An ion sits at the centre of a square pyramid of five ligands

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Character tables at work: vibrations, selection rules and splitting

Problem

An ion sits at the centre of a square pyramid of five ligands, four at

\pm x

and

\pm y

and one at

+z

, with none at

-z

. Its site symmetry is

C_{4v}

, with the

C_4

axis along

z

, the mirrors

\sigma_v

in the planes

xz

and

yz

, and the mirrors

\sigma_d

in the planes

x=\pm y

. On the classes

(E,\,2C_4,\,C_2,\,2\sigma_v,\,2\sigma_d)

the characters of

C_{4v}

are

A_1=(1,1,1,1,1)

A_2=(1,1,1,-1,-1)

B_1=(1,-1,1,1,-1)

B_2=(1,-1,1,-1,1)

and

E=(2,0,-2,0,0)

, as compiled by Cotton. ((a)) Apply one operation of each class to

xy

x^2-y^2

2z^2-x^2-y^2

and the pair

(xz,yz)

, find the species of each of the five

d

orbitals, and say into how many levels the free-ion

d

level splits at this site. ((b)) Check the result by descent from

O_h

. The classes of

C_{4v}

lie in the classes

E

6C_4

3C_2

3\sigma_h

and

6\sigma_d

O_h

. Restrict

E_g

and

T_{2g}

of the chapter's

O_h

table to

C_{4v}

and decompose them.