Methane rotation-reflection operation

A rotation-reflection operation (S ) (sometimes called improper rotation) requires rotation of 360°/n, followed by reflection through a plane perpendicular to the axis of rotation. In methane, for example, a line through the carbon and bisecting the... 

As a first example, we consider the methane molecule (CH4), whose geometry is tetrahedral. If we perform a rotation of 90°(27t/4) around the axis bisecting the angles Hi —C—H2 and H3 —C—H4, followed hy a reflection in the plane perpendicular to this axis passing through the carbon atom, we obtain a configuration that is equivalent (but not identical) to the initial one (6-11). The axis which we used is therefore an improper four fold axis (S4), and the symmetry operation described in 6-11 is written S4, or simply S4. Notice that this axis is not a C4-axis in the molecule (but a C2-axis), and that the plane used is not a symmetry element in the molecule. As in the cases of C axes, the S4 operation can be performed m times in succession, and these are written S . In the example above, it is easy to verify that S4 = C] (a two-fold axis, co-linear with S4) and that S4 = B (see Exercise 6.4). The only two operations that are unique to the S4-axis are therefore S4 and S4. 

The final symmetry operation, an improper rotation (S , the S coming from the German word spiegal for mirror) is the most complex as it is a combination of two previously defined symmetry operations a reflection through a mirror plane, and an n-fold rotation. This is one of those cases where a picture is worth a thousand words. Figure 7-4 shows an improper rotation on a molecule of methane (CH ). Note 1 rotated the molecule away from the principal axis going along the z-axis to make it easier to see. 

Fig. 6-2. Alternating axis of symmetry illustrated by an St axis of methane. The arrows show the path of one of the hydrogen atoms for the operation Sih described as rotating about the vertical axis, 1S4, by 27t/4, then reflecting through the horizontal plane containing the carbon atom.

This is a symmetry operation needed on a molecule where one of the atoms lies along the rotational axis and cannot be transformed into the position of other atoms on the molecule simply by rotation or reflection methods. In this case, the improper rotation axis must be used to bring this atom into the position of other atoms so it can be transformed (by rotation or reflection) into the position of the other atoms. Methane can be used as an example molecule. 

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