Spis treści: Molecular Symmetry


Contents
Preface xi

1 Symmetry Elements and Operations, s. 1

1.1 Introduction 1
1.2 Symmetry Elements and Operations 6 1.2.7 Proper Rotations: Cn, s. 6
1.2.2 The Plane of Symmetry: a, s. 9
1.2.3 The Inversion Centre: i, s. 13
1.3 Examples of the Impact of Geometric Symmetry on Chemistry 17 1.3.7 Oxygen Transfer via Metal Porphyrins 17 1.3.2 Nuclear Magnetic Resonance: Chemical Equivalence, s. 18
1.4 Summary, s. 22
1.5 Self-Test Questions, s. 23
Further Reading, s. 24

2 More Symmetry Operations and Products of Operations, s. 25

2.1 Introduction, s. 25
2.2 Background to Point Groups, s. 25
2.3 Closed Groups and New Operations, s. 26
2.5.1 Products of Operations, s. 26
2.5.2 Fixed Symmetry Elements, s. 29
2.3.3 The Final Missing Operation, Improper Rotations: Sn, s. 31
2.3.4 Equivalences for Improper Rotation Operations, s. 34
2.4 Properties of Symmetry Operations, s. 37
2.4.1 Equivalent Operations and Equivalent Atoms, s. 37
2.4.2 The Inverse of an Operation, s. 38
2.4.3 The Order of the Product; Operations that Commute, s. 39
2.5 Chirality and Symmetry, s. 41
2.6 Summary, s. 42
2.7 Completed Multiplication Tables, s. 43
2.8 Self-Test Questions, s. 44

3 The Point Groups Used with Molecules, s. 45

3.1 Introduction, s. 45
3.2 Molecular Classification Using Symmetry Operations, s. 45
3.3 Constructing Reference Models with Idealized Symmetry, s. 47
3.4 The Nonaxial Groups: Cs,Ch,Cl 48 3.4.1 Examples of Molecules for the Nonaxial Groups: Cs,Ch,Cl, s. 49
3.5 The Cyclic Groups: C„, S„ 50 3.5.1 Examples of Molecules for the Cyclic Groups: Cn, Sn, s. 52
3.6 Axial Groups Containing Mirror Planes: Cnh and Cnv 55 3.6.1 Examples of Molecules for Axial Groups Containing Mirror Planes: Cnh and Cm, s. 58
3.7 Axial Groups with Multiple Rotation Axes: Dn, Dnd and Dnh 59 3.7.1 Examples of Axial Groups with Multiple Rotation Axes: Dn, Dnd and Dnh, s. 61
3.8 Special Groups for Linear Molecules: Cxy and, s. 64
3.9 The Cubic Groups: Td and Oh, s. 65
3.10 Assigning Point Groups to Molecules, s. 69
3.11 Example Point Group Assignments, s. 69
3.11.1 Example 1: Conformations of Cyclohexane, s. 69
3.11.2 Example 2: Six-Coordinate Metal Complexes, s. 72
3.12 Self-Test Questions, s. 73

4 Point Group Representations, Matrices and Basis Sets, s. 75

4.1 Introduction, s. 75
4.2 Symmetry Representations and Characters, s. 75
4.2.1 Water, H20, C2v, s. 75
4.2.2 Direct Products, s. 79
4.3 Multiplication Tables for Character Representations, s. 81
4.4 Matrices and Symmetry Operations, s. 82
4.5 Diagonal and Off-Diagonal Matrix Elements 85 4.5.1 Ammonia, NH3, C3,, s. 85
4.6 The Trace of a Matrix as the Character for an Operation, s. 87
4.7 Noninteger Characters 88 4.7.1 Boron Trifluoride, BF3, D3h, s. 88
4.8 Reducible Representations 91 4.8.1 Water, H20, C2v, s. 91
4.9 Classes of Operations 93 4.9.1 [Ni(CN)4]2- D4h, s. 93
4.10 Degenerate Irreducible Representations 96 4.10.1 Ammonia, NH3 C3v, s. 98
4.11 The Labelling of Irreducible Representations, s. 100
4.12 Summary, s. 102
4.13 Completed Tables, s. 102
4.14 Self-Test Questions 102 Further Reading, s. 103

5 Reducible and Irreducible Representations, s. 105

5.1 Introduction, s. 105
5.2 Irreducible Representations and Molecular Vibrations, s. 107
5.3 Finding Reducible Representations, s. 110
5.4 Properties of Point Groups and Irreducible Representations, s. 112
5.5 The Reduction Formula 118 5.5.7 Applying the Reduction Formula, s. 120
5.6 A Complete Set of Vibrational Modes for H20, s. 122
5.7 Choosing the Basis Set 126 5.7.7 Carbonyl Stretching Modes of [Fe(CO)5 ], D3h, s. 126
5.8 The d-Orbitals in Common Transition Metal Complex Geometries 128 5.8.7 Square Planar, D4h, s. 132
5.8.2 Tetrahedral, Td, s. 137
5.8.3 Octahedral, Oh, s. 142
5.8.4 Trigonal Bipyramidal, D3h, s. 147
5.9 Linear Molecules: Groups of Infinite Order, s. 154
5.10 Summary, s. 161
5.11 Self-Test Questions, s. 162

6 Applications in Vibrational Spectroscopy, s. 163

6.1 Introduction 163
6.2 Selection Rules 165 6.2.1 Infrared Spectroscopy, s. 165
6.2.2 Infrared Absorption and the Greenhouse Gases, s. 173
6.2.3 Interstellar H2, s. 177
6.2.4 Raman Spectroscopy, s. 111
6.2.5 Comparison of Infrared and Raman Selection Rules, s.184
6.3 General Approach to Analysing Vibrational Spectroscopy, s. 186
6.3.1 Example: the C—H Stretch Bands of 1,4-Difluorobenzene, s. 187
6.4 Symmetry-Adapted Linear Combinations, s. 190
6.5 Normalization, s. 193
6.6 The Projection Operator Method 195
6.6.1 Projection Operator Applied to the C—H Stretches of 1,4-Difluorobenzene, s. 196
6.6.2 The Projection Operator and Degenerate Representation, s. 198
6.7 Linking Results for Symmetry-Inequivalent Sets of Atoms 202
6.7.1 Sets of Atoms Differing in Mass or Chemical Bond Strength, s. 203
6.8 Additional Examples, s. 206
6.8.1 Benzene, D6h, s. 206
6.8.2 The fac and mer Isomers of Transition Metal Complexes, s. 212
6.9 Summary, s. 215
6.10 Self-Test Questions 216 Further Reading, s. 217

7 Symmetry in Chemical Bonding, s. 219

7.1   Introduction, s. 219
7.7.1 Wave Phenomena and Interference, s. 220
7.7.2 The Born Interpretation of the Wavefunction, s. 222
7.2 Bond Energies, s. 225
7.2.1 The Symmetry-Adapted Linear Combinations for the Molecular Orbitals ofH2+ and H2, s. 228
7.2.2 The Chemical Bond Energy from Molecular Orbitals, s. 232
7.2.3 The Molecular Orbital Energy, s. 236
7.2.4 Bond Order, s. 238
7.3 The Relative Energies of Hydrogen-Like Atomic Orbitals, s. 239
7.3.1 Radial Behaviour of Atomic Orbitals, s. 239
7.3.2 The Relative Energies of Atomic Orbitals in Different Elements, s. 242
7.3.3 The Relative Energies of Atomic Orbitals from Electronegativity, s. 244
7.4 The Molecules Formed by Other Second-Row Elements with Hydrogen, s. 252
7.4.1 BeH2, Beryllium Hydride, s. 252
7.4.2 BH3, Boron Hydride, s. 253
7.4.3 CH4, Methane, s. 258
7.4.4 NH3, Ammonia, s. 264
7.4.5 H20, Water, s. 269
7.5 The Second-Row Diatomic Molecules, s. 270
7.5.1 Homonuclear Diatomics, s. 270
7.5.2 Heteronuclear Diatomics of Second-Row Elements, s. 276
7.6 More Complex Polyatomic Molecules, s. 278
7.6.7 Ethene, s. 278
7.7 Metal Complexes 284
7.7.1 Complexes Containing a -Donor Ligands, s. 284
7.7.2 The Jahn-Teller Effect, s. 287
7.7.3 Complexes Containing Ligand Orbitals of it-Symmetry, s. 291
7.8 Summary, s. 295
7.9 Self-Test Questions 296 Further Reading, s. 297

Appendices

Appendix 1  H20 Models for Identifying the Results of Symmetry Operation Products, s. 299
Appendix 2  Assignment of Chiral Centre Handedness using Cahn-Ingold-Prelog Rules, s. 303
Appendix 3   Model of a Tetrahedron and the Related Cube, s. 307
Appendix 4   Model of an Octahedron, s. 313
Appendix 5   Matrices and Determinants, s. 317

A5.1   Matrices as Representations of Symmetry Operators, s. 317
A5.1.1 Products of Matrices 318 A5.1.2    Products of Matrices, Expressed as Summations, s. 319
A5.2   Matrices for Solving Sets of Linear Equations, s. 321
Further Reading 324

Appendix 6   The Mathematical Background to Infrared Selection Rules, s. 325

A6.1   Model Based on Classical Mechanics, s. 325
A6.2   Model Based on Quantum Mechanics, s. 328
A6.3   Excited Vibrational States, s. 333
A6.4   Vibrational Modes for Polyatomic Molecules, s. 335
A6.5   Generalization to Arbitrary Transitions, s. 336
A6.6   Summary of Selection Rules, s. 337
Further Reading, s. 338

Appendix 7   The Franck-Condon Principle, s. 339
Appendix 8  Classical Treatment of Stokes/Anti-Stokes Absorption, s. 343
Appendix 9  The Atomic Orbitals of Hydrogen, s. 345

A9.1    Choice of Coordinate System, s.347
A9.2   Separation of Variables, s. 348
A9.3 The Angular Equation 349
A9.4   Physical Interpretation of the Angular EquationSolutions, s. 354
A9.5   Angular Momentum, s. 356
A9.6   The Radial Equation, s. 359
A9.7   The Complete Atomic Orbitals, s. 361
A9.8 Expectation Values, s. 364
A9.9   Real Combinations to Form the Familiar Atomic Orbitals, s. 367
A9.10 Cartesian Forms of the Real Angular Functions, s. 369
A9.11 Endnote on Imaginary Numbers, s. 370
Further Reading, s. 373

Appendix 10 The Origin of Chemical Bonding in, s. 375

A 10.1 Chemical Bond Formation, s.376
A 10.2 H Atom and H+Cation, s. 376
A10.3 The Virial Theorem, s. 379
A10.4 H2+Molecule, s. 381
A10.5 Choice of Coordinate System for Hp. Cylindrical Polar Coordinates, s. 383
A 10.6 H2+: the Electron Kinetic Energy, s. 386
A10.7 H2+: the Electronic Potential Energy, s. 387
A10.8 The Chemical Bond Formation Energy Based on Rigid Atomic Orbitals, s. 393
A10.9 Optimal Radial Decay of Molecular Orbitals, s. 396
Further Reading, s. 399

Appendix 11 H20 Molecular Orbital Calculation in C2v Symmetry, s. 401

Further Reading, s. 406

Appendix 12 Character Tables, s. 407

A 12.1 Non-Axial Groups, s. 407
A12.2 Axial Groups, s. 407
A12.2.1  C„ Groups, s. 407
A12.2.2 Sn Groups, s. 408
A12.2.3 Cnv Groups, s. 408
A12.2.4 Cnh Groups, s. 409
A12.2.5 Dn Groups, s. 410
A12.2.6 Groups, s. 411
A12.2.7 Dnh Groups, s. 412
A12.3 Cubic Groups, s. 413
A12.3.1  Tetrahedral, Td, s.413
A12.3.2 Rotational Subgroup of Td,T, s. 413
A12.3.3 Octahedral, Oh, s. 414
A12.3.4 Rotational Subgroup of Oh,0, s. 414
A12.4 Groups for Linear Molecules, s. 414

Index, s. 415

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