Character table is a short form of all the symmetry effects and their characters of a point group which can be utilised to solve the problems of molecular symmetry and spectroscopy.
Each character table is composed of four columns:-
Column 1:-
This column represent the Mulliken symbols for the irreducible representations of the group according to the following rules:-
Alphabets:- Four alphabets are used for Mulliken symbols:-
A ➡ For 1D IR.
B ➡ For 1D IR.
E ➡ For 2D IR.
T ➡ For 3D IR.
(IR =Irreducible Representations).
If 1D IR is symmetric with respect to principal axis (+ve charge), then alphabet will be A.
If 1D IR is antisymmetric with respect to principal axis (-ve charge), then alphabet will be B.
E and T will be used for 2D and 3D IRs respectively.
Subscript 1 and 2:- Subscript 1 will be used if IR is symmetric with respect to the subsidiary axis i.e. nC2.
Subscript 2 will be used if IR is antisymmetric with respect to the subsidiary axis.
Note:- If subsidiary axes are absent, then subscripts 1 and 2 will be assigned based on symmetry with respect to the molecular plane (a plane bisecting maximum atoms). E.g.
Subscripts g and u:- Subscript g will be assigned if IR is symmetric with respect to i (inversion centre).
Subscript u will be assigned if IR will be antisymmetric with respect to i (inversion centre).
Subscripts prime (') and double prime ("):- Subscript prime (') will be used if IR is symmetric with respect to horizontal plane.
Subscript double prime (") will be used if IR is antisymmetric with respect to horizontal plane. E.g.
Column 2:-
Column 2nd of character table represents irreducible representations of the group which can be derived by Great Orthogonality Theorem.
Column 3:-
Columnn 3rd of character table represents transformation properties of translational axis (x, y, z) and rotational axis (Rx, Ry, Rz).
Let's take an example of H2O (water) molecule and derive its transformation properties for better understanding.
Translation along the X-axis:- Translation is represented by an arrow along the respective axis for the atoms in the given molecule.
If the direction of the arrow doesn't change after operation, then it is symmetric and character is taken equal to +1. On the other hand, if the direction of the arrow changes, then -1 character will be used.
Thus, the characters of above four symmetry operations are:-
Translation along Y-axis:-
Thus, the characters of above four symmetry operations are:-
Translation along Z-axis:-
Thus, the characters of above four symmetry operations are:-
Thus, the translation along X, Y and Z-axes gives:-
Rotation along X-axis:- If the direction of the arrow (↷) changes, then character is taken equal to -1, whereas if the direction of the arrow doesn't change, then we use +1 character.
Thus, the characters of above four symmetry operations are:-
Rotation along Y-axis:-
Thus, the characters of above four symmetry operations are:-
Rotation along Z-axis:-
Thus, the characters of above four symmetry operations are:-
Thus, rotation along X, Y and Z-axes gives:-
Column 4:-
This column shows symmetry transformation properties of quadratic functions of x, y and z (squares and multiples of x, y and z).
On the basis of above calculations, operations and derivations, the character tables of some important point groups are derived below:➠➠
Character Table of C2V Point group:-
Character Table of C3V Point group:-
Character Table of C2h Point group:-
End
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