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

Point 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

axis = C2

two σv but no σh mirror planes means

point group is C2v

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.):

C3 C3v D4d D4h

A ‘C’ indicates only

one rotational axis

A ‘D’ indicates an n-fold

principal rotation axis

plus n 2-fold axes at

right angles to it

3-fold rotational has σv but 4-fold d = no ‘h’ indicates

axis no σh mirror principal σh mirror a σh mirror

planes in a C group axis plane plane

A subscript ‘h’ means that there is a σh mirror plane at

right angles to the n-fold principal axis:

Naming point groups (contd.):

D4h

C4 principal axis

σh

C3 principal axis

σv

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.

only one

of the three

σv planes

is shown

D3d

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

Td Oh Ih

C60 ‘bucky-ball’

or ‘Fullerene’

Flow chart for determining point groups.

The point group of the carbon dioxide

molecule

We start at the top of the

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.

i

C∞

D∞h

Other linear molecules:

HC≡N HI C≡O

N2 O2 F2 H2

D∞h

C∞v

i i

The top row of linear molecules all have a center of

inversion (i) and so are D∞h.

The bottom row have no

i and so are C∞v

All have a C∞

axis

The Platonic solids:

Td Oh Ih C60

‘buckyball’

tetrahedron octahedron icosahedron

The Cs point group:

Cs

σ

chloro-difluoro-iodo-

methane

I

F Cl

C

F

Most land animals have bilateral symmetry,

and belong to the Cs point group:

Mirror planes (σ) Cs

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:

Bromo-chloro-fluoro-iodo- chloro-iodo-amine

methane

I

Br

F

Cl

C

I

Cl

H

N

C1 C1

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

groups. The Cn point groups have no other symmetry elements,

the Cnv point groups have

also n mirror planes

containing the Cn rotational

axis, while the Cnh point

groups also have a σh mirror plane at right angles to the principal rotational axis.

C5

iodine

[IF6O]-

O

F

F

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

The point group of the water molecule is C2v

there is no σh plane

at right angles to

the C2 axis, but

there are two σv

planes containing

the C2 axis.

C2 C2 C2

σv σv

Other Cnv molecules:

C2v

C3v

C4v

ammonia water

σv

σv

σv

Vanadyl tetrafluoride (VOF4)

V

Some more C2v molecules:

σv

σv

σv σv

σv σv

C2 C2 C2

Phosphorus iodo- sulfur tetra- carbonyl

tetrafluoride (PF4I) fluoride (SF4) chloride (COCl2)

C S P

These have a Cn axis as their only symmetry element. Important

examples are (hydrogens omitted for clarity):

The Cn point groups:

C3 C3

C3 C3

C3

C3 triphenyl

phosphine

viewed down

C3 axis

Cobalt(III)

tris-glycinate

viewed down

C3 axis

triphenyl

phosphine

viewed from

the side

Cobalt(III)

tris-glycinate

viewed from

the side

The Dnh point groups:

C2 σh

four C2

axes at

rt. angles

to C4 axis

C2 C2

C2

C4

principal

axis

mirror plane

at rt. angles

to C4 axis

D4h

Examples of molecules belonging to Dnh point

groups:

D2h D3h D3h D3h

D4h D4h D5h D5h

C2 C3

C3 C3

C4

C4

C5 C5

C6

principal axis C2

C2

C2 C6

C2

σv σv

Benzene, an example of the D6h point group:

σh

C6

principal axis

C6

principal axis

D6h

The Dn point groups:

C2

C2

C2 principal axis

D2

these have a principal

n-fold axis, and n 2-fold axes at right

angles to it, but no

mirror planes.

[Cu(en)2]2+ complex

with H-atoms

omitted for clarity.

(en = ethylene

diamine)

Cu N

N C

Some further views of the symmetry elements

of [Cu(en)2]2+, point group D2 :

C2

[Cu(en)2]2+ complex

with H-atoms

omitted for clarity.

(en = ethylene

diamine)

C2

C2

C2

C2

C2

C2 principal

axis

C2 principal axis

C2 principal

axis

C2 principal

axis

C2 C2

D2

C2

C2

C2

C3 principal axis

Some views of the symmetry elements of

[Co(en)3]3+, point group D3.

C3

principal axis

C2

axis

view down the C3 axis

of [Co(en)3]3+ showing

the three C2 axes.

D3

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

[Co(oxalate)3]3- [Co(bipyridyl)3]

3+

C2 C2

C2

C2

C2

C2 C3

principal axis

D3 D3

Molecules belonging to the Dnd point groups

These have mirror planes parallel to the principal

axis, but not at right angles to it.

C3 axis C5 axis

Staggered form

of ethane Staggered form of ferrocene

σv planes

contain the

principal

axis

D3d D5d

The D4d point group:

C2 C2

C2 C2

C4

principal axis

C4 principal axis

C2

σv

σv

σv

σv

C4

principal axis [ZrF8]4-

Square antiprism

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.

D4d

[K(18-crown-6)]+, an example of a D3d

point group:

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.

D3d

σv

σv

K+

C3

principal axis C3 principal axis

σv

C2 C2

C2

C2

C2

C2

Some Point groups