Introduction to Group Theory.pdf

7/17/2019 Introduction to Group Theory.pdf

http://slidepdf.com/reader/full/introduction-to-group-theorypdf 1/111

Dr. Sumanta Kumar Padhi

Assistant Professor Department of Applied Chemistry

Indian School of Mines Dhanbad, 826 004, INDIA

ACC 32138

Winter Semester 2013 2314



• Symmetry

Relationship between parts of an object with respect to size,shape and position.

e.g In nature:



• Symmetry Element:

A symmetry element is a geometric entity e.g. a point, a line or aplane.

Symmetry element is a point, line or plane about which asymmetry operation is performed

• Symmetry Operator:A symmetry operator performs and action on a three

dimensional object.

A symmetry operation moves an object into an indistinguishableorientation .

Symmetry operators are like other mathematical operators (x, ÷,+, log, cos, sin, etc..)



• There are five types of symmetry operators:

Operator Symbol

Identity E

Rotation CMirror Plane σ

Inversion i

Improper rotation S



• There are five symmetry elements, which will be defined relative

to point with coordinate (x1,y1,z1):

• Identity, E

E(x1 ,y1 ,z1) = (x1 ,y1 ,z1)This operator does nothing and is required for a completeness. Itis equivalent to multiplying “1” or adding “0” in algebra.



• Proper rotation axis, Cn (where θ= 2π/n)

Convention is a clockwise rotation of the point.The symmetry element is called axis of symmetry and denotedby Cn.C2(z) (x1 ,y1 ,z1) = (-x1 ,-y1 ,z1) ; in this case θ = 180°Many molecules have more than one symmetry axis. The axis

with higher “n” values is called principal axis.

n = (2π/θ)θ→0; n = ∞; Cn = C∞



Ammonia has a C3 axis. Note that there are two

operations associated with the C3 axis. Rotation by 120o

in a clockwise or a counterclockwise direction provide

two different orientations of the molecule.

Water has a 2-fold axis of rotation. When rotated by

180o, the hydrogen atoms trade places, but the

molecule will look exactly the same.





three-fold axis three-fold axis two-fold axis two-fold axis

viewed from viewed from viewed from viewed from

above the side the side above

Note: there are 3 C2 axes

C3 C3 C2 C2

principal axis

(highest value of Cn)

.

Rotational axes of BF3



Rotational axes of BF3











• Plane of Reflection (σ)σ

xz (x1 ,y1 ,z1) = (x1 ,-y1 ,z1)

σxz

σyz

H1

H2



• Plane of Symmetry is of three types

1. Vertical plane (σ

v)2. Horizontal plane (σh)

3. Dihedral plane (σd)

• Horizontal plane (σh)

If a plane is ⊥ to the principal axis then it is called σh

• Vertical plane (σv)If the plane is along the principal axis then it is called vertical plane (σv)

• Dihedral plane (σd)

If the plane bisects the angle subtended between two similar consecutive

C2-axis





Mirror planes can contain the principal axis (σ v) or be atright angles to it (σ h). BF3 has one σ h and three σ v planes:

(v = vertical, h = horizontal)

σ vmirror plane C

3

principal axis

σ hmirror plane

C3

principal axis

σ v mirror plane

contains the C3 axis

σ h mirror plane

is at right angles to theC3 axis

Mirror planes (σ) of BF3:





C6

principal axis

C2

C2

C2C6C2

σ v σ v

σ h

C6

principal axis

C6

principal axis

Rotational axes and mirror planes of benzene





• Inversion (i)

All the points in the molecule are reflected through a single point.The point is the symmetry element for inversion. The position of the (x,y,z)

coordinate changes to thecorresponding –ve coordinate (-x,-y,-z).





center of symmetry center of symmetry



• Improper rotation (Sn)

Rotation by 2π

/n followed by reflectionσ

,⊥

to the rotation axis.Since performing σ two times is the same as doing nothing (E), therefore S

can only be performed odd number of times.



• Improper rotation (Sn)• 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

















Types of matrices:

Rectangular Matrix

Column Matrix

Row matrix

Zero null matrix

Square matrix

Diagonal Matrix

Scalar Matrix

Unit or Identity Matrix











Equal Matrices:







Direct product of two matrices:



Trace or Character of a matrix

Trace or character of matrix is the sum of the diagonal elements.It is represented by “

” .









E or Identity Matrix:

E(x1 ,y1 ,z1) = (x1 ,y1 ,z1)





σ

Matrix: σxz (x1 ,y1 ,z1) = (x1 ,-y1 ,z1)



Plane of symmetry (σ

) Matrix:



σ

Matrix:



Inversion (i) Matrix:



Inversion (i) Matrix:

Therefore, the inversion (i) matrix is:



Cn Matrix:









x2 = r [cosθ

cos + sinθ

sin ]

x2 = r [sin θ cos - cos θ sin ]









Sn Matrix:



Sn Matrix:



E C2 σxy i

The following matrices form a representation of the C2h point group

The following matrices form a representation of the C3v point group→



• The symmetry properties of an object (e.g. atoms of a molecule,set of orbitals, vibrations). The collection of objects is commonly

referred to as a basis set .

Classify objects of the basis set into symmetry operations . Symmetry operations form a group.

Group mathematically defined and manipulated by group theory.















Defini tion of a group































Group Multiplication table for C2h:































• E is always in a class by itself. It can be transformed into itself by

all the elements in the group.

• Inversion element , i, is in class by itself.

• All Cnm axes are in a class.

• Similar C2s are in a class.

• Like Cnm all Sn

m axes are in a class. If there are two or many such

types they are placed in as many classes.

• Similar vertical (σv) and similar dihedral planes (σd) are in a class.

• Horizontal plane (σh) is a special plane and is always placed in a

different class from other planes.



• In all Abelian point groups each element is in a class by itself i.e.,

the number of symmetry elements or order of the group is equal

to the number of classes.

Number of Classes = Order of the group (h)

• In non Abelian groups the number of classes is always less than

the order of the group.

• No element in the group occurs in more than one class.



• In symmetry, a point group is the collection of symmetry operations that leaves amolecule unchanged.

• Groups with very high symmetry:

1. Icosahedral, Ih

2. Octahedral, Oh

3. Tetrahedral, Td

• Groups with low symmetry:

1. C1 – molecules with only the E element

2. Cs – molecules with E and a single plane of symmetry (σ).

3. Ci – molecules with only E and a center of inversion, i.• Groups with an n-fold axis of rotation:

1. Cn – identity (E) and n-fold rotation (Cn)

2. Cnv – identity (E), n-fold rotation (Cn) and n vertical reflections (σv).

3. Cnh – identity (E), n-fold rotation (Cn) and horizontal reflection plane (σh).• Dihedral groups:

1. Dn – identity (E), n-fold rotation (Cn) and n two-fold rotations (C2) perpendicular to Cn axis (principal axis). (with no mirror planes)

2. Dnh - identity (E), n-fold rotation (Cn), n two-fold rotations (C2) perpendicular to Cn

axis and horizontal reflection plane (σh). (with a horizontal mirror plane)













B2Br 4 has the following staggered structure:

Ga2H6 has the following structure in the gas phase:



SF5Cl: C4v H2O2: C2

C3H4: D2d

C60: Ih CH2ClF:

CsOr C1h

S8: D4d [Co(Ox)3]

3- : D3

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Introduction to Group Theory.pdf