Transcript Slide 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
S
py
dz2
2) Examples of atomic orbitals symmetry analysis
Atomic orbital
s
px
py
pz
dz2
dx2-y2
dxy
dxz
dyz
C2v
Mulliken labels
B1
x, Ry
xz
B2
y, Rx
yz
a1
b1
b2
a1
a1
a1
a2
b1
b2
a1'
a1g
a1g
e'
e'
a2"
a1'
e'
e'
eu
a1
t2
t2
t2
e
e
t2
t2
t2
t1u
D3h
eg
A1’
eu
a2u
a1g
b1g
b2g
eg
eg
e"
e"
B1g
x2-y2
A2
B2g
xy
E
(xz, yz)
T1
(x, y)
xy
Oh
A1
Eu
Rz
Td
x2+y2, z2
z
A2
D4h
A1g
A2u
x2, y2, z2
D3h
t1u
t1u
eg
t2g
t2g
T2
x2+y2, z2
A2’
Rz
E’
(x,y)
(x2-y2, xy)
A1”
t2g
Td
(Rx,Ry)
z
C2v
D4h
Eg
A1
A2”
z
E”
(Rx,Ry)
(xz, yz)
Oh
x2+y2+z2
(2z2-x2-y2, x2-y2)
(Rx,Ry,Rz)
(x,y,z)
A1g
x2+y2+z2
Eg
(2z2-x2-y2, x2-y2)
T1g
(Rx,Ry,Rz)
T2g
(xz, yz, xy)
T1u
…
(xz, yz, xy)
(x,y,z)
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.
tetrahedral, AB4
square planar, AB4
sd
z
z
3
sh
1
C 3 , S3
C2, S4
y
4
2
C2'
trigonal bipyramidal, AB5
sv
sd
sv
C3
C2"
1
2
4
x
3
2
sh
4
3
C2
y
x
5
y
1
x
•
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) Symmetry adapted linear combinations of AO’s for s-bonding
square planar, AB4
sd
3
y
4
2
C2'
D4h
E
sh
1
C2
2C2’
2C2
”
i
2S4
sh
2sv
2sd
2
0
0
4
0
2
0
0
0
4
= A1g (s or dz2) + B1g (dx2-y2) + Eu (px, py)
Gr
sv
2C4
C2 "
dsp2 or d2p2
tetrahedral, AB4
x
z
C2, S4
C3
sd
Td
4
8C3
3C2
6S4
6sd
0
0
4
1
2
= A1 (s) + T2 (px, py, pz or dxy, dxz, dyz)
Gr
1
2
E
y
3
x
sp3 or d3s
trigonal bipyramidal, AB5
z
C 3, S 3
sv
2
sh
4
3
C2
x
E
2C3
3C2
sh
2S3
3sv
Gr-
2
2
0
0
0
2
= A1’ (dz2) + A2” (pz)
3
0
1
3
0
1
= A1’ (s) + E’ (px, py or dx2-y2, dxy)
axial
5
y
1
D3h
Grequ
dsp3 or d3sp
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.
tetrahedral, AB4
E
8C3
3C2
6S4
6sd
Gr
8
-1
0
0
0
= E (dz2, dx2-y2) + T1 + T2 (px, py, pz or dxz, dyz, dxy)
z
sd
Td
C 2, S 4
d2p3 or d5
x2
y2
x1
x' = x cos(2p/3)= -x/2
x'
y1
y
y
y
y
x
C3
y'
x
y' = y cos(2p/3)= -y/2
y3
x4
x3
y4
x
x
0   x    x / 2
 1 / 2


 0
 1 / 2  y   y / 2

(C3) = -1
6) Polarity
•
F
A species of high symmetry
(several rotational axes) cannot
be polar.
O
S
O
H2C
F
Cl
The polarity is a feature of
molecules belonging to the
following symmetry point groups
only: C1, Cs, Cn, Cnv.
CH
H
H
Cs
C1
•
O
C2
F
O
O
O
F
I
C4v
F
H
F
F
C
8
C2v
F
v