The determination of point groups of molecules

only one rotationalaxis = 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-foldprincipal rotation axisplus 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 mirrorplane, but only n σv mirror planes containing the principal Cn axis.

only oneof the three

σv planes

is shown

D3d

Naming platonic solids:

Platonic solids: 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 theflow-chart, and can see thatthe 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 ofinversion (i) and so are D∞h.

The bottom row have noi 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

FCl

C

F

Most land animals have bilateral symmetry, and belong to the Cs point group:

Mirror planes (σ)Cs 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

ClC

I

Cl

H

N

C1C1

The division into Cn and Dn point groups:

After we have decided thatthere is aprincipal rotat-ional axis, wecome to thered box. If thereare n C2 axesat right anglesto the principalaxis, we have aDn 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 theflow-chart, and can see thatthe water molecule is notlinear, 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, butin answer to the next question, thereare no further C2 axes at right anglesto the principal axis, which is the onlyaxis, so we proceed down the chart

The point group of the water molecule is C2v

there is no σh plane

at right angles tothe C2 axis, but

there are two σv

planes containingthe C2 axis.

C2 C2C2

σv σv

Other Cnv molecules:

C2v

C3v

C4v

ammoniawater

σv

σv

σv

Vanadyl tetrafluoride (VOF4)

V

Some more C2v molecules:

σv

σv

σvσv

σvσv

C2 C2C2

Phosphorus iodo- sulfur tetra- carbonyltetrafluoride (PF4I) fluoride (SF4) chloride (COCl2)

CSP

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

generally resemble propellers which have the front and back different. Important examples are (hydrogens omitted for clarity):

The Cn point groups:

C3C3

C3 C3

C3

C3 triphenyl phosphineviewed down C3 axis

Cobalt(III)tris-glycinateviewed down C3 axis

triphenyl phosphineviewed from the side

Cobalt(III)tris-glycinateviewed from the side

The Dnh point groups:

C2 σh

four C2

axes atrt. anglesto C4 axis

C2C2

C2

C4 principal axis

mirror planeat rt. anglesto C4 axis

D4h

Examples of molecules belonging to Dnh point

groups:

D2h D3h D3h D3h

D4h D4h D5h D5h

C2 C3C3 C3

C4

C4C5 C5

C6

principal axisC2 C2

C2C6C2

σvσv

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 n2-fold axes at rightangles to it, but nomirror planes.

[Cu(en)2]2+ complex with H-atomsomitted for clarity. (en = ethylene diamine)

CuN

NC

Some further views of the symmetry elements of [Cu(en)2]2+, point group D2 :

C2

[Cu(en)2]2+ complex with H-atomsomitted for clarity. (en = ethylene diamine)

C2

C2

C2

C2

C2

C2 principal

axis

C2 principal axis

C2 principal

axis

C2 principal

axis

C2C2

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 axisof [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

C2C3

principal axis

D3D3

Some cobalt(III) complexes belonging to the D3 point group:

NH2

Co

NH2

H2NH2N

H2N NH2

Co

N

N

N

N

N

N

CoO

O

O

O

O

O

CH3

H3C

H3C

CH3

CH3

CH3

3+ 3+

tris(ethylenediamine) tris(2,2’-bipyridyl) tris(acetylacetonato)cobalt(III) cation cobalt(III) cation cobalt(III)

D3

C2 C2C2

Comparison of C3 and D3 tris(chelates)

NH2

Co

NH2

H2NH2N

H2N NH2

O

Co

NH2

H2NH2N

O O

3+O

O

O

D3 C3C2

no C2 axis at

this point

tris(ethylenediamine)cobalt(III) tris(glycinato)cobalt(III)

Molecules belonging to the Dnd point groups

These have mirror planes parallel to the principal axis, but not at right angles to it.

C3 axisC5 axis

Staggered form of ethane Staggered form of ferrocene

σv planescontain theprincipalaxis

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-prismaticstructure. At left is seen the C4 principal axis. It has four C2 axes atright 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