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 orie