivefold axes, and the D subunits around the threefold axes (Figure 16.7). The B and C subunits are arranged so that two copies of each interact around the twofold axes in addition to two D subunits. For a T = 4 structure the twofold axes thus form pseudosixfold axes. The A, B, and C subunits interact around pseudothreefold axes clustered around the fivefold axes. There are 60 such pseudothreefold axes. The T = 4 structure therefore has a total of 80 threefold axes 20 with strict icosahedral symmetry and 60 with pseudosymmetry. 

The S domains form the viral shell by tight interactions in a manner predicted by the Caspar and Klug theory and shown in Figure 16.8. The P domains interact pairwise across the twofold axes and form protrusions on the surface. There are 30 twofold axes with icosahedral symmetry that relate the P domains of C subunits (green) and in addition 60 pseudotwofold axes relating the A (red) and B (blue) subunits (Figure 16.9). By this arrangement the 180 P domains form 90 dimeric protrusions. 

The structure of cyclo-Sio is shown in Fig. 15.5(b).The molecule belongs to the very rare point group symmetry Do (three orthogonal twofold axes of rotation as the only symmetry elements). ITie mean interatomic distance and bond angle are close to those in cyclo-Su (Table 15.5) and the molecule can be regarded as composed of two identical S5 units obtained from the S 2 molecule (Fig. 15.6). 

The rotation axis of highest order is called the principal axis of rotation it is usually placed in the vertical direction and designated the z-axis of the molecule. Planes of reflection which are perpendicular to the principal axis are called horizontal planes (h). Planes of reflection which contain the principal axis are called vertical planes (v), or dihedral planes (d) if they bisect 2 twofold axes. 

Cubic point groups have four threefold axes (3 or 3) that mutually intersect at angles of 109.47°. They correspond to the four body diagonals of a cube (directions x+y+z, -x+y-z, -x-y+z and x-y-z, added vectorially). In the directions x, y, and z there are axes 4, 4 or 2, and there can be reflection planes perpendicular to them. In the six directions x+y, x-y, x+z,. .. twofold axes and reflection planes may be present. The sequence of the reference directions in the Hermann-Mauguin symbols is z, x+y+z, x+y. The occurrence of a 3 in the second position of the symbol (direction x+y+z) gives evidence of a cubic point group. See Fig. 3.8. 

B. Screw axes with intersecting twofold axes. ... 

Fig. 2. Diagrammatic representation of the sequence of hydrogen-bonded diol molecules (denoted HOC. . . COH) comprising one turn of the spiral (pitch 2c) around the canal. Lateral twofold axes at the z-coordinates marked bisect the diol molecules 8>...

In fact there is substantial structural variability among the five instances of the helical tubuland structure described above, in the form of variable placements of the diol hosts along the twofold axes, and variable a dimensions of the lattice, both of which have marked influence on the size and shape of the canals. Full analysis of these effects is provided elsewhere 8) and the major results only are reported here. As a consequence of steric repulsions between the bridge on the syn face and methyl substituents R on adjacent molecules presenting an anti face to the canal, as shown in Fig. 8, the molecules presenting the anti face are moved along the twofold axes, farther... 

Fig. 15. Conformational map of cyclohexane. The diagram represents a partial qualitative pictorial polar projection of the conformational globe of Pickett and Strauss (106) it may be completed by rotating around 120 and 240°, respectively. Relative potential energies are given (kcal mole-1 force field of ref. 19 reference chair conformation). The lines inside the six-membered rings represent mirror planes (solid) and twofold axes (dotted), respectively.

Fig. 19 The linkages between neighboring triacontahedral clusters in the Tsai-type 1/1 ACs along a the twofold axes, and b the threefold axes. The 2/1 ACs exhibit same linkages although they have different atom identities and symmetries. All decoration atoms of the triacontahedra are omitted for clarity. (Adapted with permission from [83], Copyright 2006 American Chemical Society)...

Figure 2.22 Models of packing of chains in form I of sPP according to space groups (a) Ibca59 and (b) P2 /a.146 Arrows indicate crystallographic twofold axes, present in (a) and lost in (b). In (b) chains are rotated by nearly 5° around chain axes, according to direction indicated by the arrows.

The orthorhombic point symmetiy 222 comprises three intersecting twofold axes perpendicular to each other. It applies to the commonly observed orthorhombic space group P2,2,21. 

Point symmetry mmm (shorthand for 2lm, 2Jm, 21m) comprises three perpendicular mirror planes, which automatically generates three perpendicular twofold axes at their lines of intersection, and a center of symmetry at the origin. This point group applies to all orthorhombic centrosymmetric crystals, such as those of space group Pbca. 

One interesting aspect of the structure of [Pr(TTA)4] is the short intramolecular sulphur-oxygen distances of 2.8—2.9 A. This value is about 0.4 A shorter than the sum of the van der Waals radii indicating some interaction between these atoms. The structure of NH4[Pr(TTA)4] HaO is stabilized by the combination of water of hydration and the small cation. The line defined by HaO—Pr—NH4 corresponds to one of the twofold axes of the dodecahedron. However, the large distances between Pr and OH 3 and Pr and NH4 (3.7—3.9 A) indicate no appreciable interactions. 

The packing of the molecules (without H atoms) in both allotropic forms SOAz (I) and (II) is visualized by means of the projection on to the planes (001) and (010) (Figs. 42 and 43) for SOAz (II), half of the cell content was drawn for the sake of clarity (the helicoYdal twofold axes have not been u ). 

Axes of rotation are among the most common of molecular symmetry operations. A onefold axis is a rotation by a full turn, equivalent to the identity. A twofold rotation axis, as in the example of the water molecule, is sometimes called a dyad. Cyclopropane has a threefold axis perpendicular to the plane containing the carbon atoms it also has three twofold axes. Can you visualize them ... 

A special type of vertical plane of symmetry occurs if there are two or more twofold axes perpendicular to the principal axis, and a mirror plane bisects the angle between two of these twofold axes, as in benzene. Such a plane is called... 

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