Molecular Symmetry & Group Theory

7/28/2019 Molecular Symmetry & Group Theory

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MolecularSymmetry

&Group Theory

Project work submitted in partial fulfillment of the

requirements for the award of degree of Bachelor of

Science in mathematics of the University of Calicut.


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Group members

Basil jaseem V Midhun C Nithin P Shijith K P Sijeesh A R Vaisakh K V


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Certificate

This is to certify that the project method entitled Molecular

Symmetry is a certified record of the work done by Nithin towards

the partial fulfillment of the requirement to the award of degree in

Bsc. Mathematics during the academic year 2012-2013 under the

University of Calicut, Kerala state.

Place: Calicut Signature of the guide

Date:

HOD


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Declaration

This is to declare that this project report entitled Molecular

Symmetry submitted to Calicut University in partial fulfillment of

the requirement to the award of degree in Bsc. Mathematics is a

record of original project work done by me during my period of study

in Govt. Arts and Science College, Calicut is under supervision of Mr.

K.K. Chandrasekharan sir, department of Mathematics, Govt. Arts

and Science College, Calicut-18

Place: Calicut Signature of Candidate:


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Acknowledgement

Firstly I was deeply indebted to my internal guide Mr. K.K.

Chandrasekharan sir, department of Mathematics, for his sincere

corporation and encouragement through the duration of my project.

I would like to express my gratitude to Mrs. Jayasree miss,department of Chemistry, Mr. M.V. Sathyan sir, Mrs. Vijayakumari

miss, department of Mathematics and also gracious gratitude to all

the faculty of department of Mathematics and department of

Chemistry for their valuable advice and encouragement.

I extend my sincere gratitude to my parents and friends who

helped me to build up confidence. This project is the accumulated

guidance, the direction and the support of several important people.

I take this opportunity to express my gratitude to all whose

contribution in this project can never been forgotten.

Finally I thank the almighty, without whose blessing this project

would not been materialized.

Sincerely,

Nithin.P


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Index

Introduction Groups of symmetric polygons Polyhedral groups Molecular symmetry Symmetry elements and operations Combination of symmetry operations Inverse operations Introduction of Group theory in symmetric molecules Illustration by H2O and NH3 molecules Conclusion Reference


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Introduction

A group is a collection of mathematical objects known as

elements or members which are related to each other according to

certain rules which are called closure rule, identity rule, associative

rule and inverse rule. The elements of a group can be numbers,

matrices, vectors or symmetry operations.

We find symmetry all around us and most people at their

conscious and subconscious levels adore symmetry. Most of the

objects in nature possess varying degrees of symmetry. We find

symmetry in the shapes, patterns, and structures of all living things

and also in the various forms of material like crystals that nature

creates. Symmetry is an important aspect at the molecular level.

When we consider the geometries of molecules in their equilibrium

configurations, it can be seen that the symmetry is the major feature

associated with most molecules. Molecular symmetry, on account of

the relationship that it has with the properties of molecules, is very

important in all fields of science.

We say that some molecules are more symmetrical than others

or that some molecules have high symmetry whereas others have

low symmetry or no symmetry. But in order to make the idea of

molecular symmetry as useful as possible, we must develop some

rigid mathematical criteria of symmetry. To do this we shall first

consider the kinds of symmetry elements that a molecule may have


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and the symmetry operations generated by the symmetry elements.

We shall then show that a complete but non redundant set of

symmetry operations (not elements) constitute a mathematical

group through two examples namely H2O and NH3. Finally we shall

use the general properties of groups, aid in correctly and

symmetrically determining the symmetry operations of any molecule

we may care to consider. We shall also describe here the system of

notation normally used by chemists for the various symmetry

groups.


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Groups of symmetric polygons

TriangleConsider A={1,2,3}. Then S3 is the set of all one to one onto

mapping on A. We list the permutations of A and assign to each a

subscripted Greek letter for a name. The elements of S3 are

0= 1= 2= 1= 2= 3= Then the composition table is shown below

0 1 2 1 2 3

0 0 1 2 1 2 31 1 2 0 3 1 2

2 2 0 1 2 3 1

1 1 2 3 0 1 2

2 2 3 1 2 0 1

3 3 1 2 1 2 0

Note that S3

has minimum order for any non abilian group (it having

6 elements).

There is a natural correspondence between the elements of S3

and the ways in which two copies of equilateral triangle with vertices

(1,2,3) can be replaced one covering the other with vertices on top of

vertices. Let r0, r1, r2 are the rotations through 3600,240

0,120

0

respectively in anti clockwise direction. , , are the reflection


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about the median through upper vertex, lower right vertex, and

lower left vertex respectively.

Eg: Let the composition say is the operation follows by .

jj

The composition table for the 6 symmetries is as below.

r0 r1 r2

r0 r0 r1 r2

r1 r1 r2 r0

r2 r2 r0 r1

r0 r1 r2

r2 r0 r1

r1 r2 r0

This group is called third dihedral group denoted by D3.

SquareLet us form the dihedral group D

4of permutations

corresponding to the ways that two copies of a square with vertices

1, 2, 3 and 4 can be placed, one covering the other. D4 will then be

the groups of symmetries of the square. It is also called optic group.

Imagine a square having in its sides parallel to the axes of its co-

ordinates and its centre at the origin.

1 2

1 2


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Allow the following clockwise rotations 900, 180

0, 270

0, 360

0, say r90,

r180, r270, r360 respectively and reflections h, v about horizontal and

vertical axes and reflections d1 and d2 about diagonals.

The multiplication * on these rotations and reflections can be define

by performing two such motions in succession. Eg. r90 * h is

determined by first performing by h and then rotation r90.

The complete multiplication table for the operation is as follows.

r360 r90 r180 r270 h v d1 d2r360 r360 r90 r180 r270 h v d1 d2

v

2

3

h

d2 d1

4


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r90 r90 r180 r270 r360 d1 d2 v h

r180 r180 r270 r360 r90 v h d2 d1

r270 r270 r360 r90 r180 d2 d1 h v

h h v d1 r360 r180 r270 r90v v d1 h d2 r180 r360 r90 r270

d1 d1 h d2 v r90 r270 r360 r180

d2 d2 v d1 h r270 r90 r180 r360

Eg: v* r270 = d2

r270 v

In general Dn is the nth dihedral group of symmetries of regular n-gon

having 2n elements.

Eg: Consider one permutation in S6.

=

A cube that exactly fills a certain cubical box. As in example

the ways in which the cube can be placed into the box correspond to

a certain group of permutations of the vertices of the cube. This

group is the group of rigid motions of the cube.

1 2

34

2 3

1 4

3 2

4 1


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Polyhedral groups

Every group is a permutation group is provided by the regular

polyhedra, whose symmetry groups turned out to be important

subgroup of S4 and S5. The regular polyhedra also show us the more

literal, geometric, meaning of Symmetry. If we imagine a

polyhedron P occupying a region R in a space, the symmetries of P

can be viewed as the different ways of fitting P into R. Eachsymmetry is obtained by rotation from the initial position, and

product of symmetries is the product of rotations. The concept of

polyhedral groups can be apply in molecular symmetry and

symmetry based groups.


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Molecular symmetry

Symmetry

An object is said to posses symmetry, if it can take up two or

more spatial orientations that are indistinguishable from each other.

i.e.; it can take up two or more equivalent orientations.

Symmetry operations

Symmetry operation is a movement of a body such that, after

the movement has been carried out, every point of the body is

coinciding with an equivalent point of the body in its original

orientation. In other words, a symmetry operation is an action which,

when performed on a molecule yields a new orientation of it that is

indistinguishable from the original, though not necessarily identical

with it. This would mean that, if we were to look at the body, turn

away long enough for someone to carry out a symmetry operation,

and then look again, we would be completely unable to tell whether

or not the operation had actually been performed, because in either

case the position and orientation would be indistinguishable from

the original. Every symmetry operation is considered to be

associated with a symmetry element with respect to which that

operation is carried out.


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Symmetry element

Symmetry element is a geometrical entity such as a line, a

plane, or a point, with respect to which one or more symmetry

operation may be carried out.

Illustration of a symmetry element and symmetry

operation using H2O molecule

Consider the anticlockwise rotation of the water molecule (in

the y-z plane) through 1800

about an axis (z axis) passing through the

oxygen atom and bisecting the H-O-H angle. The new orientation,although not identical with the original one, is equivalent to it and

super imposable on it. The rotation about the axis constitutes a

symmetry operation and the axis constitutes a symmetry element,

commonly known as a proper rotation axis.

O Anticlockwise rotation through 1800 O

Ha Hb Ha Hb


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Symmetry elements and Operations

SymmetryElement

Symmetryoperations

Transformation matrix

1. Identityelement

2.Properaxis

3.Plane

4.Centre ofsymmetry

or centre

of

inversion

5. Improperaxis

Do nothing

One or more

rotations about

the axis(eg;

around z axis)

Reflection in

the plane(eg; in

xy plane)

Inversion of all

atoms through

the centre

Reflection in

mirror planefollowed by

( )=

( )

=

(

)

( )=

( )=(

)


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rotation

normal to

mirror plane

(around z axis)

( )

=()

1.The identity operation- Identity element (E)The identity operation is one in which the molecule remains

in its original configuration. It is thus effectively a do nothing

or leave the system alone operation whereby any part of the

molecule remains in its original position. In other words, the

identity operation is one that leaves the system unchanged.

The symmetry element associated with the identity operation

is called the identity element and is given the symbol E.

Obviously; all molecules possess the identity element.

2.The proper rotation operation- Proper rotationaxis(Cn)

Before discussing proper axes and rotations in a general

way, let us take a specific case. A line drawn perpendicular to

the plane of an equilateral triangle and intersecting it at its

geometric centre is a proper axis of rotation for that triangle.

Upon rotating the triangle by 1200(2/3) about this axis, the

triangle is brought into an equivalent configuration. It may be

noted that a rotation by 240

0

(2*2/3) also produces anequivalent configuration.


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A proper rotation axis or an axis of symmetry is a line

about which rotation through a certain angle brings an object

into an orientation that is indistinguishable and super

imposable on the original. Here the axis of symmetry is asymmetry element.

The general symbol for a proper axis of rotation is Cn,

where the subscript n denotes the order of the axis. By order is

meant the largest value of n such that rotation through 2/n

gives an equivalent configuration. In the above example, the

axis is a C3 axis. Another way of defining the meaning of the

order n of an axis is to say that it is the number of times that

the smallest rotation capable of giving an equivalent

configuration must be repeated in order to give a configuration

not merely equivalent to the original but also identical to it. The

meaning of identical can be amplified if we attach numbers to

each apex of the triangle in our example. Then the effects of

rotating by 2/3, 2*2/3, 3*2/3 is seen to be:

2/3

2*2/3

3*2/3

1 3 2

1

3

2 1

1 3 1 3

B

C

D=A

A

A

A


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Configurations B and C equivalent to A because without the

labels they are indistinguishable from A, although with the labels

they are distinguishable. However, D is indistinguishable from A not

only without the labels but also with them. Hence, it is not merely

equivalent; it is identical.

The C3 axis is also called a threefold axis. Moreover, we use the

symbol C3, to represent the operation of rotation by 2/3 around

the C3 axis. For the rotation by 2*2/3 we use the symbol C32

, andfor the rotation by 3*2/3 the symbol C3

3. Symbolically we can write

C34

= C3, and hence only C3, C32, and C3

3are separate and distinct

operations. However, C33

produces an identical configuration, and

hence we may write C33

= E.

After consideration of the above example, it is easy to accept

some more general statements about proper axes and proper

rotations. In general, an n-fold axis is denoted by Cn, and a rotationby 2/n is also represented by the symbol Cn. Rotation by 2/n

carried out successively m times is represented by the symbol Cnm

.

Also, in case, Cnn

= E, Cnn+1

= Cn, Cnn+2

= Cn2

and so on.

e.g. , Water molecule has two fold proper rotation axis (C2) in

the plane of the molecule, passing through the O atom and

bisecting the H-O-H angle. A rotation of the molecule through 1800

about this axis gives a configuration indistinguishable from the

original.

If a molecule possesses several types of symmetry axes, the

highest fold proper rotation axis is considered as the principal axis;

the other axes present are referred to as secondary axes.


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3.Symmetry Planes and reflectionsSymmetry must pass through a body, that is, the plane cannot

be completely outside of the body. The conditions which must be

fulfilled in order that a given plane can be stated as follows, Let us

apply a Cartesian coordinate system to the molecule in such a way

that the plane includes two of the axes (sayxand y) and is therefore

perpendicular to the third (i.e. z). The position of every atom in the

molecule may also be specified in this same coordinate system.

Suppose now, for each and every atom, we leave the x and y

coordinates fixed and change the sign of the z coordinate: thus the

ith atom, originally (xi, yi, zi), is moved to the point (xi, yi, -zi).

Another way of expressing the above operation is to say, let us

drop a perpendicular from each atom to the plane, extend that line

an equal distance on the line. If, when such an operation is carried

out on every atom in a molecule, an equivalent configuration is

obtained, the plane used is a symmetry plane.Clearly, atoms lying in the plane constitute special cases, since

the operation of reflecting through the plane does not move them

at all. Consequently, any planar molecule is bound to have at least

one plane of symmetry, namely, its molecular plane. Another

significant and immediate consequence of the definition is a

restriction on the numbers of various kinds of atoms in a molecule

having a plane of symmetry. All atoms of a given species which do

not lie in the plane must occur in even numbers; since each one

must have a given species may be in the plane. Furthermore, if

there is only one atom of a given species in a molecule, it must be in

each and every symmetry plane that the molecule may have. This

means that it must be on the line of intersection between two or

more planes or at the point of intersection of three or more planes.


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Since this atom must lie in all of the symmetry planes

simultaneously.

The standard symbol for a plane of symmetry is . The same

symbol is also for the operation of reflecting through the plane.Now we can conveniently denote the successive application of the

operation n times by writing n. We can then also write,

2=E,

where we use the symbol E to represent any combination of

operations which takes the molecule to a configuration identical

with the original one. We call E, or any combinations of operations

equal to E, the identity operation. It should be obvious that n= E

when n is even and n= when n is odd.

Examples; A tetrahedral molecule of the type AB2C2 has two

mutually perpendicular planes of symmetry. One contains AB2, and

reflection through it leaves these three atoms unshifted while

interchanging the C atoms. The other contains AC2 and reflection

through it interchanges only the B atoms.

The NH3 molecule is one example of the general class ofpyramidal AB3 molecules. Since NH3 is not planar, there can be no

symmetry plane including N and all three Hs. Hence we look for

planes including N and one H and bisecting the line between the

remaining two Hs. There are clearly three such planes. Once AB3

becomes planar there is then a fourth symmetric plane, which is a

molecular plane (ExampleBF3 Molecule).

A regular tetrahedral molecule possesses six planes of

symmetry. Symmetry planes contain the atoms AB1B2, AB1B3, AB1B4,

AB2B3, AB2B4, AB3B4.

A

B1

B

B2

B


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A regular octahedron possesses nine symmetric planes.

4.The inversion operation- Centre of symmetry orInversion centre

If a molecule can be brought into an equivalent configuration

by changing the coordinates (x, y, z) of every atom, where the origin

of coordinates lies at a point within the molecule into (-x, -y, -z),

then the point at which the origin lies is said to be a centre of

symmetry or centre of inversion. The symbol for the inversion

centre and for the operation of inversion is an italic i. Like a plane,

the center is an element which generates only one operation.

The effect of carrying out the inversion operation n times may

be expressed as in. it should be easily seen i

n=E when n is even, and

in=i when n is odd.

Some examples of molecules having inversion centers are

octahedral AB6, planar AB4, planar and trans AB2C2, linear ABA,

ethylene, and benzene. Two examples of otherwise fairly

symmetrical molecules which do not have centers of inversion are

C5H5

(plane pentagon) and tetrahedral AB4 (even though A is at the

center and Bs come in even numbers).

5.The improper rotation operation Improper rotationaxis (Sn)

An improper rotation may be thought of as taking place in

two steps; first a proper rotation then a reflection through a plane

perpendicular to the rotation axis. The axis about which this occurs

is called an axis of improper rotation or, more briefly, an improper

axis, and is denoted by the symbol Sn, where again n indicates the


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order. The operation of improper rotation by 2/n is also denoted

by symbol Sn. Obviously, if an axis Cn and a perpendicular plane exist

independently, then Sn exists. More important, however, is that a Sn

may exist when neither the Cn nor the perpendicular existseparately.

If Cn and a h exist in a molecule as independent symmetry

elements, then definitely Sn exists. E.g. BF3 molecule has an S3 axis

collinear with its C3 axis. The fact that BF3 molecule has C3 axis and a

h ensures the presence of an S3 axis. However Sn may exist even

when neither Cn nor h exists independently. E.g. CH4, is a regular

tetrahedral molecule.

Combination (Multiplication) of symmetry

operations

Performing a series s of symmetry operations in succession on

a molecule is represented algebraically as a multiplication.

Suppose we perform a symmetry operation A on a molecule

followed by another operation B. This type of combination of

symmetry operations is said to be a multiplication of the two

operations and is written as BA. Suppose the net effect of the above

multiplication is the same as what would be obtained from a singleoperation C on the molecule. Then may write: BA = C

By convention, a multiplication of symmetry operations is

written in a right to left order of their application. BA means apply A

first and then B.

If the order in which the two symmetry operations , say A and

B, are performed on a molecule is immaterial such that BA = AB, then


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it is said that the multiplication is commutative and that the

operations A and B commute.

E.g. Consider the water molecule. Suppose the yz- plane and

that its C2 axis coincides with the z-axis. Obviously, the two mirrorplanes that the molecule possesses are v(xz) and v

(yz). Consider

C2(z) operation first followed by the v (xz) on the molecule.

v(yz)

O C2(z) O v(xz) O

It is easily seen from the fig. that with respect to the water

molecule, v(xz)C2(z) = v (yz).

Now consider what the product would be if the v(xz) operation

is performed first followed by C2(z).v(yz)

O v(xz) O C2(z) O

Thus it is seen that C2(z) v(xz) = v (yz).

This means that the multiplication is commutative and v(xz) and

C2(z) commute with each other in water molecule.

i.e. C2(z) v(xz) = v(xz)C2(z)

Inverse operations

z

x

y

Ha Hb Hb Ha Ha Hb

Ha Hb Hb Ha Ha Hb

z

x

y


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For any symmetry operation that can be performed on a

molecule, there will be another symmetry operation which will

completely undo what the first operation does to the molecule; the

second operation is then said to be the inverse of the first operation.In other words, for any operation A, there exists another operations

X such that XA = E = AX.

This means that if operation A is performed first on a molecule

followed by operation X. then X returns all of the atoms of the

molecule back to their original positions. Then, X is said to be the

inverse of A and vice versa. i.e. X = A-1

thus, we write :

A-1A = AA-1 = E. It is evident that an operation and its inverse

always commute. We know that C22

= E ; 2

= E ; i2

= E.

i.e. C2-1

= C2 ; -1

= E ; i-1

= i

E.g; Inverse operations for proper rotations:

Consider a rotation of 1200

about a C3 axis in the counterclockwise

direction. Its effect is undone by rotation through 2400

(ie; C32) Thus

C32

is the inverse of C31

ie: C3-1

= C32

In general, for rotation other than C2, the relationship is:

Cn-1

Cn1

= E

ie: Cn-1

= Cnn-1

Thus, the inverse of a Cn1

operation isCn

n-1. In general the inverse of

Cnm

is Cnn-m

.

Introduction of group theory in symmetric

molecules

The Symmetry Point Groups


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A given molecule can have a number of symmetry elements

and the associated symmetry operations; some other molecules too

may have the same set of operations. In other words, a complete set

of symmetry operations will characterize a particular set ofmolecules. The symmetry operations that can be applied to a given

molecule in its equilibrium configuration form a mathematical group.

Let us first specify what we mean by a complete set of

symmetry operations for a particular molecule. A complete set is one

in which every possible product of two operations in the set is also

an operation in the set.

A very important feature of molecular symmetry is that all

symmetry elements in a molecule will intersect at a common point,

namely the centre of gravity, which is not shifted by any of the

associated symmetry operations. Therefore, these symmetry

operations are termed elements of point symmetry or point group

symmetry and a collection of symmetry operations that characterizes

a set of molecules is called a Point group.

Conditions for a point group

1.Closure rule: The productof any two elements in the groupas well as the square of each element must be an element of

the group.

2.Identity rule: In each group, there should be an identityelement which commutes with all others and leaves them

unchanged. The identity element is represented as E and

defined by the expression: AE = EA = A where A is other

element of the group.

3.Associative rule: The associative law of multiplication mustbe hold


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4.Inverse rule: Each element of a group has an inverse that isalso an element of the group. ie; For any element A, there

occurs another element X in the group such that XA = AX = E

where X=A-1 is called the inverse of A.

Illustration

1.Using H2O moleculeThe H2O molecule has the symmetry elements E, C2(z), v(xz),

v(yz). The set of four symmetry operations {E, C2(z), v(xz), v(yz)} is

said to form a point group it can easily shown that the set satisfies all

the four conditions required for a point group.

a.Adherence to the closure rule.First consider the multiplication v(xz) v(yz)

C2(z)

O v(xz) O v(yz) O

The final configuration shows that the net effect is equivalent to

performing the C2(z) operation on the molecule. The product C2(z) is

also is an element of the group.

Let us consider another multiplication, namely C2(z) v(xz)

Ha Hb Hb Ha Hb Ha


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v(yz)

O v(xz) O C2(z) O

Here the final configuration shows that the net effect is equivalent

to performing the v(yz) operation on the molecule. The product

v(yz) is also is an element of the group.

Similarly we can show that all multiplication operations are closed.

The group satisfies the closure rule.

b.Adherence to the identity ruleThe group has the identity operation as one element which

commutes with all others and leaves them unchanged. One example

is given below; C2(z)

O C2(z) O E O

C2(z)

O E O C2(z) O

ie; C2(z)E= E C2(z)= C2(z)

Thus, the group satisfies the identity rule.

Ha Hb Hb Ha Ha Hb

Ha Hb Hb Ha Hb Ha

Ha Hb Ha Hb Hb Ha


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c.Adherence to the associative ruleThe third requirement for a point group is that the associative

law of multiplication must be hold. ie; A(BC) = (AB)C

Let A= C2(z) B= v(xz) C= v(yz)Then we have,

v(xz) v(yz)= C2(z)

C2(z) v(xz)= v(yz)

(a) The multiplication A(BC), ie; C2(z)[ v(xz) v(yz)] is;v(xz) v(yz) C2(z)

O O O

(b) The multiplication (AB)C ie; [C2(z) v(xz)] v(yz) is;

v(yz)

C

2(z)

v(xz)

O O O

It is seen that the final configuration is the same in (a) as well as (b).

Therefore C2(z)[ v(xz) v(yz)] = [C2(z) v(xz)] v(yz)

Obviously, the example shows that multiplication is associative.

d.Adherence to inverse ruleWith respect to the set of symmetry operations under

consideration, we can see that each operation in the set is the

inverse of itself.

E.g; v(xz) v(xz) = E

Ha Hb Hb Ha Ha Hb

Ha Hb Ha Hb Ha Hb


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Ie; E

O v(xz) O v(xz) O

The fourth condition, namely the inverse rule, is also thus satisfied.

The point group consisting of elements E, C2(z), v(xz), and

v(yz) is given the Schoenflies symbol C2v.

The point group C2v is an abiliangroup. ie: the multiplication is

commutative for any pair of its elements E, C2(z), v(xz), and v(yz).

Group Multiplication table for the point group C2v

C2v E C2(z) v(xz) v(yz)

E E C2(z) v(xz) v(yz)C2(z) C2(z) E v(yz) v(xz)

v(xz) v(xz) v(yz) E C2(z)v(yz) v(yz) v(xz) C2(z) E

2.Using NH3 moleculeThe NH3 molecule has the symmetry elements E, C3, C3

2,v, v ,

v. The set of four symmetry operations { E, C3, C32

,v, v , v} is

said to form a point group it can easily shown that the set satisfies all

the four conditions required for a point group.

Ha Hb Hb Ha Ha Hb


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a.Adherence to the closure rule.First consider the multiplication C3v

N C3 N v N

v

The final configuration shows that the net effect is equivalent toperforming the voperation on the molecule. The product v is also

is an element of the group.

Let us consider another multiplication, namely v

Hb

Ha Hc

Hb

Hc Ha Hc Ha

HbHb

Ha Hc

Hc

Hb Ha Hb Hc

Ha


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N v

N v

N

C32

Here the final configuration shows that the net effect is

equivalent to performing the C32

operation on the molecule. The

product C32

is also is an element of the group. Similarly we can show

that all multiplication operations are closed. The group satisfies the

closure rule.

b.Adherence to the identity ruleThe group has the identity operation as one element which

commutes with all others and leaves them unchanged. One example

is given below;

N C3 N E N

C3

Hb

Ha Hc

Ha

Hb Hc Hc Hb

Ha

Hb

Ha Hc

Hc

Hb Ha Hb Ha

Hc

Hb Hb Hc


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N E N C3 N

C3

ie; C3E= E C3= C3

Thus, the group satisfies the identity rule.

c.Adherence to the associative ruleThe third requirement for a point group is that the associative

law of multiplication must be hold. ie; A(BC) = (AB)C

Let A= C3 B= vC= v

Then we have,

v v= C32

C3v= v

(a)

The multiplication A(BC), ie; C3* vv] is;

C32

C3

Ie;

N C32 N C3 N

Hb

Ha Hc

Ha

Hc Hb Ha Hc

Hb


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E

(b) The multiplication (AB)C ie; *vC3+ v is;v v

ie;

N v N v N

E

It is seen that the final configuration is the same in (a) as well as (b).

Therefore C3* vv]= [C3 v ]v

Obviously, the example shows that multiplication is associative.

d.Adherence to inverse ruleWith respect to the set of symmetry operations under

consideration, we can see that some operation in the set is the

inverse of itself. Others have inverse in the same set.

E.g.1; v v = Eie;

Hb

Ha Hc

Ha

Hb Hc Ha Hc

Hb

Hb Hc Hb


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N v N v N

E

E.g.2; C32

C3= E

ie;

N C32

N C3 N

E

The fourth condition, namely the inverse rule, is also thus satisfied.

The point group consisting of elements E, C3, C32,v, v , and

v is given the Schoenflies symbol C3v.

The point group C3v is a non abilian group. ie: the

multiplication is not commutative for some pair of its elements E, C3,

C32

,v, v , and v.

Group Multiplication table for the point group C3v

Hb

Ha Hc

Ha

Hc Hb Ha Hc

Hb


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C3v E C3 C32

v v vE E C3 C3

2 v v v

C3 C3 C32

E v v v

C32

C32

E C3 v v vv v v v E C3

2 C3

v v v v C3 E C32

v v v v C32

C3 E

Conclusion

We have, by inspection, compiled a list of all of the symmetry

elements possessed by a given molecule. We can then list all of the

symmetry operations generated by each of these elements. Our

objective in this section was to demonstrate that such a complete list

of symmetry operations satisfies the four criteria for a mathematical

group.


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Reference

Chemical applications of Group theory:- F AlbertCotton

Mathematics and its History:- John Stillwell

A first course in Abstract Algebra:- John b. Fraleigh


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Introduction to Group theory with Applications inMolecular and Solid State Physics(e book):- Karsten

Horn

Molecular Symmetry, Group theory andApplications(e book):- Claire Vallance

Molecular Symmetry & Group Theory

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Transcript of Molecular Symmetry & Group Theory