3.5: Character Tables - An Introduction

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Although the method of assigning a point group to a molecule depends on some knowledge of the symmetry elements the molecule has, it does not require the consideration of all elements. For example, the molecule $CO_{2}$ can be assigned the point group $D_{\infty h}$ using the scheme in section 3.4 by merely knowing that the molecule is linear and that it has the a center io inversion $i$. The point group assignment in this case did not require knowing that the molecule also has $\sigma_{h}$, for example. This type of knowledge is gained by examining character tables.

Definition

A character table is a table that contains the symmetry information of the molecule. This information can be used to analyze the molecule's behavior in many application, among which is spectroscopy. Each point group has its own character table. The following table is the character table of the point group $C_{2v}$:

Table 1: The character table of the point group $C_{2v}$.

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Understanding Character Tables

Symbols 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 C_nprinciple axis, if no perpendicular axis, then it is with respect to σ_v
Subscript 2	anti-symmetric with respect to the C_nprinciple axis, if no perpendicular axis, then it is with respect to σ_v
Subscript g	symmetric with respect to the inverse
subscript u	anti-symmetric with respect to the inverse
prime	symmetric with respect to $σ_{h}$
double prime	anti-symmetric with respect to $σ_{h}$

Symbols in the first row of the character tables

E	describes the degeneracy of the row (A and B= 1) (E=2) (T=3)
C_n	2pi/n= number of turns in one circle on the main axis without changing the look of a molecule (rotation of the molecule)
C_n'	2π/n= number of turns in one circle perpendicular to the main axis, without changing the structure of the molecule
C_n^"	2π/n= number of turns in one circle perpendicular to the C_n' and the main axis, without changing the structure
σ'	reflection of the molecule perpendicular to the other sigma
σ_v(vertical)	reflection of the molecule vertically compared to the horizontal highest fold axis.
σ_hor _d (horizontal)	reflection of the molecule horizontally compared to the horizontal highest fold axis.
i	Inversion of the molecule from the center
S_n	rotation of 2π/n and then reflected in a plane perpendicular to rotation axis.
#C_n	the # stands for the number of irreducible representation for the C_n
#σ	the # stands for the number irreducible representations for the sigmas.
the number in superscript	in the same rotation there is another rotation, for instance O_h has 3C₂=C₄²
other useful definitions
(R_x,R_y)	the ( , ) means they are the same and can be counted once.
x²+y², z²	without ( , ) means they are different and can be counted twice.

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