Threefold rotation axis

A systematic basis can be gained by viewing a tristchelate) complex, the most common species of chiral coordination compounds, down the threefold rotation axis. If the helix thus viewed is right-handed, the isomer is the A-isomer, and its mirror image is the A-isomer.50 The d-. l-, A-. and A-isomers may thus be portrayed as shown in the margin. Note that it is a result of these systems that A A l and A A D, furthering possible confusion. 

Threefold rotation axis Threefold screw axis 3 sub 1 ... 

S6 One sixfold mirror-rotation axis, which is, of course, equivalent to one threefold rotation axis plus center of symmetry. Example ... 

C2v, C3v, C4v, C5v, C6v,. .., Cm C2v, Two perpendicular symmetry planes whose crossing line is a twofold rotation axis C3v, One threefold rotation axis with three symmetry planes which include the rotation axis. The angle is 60° between two symmetry planes C4v, One fourfold rotation axis with four symmetry planes which include the rotation axis. The four planes are grouped in two nonequivalent pairs. One pair is rotated relative to the other pair by 45°. The angle between... 

D3 One threefold rotation axis and three twofold rotation axes perpendicular to the threefold axis. The twofold axes are at 120°, so the minimum angle between two such axes is 60° Examples ... 

D3iJ One threefold rotation axis with three twofold rotation axes perpendicular to it, and three symmetry planes. The angle between the twofold axes is 60°. The symmetry planes include the threefold axis and bisect the angles between the twofold axes. Examples Figure 3-9. 

D3h One threefold rotation axis, three symmetry planes (at 60°) which contain the threefold axis, and another symmetry plane perpendicular to the threefold axis. Examples Figure 3-10. 

Figure 4-12 illustrates different combinations of symmetry elements, for example, twofold, fourfold, and sixfold antirotation axes together with other symmetry elements after Shubnikov [15], The fourfold antirotation axis includes a twofold rotation axis, and the sixfold antirotation axis includes a threefold rotation axis. The antisymmetry elements have the same notation as the ordinary ones except that they are underlined. Antimirror rotation axes characterize the rosettes in the second row of Figure 4-12. The antirotation axes appear in combination with one or more symmetry planes perpendicular to the plane of the drawing in the third row of Figure 4-12. Finally, the ordinary rotation axes are combined with one or more antisymmetry planes in the two bottom rows of Figure 4-12. In fact, symmetry 1 m here is the symmetry illustrated in Figure 4-11. The black-and-white variation is the simplest case of color symmetry.

The same relationships hold for D /, symmetric molecules when ft, is set to zero (44). For both C3/, and D3), symmetry, the depolarization ratio is expected to be p = 1.5. However, the converse need not be true a depolarization ratio of close to 1.5 does not prove a point group with a threefold rotational axis. The molecule may be conformationally flexible then HRS detects a superposition of (ft) of the different conformers. A case in point is l,3,5-trinitro-2,4,6-triisopropylaminobenzene which exists as an interconverting mixture of strongly distorted boat and twist-boat forms (Wolff etal., 1993), with approximate C5 and C2 symmetry, respectively, but still shows a depolarization ratio of close to 1.5 (Verbiest et al., 1994 Wortmann et ai, 1997). Even in the case of conformational homogeneity it may still strongly deviate from the ideal symmetry provided the impact of conformational distortion on the electronic properties is not great. 

Orbital degeneracy occurs in molecules having at least a threefold rotational axis as in triphenylene (1 ) or possessing symmetry [382, 386]. Very... 

Furthermore, as we will see in sections 1.5.3 and 1.5.5, below, transformations performed by the three-fold inversion and the six-fold inversion axes can be represented by two independent simple symmetry elements. In the case of the three-fold inversion axis, 3, these are the threefold rotation axis and the center of inversion acting independently, and in the case of the six-fold inversion axis, 6, the two independent symmetry elements are the mirror plane and the three-fold rotation axis perpendicular to the plane, as denoted in Table 1.4. The remaining four-fold inversion axis, 4, is a unique symmetry element (section 1.5.4), which cannot be represented by any pair of independently acting symmetry elements. 

Fig. 2. The axial single crystal circular dichroism spectrum (Ae/10) of 2 A-t+XCofenlslCls NaCl 6 H2O (solid curve) and the absorption and CD spectrum of A-(+)-[Co(en)3l(a04)3 in water (upper and lower broken curves, respectively). The axial crystal CD spectrum characterises electronic transitions of the complex ion to upper states of symmetry, polarized perpendicular to the threefold rotation axis of the complex...

Figure 2-23 shows two flowers. The periwinkle (Vinca minor) has fourfold rotational symmetry cind no symmetry plane. The Norwegian tulip has threefold rotational symmetry with the axis of rotation in a symmetry plane. The threefold rotation axis will, of course, rotate not only the flower but any other symmetry element, in this case the symmetry plane, as well. The 120 ... 

Cjy. One threefold rotation axis with three symmetry planes which include the rotation axis. The angle is 60° between two symmetry planes. Examples Figure 3-12b. 

One threefold rotation axis with three twofold rotation axes perpen-... 

Note the presence of primes ( ) and double primes (") on the Mulliken symbols. They describe representations which ar, respectively, symmetric and anlisymmetric with respect to reflection in the mirror plane (ofi) perpendicular to the threefold rotation axis. 

The absolute configurations of chiral complexes are described by viewing the molecule along a threefold rotation axis of the octahedron. Fig. 5.2, and noting the handedness of the helix formed by the ligands. Left rotation of the helix is designated A, right rotation A. 

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