Point Groups and Molecular Symmetries

Identify the symmetry elements of a molecule or object and classify its point group.

Construct the character table for a simple point group (e.g., C_2v) using orthogonality relations.

Interpret the character table of a given point group (e.g., D_3) to identify its irreps and their properties.

Use character tables (e.g., for T_d, O_h) to determine the degeneracy of energy levels and the parity of wavefunctions.

Apply the direct product of representations to determine selection rules for spectroscopic transitions.

Determine the irreducible representations under which normal modes of vibration transform.

Apply group theory to predict the splitting of atomic energy levels in a crystal field.