Lecture 3

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PG510

Symmetry and Molecular Spectroscopy

Lecture no. 3

Group Theory:

Point Groups

Giuseppe Pileio

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Learning Outcomes

By the end of this lecture you will be able to:

!! Understand the concepts of Point Group

!! Classify all the possible point group from symmetry elements

!! Understand the concept of classes of symmetry operations and the tricks to arrange them in classes

!! Find molecular point groups by a simple but systematic procedure

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Symmetry Point Group

•! Rule 1 is satisfied by the meaning of complete •! Rule 2 is satisfied with the operation E as identity •! Rule 3 is satisfied since product is associative •! Rule 4 is satisfied since it is always possible to find an operation with does the opposite (inverse):

What is a Symmetry Point Group?

It is a complete set of symmetry operations where complete means that every product between operations is still a member of the group

Does a complete set of symmetry operations form a group?

!!=E; ii=E; CnmCn

n-m=E; SnmSn

n-m=E (n even, m even or odd)

Snm = Cn

m (n odd, m even); Snm = Cn

m! *Cnn-m! (n,m odd)

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Point Groups

A.! No symmetry operations but E h=1 C1

B.! Symmetry element: plane Symmetry operations: E, !" h=2 Cs (cyclic)

Asymmetry

One symmetry element

C.! Symmetry element: inversion point Symmetry operations: E, i h=2 Ci (cyclic)

one high-order axes

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D.! Symmetry element: proper axis, Cn Symmetry operations: E, Cn, Cn

2, …, Cnn-1"

h=n Cn (cyclic)

E.! Symmetry element: improper axis, Sn (n even) Symmetry operations: E, Sn, Cn/2, Sn

3, …, Snn-1

h=n Sn

F.! Symmetry element: improper axis, Sn (n odd) Symmetry operations: E, Sn, …, Sn

n-1, !h, Cn, …, Cnn-1

h=2n Cnh

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Two or more symmetry elements

G.! Symmetry elements: Cn + perpendicular C2 Symmetry operations: E, Cn, …, Cn

n-1, n C2 h=2n Dn

H.! Symmetry elements: Cn + !h Symmetry operations: E, Cn, …, Cn

n-1, !h, Sn, …, Snn-1

h=2n Cnh

I.! Symmetry elements: Cn + !v Symmetry operations: E, Cn, …, Cn

n-1, n !v (n even) E, Cn, …, Cn

n-1, n/2 !v, n/2 !d (n odd) h=2n Cnv

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J.! Symmetry elements: Cn + perpendicular C2 + !h Symmetry operations: E, Cn, …,Cn

n-1, nC2, !h, n!v, Sn, …, Sn

n-1

h=4n Dnh

K.! Symmetry elements: Cn + perpendicular C2 + !d Symmetry operations: E, Cn, …,Cn

n-1, nC2, n!d, S2n, …, S2n2n-1

h=4n Dnd

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Infinite order axes – Linear Molecules

Two equivalent halves A-A or A-X-A

Two different halves A-B or A-X-B

M. elements: C!+ !v operations: E, C!,…,C!

!-1, !!v

h=! C!v

L. elements: C!+ !h + C2(perp) operations: E, C!, …,C!

!-1, !C2, !h, !!v, Sn, …, Sn

!-1

h=! D!h

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More than one high-order axes

Platonic Solids or Regular Polyhedra

Td

Th

T

Oh

O

Ih

I

N. elements: 3S4, 4 C3, 6!d"

operations: E, 8C3, 3C2, 6S4, 6!d

h=24 Td

O. elements: 3S4, 4 C3"

operations: E, 8C3, 3C2

h=12 T

P. elements: 3S4, 4 C3, 3!h"

operations: E, 8C3, 3C2, 9S4, 3!h

h=24 Th 10

Q. elements: 3S4, 3 C4, 4 S6 …"

operations: E, 8C3, 6C4, 9C2, 6S4, 8S6, i, 6!d, 3!h

h=48 Oh

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R. elements: 3S4, 3 C4, 4 S6 …"

operations: E, 8C3, 6C4, 9C2 " h=24 O

S. elements: 6S10, 10 S6, 6 C5 …"

operations: E, 24C5, 20C3, 15C2, 24S10, 20S6, i, 15!

h=120 Ih

T. elements: 6S10, 10 S6, 6 C5 …"

operations: E, 24C5, 20C3, 15C2 " h=60 I

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Classes of symmetry operations

Group theory: a set of group element (symmetry operations in the point group case) that are conjugate to one another form a class

Operational:

A set of equivalent operations can be arranged in a class

A and B are called equivalent operations if they can be inter-converted by a third operation, C.

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Symmetry operations and classes:

i)! Inversions: only one is possible in a molecule and it always occurs in a class by itself

ii)! Reflections: !h is always in a class by itself. A set of n dihedral planes in the same class is indicated by n !d. The same for n !v if they are all in the same class otherwise the notation n !v, m !v’ etc is used

iii)! Proper Rotations: if the group is cyclic they are all in separate classes otherwise Cn

m will fall in the same class with Cn

n-m

iv)! Improper Rotations: Snm will fall in the same class

with Snn-m

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Systematic Classification

start Special Groups a)!Linear molecule: C!v, D!h

b)!Multiple high order axes: T,Th,Td,O,Oh,I,Ih

c)!No rotation at all: C1,Cs,Ci d)!Only Sn (n even): S4,S6,S8…

Cn

+ nC2

perp.

!h

N

n!v

N

N Cn Cnv

Cnh

Y

Y

Y

!h

n!d

N

N Dn Dnd

Dnh Y

Y

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What did we learn in this lecture?

•! The concept of point group and its link with group theory

•! How various combinations of symmetry elements (and operations) generate all the possible molecular point groups

•! How to arrange operation in classes

•! A systematic procedure to assign the point group to any molecular system

Pictures from: http://csi.chemie.tu-darmstadt.de/ak/immel/tutorials/symmetry/index7.html