CHEM 515Spectroscopy

Lecture # 8

Molecular Symmetry

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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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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

Reflections (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 “do 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 don’t necessarily commute.

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Operator Algebra

• Example:

For x = x and D = d/dx , does Dx = xD ?

• Associative law and distributive laws both hold for operators “see book”.

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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.

• It is important not by itself but for specific operator algebra as going to be discussed later.

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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

• 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

1

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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

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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.

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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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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

C4

σh

S4

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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)

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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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Group Multiplication Tables

• 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.

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Group Multiplication Tables

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 .

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Group Multiplication Tables

5- 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 under all symmetry operations unchanged and all of the symmetry elements meet at this point

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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

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Multiplication Table for NH3

Notice that:• 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.

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Determination of Molecular Symmetry

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Point Groups and Symmetry for Various Molecules

N

C1 Symmetry (only E)

N

Cs Symmetry

HOCl

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Point Groups and Symmetry for Various Molecules

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Point Groups and Symmetry for Various Molecules

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Point Groups and Symmetry for Various Molecules

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Special Point Groups

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Special Point Groups