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HOW TO FIND OUT POINT GROUP
Dr. Cyriac Mathew
Symmetry elements can combine in a definite number of ways. For example consider BF3
molecule. It has the C3 axis as the principal axis. Also, there are 3 C2 axes perpendicular to the C3 axis,
3 σv planes, and one σh plane. All these symmetry elements can combine. Take NH3. It also has C3 axis
as principal axis. There are 3 σv planes, but no C2 perpendicular to C3 or a σh plane. For the N2F2
molecule there exist one C2 axis and a σh plane but no σv planes.
Thus in general we can say a Cn can combine with either nC2 or no C2 perpendicular to it; it can
combine with either one σh or no σh; or it can combine with n vertical planes or no vertical planes.
These are relevant only in the case of systems where we can identify a principal axis. Such systems are
called axial systems.
In the case of tetrahedral, octahedral, cubic, icosahedral and dodecahedral objects we cannot
identify the principal axis. Solids of these structures are called platonic solids. Crystals and molecules of
these shapes are highly symmetric and can be called multi higher order axial systems. They have several
higher order axes than C2 axis. The symmetry elements combine in definite ways in these systems also.
Point Groups
Point groups are possible combinations of symmetry elements. Since symmetry elements can combine
only in a definite pattern, there will be only a finite number of point groups possible. For crystals only
32 points groups exist. Crystals cannot have axes of symmetry order 5 or higher than 6. On the other
hand molecules can have proper axes of symmetry of order 5, 7 and ∞ also. Hence for molecules some
additional point groups are possible which are not possible for crystals.
Redundant symmetry elements: symmetry elements which combine to form a point group are known
as essential symmetry elements. There exist some symmetry elements which are present as a
consequence of the essential symmetry elements. Consider BF3 molecule. The essential symmetry
elements needed, for the point group under which this molecule comes, are E, C3, 3σv, σh. The presence
of C3 and σh give rise to S3 (ie. σhC3). The following table provides the possible point groups for
molecules and crystals. The notation given is the Schoenflies notation which is applicable to molecules.
In the case of crystals the Hermann-Mauguin notation is used.
Table 1: Molecular point groups
Point group
(Schoenflies
notation)
Cn
Cs
Ci
Cnh
Cnv
Dn
Dnh
Essential symmetry elements
Only a proper axis Cn
Only a plane of symmetry
Only centre of symmetry
A proper axis Cn and σh
A proper axis Cn and n σv
A principal axis Cn and nC2
perpendicular to that
Sn
T
Th
Td
O
Oh
3C4, 4C3 , 6C2 and 3 σh
Sn, nσv
(i for even
values of n)
A proper axis Cn, nC2
perpendicular to that and σh
A proper axis Cn, nC2
perpendicular to that, and nσd
Only an Sn
4C3 and 3C2
4C3 , 3C2 and σh or i
4C3 , 3C2 and 6σd
3C4, 4C3 , and 6C2
Dnd
redundant
symmetry
elements
Sn
-
S2n
-
Values of n for
molecules
Values of n for
crystals
1,2,3,4,5,6
1,2,3,4,6
2,3,4,5,6,7,∞
2,3,4,5,6,7, ∞
2,3,4,6
2,3,4,6
2,3,4,5,6
2,3,4,6
2,3,4,5,6,7, ∞
2,3,4,6
2,3,4,5
2,3
2,4,6
2,4,6
3S4
3S4, 4S6,
6 σv and i
Cyclic point groups (Cn,Cnv, Cnh and Sn): A molecule which possesses only a Cn axis come under the Cn
point group. Here C stands for cyclic and n stands for order of the axis. Cn represents five groups,
namely, C1, C2, C3, C4, C5 and C6. C5 is a non-crystallographic point group. C1 is the point group with
no element of symmetry.
H
O
O O
Br
F
Cl
bromochlorofluoro methane
C1 point group
H
H
H
Hydrogen peroxide
Phosphoric acid
C2 point group
C3 point group
C6 point group
C5(CH3)5.
pentamethyl
cyclopentadienyl
C5 point group
S2
C6(CH3)6.
hexamethyl
cyclohexadienyl
P
O H
O
O
H
N
Quinoline
Cs point group
Br
Cl
Ci point group
i
Cl
Br
Assigning Point Group
First of all we look for an axis of symmetry and find whether it is the principal axis or not. There
can be molecules without an axis of symmetry at all and corresponding point groups are C i, Cs, S4 and
C1only. If axes of symmetry are there and the principal axis cannot be identified, the point group can be
Td, Oh or Ih. If there is a Cn axis and nC2 axes perpendicular to it the point group will be dihedral, Dn,
Dnh, Dnd. Otherwise, the point groups will be Cn, Cnh, or Cnv. Thus the following steps are involved in
assigning the point group.
Identify,
1. The presence of an axis of symmetry
2. This axis as principal axis or not
3. The existence of subsidiary axes
4. The existence of σh
5. The presence of n σv’s
6. Whether belongs to Ci, Cs, S4 or C1, if principal axis is absent.
This procedure can be simplified by asking certain questions to ourselves and finding ‘yes’ or ‘no’
answers.
Is there an axis of symmetry?
(Find the highest order axis)
Yes
No
Is it the principal axis?
Is there a plane of symmetry?
No
Yes
Is there an i?
No
No
Cs
Look for Td and Oh
point groups
Yes
Are there nC2’s
perpendicular Cn?
Yes
Is there an Sn?
Yes
No
Ci
Is there a σh?
No
Yes
Sn
Are there
nσv?
No
Yes
Cn
Cnh
Are there nσd?
No
Cnv
Yes
No
Yes
No
C1
Is there a σh?
Dn
Yes
Dnd
Dnh