Symmetry: Point Groups
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 HermannMauguin 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
Oh
3C4, 4C3 , 6C2 and 3 σh
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

1,2,3,4,6
2,3,4,5,6,7,∞
2,3,4,5,6,7, ∞
S2n
Values of n for
crystals
1,2,3,4,5,6

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
Values of n for
molecules
Sn, nσv
(i for even
values of n)
A proper axis Cn, nC2
perpendicular to that and σh
Sn
T
Th
Td
O
redundant
symmetry
elements
Sn

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 noncrystallographic 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...
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