Ethane symmetry operations

Perhaps this can best be emphasized by taking an example. Let us consider ethane in its staggered configuration. The C—C line defines a C3 axis, but certainly not a C6 axis. Yet there is an Sbi as the diagram shows. Observe that II and III are equivalent to each other but that neither is equivalent to I that is, neither a nor C is by itself a symmetry operation. But the combination of both, in either order, which we call S6, is a symmetry operation since it produces IV, which is equivalent to I. 

Fig. 2.10. Isometric structures of the ethane molecule, obtained by symmetry operations of the point group Dj and by rotation of one methyl group with respect to the other...

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. 

The set of symmetry operations is contrasted to the set of symmetry elements for eclipsed and staggered ethane in Table 13-6. Note that there are 12 symmetry operations in each case. By setting up the multipUcation table for either of these sets of 12 operations, we can show that the mathematical requirements for a group are satisfied. Thus, each of these sets of symmetry operations constitutes a separate group of order 12. 

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. 

We limit the list of operations by identifying equivalences so that there is no redundancy each arrangement of the labelled atoms in the molecule is generated only once. The hydrogen atoms of ethane are particularly useful for this. If we chose the carbon atoms, then there are only ever two arrangements the carbon atoms sit on the C3 axis and in the three mirror planes and so are unchanged by C3 or 3 rotations or any simple reflection. They are also swapped over by Si and Si any C2 rotation or i. The carbon atoms do not show the full effect of these operations because they are at special symmetry positions. For the same reason, in planar molecules like BF3, no atom sets show the full effects of symmetry operations in particular, the mirror plane containing all the atoms will always... 

Symmetry operations for ethane in the staggered conformation were covered in Section 2.3.3, including the illustration of example operations in Figure 2.5. It should now be clear that these are just the operations required to classify ethane as belonging to the Dm point group. 

Figure 2.5 Symmetry elements and example operations for ethane in the staggered conformation each example is shown both in flying wedge representation and in Newman projection. (a) The C3 principal axis, defining the vertical direction (b) C2 axis for H,—C—C—He plane there are three equivalent C2 axes, (c) one of the three equivalent

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