Character Tables and Representations
Character tables tabulate how a point group's irreducible representations behave under its symmetry operations, providing the machinery to classify orbitals and vibrations by symmetry.
Definition
Character tables and representations are the part of group theory that assigns each point-group operation a numerical character for every irreducible representation, allowing any set of molecular functions to be classified by symmetry.
Scope
This topic covers the representation theory used in chemistry: the irreducible representations of a point group and the character tables that summarize them, the construction of reducible representations from a chosen basis such as a set of bonds or atomic orbitals, the reduction formula that decomposes them, and the projection of symmetry-adapted linear combinations. It treats the formal tools, leaving their application to molecular-orbital diagrams and spectra to other topics.
Core questions
- What is an irreducible representation, and what does a character table contain?
- How is a reducible representation built from a chosen basis?
- How does the reduction formula decompose a representation?
- How are symmetry-adapted linear combinations generated?
Key concepts
- Irreducible representations
- Character tables
- Reducible representations
- Reduction (decomposition) formula
- Projection operators
- Symmetry-adapted linear combinations
Key theories
- Irreducible representations and character tables
- Each point group has a fixed set of irreducible representations whose characters under the symmetry operations are tabulated in its character table, providing labels for orbitals, vibrations, and other functions.
- Reducible representations and the reduction formula
- Choosing a basis of bonds or orbitals generates a reducible representation whose characters, fed into the reduction formula, give the number of times each irreducible representation it contains, classifying the basis by symmetry.
- Symmetry-adapted linear combinations
- Projection operators built from the character table combine equivalent basis functions into symmetry-adapted linear combinations that transform as single irreducible representations, the building blocks of molecular-orbital construction.
Clinical relevance
Representation theory is the working tool for counting and assigning infrared- and Raman-active vibrations, building molecular-orbital diagrams, and determining the symmetry labels needed throughout inorganic spectroscopy and bonding analysis.
History
The representation theory of finite groups was developed by Frobenius, Schur, and others around 1900 and applied to physics and chemistry by Wigner and Weyl in the 1920s. Cotton's textbook later made character tables and the reduction formula standard tools for practising chemists.
Key figures
- F. Albert Cotton
- Eugene Wigner
- Hermann Weyl
Related topics
Seminal works
- cottongrouptheory1990
- carter1998
- weller2018
Frequently asked questions
- What does a character in a character table actually represent?
- A character is the trace of the matrix that represents a symmetry operation acting on a basis; for a given irreducible representation it is a single number telling you how functions of that symmetry behave under the operation.
- Why do chemists reduce a reducible representation?
- Reducing a representation built from a chosen basis—such as the metal–ligand bonds—reveals which irreducible representations the basis spans, which directly tells you which combinations of orbitals can bond and which spectroscopic transitions are allowed.