1. Lecture # 8 Molecular Symmetry
CHEM 515 Spectroscopy
Lecture # 8
Molecular Symmetry

2. 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.

3. 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”

4. Symmetry Operations and Symmetry Elements
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”

5. 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

6. 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.

7. Operator Algebra Example: For x = x and D = d/dx , does Dx = xD ?
Associative law and distributive laws both hold for operators “see book”.

8. 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.

9. 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. C2
Rotation by 2π/2 radians 1 2 2 1

10. 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

11. Rotation Operator (Cn)
Successive Rotations (Cnk).
Cnk = Cn Cn … Cn (k times)
Also:
Cnn = E
Example: BF3 1 2 3 1 2 3
Rotation by 4π/3 radians C32
Rotation by 2π/3 radians C3 3 2 1
Rotation by - 2π/3 radians C3-1

12. 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. C2 σv
Reflection through σv plane 1 2 2 1

13. 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.

14. 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.

15. 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 σh C4 S4

16. 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)

17. Symmetry Operator Algebra
Symmetry operators can be applied successively to a molecule to produce new operators.
σv’’’ = σv’’ C3
σv’ = C3 σv’’

18. 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.

19. 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 .

20. 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

21. Point Group for Ammonia
The ammonia molecule has six symmetry operators.
E , C3, C3-1 (or C32), σv’ , σv’’ and σv’’’

22. Multiplication Table for NH3

23. 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.

24. 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.

25. Determination of Molecular Symmetry

26. Point Groups and Symmetry for Various Molecules
C1 Symmetry (only E)
Cs Symmetry

27. Point Groups and Symmetry for Various Molecules

28. Point Groups and Symmetry for Various Molecules

29. Point Groups and Symmetry for Various Molecules

30. Special Point Groups

31. Special Point Groups