Preface ...................................................... xvii
Acknowledgments ............................................... xix
Chapter 1 Quantum Theory ....................................... 1
1.1 Introduction ............................................... 1
1.2 Electromagnetic Radiation .................................. 1
1.2.1 Polarization of EM Radiation ........................ 2
1.2.2 Planck's Law ........................................ 2
1.2.3 Photoelectric Effect ................................ 3
1.2.4 X-Rays .............................................. 4
1.3 Electrons .................................................. 5
1.3.1 Discovery of the Electron ........................... 6
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 Schrödinger Equation ..................... 11
1.4.1 Schrödinger'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 and Eigenfunctions ..................... 18
1.5.2 Hermitean Operators ................................ 19
1.5.3 Expectation Value .................................. 19
1.5.4 Separation of Variables ............................ 20
1.6 Variation Principle: Linear Expansion ..................... 22
1.6.1 Energy Expectation Value is ≥ E0, 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
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 Schrödinger 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 Size of Orbitals ................................... 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 Self-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
Disproportionation Energy .......................... 54
2.3.5 Koopmans' Theorem .................................. 55
2.3.6 Møller-Plesset (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 States and Oxidation Potentials .......... 70
2.5.4 Hybridization of Atomic Orbitals ................... 71
Chapter 3 Molecules ........................................... 75
3.1 Introduction .............................................. 75
3.2 Chem ical 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
3.2.5 Ionic Bonds ........................................ 81
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 π-Systems ................................... 91
3.4.4 Linear π-Systems ................................... 94
3.4.5 Alternant Systems .................................. 96
3.4.6 Fullerenes ......................................... 98
3.5 Excited States ........................................... 100
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 π-Systems ............ 102
3.5.5 PPP Model ......................................... 105
3.5.6 Singlets and Triplets ............................. 106
Chapter 4 Nuclear Motion ..................................... 109
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 ................................ 111
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 ...................... 118
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
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 Helmholtz 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 Gibbs-Helmholtz Equation .......................... 153
5.4.4 Gibbs Energy for Ideal Gas ........................ 154
5.4.5 Law of Mass Action ................................ 155
5.4.6 Connection between Gø and К ....................... 156
5.4.7 Temperature Dependence of Equilibrium Constant .... 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-Dirac 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
Chapter 6 Ions in Crystals and in Solution ................... 173
6.1 Introduction ............................................. 173
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 Rates: Fermi Golden Rule .................... 201
Chapter 8 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
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 Rates 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 Adiabatic and Nonadiabatic Transfer .......... 247
10.4.2 Reorganization Energy (λ) .................... 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 Rate of ET Reactions ................................ 258
10.5.1 Gibbs Free Energy of ET Reaction ............. 258
10.5.2 Adiabatic, Asymmetric System ................. 259
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 bc1 Complex ....................................... 307
11.4.6 Cytochrome с Oxidase .............................. 307
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
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 Photokinetics ..................................... 330
12.4.2 Femtochemistry .................................... 332
12.4.3 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 π-Systems as Absorbers of Light Energy ............ 335
12.5.2 Photochemical Reactions ........................... 335
12.5.3 Cis-trans Isomerization ........................... 336
12.5.4 Polyenes .......................................... 336
12.5.5 Carotenoids ....................................... 338
12.5.6 Retinal and Vision ................................ 339
Chapter 13 Photoinduced 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 ................ 344
13.2.3 Emission from Charge-Separated States ............. 345
13.2.4 Triplet Formation by Charge Transfer .............. 345
13.3 Polari zation Energy ..................................... 347
13.3.1 Reaction Field .................................... 347
13.3.2 Rehm-Weller Equation .............................. 347
13.4 Intermolecular and Intramolecular PIET ................... 350
13.4.1 Rate of PIET ...................................... 350
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
Charges ........................................... 368
14.4 Fluorescence Resonance Energy Transfer ................... 371
14.4.1 Interaction between Spin-Singlet Excitations
(Förster) ......................................... 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 Chl and BChl ...................................... 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 Photosystem I ..................................... 385
15.5.2 Photosystem II .................................... 386
15.5.3 Binding of Carbon Dioxide: RuBisCo ................ 387
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 Effective Mass .................................... 405
16.3.5 Conductivity in Allotropic Forms of Carbon ........ 406
16.4 Localization-Delocalization .............................. 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 Holstein 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 TC 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 States ............................... 432
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 State .................... 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 π-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 Derealization Model for PA ........................ 460
18.5.3 Behavior of Three-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
Bibliography .................................................. 469
Appendices .................................................... 485
Index ......................................................... 499
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