1. Lecture 14 APPLICATIONS OF GROUP THEORY 1) Symmetry of atomic orbitals
When bonds are formed, atomic orbitals combine according to their symmetry.
Symmetry properties and degeneracy of orbitals can be learned from corresponding character tables by their inspection. Hold in mind the following transformational properties:
Atomic orbital
Transforms as s
x2+y2+z2 px x py y pz z
dz2
z2, 2z2-x2-y2
dx2-y2
x2-y2
dxy xy
dxz xz
dyz yz

2. 2) Examples of atomic orbitals symmetry analysis
C2v A1 z
x2, y2, z2 A2 Rz xy B1
x, Ry xz B2
y, Rx yz
Atomic orbital
Mulliken labels
C2v
D3h
D4h Td Oh s px py pz
dz2
dx2-y2
dxy
dxz
dyz
D3h
A1’
x2+y2, z2
A2’ Rz E’
(x,y)
(x2-y2, xy)
A1”
A2” z E”
(Rx,Ry)
(xz, yz)
D4h
A1g
x2+y2, z2
B1g
x2-y2
B2g xy Eg
(Rx,Ry)
(xz, yz)
A2u z Eu
(x, y) Td A1
x2+y2+z2 A2 E
(2z2-x2-y2, x2-y2) T1
(Rx,Ry,Rz) T2
(x,y,z)
(xz, yz, xy) Oh
A1g
x2+y2+z2 Eg
(2z2-x2-y2, x2-y2)
T1g
(Rx,Ry,Rz)
T2g
(xz, yz, xy)
T1u
(x,y,z) …

3. 3) Symmetry adapter linear combinations of atomic orbitals (SALC’s)
Hybrid orbitals can be considered as basis functions for a reducible representation Gr within a molecule point group.
Let us choose vectors originating from the central atom to represent the hybrid orbitals suitable for s-bonding as a basis function for Gr.
When constructing a reducible representation Gr, we have to consider the effect of each of the group symmetry operations on these vectors.
The character of a particular symmetry operation is equal to the number of vectors that are unshifted by the operation.

4. 4) Symmetry adapted linear combinations of AO’s for s-bonding
D4h E
2C4 C2
2C2’
2C2” i
2S4 sh
2sv
2sd Gr Td E
8C3
3C2
6S4
6sd Gr
D3h E
2C3
3C2 sh
2S3
3sv
Gr-axial
Gr-equ

5. 5) SALC’s of atomic orbitals suitable for p-bonding
Let us choose a set of vectors originating from the peripheral atoms and representing directions of the hybrid orbitals suitable for p-bonding with the central atom as a basis function for Gr. All vectors xi are directed toward z axis and all vectors yi are parallel to xy plane.
Only vectors of unshifted atoms contribute to the character of particular symmetry operations. Td E
8C3
3C2
6S4
6sd Gr

6. 6) Polarity
A species of high symmetry (several rotational axes) cannot be polar.
The polarity is a feature of molecules belonging to the following symmetry point groups only: C1, Cs, Cn, Cnv.