In group theory, in order to assign a point group, one must first identify the symmetry elements present in the molecule.
To give an example, water has:
- a rotational axis, labelled C2 since it is a two fold axis, with a π/2 rotation giving back (what appears to be)the same thing
- two mirror planes, labelled σv and σv' (with additional labels xy/xz to define what plane the mirror plane lies in)
Symmetry elements present in water
(Note that for the C2v point group, the axis are defined 'arbitrarily', different tables/books might swap x and y)
From this, it would look like water has three unique symmetry elements, however this is not the case, in fact, water has four due to the presence of an additional symmetry element, E, known as the identity operator (sometimes seen as I in certain textbooks).
This E operator corresponds to 'do nothing' (i.e. leave atoms where they are), which is easily seen in the character table for the C2v point group (to which water belongs) in which the characters under E are all '1':
$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & \color{red}{E} & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & \color{red}{1} & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & \color{red}{1} & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & \color{red}{1} & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & \color{red}{1} & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$
$\,$
C2v character table, from orthocresol's Group Theory Tables
The identity, E, consists of doing nothing; the corresponding symmetry element is the entire object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. (Taken from Atkins' Physical Chemistry).
Since transforming a molecule by the identity operator just gives back itself, the use of 'E' in a qualitative sense appears redundant, however it appears in every character table, and as such much have a purpose.
Why do we need the identity operator, E, and what use does it have in chemical group-theory?