Advanced Inorganic Chemistry/Character Tables

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Definition of a Character Table


A character table is a 2 dimensional chart associated with a point group that contains the irreducible representations of each point group along with their corresponding matrix characters. It also contains the Mulliken symbols used to describe the dimensions of the irreducible representations, and the functions for symmetry symbols for the Cartesian coordinates as well as rotations about the Cartesian coordinates.


Components of a Character Table


A character table can be separated into 6 different parts, namely:

CharacterTables.png
1. The Point Group
2. The Symmetry Operations
3. The Mulliken Symbols
4. The Characters for the Irreducible Representations
5. The Functions for Symmetry Symbols for Cartesian Coordinates and Rotations
6. The Functions for Symmetry Symbols for Square and Binary Products


1. The Point Group

The symbol for the point group is found on the uppermost left corner of the character table. It denotes a collection of symmetry operations that are present in a molecule. It is called a point group because all the symmetry elements will intersect at one point[1].


2. The Symmetry Operations/Elements

A symmetry operation is “a geometrical operation that moves an object about some symmetry element in a way that brings the object into an arrangement that is indistinguishable from the original”(Pfennig, 199)[2]. The symmetry operations are at the first row at the top of the table. They are organized into classes, with each class having an order number in front of it. For example, 2S4 represents the operation S4 with order number 2. Operations can belong to the same class when one operation may be replaced by another in a new coordinate system that is accessible by a similar symmetry operation.


Common symmetry operations that are present in character tables are:

E Cn Cn
σd σv σd
I Sn Cn


3. The Mulliken Symbols

These are symbols that occur under the first column of the character table. They are named after Robert S. Mulliken , who suggested using the symbols to describe the irreducible representations. The meanings of the symbols are as follows:

•The dimensions/degeneracy of characters are denoted by the letters A,B,E,T,G and H with each letter representing degeneracy 1,1,2,3,4 and 5 respectively i.e.


Mulliken Symbol Number of Dimensions
A,B 1
E 2
T 3
G 4
H 5

For example, the Mulliken symbol A is singly degenerate and symmetric with respect to the rotation about the principal axis whereas the symbol B is anti-symmetric with respect to rotation about the principal axis even though it is also singly degenerate[3].

•The subscripts featured with each Mulliken symbol also represent different aspects of symmetry i.e.


Subscript / Superscript Meaning
Subscript 1 symmetric with respect to the Cn principle axis, if no perpendicular axis, then it is with respect to σv
Subscript 2 anti-symmetric with respect to the Cn principle 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
Superscript prime symmetric with respect to σh
Superscript double prime anti-symmetric with respect to σh


4. The Characters for Irreducible Representations[4]

These are the rows of numbers at the center of the character table. They represent the irreducible representations of each Mulliken symbol under the point group. A representation is “a set of matrices, each corresponding to a single operation in the group, which can be combined amongst themselves similarly to how the group elements (symmetry operations) combine” .

These characters correspond to the characters of individual symmetry operations that can be described matrices, themselves. Each character can adopt a +1 or -1 or multiple of this numerical value depending on the symmetric or anti-symmetric behavior of the object undergoing a specific symmetry operation. If the object is symmetric with respect to itself after undergoing the operation, then the character is +1. If the object is anti-symmetric, then the character is -1[5].


5. The Functions for Symmetry Symbols for Cartesian Coordinates and Rotations

These are the symbols that correspond to the symmetry of the Cartesian coordinates (x, y, z) and the symmetry of the rotations about the Cartesian coordinates (Rx, Ry, Rz). They form basis representations for the group and are related to the transformation properties of the group.

For example, for the C3v point group, it can be said that z forms a basis for the A1 representation, x forms a basis for the E representation, and Rz forms a basis for the A2 representation.


6. The Functions for Symmetry Symbols for Square and Binary Products

These are the symbols for the functions that correspond to the square (x2+y2, z2, x2-y2) and binary products (xy, xy, yz) of the Cartesian Coordinates with respect to their transformation properties.

For example, for the C3v point group, it can be said that z2 forms a basis for the A1 representation, (xz,yz) forms a basis for the E representation, and there is no function for the A2 representation.


Mathematics of Character Tables


Each character table follows some main set of mathematical operations that allow for the calculation of important characteristics of the table. These operations are as follows:

a. The order of the group (h) can be calculated by taking the sum of the order of individual symmetry operations in a character table. For example, the order of the C3v point group is 6.

b. The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of the group.

c. The sum of the squares of the characters in any irreducible representation equals h.

d. The vectors whose components are the characters of two different irreducible representations are orthogonal.

e. In a given representation (reducible or irreducible) the characters of all matrices belonging to symmetry operations in the same class are identical.

f. The number of irreducible representations of a group is equal to the number of classes in the group.


Examples of Character Tables


The Character Table for the C2 Point Group

C2 E C2 Linear Functions, Rotations Quadratic Functions Cubic Functions
A +1 +1 z, Rz x2, y2, z2, xy z3, xyz, y2z, x2z
B +1 -1 x, y, Rx, Ry yz, xz xz2, yz2, x2y, xy2, x3, y3


The Character Table for the Td Point Group

Td E 8C3 3C2 6S4 d Linear Functions, Rotations Quadratic Functions Cubic Functions
A1 +1 +1 +1 +1 +1 - x2+y2+z2 xyz
A2 +1 +1 +1 -1 -1 - - -
E +2 -1 +2 0 0 - (2z2-x2-y2, x2-y2) -
T1 +3 0 -1 +1 -1 (Rx, Ry, Rz) - [x(z2-y2), y(z2-x2), z(x2-y2)]
T2 +3 0 -1 -1 +1 x, y, z xy, xz, yz (x3, y3, z3), [x(z2+y2), y(z2+x2), z(x2+y2)]


The Character Table for the D2d Point Group

D2d E 2S4 C2(z) 2C'2 d Linear Functions, Rotations Quadratic Functions Cubic Functions
A1 +1 +1 +1 +1 +1 - x2+y2, z2 xyz
A2 +1 +1 +1 -1 -1 Rz - z(x2-y2)
B1 +1 -1 +1 +1 -1 - x2-y2 -
B2 +1 -1 +1 -1 +1 z xy z3, z(x2+y2)
E +2 0 -2 0 0 (x, y),(Rx, Ry) (xz, yz) (xz2, yz2),(xy2, x2y),(x3, y3)


References

  1. Johnston, Dean H. "Symmetry @ Otterbein". http://symmetry.otterbein.edu/index.html
  2. Pfennig, Brian W. Principles of Inorganic Chemistry. , 2015. Internet resource.
  3. “Understanding Character Tables of Symmetry Groups". https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Group_Theory/Understanding_Character_Tables_of_Symmetry_Groups
  4. Rowland, Todd; Weisstein, Eric W. "Character Table". http://mathworld.wolfram.com/CharacterTable.html
  5. Jones, Richard. “Character Tables”. University of Texas, Austin. 2015. https://sites.cns.utexas.edu/jones_ch431/character-tables