Symmetry and Introduction

to Group Theory

Disclaimer: Some lecture note slides are adopted from CHEM 59-

250 - Originally by Dr. Samuel Johnson

Power point slides from Inorganic Chemistry 4th edition by Gary

L. Miessler and Donald A. Tarr

Symmetry and Point Groups

I. Introduction

A. Symmetry is present in nature and in human culture

B. Using Symmetry in Chemistry

1. Understand what orbitals are used in bonding

2. Predict IR spectra or Interpret UV-Vis spectra

3. Predict optical activity of a molecule

II. Symmetry Elements and Operations

A. Definitions

1. Symmetry Element = geometrical entity such as a line, a plane, or a point,

with respect to which one or more symmetry operations can be carried out

2. Symmetry Operation = a movement of a body such that the appearance

after the operation is indistinguishable from the original appearance (if you

can tell the difference, it wasn’t a symmetry operation)

B. The Symmetry Operations

1. E (Identity Operation) = no change in the object

a. Needed for mathematical completeness

b. Every molecule has at least this symmetry operation

2. Cn (Rotation Operation) = rotation of the object 360/n degrees about an axis

a. The symmetry element is a line

b. Counterclockwise rotation is taken as positive

c. Principle axis = axis with the largest possible n value

d. Examples:

C23 = two C3’s

C33 = E

C17 axis

3. s (Reflection Operation) = exchange of points through a plane to an

opposite and equidistant point

a. Symmetry element is a plane

b. Human Body has an approximate s operation

c. Linear objects have infinite s‘s

d. s h = plane perpendicular to principle axis

e. s v = plane includes the principle axis

f. s d = plane includes the principle axis, but not the outer atoms

O C O OH H

sh sv

sd

4. i (Inversion Operation) = each point moves through a common central

point to a position opposite and equidistant

a. Symmetry element is a point

b. Sometimes difficult to see, sometimes not present when you think

you see it

c. Ethane has i, methane does not

d. Tetrahedra, triangles, pentagons do not have i

e. Squares, parallelograms, rectangular solids, octahedra do

5. Sn (Improper Rotation Operation) = rotation about 360/n axis followed by

reflection through a plane perpendicular to axis of rotation

a. Methane has 3 S4 operations (90 degree rotation, then reflection)

b. 2 Sn operations = Cn/2 (S24 = C2)

c. nSn = E, S2 = i, S1 = s

d. Snowflake has S2, S3, S6 axes

C. Examples:

1. H2O: E, C2, 2s

2. p-dichlorobenzene: E, 3s, 3C2, i

3. Ethane (staggered): E, 3s, C3, 3C2, i, S6

4. Try Ex. 4-1, 4-2

OH H

sv

sdC2

Cl Cl

C C

H

H

H

H

H

H

III. Point Groups

A. Definitions:

1. Point Group = the set of symmetry operations for a molecule

2. Group Theory = mathematical treatment of the properties of the group

which can be used to find properties of the molecule

B. 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

a. If sh perpendicular to Cn then Dnh or Cnh

If not, go to the next step

b. If s contains Cn then Cnv or Dnd

If not, Dn or Cn or S2n

c. If S2n along Cn then S2n

If not Cn

C. Examples: Assign point groups of molecules in Fig 4.8

Rotation axes of “normal” symmetry molecules

Perpendicular C2 axes

Horizontal Mirror Planes

Vertical or Dihedral Mirror Planes and S2n Axes

Examples: XeF4, SF4, IOF3, Table 4-4, Exercise 4-3

D. Properties of Point Groups

1. Symmetry operation of NH3

a. Ammonia has E, 2C3

(C3 and C23) and 3sv

b. Point group = C3v

2. Properties of C3v (any group)

a. Must contain E

b. Each operation must

have an inverse; doing both

gives E (right to left)

c. Any product equals

another group member

d. Associative property

No symmetry – CHFClBr

Some symmetry – CHFCl2

More symmetry – CH2Cl2

More symmetry ? – CHCl3

We need to be able to specify the symmetry of molecules clearly.

F

B rCl

H

F

Cl

Cl

H

Cl

H H

Cl Cl

Cl

H

Cl

What about ?

Point groups provide us with a way to

indicate the symmetry unambiguously.

Symmetry and Point Groups

Symmetry elements are geometric entities about which a symmetry

operation can be performed. In a point group, all symmetry elements must

pass through the center of mass (the point). A symmetry operation is the

action that produces an object identical to the initial object.

Point groups have symmetry about a single point at the center of mass of

the system.

The symmetry elements and related operations that we will find in

molecules are:

Element Operation

Rotation axis, Cn n-fold rotation

Improper rotation axis, Sn n-fold improper rotation

Plane of symmetry, s Reflection

Center of symmetry, i Inversion

Identity, E

The Identity operation does nothing to the object – it is necessary for

mathematical completeness, as we will see later.

n-fold rotation - a rotation of 360°/n about the Cn axis (n = 1 to )

H(2)

O(1)

H(3) H(3)

O(1)

H(2)

In water there is a C2 axis so we can perform a 2-fold (180°) rotation to get

the identical arrangement of atoms.

H(2)

H(3)H(4)

N(1)

H(2)

H(3)

H(4)

N(1)

In ammonia there is a C3 axis so we can perform 3-fold

(120°) rotations to get identical arrangement of atoms.

H(2)H(3)

H(4)

N(1)

120° 120°

180°

H(2)

H(3)H(4)

N(1)

H(2)

H(3)

H(4)

N(1)

Notes about rotation operations:

- Rotations are considered positive in the counter-clockwise direction.

- Each possible rotation operation is assigned using a superscript integer m

of the form Cnm.

- The rotation Cnn is equivalent to the identity operation (nothing is moved).

H(2)H(3)

H(4)

N(1)

C31 C3

2

C33 = E

H(2)

H(3)H(4)

N(1)

Cl (4)

Cl (3)

Ni (1)

Cl (2)

Cl (5)

C42 = C2

1 C41

C43

Cl (4)

Cl (2) Ni (1) Cl (3)

Cl (5)

Cl (4)

Cl (2)

Ni (1)

Cl (3)

Cl (5)

Cl (4)

Cl (3) Ni (1) Cl (2)

Cl (5)

Notes about rotation operations, Cnm:

- If n/m is an integer, then that rotation operation is equivalent to an n/m -

fold rotation.

e.g. C42 = C2

1, C62 = C3

1, C63 = C2

1, etc. (identical to simplifying fractions)

Notes about rotation operations, Cnm:

- Linear molecules have an infinite number of rotation axes C because any

rotation on the molecular axis will give the same arrangement.

O(2)C(1)

C(1)O(2)

O(3) C(1) O(2)

N(2)N(1)N(1)N(2)

The Principal axis in an object is the highest order rotation axis. It is

usually easy to identify the principle axis and this is typically assigned to

the z-axis if we are using Cartesian coordinates.

Ethane, C2H6 Benzene, C6H6

The principal axis is the three-fold axis

containing the C-C bond. The principal axis is the six-fold axis

through the center of the ring.

The principal axis in a tetrahedron is a

three-fold axis going through one vertex

and the center of the object.

Reflection across a plane of symmetry, s (mirror plane)

H(2)

O(1)

H(3) H(3)

O(1)

H(2)

sv

H(2)

O(1)

H(3)

sv

H(2)

O(1)

H(3)

These mirror planes are

called “vertical” mirror

planes, sv, because they

contain the principal axis.

The reflection illustrated in

the top diagram is through a

mirror plane perpendicular

to the plane of the water

molecule. The plane shown

on the bottom is in the

same plane as the water

molecule.

Handedness is changed by reflection!

A “horizontal” mirror plane, sh, is

perpendicular to the principal axis.

This must be the xy-plane if the z-

axis is the principal axis.

In benzene, the sh is in the plane

of the molecule – it “reflects” each

atom onto itself.

Notes about reflection operations:

- A reflection operation exchanges one half of the object with the reflection of

the other half.

- Reflection planes may be vertical, horizontal or dihedral (more on sd later).

- Two successive reflections are equivalent to the identity operation (nothing is

moved).

sh

sv sv

Vertical and dihedral mirror

planes of geometric shapes.

sd sd

sh

Inversion and centers of symmetry, i (inversion centers)

In this operation, every part of the object is reflected through the inversion

center, which must be at the center of mass of the object.

i

Br

Cl

F

F

Cl

Br

Br

Cl

F

F

Cl

Br

1

1

1 1

1

1 2

2

2 2

2

2

2 2 1 1

[x, y, z]

i [-x, -y, -z]

We will not consider the matrix approach to each of the symmetry operations in this

course but it is particularly helpful for understanding what the inversion operation

does.

n-fold improper rotation, Snm (associated with an improper rotation axis or

a rotation-reflection axis) This operation involves a rotation of 360°/n

followed by a reflection perpendicular to the axis. It is a single operation and

is labeled in the same manner as “proper” rotations. F1

F2

F3

F4 S41

90°

F4

F1

F2

F3

sh F1

F2

F3

F4

Note that: S1 = s, S2 = i, and sometimes S2n = Cn (e.g. in box) this makes

more sense if you examine the final result of each of the operations.

C

H1

H3

H2H4

C

H2

H4

H3H1

C

H1

H3

H2 H4

S41

S42

C21

We can use a flow chart such as this

one to determine the point group of

any object. The steps in this process

are:

1. Determine the symmetry is special

(e.g. octahedral).

2. Determine if there is a principal

rotation axis.

3. Determine if there are rotation axes

perpendicular to the principal axis.

4. Determine if there are mirror planes.

5. Assign point group.

Identifying point groups

Identifying point groups

Identifying point groups

Perfect icosahedral (Ih) e.g. [B12H12]-2, C60

Special cases:

Perfect tetrahedral (Td) e.g. P4, CH4

Perfect octahedral (Oh) e.g. SF6, [B6H6]-2

Identifying point groups Low symmetry groups:

F1

F2

F3

F4

Only* an improper axis (Sn)

e.g. 1,3,5,7-tetrafluoroCOT, S4

Only a mirror plane (Cs)

e.g. CHFCl2 F

Cl

Cl

H

Identifying point groups Low symmetry groups:

Only an inversion center (Ci)

e.g. (conformation is important !)

No symmetry (C1)

e.g. CHFClBr

Br

Cl

F

F

Cl

Br

F

B rCl

H

Identifying point groups Cn type groups:

A Cn axis and a sh (Cnh)

e.g. B(OH)3 (C3h, conformation is important !)

O

H

H

B

O

O

H

e.g. H2O2 (C2h, conformation is important !)

H

O O

H

Note: molecule does not have to be planar

e.g. B(NH2)3 (C3h, conformation is important !)

Identifying point groups

Cn type groups:

Only a Cn axis (Cn)

e.g. B(NH2)3 (C3, conformation is important !)

e.g. H2O2 (C2, conformation is important !)

H

H

HN

B

N

N

H

H

H

H

H

O O

Identifying point groups

Cn type groups:

A Cn axis and a sv (Cnv)

e.g. NH3 (C3v)

e.g. H2O2 (C2v, conformation is important !)

H

HH

N

HH

OO

Identifying point groups

Cn type groups:

A Cn axis and a sv (Cnv)

e.g. NH3 (C3v, conformation is important !)

e.g. trans-[SbF4ClBr]- (C4v)

H

H

H

H

H

H

H

H

NH

F F

Br

Sb

Cl

FFBrF

F

Sb

F

FCl

OC

e.g. carbon monoxide, CO (Cv)

There are an infinite number of possible

Cn axes and sv mirror planes.

CO

Identifying point groups

Dn type groups:

A Cn axis, n perpendicular C2 axes

and a sh (Dnh)

e.g. BH3 (D3h)

e.g. NiCl4 (D4h)

H

B

H

H

HH BH

Cl (4)

Cl (3)

Ni (1)

Cl (2)

Cl (5)

ClCl Ni ClCl

Cl ClNiCl Cl

Identifying point groups

Dn type groups:

A Cn axis, n perpendicular C2 axes

and a sh (Dnh)

e.g. Mg(5-Cp)2 (D5h in the eclipsed conformation)

Mg

Mg

View down the C5 axis

e.g. carbon dioxide, CO2 or N2 (Dh)

There are an infinite number of possible Cn

axes and sv mirror planes in addition to the

sh.

COO O

e.g. square prism (D4h)

e.g. pentagonal prism (D5h)

Identifying point groups Dn type groups:

A Cn axis, n perpendicular C2 axes

and no mirror planes (Dn)

-propellor shapes

e.g. Ni(CH2)4 (D4)

H

H

H

H

Ni

H

H

H

H

H

HH

H

Ni

H

H

H

H

H

H

H

H

Ni

H

H

H

H

e.g. (SCH2CH2)3 (D3 conformation is important!)

e.g. Ni(en)3 (D3 conformation is important!) en = H2NCH2CH2NH2

e.g. propellor (D3)

Identifying point groups

Dn type groups:

A Cn axis, n perpendicular C2 axes

and a sd (Dnd)

e.g. ethane, H3C-CH3

(D3d in the staggered conformation)

H H

H

H

H H

H

H

H

H

H

H

dihedral means between sides or planes –

this is where you find the C2 axes

Mg AlFe M

e.g. Mg(5-Cp)2 and other metallocenes in the staggered conformation (D5d)

View down the C5 axis

These are pentagonal

antiprisms

e.g. triagular

antiprism (D3d)

e.g. square

antiprism (D4d) e.g. allene or a tennis ball (D2d)

1. Determine the symmetry is special

(e.g. octahedral).

2. Determine if there is a principal

rotation axis.

3. Determine if there are rotation axes

perpendicular to the principal axis.

4. Determine if there are mirror planes.

5. Assign point group.

Summary of point group identification