Editing Character table (section)

=== Finding Γ<sub>reducible</sub> from the Character Table of H<sub>²</sub>O molecule ===
Water (<chem>H2O</chem>) molecule falls under the point group <math>C_{2v}</math>.<ref>{{Cite journal|last1=Reimers|first1=J.R.|last2=Watts|first2=R.O.|date=1984-06-10|title=A local mode potential function for the water molecule|url=https://doi.org/10.1080/00268978400101271|journal=Molecular Physics|volume=52|issue=2|pages=357–381|doi=10.1080/00268978400101271|issn=0026-8976}}</ref> Below is the character table of <math>C_{2v}</math> point group, which is also the character table for a water molecule. 
: {| class="wikitable"
|+Character table for <math>C_{2v}</math> point group
!
!<math>E</math>
!<math>C_2</math>
!<math>\sigma_v</math>
!<math>\sigma'_v</math>
!
!
|-
|<math>A_1</math>
|1
|1
|1
|1
|<math>z</math>
|<math>x^2,y^2,z^2</math>
|-
|<math>A_2</math>
|1
|1
| −1
| −1
|<math>R_z</math>
|<math>xy</math>
|-
|<math>B_1</math>
|1
| −1
|1
| −1
|<math>R_y,x</math>
|<math>xz</math>
|-
|<math>B_2</math>
|1
| −1
| −1
|1
|<math>R_x,y</math>
|<math>yz</math>
|}
In here, the first row describes the possible symmetry operations of this point group and the first column represents the Mulliken symbols. The fifth and sixth columns are functions of the axis variables. 

Functions:

* <math>x</math>, <math>y</math> and <math>z</math> are related to translational movement and IR active bands.
* <math>R_x</math>, <math>R_y</math> and <math>R_z</math> are related to rotation about respective axis.
* Quadratic functions (such as <math>x^2+y^2</math>, <math>x^2-y^2</math>, <math>x^2</math>, <math>y^2</math>,<math>z^2</math>, <math>xy</math>, <math>yz</math>,<math>zx</math>) are related to Raman active bands.

When determining the characters for a representation, assign <math>1</math> if it remains unchanged, <math>0</math> if it moved, and <math>-1</math> if it reversed its direction. A simple way to determine the characters for the reducible representation <math>\Gamma_{\text{reducible}}</math>, is to multiply the "''number of unshifted atom(s)''" with "''contribution per atom''" along each of three axis (<math>x,y,z</math>) when a symmetry operation is carried out. 

Unless otherwise stated, for the identity operation <math>E</math>, "contribution per unshifted atom" for each atom is always <math>3</math>, as none of the atom(s) change their position during this operation. For any reflective symmetry operation <math>\sigma</math>, "contribution per atom" is always <math>1</math>, as for any reflection, an atom remains unchanged along with two axis and reverse its direction along with the other axis. For the inverse symmetry operation <math>i</math>, "contribution per unshifted atom" is always <math>-3</math>, as each of three axis of an atom reverse its direction during this operation. An easiest way to calculate "contribution per unshifted atom" for <math>C_n</math> and <math>S_n</math> symmetry operation is to use below formulas<ref>{{Cite book|last=Davidson|first=George|url=https://books.google.com/books?id=rEddDwAAQBAJ|title=Group Theory for Chemists|date=1991-06-06|publisher=Macmillan International Higher Education|isbn=978-1-349-21357-3|language=en}}</ref>
: <math>C_n = 2\cos\theta+1</math>
: <math>S_n = 2\cos\theta-1</math>
where, <math>\theta = \frac{360}{n}</math>

A simplified version of above statements is summarized in the table below
: {| class="wikitable"
!Operation
!Contribution
per unshifted atom
|-
|<math>E</math>
|3
|-
|<math>C_2</math>
| −1
|-
|<math>C_3</math>
|0
|-
|<math>C_4</math>
|1
|-
|<math>C_6</math>
|2
|-
|<math>\sigma_{xy/yz/zx}</math>
|1
|-
|<math>i</math>
| −3
|-
|<math>S_3</math>
| −2
|-
|<math>S_4</math>
| −1
|-
|<math>S_6</math>
|0
|}
''Character of <math>\Gamma_{\text{reducible}}</math> for any symmetry operation <math>=</math> Number of unshifted atom(s) during this operation <math>\times</math> Contribution per unshifted atom along each of three axis''
: {| class="wikitable"
|+Finding the characters for <math>\Gamma_{\text{red}}</math>
!<math>C_{2v}</math>
!<math>E</math>
!<math>C_2</math>
!<math>\sigma_{v(xz)}</math>
!<math>\sigma'_{v(yz)}</math>
|-
|Number of unshifted atom(s)
|3
|1
|3
|1
|-
|Contribution per unshifted atom
|3
| −1
|1
|1
|-
|<math>\Gamma_{\text{red}}</math>
|9
| −1
|3
|1
|}