Symmetry and Group Theory - NPTELnptel.ac.in/courses/104106063/Module 1/Lectures 1-3/Lectures...

NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transition elements)

Symmetry and Group Theory

K. Sridharan

Dean

School of Chemical & Biotechnology

SASTRA University

Thanjavur 613 401

Page 1 of 15 Joint Initiative of IITs and IISc Funded by MHRD


Table of Content

1SYMMETRYANDGROUPTHEORY.......................................................................................................3

1.1SYMMETRYELEMENTSANDSYMMETRYOPERATIONS........................................................................................31.1.1Whatissymmetry?.......................................................................................................................31.1.2Symmetryelements......................................................................................................................41.1.3Symmetryoperations....................................................................................................................4

2.ORDEROFAXISANDPLANEOFSYMMETRY.......................................................................................9

2.1ORDEROFAXIS.........................................................................................................................................9

2.2.1Verticalmirrorplane(v)............................................................................................................102.2.2Horizontalmirrorplane(h)........................................................................................................11

3.CENTREOFSYMMETRY,IDENTITYELEMENT,ANDIMPROPERROTATIONAXIS.................................13

3.1.CENTEROFSYMMETRY(I)........................................................................................................................133.2IDENTITYELEMENT(E).............................................................................................................................133.3IMPROPERROTATIONALAXISOFSYMMETRYORROTATIONREFLECTIONAXISOFSYMMETRY(SN)..............................13

4.REFERENCES....................................................................................................................................15


2.1.1Principalaxisofsymmetry............................................................................................................92.2PLANEOFSYMMETRY()..........................................................................................................................10


1 Symmetry and group theory It is very important to understand the symmetry and point group of orbitals and

molecules so that their behaviors under different circumstances are clearly

understood. The point groups are based on the shapes of orbitals and structures

of molecules. For example, s orbital is spherical and has a particular symmetry,

while p orbital has dumbbell shape and has different symmetry. Similarly, d

orbitals have different shapes and hence different symmetries. Methane has a

tetrahedral shape and its symmetry is Td, while benzene is hexagonal planar

and its symmetry is D6h. Water is V shaped and its point group is C2v. In order

to understand the splitting of orbitals in different environments and the spectral

characteristics of complexes, their symmetries and point groups must be

understood.

1.1 Symmetry elements and symmetry operations

1.1.1 What is symmetry?

In simple language we can say that an object has symmetry, if it has some

special characteristics, such as pleasing designs, while we look at it. As an

example, when we see the telephone posts or electric lamp posts, we say

that there is symmetry because they are arranged in a straight line at

equal distance. Similarly, when we look at the gates of houses, they will

appear symmetric because of their designs. Naturally, our eyes will compare

the design on one half of the gate with that of the other half and if they find

some characteristic feature such as mirror image or other, then we feel there

is symmetry. A suspension bridge, a butterfly, the rose petal etc. are some

examples to show the pleasing designs and hence, they are symmetric.

Page 3 of 15

Joint Initiative of IITs and IISc Funded by MHRD

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NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transition elements) Note: It can be viewed only on Acrobat 9.0 and above

C3 axis of rotation

Fig 1 1.3.2: C3 axis of rotation

The angle between any two spheres is equal to 1200. Hence, rotation by 1200

gives an indistinguishable structure. When the colors of the spheres are

removed, the two structures cannot be distinguished.

Note :



Note : It can be viewed only on Acrobat 9.0 and above

C2 axis of rotation

Fig 1 1.3.3: C2 axis of rotation in water molecule

When the above V-shaped molecule is rotated by 1800 about the axis passing through the blue sphere, red and green spheres are interchanged. If the colors of the spheres are removed, the two structures are indistinguishable.



Fig 1 1.3.4: C2 axis of rotation in a linear molecule

The angle between the blue and red spheres is 1800.Rotaion about the vertical axis by 1800 gives an indistinguishable structure, once the colors are removed.


SastraTypewritten text Can be viewed only on Acrobat 9.0 and above

NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transition elements) C6 axis of rotation

Fig 1 1.3.5: C6 axis of rotation

600 rotation about the axis perpendicular to the paper gives an indistinguishable structure, once the colors of the spheres are removed.

600 rotation



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15 Page 9 of

Principal axis for wh

NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transition elements) C4 is the principal axis because n=4 is the maximum number

2.2 Plane of symmetry () It is an imaginary plane cutting the molecule or object into two halves

which are mirror images.

2.2.1 Vertical mirror plane (v)

Fig 2.2.1: v plane of symmetry


This is the mirror plane parallel to the principal axis of symmetry.

SastraTypewritten textCan be viewed only on Acrobat 9.0 and above

NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transition elements) 2.2.2 Horizontal mirror plane (h) When the mirror plane is perpendicular to the principal axis, it is called horizontal plane of symmetry.

C2

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Fig 2.2.2: h plane of symmetry

In the first case (plane triangle), the reflection could not be distinguisdhed from the original and the mirror plane is called a horizontal mirror plane, h plane. In the other case, (V-shaped), the relection is inverted and we are able to distinguish this from the original one. Hence, it is not a h plane.




SastraTypewritten textCan be viewed only on Acrobat 9.0 and above


3. Centre of symmetry, Identity Element, and Improper rotation axis

3.1. Center of symmetry (i) If we can move in a straight line from every atom or point in a molecule or object

through a single point at the center to an identical atom or point on the other side

of the center, then the molecule or object is said to possess a center of symmetry

Fig 3.1 : Centre of symmetry

3.2 Identity Element (E) This is nothing but rotating the molecule by 3600. The original molecule is

obtained. The corresponding operation can be called as doing nothing operation.

This is important from mathematical considerations.

3.3 Improper rotational axis of symmetry or Rotation reflection axis of symmetry (Sn). Rotation by a particular angle followed by reflection in a plane perpendicular to

the rotational axis leads to an indistinguishable structure.

Example: S4 axis: rotation by 360/4 = 900

4 axis gives an indistinguishable structure.

Example: SiF4


followed by reflection in a plane

perpendicular to C


Fig 3.2 S4 axis of symmetry




4. References 1. Inorganic Chemistry: Principles

2. Chemical Applications of Group Theory, 2/e, F.Albert Cotton, Wiley

Eastern, New Delhi, 1986


of Structure and Reactivity, James

E.Huheey, Ellen A.Keiter, Richard L.Keiter, Okhil K.Medhi, Pearson

Education, Delhi, 2006

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