Square-pyramidal (oxygen at the apex

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

Square-pyramidal

\mathrm{XeOF_4}

(oxygen at the apex, four fluorines in the base) belongs to the point group

C_{4v}

, of order

8

, whose elements fall into the five classes

E

2C_4

C_2

2\sigma_v

2\sigma_d

established from its symmetry operations. Put the

z

-axis along the

C_4

axis, the

x

-axis in one

\sigma_v

plane and the

y

-axis in the other

\sigma_v

plane. ((a)) Write the

3\times3

matrices

D(C_4)

(the

90^\circ

rotation about

z

) and

D(\sigma_v)

(the reflection in the

xz

-plane) acting on

(x, y, z)

. Compute the product

D(C_4)D(\sigma_v)

(with

\sigma_v

applied first) and identify which symmetry operation of

C_{4v}

it represents. ((b)) Using the fact that the number of irreducible representations of

C_{4v}

equals its number of classes, and the equality

\sum_i d_i^2 = |G|

, find the only possible list of dimensions

d_i

for the irreducible representations of

C_{4v}