Symmetry of molecules is described in terms of symmetry elements and operations. Symmetry elements are geometric entities: axes, planes and points in space used to define symmetry operations. Symmetry operations involve moving atoms in a molecule by rotating the molecule about an axis, reflection through a plane or inversion through a point. A rotation, reflection or inversion operation will be called a symmetry operation if, and only if, the new spatial arrangement of the atoms in the molecule is indistinguishable from the original arrangement.

- Rotation about an axis
- Reflection through a mirror plane
- Inversion
- Improper rotation
- Notations of symmetry elements and operations

## Rotation about an axis

Rotation about an axis is an operation that can be performed physically and is also referred to as "proper" rotation. The symmetry element associated with it is a rotation, or symmetry, axis. The symbol *C _{n}* is used to denote a counter-clockwise rotation about an axis, at 360°/

*n*. Thus,

*C*represents rotation by 180°,

_{2}*C*represents rotation by 120°, and so on. The symbol

_{3}*C*is used for both the symmetry element (rotation axis) and the symmetry operation. If a molecule has several symmetry axes, the axis for which

_{n}*n*is greatest is termed the "principal" axis.

## Reflection through a mirror plane

A plane that bisects a molecule so that each part of the molecule is a mirror image of the other, is called a mirror plane. The symmetry operation associated with a mirror plane is reflection. Mirror planes and the reflection operations are denoted by the symbol *σ*. There are three types of reflection planes:

*σ*- a_{h}*horizontal*reflection plane that is perpendicular to the main rotation axis of the molecule*σ*- a_{v}*vertical*reflection plane that contains the main rotation axis of the molecule*σ*- a_{d}*diagonal*or*dihedral*reflection plane that contains the main rotation axis of the molecule and also bisects the angle created by the two*C*axes perpendicular to it_{2}

*σ*is a special case of a

_{d}*σ*plane - when both exist in a molecule, it is customary to use the symbol

_{v}*σ*to denote those planes that do not pass through the atoms, but rather between them.

_{d}## Inversion

The inversion operation denoted by the symbol i moves each point (*x, y, z*) in the three-dimensional space to point (*-x, -y, -z*). The symmetry element associated with the inversion operation is the inversion point, also called the symmetry center, located at the origin.

## Improper rotation

This symmetry operation is composed of two operations: rotation and reflection, and is represented by the symbol *S _{n}*. A rotation operation about an axis

*C*is followed by reflection through a plane perpendicular to the axis

_{n}*σ*(the order of the operations is unimportant). The rotation operation

_{h}*C*, and the reflection operation

_{n}*σ*, are not necessarily symmetry operations of the molecule that has

_{h}*S*symmetry.

_{n}## Notations of symmetry elements and operations

The following table summarizes the symbols of all the possible symmetry elements and operations in molecules.

Symbol | Symmetry Element | Symmetry Operation |

E | None | Identity |

C_{n} | Axis of order n (proper) | Rotation about an axis by 360°/n |

σ_{h}_{,} σ_{v}_{,} σ_{d} | Plane | Reflection through a mirror plane |

i | Point | Inversion through a point |

S_{n} | Axis of order n (improper) | Rotation about an axis by 360°/n followed by reflection through plane perpendicular to the rotation axis |