Molecular Symmetry & 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.
Group members
Basil jaseem V
Midhun C
Nithin P
Shijith K P
Sijeesh A R
Vaisakh K V
2
Group members
Basil jaseem V
Midhun C
Nithin P
Shijith K P
Sijeesh A R
Vaisakh K V
2
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
3
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:
4
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
6
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 7
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.
8
Groups of symmetric polygons
Triangle Consider 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
ρ0 =
µ1=
ρ1=
ρ2 =
µ2 =
µ3 =
Then the composition table is shown below
ρ0 ρ1 ρ2 µ1 µ2 µ3
ρ0 ρ0 ρ1 ρ2 µ1 µ2 µ3
ρ1 ρ1 ρ2 ρ0 µ2 µ3 µ1
ρ2 ρ2 ρ0 ρ1 µ3 µ1 µ2
µ1 µ1 µ3 µ2 ρ0 ρ2 ρ1
µ2 µ2 µ1 µ3 ρ1 ρ0 ρ2
µ3 µ3 µ2 µ1 ρ2 ρ1 ρ0
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 0
0
0
vertices. Let r0, r1, r2 are the rotations through 360 ,240 ,120
respectively in anti clockwise direction. α, β, γ are the reflection
9
about the median through upper vertex, lower right vertex, and lower left vertex respectively. Eg: Let the composition say αγ is the operation γ follows by α.
γ →
1
α→
jj
2
The composition table for the 6 symmetries is as below. r0 r1 r2 α β r0 r1 r2 α β r0 r1 r2 r0 γ α r1 r2 r0 r1 β γ r2 α β γ r0 r1 α β β γ α r2 r0 γ α β r1 r2 γ This group is called third dihedral group denoted by D3.
γ γ β α r2 r1 r0
Square Let us form the dihedral group D 4 of 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 coordinates and its centre at the origin. 1
2
10
0
0
0
0
Allow the following clockwise rotations 90 , 180 , 270 , 360 , say r90, r180, r270, r360 respectively and reflections h, v about horizontal and vertical axes and reflections d1 and d2 about diagonals. v
2
h 3
4
d1
d2
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
r360 r360
r90 r90
r180 r180
r270 r270 11
h h
v v
d1 d1
d2 d2
r90 r180 r270 h v d1 d2
r90 r180 r270 h v d1 d2
r180 r270 r360
r270 r360 r90 v h d2 d1
d1 h v
r360 r90 r180 d1 d2 v h
d1 v d2 r360 r180 r90 r270
d2 h d1 r180 r360 r270 r90
v d2 h r270 r90 r360 r180
h d1 v r90 r270 r180 r360
Eg: v* r270 = d2 1
2
2
3
3
r270→
4
3
2
v→
1
4
4
th
In general Dn is the n dihedral group of symmetries of regular n-gon having 2n elements.
Eg: Consider one permutation in S6.
ς=
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. 12
1
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. Each symmetry 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 0
the y-z plane) through 180 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 Ha
Anticlockwise rotation through 180
Hb
O
0
Ha
15
Hb
Symmetry elements and Operations Symmetry Element
1. Identity element
2. Proper axis
Symmetry operations
Do nothing
Transformation matrix
( ) =
One or more rotations about the axis(eg; around z axis)
( ( ) ( ) =
3. Plane
Reflection in the plane(eg; in xy plane)
4. Centre of Inversion of all symmetry atoms through or centre the centre of inversion 5. Improper axis
=
( ) ( ) =
Reflection in mirror plane followed by 16
)
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 rotation axis(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. 0
Upon rotating the triangle by 120 (2π/3) about this axis, the triangle is brought into an equivalent configuration. It may be noted that a rotation by 240 equivalent configuration. 17
0
(2*2π/3) also produces an
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 a symmetry element. The general symbol for a proper axis of rotation is C n, where the subscript n denotes the order of the axis. By order is meant the largest value of n such that rotation through 2π/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 2π/3, 2*2π/3, 3*2π/3 is seen to be:
2π/3 A
B
3
1
2
3
2*2π /3
A
C
2
1
1
3*2π/3
A
D=A
1
3
1
18
3
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 2π/3 around 2
the C3 axis. For the rotation by 2*2π/3 we use the symbol C3 , and 3
for the rotation by 3*2π/3 the symbol C3 . Symbolically we can write 4
2
3
C3 = C3, and hence only C3, C3 , and C3 are separate and distinct 3
operations. However, C3 produces an identical configuration, and 3
hence we may write C3 = 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 C n, and a rotation by 2π/n is also represented by the symbol Cn. Rotation by 2π/n m
carried out successively m times is represented by the symbol Cn . n
n+1
Also, in case, Cn = E, Cn
n+2
= Cn, Cn
2
= Cn and so on.
e.g. , Water molecule has two –fold proper rotation axis (C 2) in the plane of the molecule, passing through the O atom and bisecting the H-O-H angle. A rotation of the molecule through 180
0
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 reflections Symmetry 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 (say x and 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 i th atom, originally ( x ,i y ,i zi ), is moved to the point ( x ,i y ,i -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. 20
Since this atom must lie in all of the symmetry planes simultaneously. The standard symbol for a plane of symmetry is
. The same
σ
symbol is also for the operation of reflecting through the plane. Now we can conveniently denote the successive application of the n
2
operation σ n times by writing σ . We can then also write, σ =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 n
equal to E, the identity operation. It should be obvious that σ = E n
when n is even and σ = σ when n is odd. Examples; A tetrahedral molecule of the type AB2C2 has two mutually perpendicular planes of symmetry. One contains AB 2, 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 of pyramidal AB3 molecules. Since NH3 is not planar, there can be no symmetry plane including N and all three H’s. Hence we look for planes including N and one H and bisecting the line between the remaining two H’s. 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, AB 1B3, AB1B4, AB2B3, AB2B4, AB3B4. B1
A B
B2
21
B
A regular octahedron possesses nine symmetric planes.
4. The inversion operation- Centre of symmetry or Inversion 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 n
n
be expressed as i . it should be easily seen i =E when n is even, and n
i =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 B’s come in even numbers).
5. The improper rotation operation – Improper rotation axis (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 S n, where again n indicates the 22
order. The operation of improper rotation by 2π/n is also denoted by symbol Sn. Obviously, if an axis C n and a perpendicular plane exist independently, then Sn exists. More important, however, is that a S n may exist when neither the C n nor the perpendicular ς exist separately. If Cn and a ςh 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 ςh ensures the presence of an S3 axis. However Sn may exist even when neither Cn nor ςh 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 single operation 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
23
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 mirror ’
planes that the molecule possesses are ςv (xz) and ςv (yz). Consider C2(z) operation first followed by the ς v (xz) on the molecule.
z σ‘v(yz) y
O Ha
x
C2(z) Hb
O
σv(xz)
Hb
O Ha
Ha
Hb
It is easily seen from the fig. that with respect to the water molecule, ςv(xz)·C2(z) = ς’v (yz). Now consider what the product would be if the ςv(xz) operation is performed first followed by C2(z). σ‘v(yz)
z O y
Ha
σv(xz) Hb
O Hb
C2(z) Ha
O Ha
x
Thus it is seen that C2(z) · ςv(xz) = ς’v (yz). This means that the multiplication is commutative and ς v(xz) and C2(z) commute with each other in water molecule. i.e. C2(z) · ςv(xz) = ςv(xz)·C2(z)
Inverse operations 24
Hb
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 -1
inverse of A and vice versa. i.e. X = A thus, we write : -1
-1
A A = AA = E.
It is evident that an operation and its inverse 2
2
2
always commute. We know that C2 = E ; ς = E ; i = E. -1
-1
-1
i.e. C2 = C2 ; ς = E ; i = i
E.g; Inverse operations for proper rotations: 0
Consider a rotation of 120 about a C3 axis in the counterclockwise 0
2
direction. Its effect is undone by rotation through 240 (ie; C3 ) Thus 2
C3 is the inverse of C3 ie: C3
-1 =
1
2
C3
In general, for rotation other than C 2, the relationship is: -1 ·
Cn
Thus, the inverse of a m
1
Cn
=E -1 n-1 ie: Cn = Cn 1 n-1 Cn operation is Cn .
In general the inverse of
n-m .
Cn is Cn
Introduction of group theory in symmetric molecules The Symmetry Point Groups 25
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 of molecules. 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 product of any two elements in the group as well as the square of each element must be an element of the group. 2. Identity rule: In each group, there should be an identity element 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 must be hold 26
4. Inverse rule: Each element of a group has an inverse that is also an element of the group. ie; For any element A, there occurs another element X in the group such that XA = AX = E -1
where X=A is called the inverse of A.
Illustration 1. Using H2O molecule The H2O molecule has the symmetry elements E, C2(z), ςv(xz), ς’v(yz). The set of four symmetry operations {E, C 2(z), ςv(xz), ς’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 ςv(xz)· ς’v(yz) C2(z) O Ha
σv(xz) Hb
O Hb
σ’v(yz) Ha
O Hb
Ha
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)· ςv(xz)
27
σ’v(yz)
O
σv(xz) Hb
Ha
O
C2(z) Ha
Ha
Hb
O Hb
Here the final configuration shows that the net effect is equivalent to performing the ς’v(yz) operation on the molecule. The product ς’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 rule The 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 Ha
C2(z) Hb
O Hb
E
O Hb
Ha
Ha
C2(z)
O Ha
E Hb
O Ha
C2(z) Hb
ie; C2(z)·E= E· C2(z)= C2(z) Thus, the group satisfies the identity rule. 28
O Hb
Ha
c. Adherence to the associative rule The 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= ςv(xz) C= ς’v(yz) Then we have, ςv(xz)· ς’v(yz)= C2(z) C2(z)· ςv(xz)= ς’v(yz) (a)
The multiplication A(BC), ie; C2(z)[ ςv(xz)· ς’v(yz)] is; σv(xz)· σ’v(yz)
C2(z)
O
O Hb
Ha
(b)
O Ha
Hb
Ha
Hb
The multiplication (AB)C ie; [C2(z)· ςv(xz)]· ς’v(yz) is; σ’v(yz)
C2(z)· σv(xz)
O Ha
O Hb
Ha
O Hb
Ha
Hb
It is seen that the final configuration is the same in (a) as well as (b). Therefore C2(z)[ ςv(xz)· ς’v(yz)] = [C2(z)· ςv(xz)]· ς’v(yz) Obviously, the example shows that multiplication is associative.
d. Adherence to inverse rule With 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; ςv(xz)· ςv(xz) = E 29
Ie;
E
O Ha
σv(xz) Hb
O Hb
σv(xz)
O Hb
Ha
Ha
The fourth condition, namely the inverse rule, is also thus satisfied. The point group consisting of elements E, C2(z), ςv(xz), and ς’v(yz) is given the Schoenflies symbol C2v. The point group C2v is an abilian group. ie: the multiplication is commutative for any pair of its elements E, C2(z), ςv(xz), and ς’v(yz).
Group Multiplication table for the point group C 2v
C2v
E
C2(z)
σv(xz)
σ’v(yz)
E C2(z) σv(xz) σ’v(yz)
E C 2(z) σ v(xz) σ’ v (yz)
C 2(z) E σ’ v (yz) σ v(xz)
σ v(xz) σ’ v (yz) E C 2(z)
σ’ v(yz) σ v(xz) C 2(z) E
2. Using NH3 molecule 2
The NH3 molecule has the symmetry elements E, C3, C3 , ςv, ς’v , 2
ς’’v. The set of four symmetry operations { E, C3, C3 , ςv, ς’v , ς’’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. 30
a. Adherence to the closure rule. First consider the multiplication C3·ςv
Hb
N
Ha
Hab
Hcb
C3
Hc
N
Hbc
σv
Ha
N
Hbc
Hca
σ’v
The final configuration shows that the net effect is equivalent to performing the ς’v operation on the molecule. The product ς’ v is also is an element of the group. Let us consider another multiplication, namely ς’v ·
31
Hb
N
σ’v
Ha
Ha
Ha
N
Hc
σ’’v
Hb
N
Hc
Hc
C3
Hb
2
Here the final configuration shows that the net effect is 2
equivalent to performing the C3 operation on the molecule. The 2
product C3 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 rule The group has the identity operation as one element which commutes with all others and leaves them unchanged. One example is given below; Hb
N
Ha
Hc
Hc
C3
Hc
N
Hb
E
Ha
N
Hb
Ha
C3 Hb
Hb 32
Hc
N
E
N
C3
N
C3
ie; C3·E= E· C3= C3 Thus, the group satisfies the identity rule. c. Adherence to the associative rule
The 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= ςv C= ς’v Then we have, ςv· ς’v= C3
2
C3·ςv= ς’v
(a)
The multiplication A(BC), ie; C3·* ς’v ·ςv] is; 2
C3 · C3 Ie; Hb
Hb
Ha
N
C3
2
N
C3
N
33
Ha
Hc
Hc
Hb
Ha
Hc
E
(b)
The multiplication (AB)C ie; *ςv·C3+· ς’v is; ς’v· ς’v ie; Hb
N
Ha
Hb
Ha
σ’v
Hc
N
Hb
σ’v
Hc
N
Ha
Hc
E
It is seen that the final configuration is the same in (a) as well as (b). Therefore C3·* ςv·ς’v]= [C3· ςv ]·ς’v Obviously, the example shows that multiplication is associative.
d. Adherence to inverse rule With 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; ςv· ςv = E ie; Hb
34 Hc
Hb
N
σv
N
σv
N
E
2
E.g.2; C3 · C3= E Hb
ie;
N
Ha
Hb
Ha
C3
Hc
2
N
Hc
C3
Hb
N
Ha
Hc
E
The fourth condition, namely the inverse rule, is also thus satisfied. 2
The point group consisting of elements E, C3, C3 , ςv, ς’v , and ς’’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, 2
C3 , ςv, ς’v , and ς’’v.
Group Multiplication table for the point group C 3v 35
C3v E C3 2 C3 σv σ’v σ’’v
E E C 3 2 C 3 σ v σ’ v σ’’ v
C3 C 3 2 C 3 E σ’’ v σ v σ’ v
2
C3 2 C 3 E C 3 σ’ v σ’’ v σ v
σv σ v σ’ v σ’’ v E C 3 2 C 3
σ’v σ’ v σ’’ v σ v 2 C 3 E C 3
σ’’v σ’’ v σ v σ’ v C 3 2 C 3 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.
36
Reference
Chemical applications of Group theory:- F Albert Cotton Mathematics and its History:- John Stillwell
A first course in Abstract Algebra:- John b. Fraleigh
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