UNIT 1- Symmetry & Group Theory In Chemistry
UNIT 1- Symmetry & Group Theory in Chemistry1.0 – Introduction1.1 - Objectives1.2 – Symmetry & group theory1.2.1 -Symmetry elements1.2.2 – Symmetry operation1.2.3 - Group & Subgroups1.2.4– Relation between orders of a finite group & its subgroups1.2.5 -Conjugacy relation & classes1.2.6 – Point symmetry group1.2.7– Schonflies symbols or notations1.2.8 -Representation of Group by Matrices1.2.9 – Character of a Representation1.2.10 – The Great Orthogonality Theorem & its importance1.2.11 – Character tables & their use1.3 - Unifying Principles1.3.1– Electromagnetic Spectum1.3.2– Interaction of Electromagnetic spectrum with matter1.3.3 – Absorption of Radiation1.3.4 - Emission of Radiatuon1.3.5- Transmission of Radiation1.3.6- Reflection of Radiation1.3.7–Refraction of Radiation1.3.8–Dispersion of Radiation1.3.9 – Polarization1.3.10– Scattering of Radiation1.3.11 – The Uncertainty relation1.3.12 – Natural line width & natural line Broadening1.3.13 – Transition Probability1.3.14- Result of Time Dependent Perturbation theory1.3.15 – Transition Moment1.3.16 – Selection Rules1.3.17 –Intensity of spectral lines1.3.18 –Born Oppenheimer Approximation1.3.19- sum up1.3.20- check your progress : key1.3.21 - References1
1.0 - INTRODUCTIONGroup Theory is a mathematical method by which aspects of a molecules symmetry can bedetermined. The symmetry of a molecule reveals information about its properties (i.e., structure,spectra, polarity, chirality, etc ).Group theory can be considered the study of symmetry: the collection of symmetries of someobject preserving some of its structure forms a group; in some sense all groups arise this way.It can be grouped into three categories: Getting to know groups — It helps to group theory and contain explicit definitionsand examples of groups;Group applications — It helps to understand the applications of group theory. Themathematical descriptions here are mostly intuitive, so no previous knowledge isneeded.Group history — It focuses on the history of group theory, from its beginnings torecent breakthroughs.Electromagnetic Radiations are the radiations having electric field as well as magnetic fieldboth are perpendicular to each other & are also perpendicular to the line of propogation.There are various electromagnetic radiations like radiowaves, microwaves, x-rays, uv-rayscosmic rays etc. Theses when interact with matter give rise to various different phenomenonslike diffraction, interference, absorbtion, emission depending on the type of EMR & matter(energy).1.1 - OBJECTIVESBy studying this unit we come across many of the things which you are not aware of :1. The significance of group theory for chemistry is that molecules can be categorizedon the basis of their symmetry properties, which allow the prediction of manymolecular properties.2. The process of placing a molecule into a symmetry category involves identifying allof the lines, points, and planes of symmetry that it possesses; the symmetry categoriesthe molecules may be assigned to are known as point groups.3. It allows you to determine that Which vibrational transitions are allowed orforbidden on the basis of symmetry.4. How EMR interact to show different phenomenons like polarization, Dispersion,Refraction etc.5. What is Transition & transition probability.1.0.1 – Symmetry Elements & symmetry operation The term symmetry implies a structure in which the parts are in harmony with each other, aswell as to the whole structure i;e the structure is proportional as well as balanced.Clearly, the symmetry of the linear molecule A-B-A is different from A-A-B. In A-B-A the A-Bbonds are equivalent, but in A-A-B they are not. However, important aspects of the symmetry of H2Oand CF2Cl2 are the same. This is not obvious without Group theory.2
Symmetry Elements - These are the geometrical elements like line, plane with respect to whichone or more symmetric operations are carried out. The symmetry of a molecule can be described by 5 types of symmetry elements. Symmetryaxis: an axis around which a rotation byresults in a molecule indistinguishable from theoriginal. This is also called an n-fold rotational axis and abbreviated Cn. Examples are the C2in water and the C3 in ammonia. A molecule can have more than one symmetry axis; the onewith the highest n is called the principal axis, and by convention is assigned the z-axis in aCartesian coordinate system.Plane of symmetry: a plane of reflection through which an identical copy of the originalmolecule is given. This is also called a mirror plane and abbreviated ζ. Water has two ofthem: one in the plane of the molecule itself and one perpendicular to it. A symmetry planeparallel with the principal axis is dubbed vertical (ζv) and one perpendicular to it horizontal(ζh). A third type of symmetry plane exists: if a vertical symmetry plane additionally bisectsthe angle between two 2-fold rotation axes perpendicular to the principal axis, the plane isdubbed dihedral (ζd). A symmetry plane can also be identified by its Cartesian orientation,e.g., (xz) or (yz).Centre of symmetry or inversion center, i. A molecule has a center of symmetry when, forany atom in the molecule, an identical atom exists diametrically opposite this center an equaldistance from it. There may or may not be an atom at the center. Examples are xenontetrafluoride (XeF4) where the inversion cente is at the Xe atom, and benzene (C6H6) wherethe inversion center is at the center of the ring.Rotation-reflection axis: an axis around which a rotation by, followed by a reflection ina plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improperrotation axis, it is abbreviated Sn, with n necessarily even. Examples are present intetrahedral silicon tetrafluoride, with three S4 axes, and the staggered conformation of ethanewith one S6 axis.Identity, abbreviated to E, from the German 'Einheit' meaning Unity. This symmetry elementsimply consists of no change: every molecule has this element. It is analogous to multiplyingby one (unity).1.2.2- Symmetry Operations/ElementsA molecule or object is said to possess a particular operation if that operation when applied leaves themolecule unchanged. Each operation is performed relative to a point, line, or plane - called asymmetry element. There are 5 kinds of operations 1. Identity2. n-Fold Rotations3. Reflection4. Inversion5. Improper n-Fold Rotation1. Identity is indicated as E does nothing, has no effect i;e this operation brings back the molecule to the originalorientationall molecules/objects possess the identity operation, i.e., posses E.E has the same importance as the number 1 does in multiplication (E is needed in order todefine inverses).3
2. n-Fold Rotations: Cn, where n is an integer, rotation by 360 /n about a particular axis defined asthe n-fold rotation axis.C2 180 rotation, C3 120 rotation, C4 90 rotation, C5 72 rotation, C6 60 rotation, etc.Rotation of H2O about the axis shown by 180 (C2) gives the same molecule back.Therefore H2O possess the C2 symmetry element.However, rotation by 90 about the same axis does not give back the identical moleculeTherefore H2O does not possess a C4 symmetry axis.BF3 posses a C3 rotation axis of symmetry.This triangle does not posses a C3 rotation axis of symmetry.XeF4 is square planar. It has four DIFFERENT C2 axes . It also has a C4 axis coming out of the pagecalled the principle axis because it has the largest n. By convention, the principle axis is in the zdirection4
3. Reflection: ζ (the symmetry element is called a mirror plane or plane of symmetry)If reflection about a mirror plane gives the same molecule/object back than there is a plane ofsymmetry (ζ).If plane contains the principle rotation axis (i.e., parallel), it is a vertical plane (ζv)If plane is perpendicular to the principle rotation axis, it is a horizontal plane (ζh)If plane is parallel to the principle rotation axis, but bisects angle between 2 C2 axes, it is a diagonalplane (ζd)H2O posses 2 ζv mirror planes of symmetry because they are both parallel to the principle rotationaxis (C2)XeF4 has two planes of symmetry parallel to the principle rotation axis: ζvXeF4 has two planes of symmetry parallel to the principle rotation axis and bisecting the anglebetween 2 C2 axes : ζdXeF4 has one plane of symmetry perpendicular to the principle rotation axis: ζh4. Inversion: i (the element that corresponds to this operation is a center of symmetry orinversion center) .The operation is to move every atom in the molecule in a straight line through the inversion center tothe opposite side of the molecule.Therefore XeF4 posses an inversion center at the Xe atom.5
5. Improper Rotations: Snn-fold rotation followed by reflection through mirror plane perpendicular to rotation axis alsoknown as Rotation Reflection axis. It is an imaginary axis passing through the molecule, on whichwhen the molecule is rotated by 2π/n angle & then reflected on a plane perpendicular to the rotationaxis then an equivalent orientation is observed.Note: n is always 3 or larger because S1 ζ and S2 i.These are different, therefore this molecule does not posses a C3 symmetry axis.This molecule posses the following symmetry elements: C3, 3 ζd, i, 3 C2, S6. There is no C3 or ζh.Eclipsed ethane posses the following symmetry elements: C3, 3ζ v, 3 C2, S3, ζh. There is no S6 or i.Compiling all the symmetry elements for staggered ethane yields a Symmetry Group called D3d.Compiling all the symmetry elements for eclipsed ethane yields a Symmetry Group called D3h.6
Importance of symmetry It is an important concept in crystal morphology,crystal structure analysis.It helps in the classification of electronic states in a molecule.It is also useful in determining which atomic orbitals can combine to form molecules.It can be used in predicting the no of d-d absorption bands that are observed in coordinationcompounds.Ligand theory also depends on concept of symmetry.IR & Raman Spectroscopy used for structure illucidation also depends on symmetry.CHECK YOUR PROGRESS - 1Notes : i) Write your answer in the space given below.ii) Compare your answer with those given at the end of the unit.Q1.Name the symmetry element possessed by all the molecules.Q2.Give an example of the molecule which contain C2 axis in addition to Cn axis ofrotation.Q3. Give an example of the molecule which contain horizontal plane of symmetry inaddition to vertical plane of symmetry.Q4.All the elements possessed by a molecule is represented by .1.2.3- Groups & SubgroupsEach molecule has a set of symmetry operations that describes the molecule's overall symmetry. Thisset of operations define the group of the molecule.A group is a finite or infinite set of elementstogether with a binary operation (called the group operation) that together satisfy the four fundamentalproperties of closure, associativity, the identity property, and the inverse property. The operation withrespect to which a group is defined is often called the "group operation," and a set is said to be a group"under" this operation.The study of groups is known as group theory.A group is a set of operations which satisfies the following requirements1. Any result of two or more operations must produce the same result as application of one operationwithin the group.i.e., the group multiplication table must be closedConsider H2O which has E, C2 and 2 ζv's.i.e.,of courseetc The table is closed, i.e., the results of two operations is an operation in the group i;e the elements arecommutable.2. Must have an identity ( ) such that AE EA A for any operation A in the group.7
3. All elements must have an inverse i.e., for a given operation ( ) there must exist an operation ( )such thator AA-1 A-1A E4. Each element has follows associative lawP(QR) (PQ)Rexample, the point group for the water molecule is C2v, with symmetry operations E, C2, ζv and ζv'. Itsorder is thus 4. Each operation is its own inverse. As an example of closure, a C2 rotation followed bya ζv reflection is seen to be a ζv' symmetry operation: ζv*C2 ζv'.The group multiplication table obtained is therefore for water 'vEC2ζ'vζ'vζvC2Eσv . σv EC2 .σv σ'vC2.E E C2 C2C2 (σv.σ'v) ( C2 .σv )σ'vAnother example is the ammonia molecule, which is pyramidal and contains a three-fold rotation axisas well as three mirror planes at an angle of 120 to each other. Each mirror plane contains an N-Hbond and bisects the H-N-H bond angle opposite to that bond. Thus ammonia molecule belongs to theC3v point group which has order 6: an identity element E, two rotation operations C3 and C32, andthree mirror reflections ζv, ζv' and ζv".Classification Of Group1. Abelian Group – All elements are commutable. Example Water2. Non Abelian Group- All elements do not commute with one another.Example- Phosphine symmetry operations are E,C13, C34, ζv1, ζv2C3 . ζv ζv.C33.Cyclic group- In cyclic group all the elements of a group can be generated from one element .It isdenoted by An. A represents identity element & n represents total no of elements & is called as orderof group. Each cyclic group is abelian but each abelian group is not cyclic.Example Trans 1,2 dichlorocyclopropane.Classification of group on the basis of element1.Monoid - A group is a monoid each of whose elements is invertible.2. Trivial Group- A group must contain at least one element, with the unique (up to isomorphism)single-element group known as the trivial group.3. Finite group - If there are a finite number of elements, the group is called a finite group8
SubgroupsAny subset of element which form a group is called as subgroup.A subgroup is a subset of group elements of a group that satisfies the four group requirements. Itmust therefore contain the identity element. " is a subgroup of " is written, or sometimes. A subset of a group that is closed under the group operation and the inverse operation is calleda subgroup.The elements of a subgroup should obey the following conditions-If g is the order of the group & s isthe order of the subgroup ,then g/s is a natural number. Example- water molecule has symmetryelements- E,C2,ζv, ζv1GROUP - E,C2,ζv, ζv1SUBGROUPS EE,C2E,ζv,E, ζv1CLASSES – This is the subdivision of a group.Two elements A & B in a group form a class if they are conjugate to each other. Conjugate elementsare related by the equationX-1AX BWhere X is similarity transformation element .It is used to find whether a set of elements form a class.Example- water molecule has symmetry elements- E,C2,ζv, ζv1GROUP - E,C2,ζv, ζv1CLASSES - E-1 C2E C2ζv -1C2 ζv C2ζv-1 C2 ζv1 C2C2-1 C2 C2 C2ORDER- The order of a class of a group must be an integral factor of the order of a group and thenumber of elements is called the group order of the group.Method to find the class –1.Symmetry operations which commutes with all symmetry operations forms a class.E, ζh, I belongs to separate class2.Rotation operation & its inverse forms a class like C2-1 & C23. Improper axis & inverse forms a class S1S1-1.4. Two rotation about different axis forms a class if there is a third operation which interchange thepoints of the axis.5. Two reflection about different planes belongs to the same class if there is a third operation whichinterchange points on the two plane.Example- Square Planar AB4 molecule hasSymmetry operations- 16- E, i , ζh, C21 C41 C43 S41 S43 4C214 ζvNo Of Elements - 13Classes- (i) E, i , ζh, C2(iv) 2 C21operations about C2 axis (reflection)13(ii ) C4 C4(v) 2 C21operations about C2‘ axis (reflection)13(iii) S4 S4(vi) 2 reflection operations in two ζv planes(vii) 2 reflection operations in two ζ‘v planes9
1.2.4 - Relation between orders of a finite group & its subgroup –If there are a finite number of elements, the group is called a finite group and the number of elementsis called the group order of the group. A subset of a group that is closed under the group operation and the inverse operation iscalled a subgroup. Subgroups are also groups, and many commonly encountered groups are infact special subgroups of some more general larger group. A finite group is a group having finite group order. Examples of finite groups are the modulomultiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups,alternating groups, and so on. The finite (cyclic) group forms the "Finite Simple Group of Order 2" A basic example of a finite group is the symmetric group , which is the group ofpermutations (or "under permutation") of objects.Check your progress 2Notes : i) Write your answer in the space given below.ii) Compare your answer with those given at the end of the unit.1. n-fold rotation followed by reflection through mirror plane perpendicular to rotation axisalso known as ----------- .2. Any subset of element which form a group is called as -------------- .3. The order of a class of a group must be an integral factor of the order of a group and thenumber of elements is called the -------- of the group.1.2.5 - Conjugacy Relation & ClassA complete set of mutually conjugate group elements. Each element in a group belongs to exactly oneclass, and the identity element () is always in its own class. The conjugacy class orders of allclasses must be integral factors of the group order of the group. A group of prime order has one class for each element.In an Abelian group, each element is in a conjugacy class by itself.Two operations belong to the same class when one may be replaced by the other in a newcoordinate system which is accessible by a symmetry operation . These sets corresponddirectly to the sets of equivalent operations.Two elements A & B in a group form a class if they are conjugate to each other.Conjugateelements are related by the equationX-1AX BWhere X is similarity transformation element .It is used to find whether a set of elementsform a class.conjugacy is an equivalence relation. Also note that conjugate elements have the same order.The set of all elements conjugate to a is called the class of a.To find conjugacy class similarity transformationson . Applying asimilarity transformation gives(6)(7)(8)(9)(10)10
soform a conjugacy class.1.2.6 - Point Symmetry Groups - Each molecule has a set of symmetry operations thatdescribes the molecule's overall symmetry. This set of operations define the point group of themolecule. Since all the elements of symmetry present in the molecule intersect at a common point &this point remains fixed under all symmetry operations of the molecule and is known as pointsymmetry groups.1.2.7 - Schonflies notationThe point groups are denoted by their component symmetries. There are a few standard notations usedby crystallographers. The Schoenflies notation or Schonflies notation, named after the Germanmathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describecrystallographic point groups. This notation is used in spectroscopy. The other convention is theHermann-Mauguin notation, also known as the International notation. A point group in theSchoenflies convention is completely adequate to describe the symmetry of a molecule; this issufficient for spectroscopy. The Hermann-Maunguin notation is able to describe the space group of acrystal lattice, while the Schoenflies notation isn't. Thus the Hermann-Mauguin notation is used incrystallography.Schönflies notationIn Schönflies notation, point groups are denoted by a letter symbol with a subscript. The symbols usedin crystallography mean the following: The letter O (for octahedron) indicates that the group has the symmetry of an octahedron (orcube), with (Oh) or without (O) improper operations .The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. Tdincludes improper operations, T excludes improper operations, and Th is T with the addition ofan inversion.The letter I (for icosahedron) indicates
UNIT 1- Symmetry & Group Theory in Chemistry 1.0 – Introduction 1.1 - Objectives 1.2 – Symmetry & group theory 1.2.1 -Symmetry elements 1.2.2 – Symmetry operation 1.2.3 - Group & Subgroups 1.2.4– Relation between orders of a finite group & its subgroups 1.2.5 -Conj
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Symmetry Point Groups Symmetry of a molecule located on symmetry axes, cut by planes of symmetry, or centered at an inversion center is known as point symmetry . Collections of symmetry operations constitute mathematical groups . Each symmetry point group has a particular designation. Cn, C nh, C nv Dn, D nh, D nd S2n C v ,D h
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