Symmetry Operations
E: Identity operation: The E operation
does nothing. All molecules have this
element
Cn: Proper rotation: Rotation of molecule
by an angle 360/n about an axis through the
center of mass (COM) gives no change. C2
rotates 180°, C6 rotates 60°. The rotation
axis with the largest n is the principal axis.
H
CH3
H3C
i: Inversion: All atoms are inverted
through the center of mass to the opposite
side of the molecule.
Cl
Cl
W
Cl
CH3
H3C
H
Cl
H
C2
H
i
Cl
Cl
Cl
Cl
Cl
W
Cl
Cl
Cl
C4
σh: Horizontal mirror plane: Mirror plane
that is perpendicular to the principal
rotation axis (largest Cn).
H
H
H
H
H
H
H
σh
H
H
H
H
H
H
H
H
H
C2
σv: Vertical mirror plane: Mirror plane that
is parallel to, and encompasses, the
H H
H H
σv
principal rotation axis.
Br
Br
Br
Br
σd: Dihedral mirror plane: A σv that
bisects angle between C2 axes ⊥ to principal
axis.
Sn: Improper rotation: This operation is the product of a Cn rotation followed by a σh. Note that
S2, which is the product of C2 and σh, is identical to an inversion (i).
S4
H3 H4
H1
H2
H3
Me
Me
H4
σh
C2
Me
Me
Me
H2 H1
H3
H1 H2
S2 or i
center of inversion
Me
Me
C4
Me
σh
Me
H4
Me
Me
Me
A molecule has a given symmetry element if it can undergo the symmetry operation without
changing its structure (i.e. groups related by the symmetry element will exchange positions).
Point Group Designations
Point Group
C1: E. No symmetry elements are present.
Cs: E, σ. Molecule has a mirror plane of
symmetry
Ci: E, i. Molecule has an inversion center
Sn: E, Sn. In many cases there will also be a
Cn/2 rotation axis, although this is not a
requirement. Note that S1 is the same as a
mirror plane (σ) and should be called Cs. S2 is
the same as i and should be called Ci.
Example
F
C H
Cl
Br
C1
Br
Br
Br
H
H
H
Cs
F Cl
H
H
F Cl
Ci
Br H
Br
H
H
Br
H
Br S
4
CH3
Cn: E, Cn. Molecule has only a single rotation
axis, with no mirror planes.
Cnv: E, Cn, and n σv. Molecule vertical mirror
planes corresponding to the Cn rotation.
C∞v: linear unsymmetric molecule. A special
case of Cnv.
CH3
C2
N H
H
H
C3v
O C S
C∞v
Dn: E, Cn, and n C2 axes ⊥ to Cn axis: Thus a
D3 molecule would have 3 C2 ⊥ to the Cn axis.
Dnh: E, Cn, n C2 axes, σh. In addition to the n
C2 axes there is a horizontal mirror plane.
D∞h: Special case of Dnh for linear symmetric
molecules.
Dnd: E, Cn, n C2 axes, and n σv (also known as
σd).
D2
D6h
O C O
H
H
D∞h
H
H
D3d
H
H
H
C H
H
H
Td: tetrahedral symmetry. There are 4 C3
rotation axes.
Td
O
O
Oh: octahedral symmetry. There are six C4
axes.
O
C
C
C
W
C
O
Oh
Ih: Icosahedral symmetry. There are 12 C5
rotation axes.
Ih
C
C
O
O
Point Group Decision Tree
Yes
Cnh
No
σh?
C∞v or
D∞h
Cubic
T, O, I
Yes
S2n colinear
w/ Cn?
Yes
Cn
Yes
No
More than one
Cn (n ≥ 3)
No
σv?
Cnv
No
No
Linear?
Find
principal
axes
Cn is the
principal
axis?
Yes
None
Cs, Ci
or C1
σh?
Yes
Dnh
Adapted from: Physical Chemistry,
Joseph H. Noggle, 2nd ed., Scott
Foresman & Co, Glenview, IL, 1996,
pg 840.
n vertical
mirror planes
nC2 ⊥
to Cn?
No
No
Yes
Yes
σv?
No
Dn
Dnd
S2n