Editing Character table (section)

== Finding the vibrational modes of a water molecule using character table ==
To find the total number of vibrational modes of a water molecule, the irreducible representation Γ<sub>irreducible</sub> needs to calculate from the character table of a water molecule first.

=== 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
|}

=== Calculating the irreducible representation Γ<sub>irreducible</sub> from the reducible representation Γ<sub>reducible</sub> along with the character table ===
From the above discussion, a new character table for a water molecule (<math>C_{2v}</math> point group) can be written as
: {| class="wikitable"
|+New character table for <chem>H2O</chem> molecule including <math>\Gamma_{\text{red}}</math>
!
!<math>E</math>
!<math>C_2</math>
!<math>\sigma_{v(xz)}</math>
!<math>\sigma'_{v(yz)}</math>
|-
|<math>A_1</math>
|1
|1
|1
|1
|-
|<math>A_2</math>
|1
|1
| −1
| −1
|-
|<math>B_1</math>
|1
| −1
|1
| −1
|-
|<math>B_2</math>
|1
| −1
| −1
|1
|-
|<math>\Gamma_{\text{red}}</math>
|9
| −1
|3
|1
|}
Using the new character table including <math>\Gamma_{\text{red}}</math>, the reducible representation for all motion of the <chem>H2O</chem> molecule can be reduced using below formula

: <math>N = \frac{1}{h}\sum_{x}(X^x_i \times X^x_r\times n^x)</math> 

where, 

: <math>h =</math> order of the group, 
: <math>X^x_i =</math> character of the <math>\Gamma_{\text{reducible}}</math> for a particular class, 
: <math>X^x_r =</math> character from the reducible representation for a particular class, 
: <math>n^x =</math> the number of operations in the class

So,

<math>N_{A_1} = \frac{1}{4}[(9\times 1\times 1)+((-1)\times 1\times 1)+(3\times 1\times 1)+(1\times 1\times 1)] = 3</math>

<math>N_{A_2} = \frac{1}{4}[(9\times 1\times 1+((-1)\times 1\times 1)+(3\times(-1)\times 1)+(1\times(-1)\times 1)] = 1</math>

<math>N_{B_1} = \frac{1}{4}[(9\times 1\times 1)+((-1)\times(-1)\times 1)+(3\times 1\times 1)+(1\times(-1)\times 1)] = 3</math>

<math>N_{B_2} = \frac{1}{4}[(9\times 1\times 1)+((-1)\times(-1)\times 1)+(3\times(-1)\times 1)+(1\times 1\times 1)] = 2</math>

So, the reduced representation for all motions of water molecule will be

<math>\Gamma_{\text{irreducible}} = 3A_1 + A_2 + 3B_1 + 2B_2</math>

=== Translational motion for water molecule ===
Translational motion will corresponds with the reducible representations in the character table, which have <math>x</math>, <math>y</math> and <math>z</math> function
: {| class="wikitable"
|+For <chem>H2O</chem>molecule
!
!
|-
|<math>A_1</math>
|<math>z</math>
|-
|<math>A_2</math>
|
|-
|<math>B_1</math>
|<math>x</math>
|-
|<math>B_2</math>
|<math>y</math>
|}
As only the reducible representations <math>B_1</math>, <math>B_2</math> and <math>A_1</math> correspond to the  <math>x</math>, <math>y</math> and <math>z</math> function, 

<math>\Gamma_{\text{translational}} = A_1 + B_1 + B_2</math>

=== Rotational motion for water molecule ===
Rotational motion will corresponds with the reducible representations in the character table, which have <math>R_x</math>, <math>R_y</math> and <math>R_z</math> function
: {| class="wikitable"
|+For <chem>H2O</chem> molecule
!
!
|-
|<math>A_1</math>
|
|-
|<math>A_2</math>
|<math>R_z</math>
|-
|<math>B_1</math>
|<math>R_y</math>
|-
|<math>B_2</math>
|<math>R_x</math>
|}
As only the reducible representations <math>B_2</math>, <math>B_1</math> and <math>A_2</math> correspond to the  <math>x</math>, <math>y</math> and <math>z</math> function, 

<math>\Gamma_{\text{rotational}} = A_2 + B_1 + B_2</math>

=== Total vibrational modes for water molecule ===
Total vibrational mode, <math>\Gamma_{\text{vibrational}} = \Gamma_{\text{irreducible}} - \Gamma_{\text{translational}} - \Gamma_{\text{rotational}}</math>

<math>= (3A_1 + A_2 + 3B_1 + 2B_2) - (A_1 + B_1 + B_2) - (A_2 + B_1 + B_2)</math>

<math>= 2A_1 + B_1</math>

So, total <math>2+1 = 3</math> vibrational modes are possible for water molecules and two of them are symmetric vibrational modes (as <math>2A_1</math>) and the other vibrational mode is antisymmetric (as <math>1B_1</math>) 

=== Checking whether the water molecule is IR active or Raman active ===
There is some rules to be IR active or Raman active for a particular mode. 

* If there is a <math>x</math>, <math>y</math> or <math>z</math> for any irreducible representation, then the mode is IR active
* If there is a 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> or <math>xz</math> for any irreducible representation, then the mode is Raman active
* If there is no  <math>x</math>, <math>y</math>, <math>z</math> nor quadratic functions for any irreducible representation, then the mode is neither IR active nor Raman active

As the vibrational modes for water molecule <math>\Gamma_{\text{vibrational}}</math> contains both <math>x</math>, <math>y</math> or <math>z</math> and quadratic functions, it has both the IR active vibrational modes and Raman active vibrational modes.

Similar rules will apply for rest of the irreducible representations <math>\Gamma_{\text{irreducible}}, \Gamma_{\text{translational}}, \Gamma_{\text{rotational}}</math>