Symmetry Operations

The classification of objects according to symmetry elements corresponding to operations that leave at least one common point unchanged gives rise to the point groups. These are five kinds of symmetry operations and five kinds of symmetry elements of this kind. These symmetry operations are as follows.

• The identity, E, consists of doing nothing: the corresponding symmetry element is an entire object. In general, any object undergo this symmetry operation. The example of the molecule which has only the identity symmetry operation is $C_3H_6O_3$, $DNA$, and $CHClBrF$.
• The n-fold rotation about an n-fold axis of symmetry, $C_n$ is a rotation through the angle $360^o/n$. Particularly, the operation $C_1$ is a rotation through $360^o$ which is equivalent to the identity E. $H_2O$ molecule has one twofold axis, $C_2$. $NH_3$ molecule has one threefold axis, $C_3$ which is associated with two symmetry operations: $120^o$ rotation $C_3$ and $240^o$ (or $-120^o$) rotation $C_3^2$. $C_6H_6$ molecule has one sixfold axis $C_6$ and six twofold axes $C_2$. If a molecule possess several rotational axes, then the one of them with the greatest value of $n$ is called the principal axis. All linear molecules including all diatomics has $C_\infty$ axis because rotation on any angle remains the molecule the same.
• The reflection in a mirror plane, $\sigma$ may contain the principal axis of a molecule, or be perpendicular to it. If the plane contains the principal axis, it is called vertical and denoted $\sigma_v$. For instance, $H_2O$ molecule has two vertical planes of symmetry and $NH_3$ molecule has tree. A vertical mirror plane which bisects the angle between two $C_2$ axes is called a dihedral plane and is denoted by $\sigma_d$. If the plane of symmetry is perpendicular to the principal axis, it is called horizontal and denoted $\sigma_h$. For instance, $C_6H_6$ molecule has a $C_6$ principal axis and a horizontal mirror plane.
• The inversion through the center of symmetry is the operation which transforms all coordinates of the object according to the rule: $(x,y,z) \rightarrow (-x,-y,-z)$. For instance, a sphere, or a cube has a center of inversion, but $H_2O$, and $NH_3$ have not. $C_6H_6$ molecule has a center of inversion.
• The n-fold improper rotation about an n-fold axis of symmetry, $S_n$ is a combination of two successive transformations. The first transformation is a rotation through $360^o/n$ and the second transformation is a reflection through a plane perpendicular to the axis of the rotation. Note, that neither operation alone needs to be a symmetry operation. For instance, $CH_4$ molecule has three $S_4$ axes.

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