**Proper Rotations**** - Rotation by ****360**^{o}/n

This is simply rotation about an axis, which passes through the molecule by an angle of 360^{o}/n (or 2π/n). When repeated n times, the molecule returns to the original orientation. The appearance of the molecule must be exactly the same after the operation.

Example 1. H_{2}O

H_{2}O has a C_{2} axis - rotation by 360^{o}/2 (180^{o}). The axis passes through O and bisects the line between the H atoms. This operation interchanges the H atoms as well as the O-H bonds. Since these atoms and bonds are equivalent, there is no detectable difference after the operation.

For HOD the C_{2} operation results in the molecule having a different orientation. Therefore, for HOD, C_{2} is **NOT** a valid symmetry operation.

Example 2. Ammonia, NH_{3}

This molecule has a 3-fold rotation axis (a C_{3} axis) where rotation by 360^{o}/3 (120^{o}) produces an indistinguishable molecule (C_{3}). Rotation by 2 x 120^{o} also produces a configuration which is physically indistinguishable from the original (C_{3}^{2}). Finally, rotation by 3 x 120^{o} (360^{o}) returns the molecule to its initial position (C_{3}^{3}). This is equivalent to performing no operation at all, and we can say that C_{3}^{3} = E where E is the identity operation. We include E for mathematical reasons. There are therefore two symmetry operations associated with the C_{3} axis (C_{3} and C_{3}^{2}).

When only a single rotation axis is present, it is assigned to the Z axis by convention. If more than one rotation axis exists, the C_{n} of highest order (highest value of n) is assigned as the Z axis.

Note - the proper rotation operation (C_{n}) is the **ONLY** operation we can actually perform on either a real molecule or its macroscopic model. The remaining operations (σ, i, S_{n}) are non-feasible and we have to use our imaginations a little to see how they work - they have to be *visualized*.

Next: Reflections

Copyright © 2015 Richard Jones. All Rights Reserved.