Equivalences for Improper Rotation Operations

= i for nl2 odd, = C2, for n/2 even that is, when the improper rotation is carried out half as many times as the order of the axis there is an equivalent simple operation. Here, there are two scenarios, (i) If n/2 is odd, the repetition of the improper rotation implies an odd number of reflections and a C2 rotation the result is then the same as the inversion operation i. (ii) If n/2 is even, the molecule will not appear to have been reflected through oi, and so is simply an instance of (1). 

S = E that is, carrying out the even ordered improper rotation the same number of times as its order results in a configuration identical to the starting point. 

These observations also imply that, if an improper axis of even order is found, there must be a rotation axis with half its order and an inversion centre. Also, for a molecule with an inversion centre, there must be an improper axis of rotation. In the case of a molecule with an inversion centre but without any simple rotation axes, the improper rotation is an S2 axis. However, this S2 axis is not usually quoted, since 5 2 = i and 82 = E, so there are no unique operations associated with an S2 axis. 

After even numbers of operations the arrow returns to the top face of the molecule, and so there is an equivalent simple rotation that can give the same result. Since the order of the axis is odd, does not return the molecule to its start point although all the atoms are in the same place as in the original configuration, the arrow is pointing down, and so 83 = CTh. Only after six applications of the operation do we find that 83 = E. In this case there are only two unique operations arising from the S3 axis S3 and 83.  

Sn = C/, where m runs from 1 to (n - 1) and p = 2mfor 2m n or p = 2m - n for 2m n that is, an even number of applications of an odd-order improper rotation is the same as a related simple rotation. The operations with rotations less than 360° can also be described as the same number of simple rotations, and improper rotations involving angles greater than 360° are related to odd numbers of simple rotations. 

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