How to build Character Tables

In this post I'm going to explain you How to Make a Character Table using some basic knowledge of symmetry and matrices. Before starting I would Strongly Recommend you to Read How Symmetry Operations are carried out, you can follow Some Standard Books for this  reading, or you can read My post on Symmetry operations and Group Theory.

What is Character Table

a character table is a two-dimensional table whose rows correspond to irreducible group representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the trace of the matrices representing group elements of the column's class in the given row's group representation.

In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons.

How To Make Character Table:

A character table consist of four parts

part-I	part-II	part-III	part-IV
mulliken symbol or irreducible representations	characters of irreducible representations	this part includes: x, y, z, Rx, Ry, Rz	this part includes double coordinates: x2, y2, z2, x2-y2, xy, yz, zx

To start making a character table we first start with building part-II then part-I, part-III and part-IV respectively.

Lets understand some terms which will be further used in a character table:

A character table is understood only with the help of matrices and you must have a basic knowledge of matrices, so for current scenario I’m considering that you know what matrices are and how to solve them.

So here are some standard matrices which are taken into consideration.

Standard Matrix for Identity Element (E) is:

Standard Matrix for symmetry element C_n is:

Standard Matrix for symmetry element S_n is:

Standard Matrix for symmetry element σ_xy is:

Standard Matrix for symmetry element σ_yz is:

Standard Matrix for symmetry element σ_zx is:

Example:

Now We are considering an example of H₂O molecule

For H₂O molecule having C₂v point group, we have to build character table:

In  C₂V point group, there are following symmetry operations:

E, C_2, σ_{yz and} σ_zx

at first, we have to make seperate matrices for all oprerations

•  matrix for identity element (E) is same , there will be no change 
  
• for building matrix of symmetry element C₂ , Here θ is 180 or 𝜋

• matrix for σ_yz and σ_xz symmety operations will be same as mentioned above

now taking values of x,y & z coordinates from all the above four matrices we get following table:

	E	C2	σyz	σzx
x	1(from first row of E matrix)	-1(from first row of C2 matrix)	-1(from first row of σyz matrix)	1(from first row of σzx matrix)
y	1(from second row of E matrix)	-1(from second row of C2 matrix)	1(from second row of σyz matrix)	-1(from second row of σzx matrix)
z	1(from third row of E matrix)	1(from third row of C2 matrix)	1(from third row of σyz matrix)	1(from third row of σzx matrix)
filled with orthogonality rule	1	1	-1	-1

 table (i)

 [NOTE: the fourth row in above table is filled according to Orthogonality Rule i.e. product of corresponding element of row with any other row gives Zero. 

for eg: if we check orthogonality with first row, (1)(1)+(1)(-1)+(-1)(-1)+(-1)(1)=0

if we check orthogonality with second row, (1)(1)+(1)(-1)+(-1)(1)+(-1)(-1)=0

if we check orthogonality with third row, (1)(1)+(1)(1)+(-1)(1)+(-1)(1)=0 ]

 Making Of part I of Character table:

Symbols used under the first column of the character tables

A (Mulliken Symbol)	(singly degenerate or one dimensional) symmetric with respect to rotation of the principle axis
B (Mulliken Symbol)	(singly degenerate or one dimensional) anti-symmetric with respect to rotation of the principle axis
E (Mulliken Symbol)	(doubly degenerate or two dimensional)
T (Mulliken Symbol)	(thirdly degenerate or three dimensional )
Subscript 1	symmetric with respect to the Cnprinciple axis, if no perpendicular axis, then it is with respect to σv
Subscript 2	anti-symmetric with respect to the Cnprinciple axis, if no perpendicular axis, then it is with respect to σv
Subscript g	symmetric with respect to the inverse
subscript u	anti-symmetric with respect to the inverse
prime (')	symmetric with respect to \(σ_h\) (reflection in horizontal plane)
double prime (")	anti-symmetric with respect to \(σ_h\) ( opposite reflection in horizontal plane)

 table (ii)

 since Here we are considering example of C₂V molecule 

so according to table (i) and table(ii) we get following table:

	E           C2                      σyz                        σzx
B2	1           -1                       -1                 1
B1	1            -1                       1                -1
A1	1             1                        1                 1
A2	1             1                      -1                -1

 [Explanation: considering 1st row, value 1 of E signifies it is 1 Dimensional (either A or B mulliiken symbol), now next value -1 of C₂ signifies axis is inverted on C₂ operation, (so B mulliken symbol confirmed), now next value is  -1 means molecule is antisymmetric with respect to _{σyz  (vertical plane) (subscript 2 is confirmed)}

 so combining all the aspects we get B₂ mulliken symbol for first row.

Attain other mulliken symbols in similar manner]

Now arrange Mullikan Symbols in Alphabatical order (rearranging rows according to alphabatical order of mulliken symbols)

	E           C2                      σyz                        σzx
A1	1             1                        1                 1
A2	1             1                      -1                -1
B1	1            -1                       1                -1
B2	1           -1                       -1                 1

Third part of Character Table:

 The third column block shows which irreducible representation (Mulliken Symbol) correspond to translations (unit x, y, z vectors) and rotations (R_x, R_y, Rz)

where R_x, R_y, R_z are rotations along X, Y and Z axes, if rotation is symmetrc for a given operation we write 1 and if it is antisymmetric we write -1, in this way a set of 1 and -1 is made and is matched with horizontal column of obtained table and write that symbol against it.

x,y & z are notations for axes, if during an operation axis remains same we  write 1 and if axis is reversed we write -1,

concluding above both tables alongwith table obtained in First part we get third part of character table.

 Hence the we get following table:

Fourth part of Character table:

The fourth column block gives the corresponding information
for quadratic functions,

Note: If all the values are 1 we get x², y², z² 

rotation along Z axis  gives xy

rotation along X axis gives yz  

rotation along Y axis gives xz

After filling all the values we get Required Character Table as follows:

	E           C2              σyz                     σzx
A1	1             1                1                 1	z	x2, y2, z2 , x2-y2
A2	1             1               -1               -1	Rz	xy
B1	1            -1               1                -1	y, Rx	yz
B2	1           -1               -1                 1	x, Ry	xz

In The Similar Way You can Make Character Table for  C3V
Click Here for method for building character table for C3V point group.

Click Here to Download some more Related Notes on Forming Character Tables.

For Further Reading You Can Refer to These Books:

1. Atkins - Physical Chemistry 
2. Atkins - Molecular Quantum Mechanics 
3. Ogden – Introduction to Molecular Symmetry (Oxford Chemistry Primer) 
4. Cotton – Chemical Applications of Group Theory 
5. Davidson – Group Theory for Chemists 
6. Kettle – Symmetry and Structure 
7. Shriver, Atkins and Langford – Inorganic Chemistry 
8. Alan Vincent – Molecular Symmetry and Group Theory (Wiley) 

[For PDF format of some of these Books you can click here ]

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