Rotation-reflection operations

The remaining fifty-eight magnetic point groups include the time reversal operator only in combination with rotation and rotation-reflection operators. The representations of these groups may be obtained from Eq. (12-27). 

Improper rotation axis. Rotation about an improper axis is analogous to rotation about a proper synunetry axis, except that upon completion of the rotation operation, the molecule is mirror reflected through a symmetry plane perpendicular to the improper rotation axis. These axes and their associated rotation/reflection operations are usually abbreviated X , where n is the order of the axis as defined above for proper rotational axes. Note that an axis is equivalent to a a plane of symmetry, since the initial rotation operation simply returns every atom to its original location. Note also that the presence of an X2 axis (or indeed any X axis of even order n) implies that for every atom at a position (x,y,z) that is not the origin, there will be an identical atom at position (—x,—y,—z) the origin in such a system is called a point of inversion , since one may regard every atom as having an identical... 

This denotes an axis about which a rotation-reflection (or improper rotation) operation may be carried out. The rotation-reflection operation Sn involves a... 

Enantiotopic groups are those which can be interchanged one with the other, by a rotation-reflection operation. If two such groups, a and a" in Ca a"bc are separately substituted by an achiral group d, the products are the two enantiomers of Cabcd. The central carbon atom is described as prochiral. The two H atoms in... 

A rotation-reflection operation (S ) (sometimes called improper rotation) requires rotation of 360°/n, followed by reflection through a plane perpendicular to the axis of rotation. In methane, for example, a line through the carbon and bisecting the... 

Improper rotation-reflection operation is the rotation about the improper axis S through an angle luk/n, combined with reflection k times in a plane normal to this axis. 

The single generating operation can also be a rotation reflection operation, S2n, consisting of a rotation through 2nl2n followed by a reflection in a plane perpendicular to the rotation axis. The group is S2n. The operation S2 is identical to the inversion operation, I, and for n odd S2n can also be generated by the two operations Cn and I. 

Fig. 9. Schematic representation of non-coordinated and coordinated CCh ion and the corresponding point group symmetry elements. The changes in the Vj and V3 IR vibrations of the COs " ion upon coordination are also shown. For simplicity, only monodentate coordination is presented. Notations I - identity, Cn - n-fold axis of rotation, Oh, a, - mirror planes perpendicular and parallel to the principal axis, respectively, Sn - n-fold rotation-reflection operation. The number preceding the symmetry operation symbol refers to number of such symmetry elements that the molecule possesses. For further details consult Nakamoto, 1997.

We can also describe the symmetry of a molecule using the compound rotation-reflection operation, i.e. rotate about an axis by lirln and then reflect in the plane perpendicular to that axis . This type of operation is given the symbol The symmetry element consists of an axis and a plane. Examples of rotation-reflection operations are shown in Figure 2.7. It is crucial to note that the reflection plane must be perpendicular to the rotation axis. Also, a rotation-reflection axis of order 2n wiU be associated with a pure-rotation axis of order n. 

Rotation-reflection operations in (a) SiCfi, drawn looking down the S4 axis note that in this example neither the rotation hy 2tt/4 hy itself nor the reflection in a plane perpendicular to this axis generate a configuration equivalent to the initial one, hut the combined operation does (b) Sn(T -C5Ph5)2, showing an almost perfect Sio operation, looking down the axis. Reprinted with permission from [2]. Copyright 1984 American Chemical Society. 

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