Ethane improper rotation operation

Figure 2.7 Summary of the improper rotation operation for ethane in the staggered conformation.

The missing symmetry operations for ethane in the staggered conformation are improper rotations. Ethane has an order 6 improper rotation axis, S, which is illustrated along with the operation, in Figure 2.7. After describing the operations that the Se axis leads to, we will use them to close the symmetry point group of ethane. 

The number of operations from an improper rotation axis depends on the order of the axis and on the removal of any operations that can be written more simply. The number of operations can be seen by repeatedly carrying out the improper rotation until the original configuration is obtained. For example, the result of six successive Se operations on ethane returns the molecule to its starting configuration, as shown in Figure 2.8. 

Improper axes can also be associated with several symmetry operations. We noted earlier that e, applied twice in succession, results in a simple 2tt/3 rotation about the S6 axis. In other words, we can write the set 5e, 5g, Sl, S, as Se, C3, S2, C. We stop at s because S = E due to the combination of C and an even number of reflections. 5g is equivalent to S2 because it contains three rotations by 2 r/6 and an odd number of reflections, and S2 means one rotation by n and one reflection. The operation S2, however, is easily shown to be equivalent to an inversion, and so we have, using a compressed notation, 25e, 2C3, i associated with the Se axis. Since we have already explicitly listed the elements C3 and i in our set of elements for staggered ethane (or any other system containing an Se axis) only the ISe operations are unique to the Se axis. The generalization of this case is that an S2n axis with odd n implies that elements C and i are also present. Of the 2 — 1 operations associated with S2 , — 1 are preempted by the C axis and 1 by the element i leaving — 1 operations to be attributed to the S2a axis. [If = 1, we have S2, C, and i as elements. But Ci = E, so we ignore it. There is only one operation here (21 — 1 = 1) and it is preempted by i. Therefore, S2 has no unique operations and it is not listed as a symmetry element.] For S2n with n even, i is not implied and there are n unique operations. 

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