CH4 – four-fold improper
rotation
Special Features of Improper
Rotations
• Certain improper rotations are equivalent to other
symmetry operations.
• Improper rotations are considered to be the
lowest possible symmetry operations so when
one of them is equivalent to another symmetry
operation the symbol for the other symmetry
operation is used.
i
Since S2 is equivalent
z
to the inversion
y
operation, i, it is
x
always designated
C2(z)
x,y
“i”, not S2.
S2 = x,yC2(z)
Improper Rotations – A Combination of Rotation
and Reflection
2
2
1
1
S4
h
C4
2
1
S
2
4
2
S
2
 C2
1
1
3
4
h
C4
1
2
1
1
2
h
2
C4
1
2
2
S
4
4
h
C4
1
2
2
1
1
E
• Consider the operation Snm for even values of m.
– The reflection operation in S is always done an even
number of times so Snm = Cnm E = Cnm when m is even.
• Consider the operation Snn for odd values of n.
– In this specific case where n = m, the rotation operation
has been carried through an angle of 2.
– Since n is odd, the reflection operation is carried out an
odd number of times so n = .
– The result is that in the specific case where n is odd, Snn
= .
• Consider Snm generally when n is odd.
– When n is odd, Snn =  and Snn+1 = Cn.
– When n is even, Snn = E and Snn+1 = Sn and cannot be
reduced except for S2 which is the same as the
inversion.
– We conclude that when n is odd, the existence of a Sn
axis requires the existence of both a Cn axis and a 
plane.
Point Groups and Multiplication Tables
Point groups
• So called because all the symmetry elements pass
through one common point
• It is useful to be able to classify the molecular point
group so that we can easily identify all the symmetry
elements
• Symmetry classification can be used to discuss
molecular properties.
• We can use symmetry transformations of orbitals to
decide which atomic orbitals contribute to the
formation of molecular orbitals, and select linear
combinations of atomic orbitals that match the
symmetry of the molecule.
C1
• If only the identity
element is present,
a molecule is in the
C1 point group.
Ci
• If the only additional
element is inversion,
the point group is Ci.
An example is
meso-tartaric acid.
Cs
• Molecules in the Cs
group, e.g.,
fluoroethane, have
only one symmetry
element other than
E - a mirror plane.
Cn
• If the only
symmetry element
other than E is an n
-fold axis, the point
group is Cn . For
example, H2O2 is
C2 .
Cnv
• If in addition there
are n vertical mirror
planes, it belongs to
the group Cnv . We
saw that water has a
C2 axis and two v
planes, so its point
group is C2v .
Cv
• All heteronuclear
diatomics and
linear molecules
with different atoms
on the ends, are
symmetrical for any
rotations around
and reflections
across the nuclear
axis, so are Cv .
Cnh
• If there is a
horizontal mirror
plane, the point
group is Cnh . For
C2h there is also
an implied center
of inversion.
Dn
H
• The Dn group has
the symmetry
elements of the Cn
group, as well as n
C2 axes
perpendicular to the
principal axis.
H
H
H
H
H
Gauche Ethane, neither staggered
nor eclipsed, is D3
Dnh
• If in additional there is a horizontal mirror
plane, the group is Dnh
Dh
• All homonuclear
diatomics and linear
molecules which are
symmetrical about the
center point, have
symmetry elements
for any rotation about
the nuclear axis and
for end-to-end
rotation and end-toend reflection.
Dnd
• If in addition to the
elements of Dn there
are n dihedral mirror
planes the point
group is Dnd . An
example is
staggered ethane,
which is D3d .
Sn
• Molecules which do not fit one of the above
classifications, but which possess one Sn axis,
belong to the Sn group.
• n is a multiple of two, but S2 is equivalent to Ci ,
and the latter designation takes precedence.
• Members of the Sn group also have a Cn/2 axis;
e.g., an S4 molecule will have a C2 axis.
The cubic groups
• So far we have seen molecules with one
principal axis (if any).
• Some highly symmetrical molecules have
more than one principal axis, and most of
these belong to the cubic groups.
Td
• Molecules in the
shape of a regular
tetrahedron, e.g.,
CH4, are in the
group Td
Th
• If in addition to the
symmetry of T there
is an inversion center,
the group is Th
Oh
• Molecules with a
regular octahedron
shape are in the
group Oh
• An example is SF6
Ih
• Icosahedral (20faced) molecules with
the maximum
symmetry for that
arrangement belong
to the point group Ih .
Examples are some
of the larger boranes
and C60.
Td
Oh
Ih
Linear molecules
R3
• An atom or a sphere has an infinite
number of rotation axes in three
dimensions and for all possible values of
n. In these cases the point group is R3
Td, Oh, or Ih
D∞h
Start
yes
yes
Has the molecule
Td, Oh, Ih symmetry?
no Is the molecule linear?
Is there a center of
inversion (symmetry)?
yes
no
no
no
Is there a principal Cn axis?
no
yes
Is there a center of
inversion?
Are there n C2 axes
Is there a σh plane
yes
no
perpendicular to the
(perpendicular to the
Cn axis?
Cn axis)?
no
Is there a σh plane
(perpendicular to the yes
Cn axis)?
no
Are there n σv planes
(containing the Cn axis)?
yes
Dnh
Cnh
yes
Cnv
no
yes
Is there a mirror plane?
yes
no
C∞v
Cs
Ci
C1
Are there n σv planes
(containing the Cn axis)?
(These σv planes are of
the σd type.)
yes
no
Dnd
Dn
yes
Is there an
no
S2n improper rotation axis?
S2n
Cn