A tripod-shaped complex ion has three identical ligands at the vertices of an…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry
Invariance, Ch Representations, Schur's lemma and characters
Problem
A tripod-shaped complex ion has three identical ligands at the vertices of an equilateral triangle, twisted so that no mirror plane maps the ion to itself: its point group is D3, generated by rotations only. The classes are E (size 1), 2C3 (size 2) and 3C2 (size 3, each C2 axis passing through one ligand and the midpoint of the opposite edge). The irreducible representations have characters on (E,2C3,3C2): A1: (1,1,1)A2: (1,1,−1)E: (2,−1,0) Let Γ be the representation generated by the permutation of the three ligand labels.
((a)) Find χ(E), χ(C3) and χ(C2) for Γ, noting that each C2 axis passes through exactly one ligand.
((b)) Decompose Γ into irreducible representations of D3.