4.3: Introduction to Character Tables
- Page ID
- 326174
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Although the flow chart method of assigning a molecule to a point group requires some knowledge of the symmetry elements the molecule has, it does not require the consideration of all elements. For example, the molecule PF5 can be assigned the point group D3h using the flow chart by knowing that the molecule has a C3 principal rotation axis, ⟂C2 axes, and a σh reflection plane. The point group assignment in this case did not require knowing that the molecule also has an improper rotation axis, for example. This type of knowledge is gained by examining character tables. A character table is a table that contains all the symmetry information of the point group. This information is essential for applying symmetry and group theory to chemical problems. In this course our goal is to become familiar with the information contained in character tables, in Advanced Inorganic we will use character tables for different chemical applications. As an example, the following table is the D4h character table, a complete listing of character tables can be found under resources.
Table 1: The character table of the point group D4h.
| D4h | E | 2C4 | C2 | 2C2' | 2C2'' | i | 2S4 | σh | 2σv | 2σd | ||
| A1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | \(x^2+y^2\), \(z^2\) | |
| A2g | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | Rz | |
| B1g | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | \(x^2-y^2\) | |
| B2g | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | \(xy\) | |
| Eg | 2 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | (Rx, Ry) | (\(xz,yz\)) |
| A1u | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | ||
| A2u | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | \(z\) | |
| B1u | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | ||
| B2u | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | ||
| Eu | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | (\(x,y\)) |
Components of Character Tables
Symmetry Operations
Across the top of each character table is a list of all the symmetry operations in the point group. Rather than being listed individually, they are grouped into classes. Symmetry operations in the same class will result in equivalent transformations.
Irreducible Representations
Down the left side of each character table is a list of irreducible representations. An irreducible representation describes how something (an atom, bond, orbital, etc.) changes when the different symmetry operations are performed. It could stay the same (1), become inverted (-1), or change completely (0). Representations with A and B labels are non-degenerate. Representations with E labels are two-fold degenerate, a set of two things that are the same energy like π bonding MOs formed from px and py atomic orbitals. Representations with T labels, found in high symmetry point groups, are three-fold degenerate. Representations with a subscript g are even (gerade) and stay the same after the inversion operation and representations with a subscript u are odd (ungerade) and are inverted after the the inversion operation.
Mathematical Functions
The last two columns contain mathematical functions, indicating which irreducible representation the functions belong to. When two or more of these functions are listed together in parentheses, (\(x,y\)) it means these functions are degenerate. In chemistry applications these mathematical functions can be used to assign orbitals to irreducible representations. For example, in the D4h point group the pz orbital will have the same symmetry as the mathematical function \(z\) and will be in the A2u irreducible representation and the dxy orbital will have the same symmetry as the mathematical function \(xy\) and will be in the B2g irreducible representation