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UNIT 1- Symmetry & Group Theory in Chemistry

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UNIT 1- Symmetry & Group Theory in Chemistry 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 -Conjugacy relation & classes 1.2.6 – Point symmetry group 1.2.7– Schonflies symbols or notations 1.2.8 -Representation of Group by Matrices 1.2.9 – Character of a Representation 1.2.10 – The Great Orthogonality Theorem & its importance 1.2.11 – Character tables & their use 1.3 - Unifying Principles 1.3.1– Electromagnetic Spectum 1.3.2– Interaction of Electromagnetic spectrum with matter 1.3.3 – Absorption of Radiation 1.3.4 - Emission of Radiatuon 1.3.5- Transmission of Radiation 1.3.6- Reflection of Radiation 1.3.7–Refraction of Radiation 1.3.8–Dispersion of Radiation 1.3.9 – Polarization 1.3.10– Scattering of Radiation 1.3.11 – The Uncertainty relation 1.3.12 – Natural line width & natural line Broadening 1.3.13 – Transition Probability 1.3.14- Result of Time Dependent Perturbation theory 1.3.15 – Transition Moment 1.3.16 – Selection Rules 1.3.17 –Intensity of spectral lines 1.3.18 –Born Oppenheimer Approximation 1.3.19- sum up 1.3.20- check your progress : key 1.3.21 - References 1 1.0 - INTRODUCTION Group Theory is a mathematical method by which aspects of a molecules symmetry can be determined. 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 some object 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 definitions and examples of groups; Group applications — It helps to understand the applications of group theory. The mathematical descriptions here are mostly intuitive, so no previous knowledge is needed. Group history — It focuses on the history of group theory, from its beginnings to recent breakthroughs. Electromagnetic Radiations are the radiations having electric field as well as magnetic field both are perpendicular to each other & are also perpendicular to the line of propogation. There are various electromagnetic radiations like radiowaves, microwaves, x-rays, uv-rays cosmic rays etc. Theses when interact with matter give rise to various different phenomenons like diffraction, interference, absorbtion, emission depending on the type of EMR & matter (energy). 1.1 - OBJECTIVES By 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 categorized on the basis of their symmetry properties, which allow the prediction of many molecular properties. 2. The process of placing a molecule into a symmetry category involves identifying all of the lines, points, and planes of symmetry that it possesses; the symmetry categories the molecules may be assigned to are known as point groups. 3. It allows you to determine that Which vibrational transitions are allowed or forbidden 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, as well 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-B bonds are equivalent, but in A-A-B they are not. However, important aspects of the symmetry of H2O and 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 which one or more symmetric operations are carried out. The symmetry of a molecule can be described by 5 types of symmetry elements. Symmetry axis: an axis around which a rotation by results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated Cn. Examples are the C2 in water and the C3 in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is assigned the z-axis in a Cartesian coordinate system. Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated ζ. Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel 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 bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed 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, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center. Examples are xenon tetrafluoride (XeF4) where the inversion cente is at the Xe atom, and benzene (C6H6) where the inversion center is at the center of the ring. Rotation-reflection axis: an axis around which a rotation by , followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn, with n necessarily even. Examples are present in tetrahedral silicon tetrafluoride, with three S4 axes, and the staggered conformation of ethane with one S6 axis. Identity, abbreviated to E, from the German 'Einheit' meaning Unity. This symmetry element simply consists of no change: every molecule has this element. It is analogous to multiplying by one (unity). 1.2.2- Symmetry Operations/Elements A molecule or object is said to possess a particular operation if that operation when applied leaves the molecule unchanged. Each operation is performed relative to a point, line, or plane - called a symmetry element. There are 5 kinds of operations - 1. Identity 2. n-Fold Rotations 3. Reflection 4. Inversion 5. Improper n-Fold Rotation 1. Identity is indicated as E does nothing, has no effect i;e this operation brings back the molecule to the original orientation all 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 to define inverses). 3 2. n-Fold Rotations: Cn, where n is an integer, rotation by 360°/n about a particular axis defined as the 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 molecule Therefore 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 page called the principle axis because it has the largest n. By convention, the principle axis is in the z- direction 4 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 of symmetry (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 diagonal plane (ζd) H2O posses 2 ζv mirror planes of symmetry because they are both parallel to the principle rotation axis (C2) XeF4 has two planes of symmetry parallel to the principle rotation axis: ζv XeF4 has two planes of symmetry parallel to the principle rotation axis and bisecting the angle between 2 C2 axes : ζd XeF4 has one plane of symmetry perpendicular to the principle rotation axis: ζh 4. Inversion: i (the element that corresponds to this operation is a center of symmetry or inversion center) . The operation is to move every atom in the molecule in a straight line through the inversion center to the opposite side of the molecule. Therefore XeF4 posses an inversion center at the Xe atom. 5 5. Improper Rotations: Sn n-fold rotation followed by reflection through mirror plane perpendicular to rotation axis also known as Rotation Reflection axis. It is an imaginary axis passing through the molecule, on which when the molecule is rotated by 2π/n angle & then reflected on a plane perpendicular to the rotation axis 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.
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