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Table of Contents Chapter I. Symmetry and Group Theory I.1 Symmetry operations and symmetry elements 1 I.2 Groups 4 I.3 Similarity Transformations 5 I.4 Point Groups 6 I.5 Matrix Representations of Groups 8 I.6 Point Group Representations 10 I.7 Decomposing Reducible Representations 15 I.8 Direct Products 16 I.9 Symmetry Adapted Linear Combinations 18 Chapter II. Molecular Orbital Theory II.1 Quantum Theory – a brief tour 21 II.2 Wavefun
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  Table of Contents Chapter I. Symmetry and Group Theory I.1 Symmetry operations and symmetry elements 1 I.2 Groups 4 I.3 Similarity Transformations 5 I.4 Point Groups 6 I.5 Matrix Representations of Groups 8 I.6 Point Group Representations 10 I.7 Decomposing Reducible Representations 15 I.8 Direct Products 16 I.9 Symmetry Adapted Linear Combinations 18 Chapter II. Molecular Orbital Theory II.1 Quantum Theory – a brief tour 21 II.2 Wavefunctions as Bases for Irreducible Representations 22 II.3 Quantum Mechanical Approach to Molecular Orbitals 23 II.4 Homonuclear Diatomic Molecules 26 II.5 Orbital Mixing in the Nondegenerate Case 29 II.6 Orbital Energies 29 II.7 Polyatomic Molecules 31 II.8 Crystal Field Theory 35 II.9 Molecular Orbitals in Inorganic Complexes 38 II.9a Sigma Bonding in Octahedral ML 6  38 II.9b Sigma Bonding in Tetrahedral ML 4  40 II.9c Pi Bonding in Octahedral ML 6  41 II.9d Pi Bonding in Tetrahedral ML 4  42 II.9e Ferrocene 43 II.9f Electron Deficient Bonds 45 II.9g Linear Chains 46 Chapter III. General Spectroscopic Considerations III.1 Electromagnetic Radiations 50 III.2 Instrumentation 51 III.3 Time Dependent States 53 III.4 Experimental Quantities 54 Chapter IV. Vibrational Spectroscopy – Part I. Theory IV.1 The Harmonic Oscillator – a Classical view 56 IV.2 Quantum Mechanical Description of the Harmonic Oscillator 58 IV.3 Selection Rules for the Harmonic Oscillator 59 IV.4 The Anharmonic Oscillator 59 IV.5 The Wilson FG Matrix Forumulation of Molecular Vibrations 61 IV.6 Symmetry Coordinates 62 IV.7 The Linear Modes of X-Pt-Pt-X of [Pt 2 (pop) 4 X 2 ] 4-  64 IV.8 The Potential Energy Distribution 68 IV.9 Overtones and Combinations 69 IV.10 Raman Scattering 71 IV.11 Raman Selection Rules 73 IV.12 Infrared and Raman Intensities 74 IV.13 A Complete Vibrational Assignment of SbCl 5  75 IV.14 Pseudorotation in a Trigonal Bipyramid 78 IV.15 Multi-minima Potential Functions 79 IV.16 Use of IR and Raman to Deduce Isomeric Forms 81 IV.17 Vibrational Spectra of Solids 83    Chapter V. Vibrational Spectroscopy Part II. Examples V.I The Effect of Mass 85 V.2 The Effect of the Oxidation State of the Metal 86 V.3 Amines 86 V.4 Nitro and Nitrito Complexes 87 V.5 Carbonyl Complexes 87 V.6 Cyano Complexes 88 V.7 Nitrosyl Complexes 89 V.8 Olefin Complexes 90 V.9 Metal-Ligand Multiple Bonds 92 V.10 Metal-Metal Bonds 93 Chapter VI. Electronic Spectroscopy. Part I. Theory VI.1 Electron Spin 94 VI.2 States 95 VI.3 Selection Rules 100 VI.4 Electronic Transitions 101 VI.5 Vibronic Coupling 104 VI.6 Jahn-Teller Distortions 105 VI.7 Spin Orbit Coupling 107 VI.8 Resonance Raman 109 VI.9 Relaxation Pathways 111 VI.10 Bandshapes 113 VI.11 Emission vs. Absorption 114 VI.12 Ligand Field Spectra 115 VI.13 Charge Transfer Spectra 121 Chapter VII. Electronic Spectroscopy. Part II. Examples VII.1 Metal-ligand Multiple Bonding in trans -Dioxorhenium(V) 124 VII.2 Metal-Metal Bonding in Pt 2 ( µ -P 2 O 5 H 2 ) 4+  129 VII.3 Mixed Valence Species 132 VII.4 Spatially Isolated Orbitals in Ru(bpy) 32+  136 VII.5 Metalloporphyrins 139 Appendices A. Character Tables of Selected Point Groups B. Direct Products for Selected Point Groups C. Fundamental Constants D. Calculated Valence Orbital Energies, H ii    Chapter I. Symmetry and Group Theory 1 I. Symmetry and Group Theory. All of the topics covered in this course make extensive use of molecular symmetry and group theory - topics of entire books by themselves. It is the aim of this text to present group theory in sufficient detail so that the student can solve most day to day spectroscopic problems and read the spectroscopy literature. A symmetry operation  is a rotation and/or reflection which leaves the molecule unchanged. It is performed about a symmetry element : a point, a line or a plane. The square  planar PtCl 42-  ion is said to be a higly symmetric ion because it contains a large number of symmetry elements. All of the symmetry operations that apply to a molecule or ion constitute its  point group . It is the point group to which the molecule belongs that designates its symmetry. In the next section, it will be shown that sixteen symmetry operations can be performed on the PtCl 42-  ion. These sixteen operations constitute what is known as the D 4h  point group, i.e ., PtCl 42-  has D 4h  symmetry. In a treatment of the spectroscopy or bonding of an inorganic compound, one must first determine the point group to which the system belongs. A full appreciation of what D 4h  symmetry implies requires the visualization of the symmetry elements and operations as well as an understanding of groups. In the following two sections, the various symmetry operations and groups will be discussed while the point group definitions are given in section I.4. The remainder of the chapter is devoted to representations of the point groups and to some simple applications of group theory to spectroscopy and bonding. I.1 Symmetry operations and symmetry elements There are only five types  of symmetry operations required for the systems covered in this course, and the PtCl 42-  ion contains examples of each. In this section, each type of operation will  be discussed in terms of its effect on this ion. The effect of the operations on the chlorine p z  orbitals will also be considered in order to better illustrate the effects of the operations. Thus, in the figures that follow, !  will represent chlorine atom 1 with the positive lobe of the p z  orbital out of the plane of the paper while  will imply that the negative lobe of the p z  orbital is out of the plane of the paper for chlorine atom 1. 1. The Identity Operation  ( E ) does nothing and has no symmetry element but it is a required member of each symmetry group. Thus, operation with E  will change neither the  positions of the atoms nor the phases of the p z  orbitals. Figure I-1. The effect of the identity operation on the atoms and the chlorine p z  orbitals in PtCl 42- . Also the definitions of the X, Y, a and b axes relevant to other discussions are defined. 2. An n-fold rotation ( C n ) is a rotation of 2 ! /n radians about an axis. The axis with the highest value of n is the principle axis  and is designated as the z-axis. Thus the z-axis in PtCl 42- is perpendicular to the plane of the ion. This axis is actually three symmetry elements since rotations by ! /2, ! , and 3 ! /2 about this axis all result in no change in the molecule. These  Chapter I. Symmetry and Group Theory 2 three axes are referred to as C 4 , C 42 = C 2  and C 43 , respectively. 1  Rotation about the z-axis will not change the phase of the p z  orbitals. The X, Y, a and b axes defined in figure I-1 are also C 2  rotational axes. It will be shown later that the C 2  rotations in this group can be grouped into three classes  which are differentiated with the use of ' and {C 2 (Z)},{C 2 '(X) & C 2 '(Y)}, {C 2 (a) & C 2 (b)}. Since the C 2 ' and C 2 axes are perpendicular to the z-axis, rotation about any one of them will invert the p z  orbitals as shown in figure I-2. Figure I-2. The effect of some of the C n  operations on the atoms and the chlorine p z  orbitals in PtCl 42- . Thus the ion contains C 4  , C 43  , C 42 = C 2 , C 2 '(x) ,C 2 '(y), C 2 (a) and C 2 (b) or 7 rotational axes. 3. Reflections can be made through three different types of planes:   vertical planes (   v ) contain the principle axis , horizontal planes (   h ) are perpendiuclar to the principle axis and dihedral planes (   d ) contain the principle axis and bisect two C 2  axes. The distinction between vertical and dihedral is often unclear. Where appropriate, planes bisecting bond angles will be designated as dihedral while those containing bonds will be designated as vertical. See figure I-3. 1  Since the clockwise C 43  operation is equivalent to a counterclockwise C 4  rotation, the C 4  and C 43  operations are also referred to as the C 4+  and C 4- operations, respectively.

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Jul 23, 2017

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