# 3.2: Symmetry Operations and Elements

When a molecule or ion is rotated or is gone through symmetry operation the configuration of it will not be different from the original configuration. Symmetry element refers to the axis of rotation or the plane mirror containing the molecule.

## Introduction

Symmetry of a molecule consisted of symmetry operations and symmetry elements. Symmetry operation is like rotating a water molecule at an angle of 180 degree around an axis through oxygen and superimposing it on the original molecule, it is going to be the same as the original element. There are five types of operational elements for symmetry such as identity operation, rotation, reflection through a mirror plane, center of inversion and lastly the rotation about axis then reflected through a plane.

A symmetry operation is an operation that is performed to a molecule which leaves it indistinguishable and superimposable on the original configuration. Symmetry operations are performed with respect to points, lines, or planes.

Table 3.2.1: Table of elements and operations
Element Operation Symbol
Identity identity E
Symmetry plane reflection in the plane σ
Inversion center inversion of a point x,y,z to -x,-y,-z i
Proper axis rotation by (360/n)o Cn
Improper axis rotation by (360/n)o, followed by reflection in plabe perpendicular to the rotation axis Sn

## Identity (E)

All molecules have this element.

## Symmetry plane (σ)

Mirror planes of the molecules:

1. σh(horizontal): horizontal plane perpendicular to principal axis

2. σd (dihedral): σ parallel to Cn and bisecting two C2' axes

3. σv (vertical): vertical plane parallel to principal axis

## Inversion (i)

Inversion is a center of symmetry of 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 tetrahedral, triangles, pentagons lack of an inversion centre.

## Proper rotation (Cn)

Rotation with respect to an axis of rotation (the highest axis of rotation).

## Improper rotation (Sn)

Improper rotation is a combination of a rotation eith respect to an axis of rotation (Cn), followed by with a reflection through a plane perpendicular to the Cn axis (σh). In short, Cn followed by σh.

σ.Cn=Sn

## References

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