Skip to main content
Chemistry LibreTexts

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.


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.



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


 1. Water molecule H2O was used as an example and was mentioned that water was rotated 180 degree around an axis bisecting the oygen, then the molecule was superimposable on the original water molecule, how about CO2? Is it going to be like water molecule, anyway CO2 also has 2 atoms of oxygen.

Of course not, since every molecule has different molecular shape and for recognizing the symmetry of any molecule the structure and the molecular shape of that molecule should be defined. Water molecule is bent but CO2 is not and if it is rotated 360 degree around the axis bisecting the C atom then it can be superimposed on the original molecule and then see the symmetry for the CO2.

2. Why should all of the five symmetry elements be done on a molecule in order to find the point group the molecule belongs to, why performing only one or two of the symmetry element is not enough for recognizing the point group?

One or two of the symmetry element will not be able to tell us everything about the molecule's symmetry since those one or two properties do not tell us everything about the molecule. Also, maybe different molecules have one or two symmetrical properties in common but the five properties will not be the same for all molecules.

3.  What symbol Cn stands for and what n is? Why is it important to identify it? 

C is the axis of rotation and n is the order of the axis. 

4. How the character tables are helpful?

The character table tells us about all the operational elements performed on the molecule and if have  forgotten to perform any of the symmetry elements the Table will remind us since all the operational elements should be done on the molecule in order to find the point group of the molecule.

5. Why is important to find symmetry in molecules?

Symmetry tells us about bounding for that molecular bonding.