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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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32
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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33
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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34
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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35
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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36
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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37
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
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