## Introduction

The symmetry of a molecule consists of **symmetry operations** and **symmetry elements**. A symmetry operation is an operation that is performed to a molecule which leaves it indistinguishable and superimposable on the original position. Symmetry operations are performed with respect to symmetry elements (points, lines, or planes).

An example of a symmetry operation is a 180° rotation of a water molecule in which the resulting position of the molecule is indistinguishable from the original position (see Figure \(\PageIndex{1}\)). In this example, the symmetry operation is the rotation and the symmetry element is the axis of rotation.

There are five types of symmetry operations including **identity**, **reflection**, **inversion**, **proper rotation**, and **improper rotation**. The improper rotation is the sum of a rotation followed by a reflection. The symmetry elements that correspond to the five types of symmetry operations are listed in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\): Table of elements and operations
**Element** |
**Operation** |
**Symbol ** |

Identity |
identity |
E |

Proper axis |
rotation by (360/n)^{o} |
C_{n} |

Symmetry plane |
reflection in the plane |
σ |

Inversion center |
inversion of a point at (x,y,z) to (-x,-y,-z) |
*i* |

Improper axis |
rotation by (360/n)^{o}, followed by reflection in the plane perpendicular to the rotation axis |
S_{n} |

## Symmetry Operations and Elements

### Identity (E)

All molecules have the identity element. The identity operation is doing nothing to the molecule (it doesn't rotate, reflect, or invert...it just is).

### Proper Rotation and Proper Axis (C_{n})

A "proper" rotation is just a simple rotation operation about an axis. The symbol for any proper rotation or proper axis is C_{(360/n)}, where n is the degree of rotation. Thus, a 180° rotation is a C_{2} rotation around a C_{2} axis, and a 120° rotation is a C_{3} rotation about a C_{3} axis.

**PRINCIPLE AXIS**: The principle axis of a molecule is the highest order proper rotation axis. For example, if a molecule had C_{2} and C_{4} axes, the C_{4} is the principle axis.

### Reflection and Symmetry Planes (σ)

Symmetry planes are mirror planes within the molecule. A reflection operation occurs with respect to a plane of symmetry. There are three classes of symmetry elements:

**σ**_{h }(horizontal): horizontal planes are perpendicular to principal axis
**σ**_{v }(vertical): vertical planes are parallel to the principal axis
**σ**_{d }(dihedral): dihedral planes are parallel to the principle axis and bisecting two C_{2}' axes

### Inversion and Inversion Center (i)

The inversion operation requires a point of symmetry (a center of symmetry within a molecule). In other words, a point at the center of the molecule that can transform (x,y,z) into (-x,-y,-z) coordinate. Structures of tetrahedrons, triangles, and pentagons lack an inversion center.

### Improper rotation (S_{n})

Improper rotation is a combination of a rotation with respect to an axis of rotation (C_{n}), followed by a reflection through a plane perpendicular to that C_{n} axis. In short, an S_{n} operation is equivalent to C_{n} followed by \(\sigma_h\).