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Molecular symmetry

and group theory

Natural symmetry in plants

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Symmetry

in animals

Symmetry in the human body

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Symmetry in modern art

M. C. Escher

Symmetry in Arabic architecture

La Alhambra, Granada (Spain)

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Symmetry in baroque art

Gianlorenzo Bernini

Saint Peter’s Church

Rome

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Platonic solids (5 of the 8 shown)

Archimedean solids (3 of the 8 shown)

cuboctahedron, icosidodecahedron, truncated octahedron

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Molecular Symmetry

• Group theory is an important aspect for

spectroscopy. It is used to explain in details

the symmetry of molecules.

• Group theory is used to:

– label and classify molecule’s energy levels /

molecular orbitals (electronic, vibrational and

rotational)

– look up the possibility of molecular and electronic

transitions between energy levels / molecular

orbitals.

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Point Groups

Molecules are classified and grouped

based on their symmetry. Molecules with

similar symmetry are but into the same

point group. A point group contains all

objects that have the same symmetry

elements.

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Symmetry Operations

• A symmetry operation is geometrical action

that leaves the nuclei in a molecule in

equivalent positions. (leaves them

indistinguishable).

• Five main classes of symmetry operations:

– Reflections (σ).

– Rotation (Cn).

– Rotation-reflection “Improper rotation” (Sn).

– Inversion (i).

– Identity (E). “do nothing”

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Symmetry Operations and Symmetry

Elements

Symmetry Operation Symmetry Element

Reflection (σ) Plane of reflection (σh, σv, σd)

Rotation (Cn) Axis of rotation (principal and

non-principle)

Improper rotation (Sn) Rotation followed by reflection

Inversion (i) Center of inversion

Identity (E) E itself “does nothing”

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Operator Algebra

• Operator algebra is similar in many aspects

to ordinary algebra.

• For: Af1 f2 ,

operator A is said to transform functions f1 to

f2 by a sort of operation.

• Addition of operators:

Cf = (A + B)f = Af + Bf

or

C = (A + B) = A + B

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Operator Algebra

• Multiplication of operators:

Cf = (AB)f = A(Bf)

or

C = (AB) = AB

However, it is important to note that:

A(Bf) is not necessarily equivalent to B(Af).

We say operators A and B do not necessarily

commute.

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Identity Operator (E)

• The identity operator leaves a molecule

unchanged. It is applied for all molecule with

any degree of symmetry or asymmetry.

• The identity operator does nothing. Why are

we still dealing with it?

• It is important not by itself but for specific

operator algebra as going to be discussed

later.

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Identity Operator (E)

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Rotation Operator (Cn)

• Cn rotates a molecule by an angle of 2π/n

radians in a clockwise direction about a Cn axis.

• If a rotation of 2π/n leaves out the molecule

indistinguishable, the molecule is said to have

an n-fold axis of rotation.

1 2

C2

Rotation by

2π/2 radians

2 1

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Rotation Operator (Cn)

• When a molecule has several rotational axes

of symmetry, the one with the largest value

of n is called the principle axis.

Example: Trifluoroborane

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Rotation Operator (Cn)

• Successive Rotations (Cnk).

Cnk = Cn Cn … Cn (k times)

Also:

Cnn = E Cn

n+1 = Cn

• Example: BF3

Rotation by

2π/3 radians

C3

Rotation by

4π/3 radians

C32

Rotation by - 2π/3 radians C3-1

1 2

3

3 1

2

2 3

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Rotation angle Symmetry operation

60º C6

90º C4

120º C3 (= C62)

180º C2 (= C63 = C4

2)

240º C32 (= C6

4)

270º C43

300º C65

360º E

Rotation Operator (Cn)

Cn is shorthand for (n1) rotation operators.

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Reflection Operator (σ)

• σ reflects a molecule through a plane passing

through the center of the molecule. The

molecule is said to have a plane of symmetry.

1 2

C2

Reflection

through σv

plane

2 1

σv

Some simple algebra: σ2 = E

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Reflection Operator (σ)

• There are three types of mirror

planes:

– σv vertical mirror plane which

contains the principle axis.

– σh horizontal mirror plane

which is perpendicular to the

principle axis.

– σd dihedral mirror plane which

is vertical and bisects the

angle between two adjacent

C2 axes that are perpendicular

to the principle axis.

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s

s

Reflection Operator (σ)

H2O

s

Reflection Operator (σ)

NH3

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Inversion Operator (i)

• This operator inverts all atoms through a

point called “center of inversion” or “center

of symmetry”.

i (x,y,z) (-x,-y,-z)

Inversion not the same as C2 rotation !!

Inversion Operator (i)

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Inversion Operator (i)

Figures with center of inversion

Figures without center of inversion

Inversion Operator (i)

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Improper Rotation Operator (Sn)

• This operator applies a clockwise rotation on

the molecule followed by a reflection in a

plane perpendicular to that axis of rotation.

Sn = σhCn

• Example: Methane

C4σh

S4

The staggered

conformation of

ethane has an S6

axis that goes

through both carbon

atoms.

Improper Rotation Operator (Sn)

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Note that an S1

axis does not exist;

it is same as a

mirror plane.

Improper Rotation Operator (Sn)

Likewise, an S2

axis is a center of

inversion.

Improper Rotation Operator (Sn)

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An S3 is equivalent

to simultaneous C3

and s.

S32 = C3

2

S33 = s

S34 = C3

Improper Rotation Operator (Sn)

S42 = C2 S4

4 = E S2 = i S1 = s

Improper Rotation Operator (Sn)

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Symmetry Operator Algebra

• Symmetry operators can be applied

successively to a molecule to produce new

operators.

σv’’’ = σv’’ C3 σv’ = C3 σv’’

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A group multiplication must satisfy the following

conditions in regard with the group’s elements:

1 Closure: If P and Q are elements of a group and

PQ = R , then R must be also an element of that

group.

2 Associative Law: The order of multiplication is

not important. (PQ)R = P(QR).

3 Identity Element: There must be an identity

element (E) in the group so that: RE = ER = R.

4 Inverse: Every element has an inverse in the

group so that: RR-1 = R-1R = E

More Operator Algebra

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If the group elements commute, i.e. PQ = QP,

then the group is said to be “Abelian group”.

For point symmetry groups, we have non-

Abelian groups.

Point groups retain the center of mass of the

molecules unchanged under all symmetry

operations and all of the symmetry elements

meet at this point

More Operator Algebra

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Point Group for Ammonia

• The ammonia molecule has six symmetry

operators.

E , C3, C3-1 (or C3

2), σv’ , σv’’ and σv’’’

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Multiplication Table for NH3

Each operator appears just once in a given row

or column in the table but in a different position.

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Classes

• The members of a group can be divided into

classes.

The members of a class within a group have

a certain type of a geometrical relationship.

For ammonia with the C3v symmetry, the

three classes are:

E , C3 and σv

• The point group C3v will contain E , 2C3 and

3σv elements.