Molecular Symmetry
(Structural Classifications)
Molecular structures
Linear
Planar triangle
Tetrahedral
Trigonal bipyramidal
Octahedral
Pentagonal bipyramidal
Square antiprism
O=C=O
SO3
CH4
PCl5
SF6
IF7
TaF83-
Symmetry Operations:
an action such as reflection, rotation, …
Symmetry Elements:
associated with each symmetry operation, a plane, a line , a point, …
A symmetry element is an axis (line), plane, or point about which symmetry
operations are performed.
Aspects of Symmetry and Point Groups
Symmetry operations:
1.
1.
2.
3.
4.
Proper rotation, Cn
Reflection, 
Inversion, i
Improper rotation, Sn
Proper rotation, Cn
simple rotation about an axis passing through the molecule by an angle of 360/n.
This operation is called a proper rotation and is symbolized by Cn
*
180º rotation
C2
*
For a H2O molecule, please locate the C2 axis
*
*
C3
C3
*
2
C3
For a BF3 molecule, please locate the C3 axis
2.
Reflection (mirror) plane, 
Reflection of all atoms through a plane
Please locate the two vertical reflection planes in H2O
3.
Inversion (Center of Inversion), I
Reflection of all atoms through a point
Please locate the center of inversion in SF6, XeF4, etc.
4.
Improper rotation, Sn
The combination, in either order, of rotating the molecule about an axis
passing through it by 2/n and reflecting all atoms through a plane that is
perpendicular to the axis of rotation
h
1
4
4
C4
2
3
2
3
4
h
1
through an S 4 operation
An S4 improper rotation axis
How many S4 rotation axes for a CH4 molecule?
2
1
3
Symmetry element
Symmetry
Operation
examples
None
Identity, E
CHFClBr
Mirror plane
Reflection, 
H2O
Rotation axis
Rotation
Inversion center
Inversion, i
Improper axis
Rotation
-reflection
C2
p-C6H4Cl2
C3
NH3
C4
[PtCl4]2-
C5
C5H5-
C6
benzene
Ferrocene (staggered)
S4
CH4
S6
Ethane(staggered)
S10
Ferrocene (staggered)
Some important points
1. Do not confuse an inversion operation with a two-fold rotation
A
D
B
C
A
B
i
C
D
B
A
C
D
2. S1 operation is equivalent to the mirror reflection 
3. S2 is equivalent to a center of inversion
Exercises:
B
A
D
C
C2
1. Which conformation of CH3CH3 has an S6?
2. Identify a C3 axis of an NH4+ ion. How many C3 axes?
Summary of important symmetry operations and symmetry elements
Symmetry element
Symmetry operation
Symbol
Identity
E
n-Fold symmetry axis
Rotation by 2/n
Cn
Mirror plane
Reflection

Center of inversion
Inversion
I
n-Fold axis of
improper rotation
Rotation by 2/n
Sn
followed by reflection
perpendicular to rotation axis
One molecule may have more than one symmetry elements
NH3
H2O
SO2Cl2
SF6
To summarize the collection of symmetry elements a molecule has, we introduce the
concept of point group.
Point Groups (or Symmetry Groups)
The complete set of symmetry operations than can be performed on a molecule is
called the symmetry group for the molecule
Point group
symmetry elements
examples
C1
C2
Cs
C2v
C3v
Cv
D2h
D3h
D4h
Dh
Td
Oh
E
E, C2
E, 
E, C2, 2v
E, C3, 3v
E, C
E, 3C2, 2v, h, i
E, C3, 3C2, 3v, h
E, C4, 4C2, 4v, h, i, S4, …
E, C, v, …
E, 3C2, 4C3, 6, 3S4
E, 6C2, 4C3, 4S6, 3S4, i, …
SiBrClFI
H2O2
NHF2
H2O, SO2Cl2
NH3, PCl3, POCl3
CO, HCl, OCS
N2O4, B2H6
BF3, PCl5
XeF4, trans-MA4B2
H2, CO2, C2H2
CH4, SiCl4
SF6
Classification of Point Groups
1. Groups of low symmetry
a) C1
no symmetry at all
CHFClBr
b) Cs
only one mirror plane
H2C=CClBr
c) Ci
only an inversion center
HClBrC-CBrClH
2. Cn groups (n=2, 3, 4, …)
Contain only one Cn rotational axis
Example: H2O2
3. Cnv groups (n=2, 3, 4, …)
Contain Cn and nv
(a set of n vertical mirror planes: these mirror planes contain
the Cn axis)
Examples:
H2O
NH3
SO2Cl2
SClF5
ONNi(5-C5H5)
4. Cnh groups (n=2, 3, …)
Cn + h
h:
mirror plane, perpendicular to Cn
O
Me
C
Examples:
O
trans-HClC=CHCl
S
Mo
R 3P
O
O
X
Mo
X
O
O
C
Me
O
PR3
O
O
S
O
Re
O
O
O
O
S O
Re
O
O
O
S O
O
5. Dn groups (n=2, 3, …)
Cn + n C2
A Cn principal rotational axis is accompanied by a set of n C2 axes
perpendicular to it
example: Co(H2NCH2CH2NH2)3
6. Dnh groups (n=2, 3, 4, …)
Dn + h
h: horizontal mirror plane, perpendicular to the principal axis Cn
 generating nv mirror planes
Examples:
trigonal prism
trigonal-bipyramidal
pentagonal prism,
pentagonal-bipyramidal
PtCl42C6H6
BF3
PCl5
7. Dnd groups (n=2, 3, 4, …)
Dn + a set of n vertical mirror planes (v)
do not have h mirror plane
Examples:
R3WWR3
Me
H
H
Me
Ferrocene
XB
8. Sn groups (n=4, 6, 8, …)
N
contain only Sn axis
S1   Cs
S 2  i  CI
RN
NR
BX
BX
NR
H
Me
Me
H
B
X
N
R
When n = odd, Sn  Cnh
n can only be even
9. High symmetry point groups
1) Tetrahedral point groups
T point group:
4C3 + 3C2
Td point group:
T + 6d
Th point group :
T+ i
(CH4, SiCl4, etc…)
2) Octahedral point groups
O point group:
4C3 + 6C2
Oh point group:
O+i
3) Icosahedral point groups
I point group:
6C5
Ih point group:
I+i
10. Point groups for linear molecules
A-B
A-B-C
A-A
A-B-A
CO, CO2, N2
Summary:
Some typical shapes of point groups
Procedure in identifying point groups
n=
2
Cn
Dn
Cnv
Cnh
Dnh
(plane or bipyramidal)
Dnd
S2n
3
4
5
6
Important points:
The symmetry elements used to identify a point group do not form a complete set of
symmetry elements for the molecule under consideration
Using the highest symmetry point to assign the molecular symmetry
Exercises: Mer-[Fe(CN)3Cl3]3-
IF7
Mo(CN)84-
P4
XeO2F2
TeCl4
ClSbO
trans-HClC=CClH
Cyclopropane
cyclopropene
C8H8
C8H4F4
Applications of symmetry
1. Molecular polarity
A polar molecule is a molecule with a permanent electric dipole moment
Cannot have a permanent dipole perpendicular to any mirror plane
Cannot have a permanent dipole perpendicular to any axis of symmetry
Molecules having both a Cn axis and a perpendicular C2 axis or h cannot have a dipole
in any direction.
Molecules belonging to any D, T, O or I groups cannot have permanent dipole moment
+
C2
-

+
The presence of a mirror plane or a C2 axis rules out a dipole in the direction shown.
Exercises:
RuCp2 and FeCp2, are they polar?
2. Molecular chirality
A chiral molecule is a molecule that is distinguished from its mirror image in the same
way that left and right hands are distinguishable
A molecule that has no axis of improper rotation (Sn) is chiral
Sn: including S1= and S2=i
Exercises:
(1) [Cr(ox)3]3-, is it chiral?
(2) The skew form of H2O2, is it chiral?