# Improper Rotation of Molecules Improper rotation is an important symmetry operation used in molecular symmetry and group theory. It is also called a rotation-reflection operation and is represented by the symbol (S_n), where (n) is the order of the improper rotation. An improper rotation consists of two successive symmetry operations: first, rotation of the molecule through an angle of (360^\circ/n) about an axis, followed by reflection through a plane perpendicular to that axis. The molecule is said to possess an improper rotation axis if, after these two operations, it becomes indistinguishable from its original configuration. The improper rotation operation is therefore different from a simple proper rotation. In a proper rotation, only rotation is carried out, whereas in an improper rotation, rotation is immediately followed by reflection. The corresponding symmetry element is called an improper rotation axis or rotation-reflection axis. It is denoted by (S_n). For an improper rotation (S_n), the molecule is first rotated through an angle [ \theta=\frac{360^\circ}{n} ] about the specified axis. After this rotation, reflection is performed through a plane perpendicular to the same axis. If the final arrangement is identical with the initial arrangement, the molecule possesses an (S_n) symmetry element. For example, consider the (S_4) operation. In this case, (n=4), so the molecule is first rotated through [ \frac{360^\circ}{4}=90^\circ. ] After rotation through (90^\circ), the molecule is reflected through a plane perpendicular to the (S_4) axis. If the resulting molecular arrangement is indistinguishable from the original arrangement, the molecule has an (S_4) axis. A useful example of improper rotation is provided by the staggered conformation of ethane. The staggered ethane molecule has an (S_6) axis along the C–C bond. An (S_6) operation involves rotation through [ \frac{360^\circ}{6}=60^\circ ] followed by