Improper rotation axes

One example of a quantitative measure of molecular chirality is the continuous chirality measure (CCM) [39, 40]. It was developed in the broader context of continuous symmetry measures. A chital object can be defined as an object that lacks improper elements of symmetry (mirror plane, center of inversion, or improper rotation axes). The farther it is from a situation in which it would have an improper element of symmetry, the higher its continuous chirality measure. 

If your octahedral molecule has a center of symmetry, it also has nine planes of symmetry (three horizontal and six diagonal ), as well as a number of improper rotation axes or orders four and six. Can you find all of them If so, you can conclude that your molecule is of symmetry (9%. 

Point of inversion. The action of a point of inversion is described above in the context of improper rotation axes. Note that planes of symmetry and points of inversion are somewhat redundant symmetry elements, since they are already implicit in improper rotation axes. However, they are somewhat more intuitive as separate phenomena than are S axes, and thus most texts treat them separately. 

Since improper rotation axes include 5, = <7, and S2 — /, the more familiar (but incomplete ) statement about optical isomerism existing in molecules that lack a plane or center of symmetry is subsumed in this more general one. In this connection, the tetramethylcyclooctatetraene molecule (page 37) should be examined more closely. This molecule possesses neither a center of symmetry nor any plane of symmetry. It does have an SA axis, and inspection will show that it is superimposable on its mirror image. 

In molecules, there are five symmetry elements identity, mirror planes, proper rotation axes, improper rotation axes and inversion. A full explanation of these symmetry elements and their corresponding operators may be found in any standard chemistry textbook, and are shown diagramatically in Figure 8.13. 

Figure 8.55 Ways of packing monoalcohols with variable thickness of substituent (a) thin substituents pack by translation or by 2, screw or glide operations, (b) with thicker substituents rings are favoured, (c) thicker substituents may also pack with more than one molecule in the asymmetric unit and (d) very thick substituents result in packing about higher order screw or improper rotation axes.84...

An object is chiral if it cannot be brought into congruence with its mirror image by translation and rotation. Such objects are devoid of symmetry elements which include reflection mirror planes, inversion centers, or improper rotational axes. 

One should be careful of any improper rotation axes that may be implied by the group rather than just written explicitly. For example, the group Cnh include sn i n a concealed form because they include Cn and on. Any group containing the inversion as an element also possess at least the element S,y because an inversion can be envisaged as a 180 rotation followed by a ah reflection. It follows that molecules with centres of inversion cannot be optically active. However, not all molecules without a centre of inversion are optically active. For instance, if their symmetry is S4 they lack an i element but possess S4 which implies inactivity. 

Apart from the symmetry elements described in Chapter 3 and above, an additional type of rotation axis occurs in a solid that is not found in planar shapes, the inversion axis, n, (pronounced n bar ). The operation of an inversion axis consists of a rotation combined with a centre of symmetry. These axes are also called improper rotation axes, to distinguish them from the ordinary proper rotation axes described above. The symmetry operation of an improper rotation axis is that of rotoinversion. Two solid objects... 

The operation of some of the other improper rotation axes can be illustrated with respect to the five Platonic solids, the regular tetrahedron, regular octahedron, regular icosahedron, regular cube and regular dodecahedron. These polyhedra have regular faces and vertices, and each has... 

The symbol E represents the identity operation, that is, the combination of symmetry elements that transforms the object (molecule say) into a copy identical in every way to the original. There is one important feature to note. The improper rotation axes defined here are not the same as the improper rotation axes defined via Hermann-Mauguin symbols, but are rotoreflection axes (see Section 4.3 for details). 

The additional symmetry elements which are necessary for the 230 space groups to define the symmetry of all crystals (i.e. enantiomorphic and non-enantiomorphic) are glide planes (i.e. mirror reflection + translation) and improper rotation axes (rotation axis + inversion). 

Improper rotation axes are simply the combination ofaproper rotation axis with a Oj, mirror plane. It does not matter in which order these two symmetry operations are applied, as shown in Fig. 5.9 for the S4 axis of perpendicular B2H4. 

Note that neither the C4-axis nor the ajj mirror plane are correct symmetry elements for the molecule. Only their combination is valid. Improper rotation axes with the order 2n are often found to coincide with a proper rotation axis of order n. 

Using the flowchart in order to determine the molecular point group of NH3, there are no rotational axes (a axis will only occur in a linear molecule). The ammonia molecule does, however, have a principal proper rotational axis, C3. It does not have any Cj or C4 axes and there is only one (not four) C3 axes. Thus, the molecule belongs to one of the point groups in the lower box of Figure 8.9. There are no C2 axes in NH3 that are perpendicular to the principal axis and there are no improper rotational axes. There are also no horizontal mirror planes, but there are n = 3 vertical mirror planes. Thus, the molecular point group for NH3 is 3. ... 

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