Symmetry of d electron in different environment

1) Isotropic potential (free atom)

This corresponds to the original definition of d orbitals with

L=2 and

m=-2,-1,0,1,2

The symmetry of the states expressed in the form of transformation in terms of x,y and z is shown in the table.

2) Oh point group symmetry

3) D3h point group symmetry

Isotropic potential

m=0

zz

m=1

xz+iyz

m=-1

xz-iyz

m=2

xx-yy+2ixy

m=-2

xx-yy-2ixy

Oh

t2g

xy,xz,yz

eg

zz, xx-yy

D3h

a1’

zz

e’

xx-yy, xy

e’’

xz, yz

It is obvious that the <psi|m|psi>=m for |psi>= xy, xz, yz, zz, xx-yy. In other words, the orbital moments are quenched.