Symmetry operations rotation

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

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... 

Several of the molecular point groups exhibit character tables involving complex traces for certain classes of rotational symmetry operation. For example, the character table display [compare Figure 1.4] for the GT Calculator file, C3h.xls, is shown in Figure 1.27. 

Molecules that appear to have no S5anmetry at all, e.g. 3.9, must possess the symmetry element E and effectively possess at least one C axis of rotation. They therefore belong to the C point group, although since C = E, the rotational symmetry operation is ignored when we list the S5anmetry elements of this point group. 

Fig.1.3. The Brillouin zones for the bcc and fee structures. The conventional symmetry labels and the irreducible wedges, which by means of the rotational symmetry operations, i.e., the point group, may generate the full zone, are i ndicated...

Note that the two rotations (symmetry operations) are associated with the same rotational axis (symmetry element) see footnote 3. 

The set of symmetry elements and operations that characterize the symmetry of an individual molecule defines its point group. If only one rotational symmetry operation (besides E) is possible then the point group bears the same name (C2, S3, etc.) Otherwise, if there is just one symmetry element, the point group is called Cs if there is a mirror plane, a, and Q if there is an inversion center, i. Finally, if no other symmetry elements are present then the point group is Ci. 

Combination of inversion and rotation to give inversion-rotation symmetry operation. Pj (—x,—y,z) is an intermediate point in the combined inversion and rotation operation. 

Cyclic groups contain only operations derived from the repeated application of a single rotational symmetry operation. The point group is C if the repeated operation is a simple rotation, and we have the point group S if it is an improper rotation axis. In both cases the subscript denotes the order of the axis. 

Figure 4. Rotational symmetry operations in chiral materials.

Thus, dissymmetric molecules commonly have a simple axis of symmetry, and in asymmetric molecules this axis is absent however, both species are usually optically active. In liquid crystal systems both types of material are capable of exhibiting chiral properties. Table 1 summarizes the relationships between optical activity, molecular structure, and rotational symmetry operations [1]. 

Table 1. Relationships between optical activity, molecular structure, and rotational symmetry operations (after Eliel [1]).

---