Point Groups
Point Group = the set of symmetry operations for a molecule
Group Theory = mathematical treatment of the properties of the group which
can be used to find properties of the molecule
Assigning the Point Group of a Molecule
1. Determine if the molecule is of high or low symmetry by inspection
A. Low Symmetry Groups
B. High Symmetry Groups
2.
If not, find the principle axis
3.
If there are C2 axes
perpendicular to Cn the
molecule is in D If not, the
molecule will be in C or S
4.
If h perpendicular to Cn then
Dnh or Cnh If not, go to the next
step
5.
If  contains Cn then Cnv or Dnd
If not, Dn or Cn or S2n
6.
If S2n along Cn then S2n
7.
If not Cn
The determination of point groups of
molecules
only one rotational two σv but no σh mirror planes means
axis = C2
point group is C
2v
The point group of the water molecule is C2v
Naming point groups:
The name of the point group has information about the
symmetry elements present. The letter is the rotational group
and the subscript number after the letter indicates the order
of the principal rotational axis (e.g. 3-fold or 4 fold etc.):
A ‘C’ indicates only
one rotational axis
C3
C3v
3-fold rotational has σv but
axis
no σh mirror
planes in a C group
A ‘D’ indicates an n-fold
principal rotation axis
plus n 2-fold axes at
right angles to it
D4d
4-fold
principal
axis
D4h
d = no
σh mirror
plane
‘h’ indicates
a σh mirror
plane
Naming point groups (contd.):
A subscript ‘h’ means that there is a σh mirror plane at
right angles to the n-fold principal axis:
C4
principal axis
C3
principal axis
only one
of the three
σv planes
is shown
σh
D4h
σv
D3d
A subscript ‘d’ (or v for C groups) means there is no σh mirror
plane, but only n σv mirror planes containing the principal Cn axis.
Naming platonic solids:
Platonic solids: Five special polyhedra: Tetrahedron, Cube,
Octahedron, Dodecahedron, and
Icosahedron
Their faces are all exactly the same.
The same number of faces meet at each vertex.
T = tetrahedral = 4 three-fold axes
O = octahedral = 3 four-fold axes
I = icosahedral = 6 five-fold axes
C60
‘bucky-ball’
or ‘Fullerene’
Td
Oh
Ih
Flow chart for determining point groups.
The point group of the carbon dioxide
molecule
i
We start at the top of the
C∞
D∞h
flow-chart, and can see that
the CO2 molecule is linear,
and has a center of inversion
(i) so it is D∞h. Note the C∞
principal rotation axis.
Other linear molecules:
The top row of linear molecules all have a center of
inversion (i) and so are D∞h.
D∞h
i
N2
O2
HC≡N
The bottom row have no
i and so are C∞v
i
F2
HI
C∞v
H2
C≡O
All have a C∞
axis
The Platonic solids:
tetrahedron
Td
octahedron
Oh
icosahedron
Ih
C60
‘buckyball’
The Cs point group:
σ
I
Cs
C
F
F
Cl
chloro-difluoro-iodomethane
Most land animals have bilateral symmetry,
and belong to the Cs point group:
Cs
Mirror planes (σ)
The C1 point group:
Molecules that have no symmetry elements at all except
the trivial one where they are rotated through 360º and
remain unchanged, belong to the C1 point group. In
other words, they have an axis of 360º/360º = 1-fold, so
have a C1 axis. Examples are:
I
I
N
Cl
Br
C1
C
H
Cl
F
Bromo-chloro-fluoro-iodomethane
C1
chloro-iodo-amine
The division into Cn and Dn point groups:
After we have decided
that there is a principal
rotation axis, we come
to the red box. If there
are n C2 axes at right
angles to the principal
axis, we have a Dn point
group, If not, it is a Cn
point group.
Dn
Cn
The Cn point groups:
The Cn point groups all have only a single rotational
axis, which can theoretically be very high e.g. C5 in
the complex [IF6O]- below. They are further divided
into Cn, Cnv, and Cnh point
C5
groups. The Cn point
groups
have no other
O
symmetry
elements,
iodine
the Cnv point groups
have
also n mirror planes
F containing the Cn rotational
axis, while the Cnh point
groups also have a σh mirror
F
plane at right angles to the
[IF6O]- principal rotational axis.
The point group of the water molecule
We start at the top of the
flow-chart, and can see that
the water molecule is not
linear, and is not tetrahedral (Td),
octahedral (Oh), or icosahedral,
(Ih) so we proceed down the chart
C2
Yes, there is a principal Cn axis,
so we proceed down the chart, but
in answer to the next question, there
are no further C2 axes at right angles
to the principal axis, which is the only
axis, so we proceed down the chart
C2
C2
C2
σv
σv
there is no σh plane
at right angles to
the C2 axis, but
there are two σv
planes containing
the C2 axis.
The point group of the water molecule is C2v
Other Cnv molecules:
water
ammonia
σv
C2v
C3v
C4v
V
σv
Vanadyl tetrafluoride (VOF4)
σv
Some more C2v molecules:
C2
σv
P
C2
σv
S
σv
Phosphorus iodotetrafluoride (PF4I)
σv
C
σv
sulfur tetrafluoride (SF4)
C2
σv
carbonyl
chloride (COCl2)
The Cn point groups:
These have a Cn axis as their only symmetry element. Important
examples are (hydrogens omitted for clarity):
C3
triphenyl
phosphine
viewed down
C3 axis
C3
C3
C3
C3
triphenyl
phosphine
viewed from
the side
Cobalt(III)
tris-glycinate
viewed down
C3 axis
C3
Cobalt(III)
tris-glycinate
viewed from
the side
The Dnh point groups:
C4
principal
axis
four C2
axes at
rt. angles
to C4 axis
C2
C2
C2
mirror plane
at rt. angles
to C4 axis
σh
C2
D4h
Examples of molecules belonging to Dnh point
groups:
C2
C4
C3
C3
D2h
D3h
D3h
C5
C4
D4h
C3
D4h
D3h
C5
D5h
D5h
Benzene, an example of the D6h point group:
C6
principal axis
C6
C2
σh
C2
C2
C2
σv
σv
D6h
C6
principal axis
C6
principal axis
The Dn point groups:
these have a principal
n-fold axis, and n
2-fold axes at right
angles to it, but no
mirror planes.
C2 principal axis
N
C2
N
C2
C
[Cu(en)2]2+ complex
with H-atoms
omitted for clarity.
(en = ethylene
diamine)
Cu
D2
Some further views of the symmetry elements
of [Cu(en)2]2+, point group D2 :
C2 principal axis
D2
C2 principal
axis
[Cu(en)2]2+ complex
with H-atoms
omitted for clarity.
(en = ethylene
diamine)
C2
C2
C2
C2
C2 principal
C2 principal
axis
axis
C2
C2
C2
C2
Some views of the symmetry elements of
[Co(en)3]3+, point group D3.
C2
C3 principal axis
C2
C2
D3
C3
principal axis
view down the C3 axis
of [Co(en)3]3+ showing
the three C2 axes.
C2
axis
view down one of the
three C2 axes of [Co(en)3]3+
at right angles to C3
Other examples of the D3 point group
C2
C3
C2
C2
principal axis
C2
C2
C2
D3
D3
[Co(oxalate)3]3-
[Co(bipyridyl)3]3+
Molecules belonging to the Dnd point groups
These have mirror planes parallel to the principal
axis, but not at right angles to it.
C5 axis
C3 axis
σv planes
contain the
principal
axis
D3d
Staggered form
of ethane
D5d
Staggered form of ferrocene
The D4d point group:
C2
C4 principal axis
C2
C2
σv
σv
σv
C2
σv
C2
C4
principal axis
]4-
[ZrF8
Square antiprism
C4
principal axis
D4d
As predicted by VSEPR, the [ZrF8]4- anion has a square anti-prismatic
structure. At left is seen the C4 principal axis. It has four C2 axes at
right angles to it, so it has D4 symmetry. One C2 axis is shown side-on
(center). There are four σv mirror planes (right), but no mirror plane at
right angles to C4, so the point group does not rate an h, and is D4d.
[K(18-crown-6)]+, an example of a D3d
point group:
C3
C2
C2
principal axis
C3 principal axis
K+
σv
σv
C2
C2
C2
C2
σv
D3d
The complex cation [K(18-crown-6)]+ above is an important structure that
has D3d symmetry. It has a C3 principal axis with 3 C2 axes at right
angles to it, as well as three σv mirror planes that contain the C3 axis,
but no σh mirror plane (because it’s not flat, as seen at center), so is D3d.
Some Point groups