Constructing the Character Table for C4v

Symmetry Operations grouped by class:

Mulliken Notation: Symbols for the Irreducible Representations (ΓI)

A – 1 dimensional representation sym. to rotation about Cn

B – 1 dimensional representation antisym. to rotation about Cn

E – 2 dimensional representationsT – 3 dimensional representations

Subscripts: 1 – representation is symmetric to ⊥ C2 (or to σv)2 – representation is antisym. to ⊥ C2 (or to σv)g – representation is symmetric to inversionu – representation is antisym. to inversion

Superscripts: ′ - representation is symmetric to σh

′′ - representation is antisym. to σh

Character Tables and Bonding

We can use character tables to determine the orbitals involved in bonding in a molecule. This process is done a few easy steps.

1. Determine the point group of the molecule.

2. Determine the Reducible Representation, Γ, for the type of bonding you wish to describe (e.g. σ, π). Picking the correct basis set is critical! The Reducible Representation indicates how the bonds are affected by the symmetry elements present in the point group.

3. Identify the Irreducible Representations that provides the Reducible Representation; there is a simple equation to do this. The Irreducible Representation (e.g. 2A1+ B1+ B2) is the combination of symmetry representations in the point group that sum to give theReducible Representation.

4. Identify which orbitals are involved from the Irreducible Representation and the character table.

MSGT Application: Find ΓR for the σ Bonds in Methane

6σd6S43C28C3ETd

I. CH4: Td

Reduce ΓR for MethaneI. CH4: Td

General Procedure to Get Molecular Orbital Diagram(for cental atom + ligands)

1. Assign the Point Group and use the appropriate ligandatomic orbitals to find ΓR, express as sum of ΓI, interpret the results (using steps on the previous slide).

2. Classify the central atom atomic orbitals according to the ΓI’s in the character table.

3. Combine central atom atomic orbitals and ligand LCAOshaving the same symmetry

→This will lead to a molecular orbital diagram

I. CH4: Td

I. CH4: Td

II. H2O: C2v

II. H2O: C2v

II. H2O: C2v

II. H2O: C2v

II. H2O: C2v

II. H2O: C2v

III. BF3: D3h

III. BF3: D3h

III. BF3: D3h