Improper Rotational Axes

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. 

Example 12.6-1 The point group C2v = E C2z perpendicular axes, all operations except E are irregular and there is consequently only one doubly degenerate spinor representation, Ei/2. Contrast C 21, = E C2z / [ in which rrh is az = IC2z and thus an improper binary rotation about... 

Pseudoscalars with the property T T (where the positive sign applies to proper rotations and the negative sign applies to improper rotations) are also called axial tensors of rank 0, 7 (0)ax. A quantity T with three components 7 7 2 7) that transform like the coordinates x x2 x3 of a point P, that is like the components of the position vector r, so that... 

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