Chemical physics : electrons and excitations · CHEMICAL PHYSICS ELECTRONS and EXCITATIONS Sven...
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CHEMICAL PHYSICS ELECTRONS and EXCITATIONS Sven Larsson C Kress ujl*' J Taylor & Francis Croup Boca Raton London NewYork CRC Press is an imprint of the Taylor & Francis Group, an informa business
Transcript of Chemical physics : electrons and excitations · CHEMICAL PHYSICS ELECTRONS and EXCITATIONS Sven...
CHEMICALPHYSICS
ELECTRONS and EXCITATIONS
Sven Larsson
C Kress
ujl*' J Taylor & Francis CroupBoca Raton London NewYork
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Contents
Preface *v"
Acknowledgments xix
Chapter 1 Quantum Theory 1
1.1 Introduction I
1.2 Electromagnetic Radiation I
1.2.1 Polarization ol'HM Radiation 2
1.2.2 Planck's Law 2
1.2.3 Photoelectric Effect 3
1.2.4 X-Rays 4
1.3 Electrons 3
1.3.1 Discovery of the Electron f>
1.3.2 Quantum Conditions in the Atom 6
1.3.3 Old Quantum Theory 7
1.3.4 Matter Waves 8
1.4 Time-Independent Sehriidinger Equation 11
1.4.1 Scliriidinger's Standing Waves 11
1.4.2 Particle in a Box 12
1.4.3 Finite Walls, Tunneling 14
1.4.4 Interpretation of the Wave Function 15
1.5 Mathematical Background 18
1.5.1 Eigenvalues aiul Eigenfunctions 18
1.5.2 I-Iermileau Operators 19
1.5.3 Expectation Value ll->
1.5.4 Separation of Variables 20
I .o Variation Principle: Linear Expansion 22
1.6.1 Energy Expectation Value is >E„, the Lowest
Eigenvalue of H 22
1.6.2 Linear Expansion 23
1.7 Spin 23
1.7.1 Spin of a Single Electron 24
1.7.2 Properties of Spin Functions 25
1.7.3 Spin Multiplicity 28
1.8 Many-Electron Theory 30
1.8.1 Wave Function for Many Electrons 30
1.8.2 Pauli Exclusion Principle 31
1.8.3 Independent Electron Model 32
1.8.4 Correlation Hole 34
1.8.5 Correlation Energy 35
v
vi Contents
1.8.6 Configuration Interaction 36
1.8.7 Electronic Density Matrix 36
1.8.8 Correlation Error 38
Chapter 2 Atoms 41
2.1 Atomic Units 41
2.2 Hydrogen Atom 42
2.2.1 Time-Independent Schrodinger Equation for the
Hydrogen Atom 42
2.2.2 Angular Function 43
2.2.3 Radial Function 46
2.2.4 Energy Spectrum 47
2.2.5 SizeofOrbitals 48
2.2.6 Slater's Screening Rules: Size of Atoms 49
2.3 Equation of Motion for Single Electrons 50
2.3.1 Hartree's Sell-Consistent Field (SCF) Method 50
2.3.2 Hartree-Fock 51
2.3.3 Brillouin's Theorem 53
2.3.4 Ionization Energy, Electron Affinity, and
Disproportionate)! Energy 54
2.3.5 Koopmans' Theorem 55
2.3.6 M0ller-Plessel (MP) Theorem 57
2.3.7 Best Overlap Orbitals 58
2.3.8 Exchange Hole 59
2.3.9 Local Exchange and Density Functional Theory 61
2.3.10 DFT Method as a Practical Calculation Method 62
2.4 Correlation and Multiplet Theory 63
2.4.1 Hylleraas'Method 63
2.4.2 Central Field Approximation 65
2.4.3 Correlation 67
2.5 Atoms in Chemistry 68
2.5.1 Periodic Table of the Elements 68
2.5.2 Hydrogen Atom 69
2.5.3 Oxidation Slates and Oxidation Potentials 70
2.5.4 Hybridization of Atomic Orbitals 71
Chapter 3 Molecules 75
3.1 Introduction 75
3.2 Chemical Bonding 75
3.2.1 Hydrogen Molecule, H2 75
3.2.2 Representation of MO 78
3.2.3 Homonuclear Diatomic Molecules 78
3.2.4 Heteronuclear Diatomic Molecules 80
Contents VM
3.2.5 Ionic Bonds SI
3.2.6 Bond Distance Depends on Occupation 82
3.3 Polyatomic Molecules 83
3.3.1 Water Molecule 84
3.3.2 Saturated Hydrocarbons 85
3.3.3 Aromatic (Unsaturated) Hydrocarbons 86
3.4 Hiickel Model for Aromatic Hydrocarbons 87
3.4.1 Hiickel Model 87
3.4.2 Bond-Length-Dependent Couplings 91
3.4.3 Cyclic jc-Syslems 91
3.4.4 Linear 7t-Systems 94
3.4.5 Alternant Systems 96
3.4.6 Fullerencs 98
3.5 Excited Slates I""
3.5.1 Diatomic Molecules 101
3.5.2 Aromatic Molecules 101
3.5.3 Transition Moment 102
3.5.4 Spectra of Cyclic and Linear jr-Syslems 102
3.5.5 PPP Model 105
3.5.6 Singlets and Triplets 106
Chapter 4 Nuclear Motion '09
4.1 Introduction 109
4.2 Separation of Electronic and Nuclear Coordinates 109
4.2.1 Born-Oppenheimer Approximation 109
4.2.2 Nuclei Move on PES 110
4.2.3 Calculation of PES HI
4.2.4 Isotope Effects and Isotope Separation 112
4.2.5 Hellman-Feynman Theorem 113
4.2.6 Car-Parinello Approach 115
4.3 Classical Molecular Dynamics 115
4.3.1 Classical Harmonic Oscillator 116
4.3.2 Anharmonic Motion 118
4.3.3 Small Molecular Oscillations H8
4.3.4 Eigenvalue Equation 120
4.3.5 Molecular Dynamics Simulation 120
4.4 Quantization of Vibrations 122
4.4.1 Harmonic Oscillator 122
4.4.2 Small Vibrations 125
4.5 Vibrational Spectra 126
4.5.1 Intensity in Infrared Spectra 126
4.5.2 IR Frequency Depends on Type of Bond 128
4.5.3 Raman Spectra 129
4.5.4 Rotation Spectra 129
Contents
4.6 Vibrations in Electronic Spectra 132
4.6.1 Vibrational Broadening 132
4.6.2 Franck-Condon Factors 133
4.7 PES Crossing 134
4.7.1 Avoided Crossing 134
4.7.2 Vibration Spectrum in Double Minimum PES 136
Chapter 5 Statistical Mechanics 139
5.1 Introduction 139
5.2 Partition Function and Thermodynamic Properties 140
5.2.1 Boltzmann Distribution 140
5.2.2 Partition Function 142
5.2.3 Internal Energy 142
5.2.4 Entropy 143
5.2.5 Hclmhollz Free Energy 144
5.2.6 Pressure 145
5.2.7 Enthalpy 145
5.2.8 Gibbs' Free Energy 146
5.2.9 Maxwell Relations 147
5.3 Internal Energy and Heat Capacity in Gas Phase 148
5.3.1 Translational Contribution 148
5.3.2 Internal Energy and Heat Capacity due to
Vibrations 150
5.4 Chemical Reactions 152
5.4.1 Chemical Potential 152
5.4.2 Gibbs-Duhem Equation 153
5.4.3 Gibhs-Hclmhollz Equation 153
5.4.4 Gibbs Energy for Ideal Gas 154
5.4.5 Law of Mass Action 155
5.4.6 Connection between Giy and K 156
5.4.7 Temperature Dependence of EquilibriumConstant 156
5.5 Equilibrium Statistical Mechanics Using Ensembles 157
5.5.1 Phase Space 157
5.5.2 Problems in the Earlier Derivation 158
5.5.3 Canonical Ensemble 159
5.5.4 Grand Canonical Ensemble 160
5.5.5 Fermi-Diiac and Bose-Einstein Statistics 163
5.5.5.1 FD Statistics 164
5.5.5.2 BE Statistics 165
5.6 Nonequilibrium Statistical Mechanics 166
5.6.1 Maxwell Velocity Distribution 166
5.6.2 Kinetic Theory of Gases 167
5.6.3 Molecular Dynamics: The Entropy Problem 169
5.6.4 Diffusion 170
Contents IX
Chapter 6 Ions in Crystals and in Solution '73
6.1 Introduction '73
6.2 Ions in Aqueous Solution 173
6.2.1 Solvent Structure around Ions 174
6.2.2 Born Equation 175
6.3 Crystals 176
6.3.1 Crystal Directions and Planes: Unit Cell and
Reciprocal Space 176
6.3.2 Crystal Systems 179
6.3.3 Lattice Enthalpy 180
6.4 Crystal Field Theory for Transition Metal Ions 182
6.4.1 Transition Metal Oxides and Salts 183
6.4.2 Energy Levels 183
6.4.3 High Spin and Low Spin 184
6.4.4 Problems with Crystal Field Theory 185
6.5 Ligand Field Theory 186
6.5.1 Extension to LFT 186
6.5.2 Localized or Delocalized Excitations 188
6.5.3 First-Order Jahn-Teller Effect 188
6.5.4 L -» M Charge Donation 189
Chapter 7 Time-Dependent Quantum Mechanics 191
7.1 Introduction 191
7.2 Wave Equation 191
7.2.1 Time-Independent Energy Levels and Coefficients 191
7.2.2 Time-Dependent Energy Levels 194
7.2.3 Electron Transfer Dynamics 197
7.2.4 Landau-Zener Approximation 199
7.3 Time Dependence as a Perturbation 200
7.3.1 Time-Dependent Perturbation Theory 200
7.3.2 Decay Rales: Fermi Golden Rule 201
Chapters Chemical Kinetics 207
8.1 Introduction 207
8.2 Rate of Chemical Reactions 207
8.2.1 Reversible and Irreversible Reactions 207
8.2.2 Activation Energy 208
8.2.3 Elementary Reactions 209
8.2.4 Rate Measurements 211
8.3 Integrated Rate Equations 212
8.3.1 Irreversible Reactions of First Order 212
8.3.2 Irreversible Reactions of Second Order 212
8.3.3 Irreversible Reactions of Zeroth Order 214
8.3.4 Unimolecular Reversible Reaction 214
X Contents C
8.4 Consecutive Reactions 216
8.4.1 Rate Derivation 216
8.4.2 Steady State Assumption 218
Chapter 9 Proton Transfer 219
9.1 Introduction 219
9.1.1 Acid-Base Concept of Br0nsted 219
9.1.2 Acid-Base Equilibrium in Water 220
9.1.3 Proton Affinity 222
9.1.4 Hydration 223
9.2 Hydrogen Bonding 223
9.2.1 Typical Hydrogen Bonds 224
9.2.2 Hydrogen Bonds in Proteins 225
9.2.3 Strength of a Hydrogen Bond 226
9.2.4 Potential Energy Surface 226
9.2.5 Coordinate System 229
9.2.6 Parabolic Model (Marcus Model for Proton
Transfer) 231
9.3 Proton Transfer 233
9.3.1 Rales of PT Reactions 233
9.3.2 Proton Tunneling 234
9.3.3 Grotthuss Effect 234
Chapter 10 Electron Transfer Reactions 237
10.1 Introduction 237
10.2 Homogeneous ET Reactions 237
10.2.1 Inner and Outer Sphere ET Reactions ..237
10.2.2 Electron Transfer Coupled to Proton Transfer 240
10.3 Electrochemistry 241
10.3.1 Electrochemical Cells 242
10.3.2 Thermodynamics of the Cell 243
10.3.3 Electrochemical Series 244
10.3.4 Latimer and Frost Diagrams 245
10.4 Marcus Parabolic Model for ET 246
10.4.1 Adiabalie and Nonadiabatic Transfer 247
10.4.2 Reorganization Energy (X) 250
10.4.3 Localized and Delocalized Mixed Valence 253
10.4.4 Wave Functions 256
10.4.5 Intensity of Intervalence Transition 257
10.5 Rale of ET Reactions 258
10.5.1 Gibbs Free Energy ofET Reaction 258
10.5.2 Adiabalie, Asymmetric System 259
Contents xi
10.5.3 ET Rate for Nonadiabatic Reaction 260
10.5.4 Electronic Factor k 261
10.5.5 Adiabatic and Nonadiabatic Limits 264
10.5.6 Miller's Experiment 264
10.6 Electronic Coupling 265
10.6.1 Gamow Model 266
10.6.2 Orbital Interaction Model 267
10.6.3 State Interaction Model 269
10.6.4 Direct Calculation of Electronic Coupling 270
10.6.5 Pathway Model •273
10.6.6 Nonexponential Decrease 274
10.6.7 Electron Transfer through a Solvent 275
10.7 Disproportionation 276
10.7.1 Examples of Disproportionation 276
10.7.2 Day-Hush Disproportionation Model 277
10.8 Quantized Nuclear Motion 279
10.8.1 PKS Model 280
10.8.2 Nuclear Tunneling 280
10.8.3 Vibrational Model for ET in the Limit of Low
Barrier 281
Chapter 11 Biological Electron Transfer 285
11.1 Introduction 285
11.2 The Living System 285
11.2.1 Formation of Life 286
11.2.2 Cells, Mitochondria, and Cell Membranes 288
11.2.3 Membrane Proteins 289
11.3 Electron Carriers and Other Functional Groups 290
11.3.1 Functional Groups 290
11.3.2 Carbohydrates and Lignin 292
11.3.3 Lipids 294
11.3.4 Nicotinamide Adenine Dinucleotide (NAD+) 295
11.3.5 Flavins 296
11.3.6 Quinones 297
11.3.7 Hemes and Cytochromes 298
11.3.8 Iron-Sulfur Proteins 298
11.4 Biological Electron Transfer 300
11.4.1 Electrons in the Electron Transport Chain 301
11.4.2 Electron Transfer Steps 303
11.4.3 More on Activation Energy 304
11.4.4 More on Coupling 305
11.4.5 be. Complex 307
11.4.6 Cytochrome c Oxidase 307
xii Contents t
Chapter 12 Photophysics and Photochemistry 309
12.1 Introduction 309
12.2 Photophysics 309
12.2.1 Absorption and Reflection of Light in Matter 309
12.2.2 Refraction and Diffraction 311
12.2.3 Lambert-Beer's Law 312
12.2.4 Laser Radiation 313
12.2.5 Absorption of Radiation in Atoms and
Molecules 315 C
12.2.6 Rate of Spontaneous Emission 319
12.3 Molecular Photophysics 323
12.3.1 Fluorescence: Stokes Shift 323
12.3.2 Internal Conversion 326
12.3.3 Spin-Orbit Coupling and Intersystem Crossing 326
12.3.4 Phosphorescence 327
12.3.5 Types of Spectra 328
12.3.6 Spectral Narrowing 329
12.4 Rate Measurements 329
12.4.1 Photokineties 330
12.4.2 Femtochemislry 33212.4.3 Laser Light in Chemistry 332
12.4.4 Transient Absorption Spectroscopy 333
12.4.5 Time-Resolved Resonance Raman Spectroscopy.... 333
12.4.6 Time-Resolved Emission Spectroscopy 334
12.5 Photochemistry: Mechanisms 334
12.5.1 K-Systems as Absorbers of Light Energy 335
12.5.2 Photochemical Reactions 335
12.5.3 Cis-lrans Isomerization 336
12.5.4 Polyenes 336 <
12.5.5 Carotenoids 338
12.5.6 Retinal and Vision 339
Chapter 13 Pholoinduccd Electron Transfer 343
13.1 Introduction 343
13.2 Charge Transfer Transition in Spectra 343
13.2.1 Charge Transfer States as Excited States 343
13.2.2 Mulliken Charge Transfer Complexes 34413.2.3 Emission from Charge-Separated States 345
13.2.4 Triplet Formation by Charge Transfer 34513.3 Polarization Energy 347
13.3.1 Reaction Field 34713.3.2 Rehm-Weller Equation 347
13.4 Inlermolecular and Intramolecular PIET 35013.4.1 Rate of PIET 350
Contents
13.4.2 Intermolecular PIET in Solution and in Glass 352
13.4.3 Charge Recombination 353
13.4.4 Intramolecular PIET 353
13.4.5 Intramolecular Charge Transfer 353
13.4.6 Fullerene Systems 354
13.4.7 Other Intramolecular PIET Experiments 355
13.5 Molecular Photovoltaics 356
Chapter 14 Excitation Energy Transfer 359
14.1 Introduction 359
14.2 Excited States of Bichromophores 359
14.2.1 Chromophores 359
14.2.2 Wave Functions and Matrix Elements of
Bichromophores 363
14.2.3 Covalent Bonding in Ground and Excited States 365
14.3 Transition Moments 366
14.3.1 Transition Densities 366
14.3.2 Energy Order of Dimer Exciton States 367
14.3.3 Distant Chromophores Interact via Transition
Charges368
14.4 Fluorescence Resonance Energy Transfer 371
14.4.1 Interaction between Spin-Singlet Excitations
(Forster) 371
14.4.2 The Mysterious Factor of Two 373
14.4.3 Dexter Coupling 373
14.4.4 Rate Equations for EET 374
Chapter 15 Photosynthesis 375
15.1 Introduction 375
15.2 Molecules of Photosynthesis 375
15.2.1 ChlandBChl 375
15.2.2 Carotenoids 377
15.2.3 Phycocyanobilins 377
15.3 Antenna Systems 379
15.3.1 Purple Bacteria Antenna Systems 379
15.3.2 Green Plant Antenna Systems 381
15.4 Bacterial Reaction Centers 382
15.4.1 Structure 382
15.4.2 Charge Separation and ET 383
15.5 Green Plant Photosynthesis 385
15.5.1 PhotosystemI 385
15.5.2 Photosystem II 386
15.5.3 Binding of Carbon Dioxide: RuBisCo 387
x|v Contents
Chapter 16 Metals and Semiconductors 389
16.1 Introduction 389
16.2 Free Electron Models and Conductivity 389
16.2.1 Resistivity and Conductivity 390
16.2.2 Drude Model 391
16.2.3 Atomic Orbital Overlap 393
16.2.4 Free Electron Model in One Dimension 395
16.2.5 Bethe-Sommerfeld Model 396
16.2.6 Conductivity in a Periodic Potential at T = 0 398
16.2.7 Conductivity at Elevated Temperature 400
16.3 Tight-Binding Model 400
16.3.1 One-Dimensional Model 401
16.3.2 Peierl's Distortion 402
16.3.3 Bloch Band Model 404
16.3.4 Effeclive Mass 405
16.3.5 Conductivity in Allolropic Forms of Carbon 406
16.4 Localization-Delocali/ation 407
16.4.1 Metal-Insulator Transition 407
16.4.2 Polarons 409
16.4.3 Mott Insulators 409
16.4.4 Simple Model for Metal-Insulator Transition 411
16.4.5 Holslcin Model 412
16.5 Semiconductors 413
16.5.1 Bonding Conditions in Diamond 413
16.5.2 Conductivity and Doping in Semiconductors 415
16.5.3 p-n Junctions 416
16.5.4 Solar Cells 417
16.6 Phonons 418
Chapter 17 Conductivity by Electron Pairs 419
17.1 Introduction 419
17.2 Superconductivity 419
17.2.1 Experiment and Theory 419
17.2.2 Meissner Effect 421
17.2.3 Metal-Ammonia Solution 421
17.2.4 Cooper Pairs and the BCS Model 423
17.2.5 High Tr Superconductivity 425
17.3 Coupling and Correlation in Electron Pair Transfer 426
17.3.1 Mott's Justification of Hubbard U 426
17.3.2 Application of the MV-3 Model 427
17.3.3 Intermetal Coupling 429
17.3.4 Stable Charged States 430
17.3.5 Spin-Coupled Slates 432
Contents xv
17.4 MV-3 Systems in the State Overlap Region 433
17.4.1 Calculation of Hubbard U: Born Effect 433
17.4.2 Fullerene Superconductivity 434
17.4.3 Cuprate Superconductivity 435
17.4.4 Bismuthates 436
17.5 Pair Conductivity in the Ground Stale 436
17.5.1 Cyclobutadiene with Equal Bond Lengths 436
17.5.2 Wave Functions at the van Hove Singularity
(x = 0) 438
17.5.3 Vibronic Wave Functions 442
17.5.4 Final Wave Function 443
Chapter 18 Conductivity in Organic Systems 445
18.1 Introduction ' 445
18.2 Organic Semiconductors 445
18.2.1 Electrons and Excitations in Organic Molecular
Crystals 445
18.2.2 Conductivity in Organic Systems 447
18.2.3 Charge Transfer Spectra 448
18.2.4 Organic Light-Emitting Diodes 451
18.3 Stacked, Conducting re-Systems 452
18.3.1 TTF-TCNQ 452
18.3.2 Bechgaard Salts: Organic Superconductors 453
18.4 Conducting Polymers 454
18.4.1 (SN)x 454
18.4.2 Polyacetylene 455
18.4.3 Polyaniline 456
18.4.4 Other Conducting Polymers 457
18.5 Electronic Structure of One-Dimensional Crystals 460
18.5.1 Su-Schrieffer-Heeger Model 460
18.5.2 Delocalization Model for PA 460
18.5.3 Behavior ofThree-Quarter or One-Quarter
Filled Bands 461
18.5.4 Mobility of Electrons 462
18.5.5 Conductivity in DNA? 464
18.5.6 Conductivity at Low Temperatures 467
Bibliography469
Appendices485
Index 499
