Ozone, , is a bent triatomic molecule of point group lying entirely in the plane,…

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

Ozone,

\mathrm{O_3}

, is a bent triatomic molecule of point group

C_{2v}

lying entirely in the

yz

plane, with the central oxygen on the twofold axis

z

and the two terminal oxygens (labelled

2

and

3

) related by the symmetry operations; the central oxygen is labelled

1

. The four classes, each of size

1

, are

E

C_2

(rotation by

180^\circ

about

z

\sigma_v(xz)

(the plane through

z

perpendicular to the molecular plane) and

\sigma_v'(yz)

(the molecular plane). The irreducible representations have characters, on the classes

(E, C_2, \sigma_v(xz), \sigma_v'(yz))

A_1

(1,1,1,1)

A_2

(1,1,-1,-1)

B_1

(1,-1,1,-1)

B_2

(1,-1,-1,1)

Let

\Gamma

be the representation obtained from the permutation of the three oxygen labels

1,2,3

under these operations. (Every atom lies in the plane

\sigma_v'(yz)

, which therefore fixes all three;

C_2

and

\sigma_v(xz)

each swap the two terminal oxygens and fix the central one.) ((a)) Write down the character of

\Gamma

on each class. ((b)) Decompose

\Gamma

into irreducible representations.