# 5.1: Symmetry Elements and Operations

 Figure $$\PageIndex{1}$$: An example of a symmetry operation is a 180° rotation where the resulting position is indistinguishable from the original. A 180° rotation is called a C2 operation; the axis of rotation is the symmetry element.

## 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, reflectioninversion, 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 Cn
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 Sn

## 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 (Cn)

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 C2 rotation around a C2 axis, and a 120° rotation is a C3 rotation about a C3 axis.

PRINCIPLE AXIS: The principle axis of a molecule is the highest order proper rotation axis. For example, if a molecule had C2 and C4 axes, the C4 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:

1. σ(horizontal): horizontal planes are perpendicular to principal axis
2. σv (vertical): vertical planes are parallel to the principal axis
3. σd (dihedral):  dihedral planes are parallel to the principle axis and bisecting two C2' 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 tetrahedron, triangles, and pentagons lack an inversion center.

### Improper rotation (Sn)

Improper rotation is a combination of a rotation with respect to an axis of rotation (Cn), followed by a reflection through a plane perpendicular to that Cn axis. In short, an Sn operation is equivalent to Cn followed by $$\sigma_h$$.

## References

1. Introduction to Molecular Symmetry by J. S Ogden
2. Inorganic Chemistry by Catherine Housecroft And Alan G. Sharpe.