Conical Intersections: Electronic Structure, Dynamics & Spectroscopy

CONTENTS......Page 16
Introduction......Page 6
Preface......Page 8
Historical Introduction J. Michl......Page 10
PART I. Fundamental Concepts and Electronic Structure Theory......Page 20
1. Born–Oppenheimer Approximation and Beyond L.S. Cederbaum......Page 22
1. Introduction......Page 23
2. The Born–Oppenheimer Expansion......Page 24
3.1. Separation of Electronic and Nuclear Motion......Page 29
3.2. Group-Born–Oppenheimer Approximation......Page 30
4.1. Gauge Invariance of Group-Born–Oppenheimer Approximation......Page 33
4.2. Strictly Diabatic and Quasidiabatic Electronic States......Page 35
5.1. General......Page 42
5.2. Traditional Approach for Quasi-Rigid Molecules......Page 46
6.1. Born–Oppenheimer Approximation in Magnetic Fields......Page 49
6.2. Conical Intersections and Molecular Rotations......Page 51
6.3. Born–Oppenheimer Approximation for Continuum Electronic States?......Page 54
References......Page 56
2. Conical Intersections: Their Description and Consequences D.R. Yarkony......Page 60
1. Introduction......Page 62
1.1.1. By Electronic State Symmetry: The Noncrossing Rule......Page 63
1.1.2. By Topography......Page 64
1.2. Derivative Couplings......Page 65
2.1. Born–Huang Approach......Page 66
2.2. Adiabatic Electronic States......Page 67
2.3.1. A Model Hamiltonian......Page 68
2.3.2. Intersection Adapted Coordinates......Page 69
2.3.3. Describing Conical Topographies: Linear Terms — Pitch, Asymmetry and Tilt......Page 70
2.3.4. Describing Conical Topographies: Second Order Terms — Branches, Confluences and Seam Curvature......Page 71
2.3.5. Illustrative Calculations......Page 72
2.3.6.2. Geometric Phase Effect for Real-Valued Wave Functions......Page 74
2.3.6.3. The η = 3 and 5 Cases......Page 75
2.3.7.1. Derivative Couplings from Model Hamiltonians......Page 76
2.3.7.2. Derivative Couplings and the Geometric Phase......Page 78
2.3.7.3. A Useful Approximation......Page 79
2.4. The Nuclear Schrodinger Equation......Page 80
3. Perturbation Theory......Page 84
3.1. Crude Adiabatic Basis......Page 85
3.2. First Order: W(1),f......Page 87
3.4. Intersection-Adapted Hyperspherical Coordinates......Page 89
3.4.1. Eigenvalues in Terms Hyperspherical Coordinates......Page 91
3.5. Orthogonal Intersection-Adapted Coordinates......Page 92
3.6. Eigenvectors......Page 93
3.7. Second Order: W(2)......Page 94
3.7.1. Parametrization......Page 95
3.7.2. Confluences......Page 96
3.7.3. Portion of the Wave Function not Determined by the Schrodinger Equation......Page 98
3.8.1. First Order......Page 99
3.8.1.3. η = 5......Page 100
3.8.2. Second Order......Page 102
3.9. The Geometric Phase......Page 103
4. Numerical Examples......Page 104
4.1.1. Motivation......Page 105
4.1.3. Energies and Derivative Couplings......Page 106
4.1.3.1. Energy and Derivative Couplings from Perturbation Theory......Page 107
4.1.3.2. Conical Parameters Along the Seam......Page 109
4.1.3.3.1 cis Confluences......Page 111
4.1.3.3.2 trans Confluences......Page 113
4.1.4. Implications for the S1 S0 Internal Conversion......Page 114
4.2.1. Motivation......Page 115
4.2.2. Wave Functions......Page 116
4.2.3. Orthogonal Intersection Adapted Coordinates......Page 117
5.1. Seam Curvature and Higher Order Effects......Page 122
5.2. Three State Intersections: Coalescences......Page 123
Appendix A. Time-Reversal Adapted Configuration State Functions......Page 124
Appendix B. Eigenvalues and Eigenvectors of a 2 × 2 Hermitian Matrix......Page 125
B.1. The General Case......Page 126
B.2. The Linear Hamiltonian......Page 127
Appendix C. Eigenvalues and Eigenvectors of a 4 × 4 Hermitian Matrix in a Time Reversal Adapted Basis......Page 128
D.1. Scaled Coordinates......Page 131
D.2. Unscaled Coordinates......Page 132
Appendix E. Degenerate Perturbation Theory......Page 133
Appendix F. Orthogonal Intersection Adapted Coordinates......Page 134
F.1.1. η = 5......Page 135
F.2. Solutions......Page 136
Appendix G. The Phase of the Electronic Wave Function......Page 138
Appendix H. Evaluation of As k,l s = x, y, z, v,w......Page 139
References......Page 142
3. Determination of Potential Energy Surface Intersections and Derivative Couplings in the Adiabatic Representation D.R. Yarkony......Page 148
1.1. Background......Page 149
1.2. Scope......Page 152
2.2. Nonrelativistic Wave Functions......Page 153
2.2.1. Multireference Configuration Interaction Wave Functions......Page 154
3. Locating Conical Intersections of Born–Oppenheimer Potential Energy Surfaces......Page 155
3.1.1. The Energy Difference Gradient......Page 157
3.1.2. The Algorithm......Page 158
3.2. A Lagrange Multiplier Based Approach......Page 159
4. Matrix Elements and Generalized Density Matrices......Page 163
5.1. State-Averaged Multiconfigurational Self-Consistent Field Method......Page 165
5.2.1. Variational and Orthogonality Contributions......Page 169
5.2.2. Additional Constraints on the Molecular Orbitals......Page 173
6.1. Evaluation of gi,,j , hi,,j and CIai,j......Page 174
6.2. Computational Efficiencies......Page 176
7.1.1. CIAi,j......Page 177
7.1.2. CSFAi,j......Page 178
7.2. Second Derivative Coupling......Page 179
7.2.1. Evaluation of CSFk(X)......Page 181
7.2.2. Evaluation of CIk and CSF–CIk: the Coupled Perturbed-CI Equations......Page 182
8.1. Locating Conical Intersections in the η = 2 Case......Page 183
8.2. Locating Conical Intersections in the η = 3 Case......Page 184
8.3. The Adiabatic Correction: LiH(X1Σ+)......Page 186
9. Summary and Future Directions......Page 189
References......Page 190
1. Introduction......Page 194
2.1. Adiabatic and Crude Adiabatic States......Page 197
2.2. Diabatic Electronic States......Page 199
3.1. Derivative-Based Methods......Page 203
3.2. Property-Based Methods......Page 206
3.3. Energy-Based Methods......Page 208
4. The Concept of Regularized Diabatic States......Page 211
5. Concluding Remarks......Page 219
Acknowledgment......Page 220
References......Page 221
5. Modeling and Interpolation of Global Multi-Sheeted Potential Energy Surfaces A.J.C. Varandas......Page 224
1. Introduction......Page 225
2. The Many-Body Problem......Page 227
2.1. Approximations and Concept of Potential Energy Surface......Page 228
2.2. Geometric Phase, Adiabatic Potentials and Derivative Couplings......Page 230
3. Strategies for Multi-Sheeted Potential Energy Surfaces......Page 235
3.1. General Rules......Page 236
3.2.1. Diatomics-in-Molecules......Page 240
3.2.2. Double Many-Body Expansion......Page 243
3.2.3. Reproducing Kernel Hilbert Space......Page 246
3.2.4. Energy-Switching......Page 248
3.3.1. Shepard Interpolation......Page 249
3.3.2. Moving Least-Squares......Page 250
4.1. Three-Atom Three-Electron Systems......Page 252
4.1.1. H3......Page 254
4.1.2. Li3......Page 258
4.2. The NO2 Molecule......Page 261
4.3. H3O: A Prospective on Tetratomic Systems and Beyond......Page 273
4.4. Other Systems......Page 276
5. Concluding Remarks......Page 277
Appendix A. The Geometric Phase......Page 278
Appendix B. Diatomic Curves: The EHFACE2U Model......Page 279
Appendix C. Spin Wave Functions and Recoupling Matrices for H3......Page 280
Appendix D. Note Added in Proof......Page 282
References......Page 283
6. Conical Intersections and Organic Reaction Mechanisms A. Migani and M. Olivucci......Page 290
1. Introduction: The Search for the Photochemical Funnel......Page 291
2.1. General Structure of the Photochemical Reaction Path......Page 295
2.2. Bond Making, Bond Breaking, Group Exchange, Electron Transfer......Page 303
2.3. Chemical Reaction, Quenching and Deactivation......Page 308
3.1. Tracing and Recording Conical Intersections......Page 312
3.2. Types of Classification......Page 314
3.3.1. Conjugated Dienes and Polyenes......Page 318
3.3.2. Cyclocta-1,3,5,7-tetraene......Page 320
3.3.3. Conjugated Enones......Page 322
3.3.4. Benzene and [18]annulene......Page 323
3.3.5. Indole......Page 326
3.3.6. Tryptophan......Page 327
3.3.7. Cytosine......Page 329
3.3.8. Benzopyran......Page 331
4. Conclusions......Page 332
References......Page 334
PART II. Conical Intersections in Photoinduced and Collisional Dynamics......Page 340
7. The Multi-Mode Vibronic-Coupling Approach H. Koppel, W. Domcke and L.S. Cederbaum......Page 342
1. Introduction......Page 343
2.1. General Concepts......Page 345
2.2. Specific Models......Page 349
2.3. Ab initio Determination of Electronic-Vibrational Coupling Parameters......Page 353
3.1.1. Vibronic Eigenvalue Problem and Spectra......Page 355
3.1.2. Lanczos Algorithm......Page 357
3.2.1. Observables of the Time-Dependent Dynamics......Page 360
3.2.2. Integration of the Time-Dependent Schrodinger Equation......Page 362
4. Representative Applications......Page 365
4.1. Photoelectron Spectra of Triatomics: NO2 and O+3......Page 366
4.2. The Photoelectron Spectrum of Allene......Page 370
4.3. Absorption, Resonance Raman and Photoelectron Spectra of Pyrazine......Page 372
4.4. Absorption Spectra of Trans-Butadiene and Trans-Hexatriene......Page 377
5. Conclusions......Page 381
References......Page 382
8. Model Studies of the Dynamics at Conical Intersections A. Lami and G. Villani......Page 388
2. Basic Concepts and Simple Interpretation Schemes......Page 389
2.1. The Minimal Model for a CI......Page 390
2.2. The Vibronic Model......Page 391
2.3. The Time-Dependent One-Dimensional Model......Page 395
3. Exact Time-Dependent Results......Page 396
3.1. Diabatic vs. Adiabatic Populations......Page 397
3.2. Some Numerical Results and their Interpretation......Page 399
3.3. Initial Conditions and the Geometric Setup of Surfaces......Page 404
4. Model Studies of Photoisomerization Occurring via a CI......Page 405
4.2. The Solvent Role: A Stochastic Model for CT Associated with Large Amplitude Motions......Page 407
References......Page 411
1. Introduction......Page 414
2. Time-Dependent Quantum Wave-Packet Dynamics and Reduced Density-Matrix Dynamics......Page 417
3. Electronic Decay Dynamics......Page 418
4. Vibrational Energy Redistribution and Dephasing......Page 431
5. Wave Packets and Vibrational Probability Densities......Page 436
6. Photochemical Dynamics......Page 440
7. Conclusions......Page 442
References......Page 443
10. Jahn–Teller and Pseudo-Jahn–Teller Intersections: Spectroscopy and Vibronic Dynamics H. Koppel......Page 448
1. Introduction......Page 449
2.1. General......Page 451
2.2. The Single-Mode E  e Jahn–Teller Effect......Page 452
2.3. Influence of Additional Modes......Page 457
2.3.1. Additional e Vibrational Modes......Page 458
2.3.2. Influence of a1 Vibrational Modes......Page 460
2.4. The E  b Jahn–Teller Effect......Page 461
2.5. The Pseudo-Jahn–Teller Effect......Page 463
3.1. Brief General Overview......Page 467
3.2. Triatomic Hydrogen......Page 471
3.3. The Sodium Trimer in the B-State......Page 473
3.4. The Benzene Radical Cation......Page 477
4. Concluding Remarks......Page 485
References......Page 486
1. Introduction......Page 492
2. Basic Equations and Numerical Aspects......Page 495
3. Direct Dissociation......Page 498
4. Delayed Dissociation — Symmetric Stretch Excitation......Page 505
5. Delayed Dissociation — Bending Excitation......Page 516
6. Renner–Teller Induced Dissociation......Page 526
7. Outlook......Page 530
References......Page 533
1. Introduction......Page 540
2.1. Theory......Page 543
2.2. Numerical Techniques......Page 548
2.3. Applications......Page 551
3. Conclusions......Page 566
References......Page 570
13. Quantum Reaction Dynamics on Coupled Multi-Sheeted Potential Energy Surfaces S. Mahapatra......Page 574
1. Introduction......Page 575
2.1. The Flux Operator......Page 578
2.2.1. The Hamiltonian......Page 579
2.2.2. The Flux Operator and Reaction Probability......Page 580
2.3.2. The Flux Operator and Reaction Probability......Page 582
3.1. General Consideration......Page 584
3.2. Potential Energy Surface......Page 585
3.3. Diagonal Correction to the Adiabatic Hamiltonian......Page 587
3.4. Computational Details......Page 588
3.5. Results and Discussion......Page 591
4. Summary and Outlook......Page 595
References......Page 597
14. Multidimensional Dynamics Involving a Conical Intersection: Wavepacket Calculations Using the MCTDH Method G.A. Worth, H.-D. Meyer and L.S. Cederbaum......Page 602
1. Introduction......Page 603
2. Vibronic Coupling Hamiltonian......Page 604
3.1. Introduction......Page 607
3.2. Equations of Motion......Page 608
3.3. Representation of the Hamiltonian......Page 611
3.4. Nonadiabatic Systems......Page 612
3.5. Mode Combination......Page 613
4.1. Spectral Intensities......Page 615
4.2. The Butatriene Cation X / A Manifold......Page 616
4.3. The Allene Cation A / B Manifold......Page 621
4.4. The Pyrazine S1/S2 Manifold......Page 626
4.5. The Pyrazine Molecule Viewed as a System Coupled to a Bath......Page 627
5. Conclusions......Page 632
References......Page 635
15. Mixed Quantum-Classical Description of the Dynamics at Conical Intersections G. Stock and M. Thoss......Page 638
1. Introduction......Page 639
1.1. Classification of Methods......Page 640
1.2. Application to the Dynamics at Conical Intersections......Page 643
1.3. Outline of the Chapter......Page 644
2.1. Model Hamiltonian......Page 645
2.2. Observables of Interest......Page 647
2.3. Model Systems......Page 649
2.3.1. Model I: S2  S1 Internal Conversion in Pyrazine......Page 650
2.3.2. Model II: C  B  X Internal-Conversion Process in the Benzene Cation......Page 651
3.1. Classical-Path Approximation......Page 654
3.2. Mean-Field Trajectory Scheme......Page 657
3.3. Results......Page 658
3.4. Discussion......Page 659
4.1. General Idea......Page 661
4.2. Theory and Numerical Implementation......Page 662
4.3. Results and Discussion......Page 665
5.1. General Idea......Page 669
5.2. Theory and Numerical Implementation......Page 672
5.3. Results......Page 675
6.1. Theoretical Formulation......Page 678
6.2. Classical Dynamics......Page 680
6.3. Relation to Other Formulations......Page 681
6.4. Zero-Point Energy Problem......Page 684
6.5. Results......Page 687
6.6. Discussion......Page 694
7. Semiclassical Description of Nonadiabatic Quantum Dynamics......Page 695
7.1. Semiclassical Propagator......Page 696
7.2. Nonadiabatic Dynamics......Page 698
7.3. Results......Page 700
8. Conclusions......Page 705
References......Page 709
PART III. Detection and Control of Chemical Dynamics at Conical Intersections......Page 716
16. Absorption, Emission, and Photoelectron Continuous-Wave Spectra A. Lami, C. Petrongolo and F. Santoro......Page 718
1. Introduction......Page 719
2.1. Electric-Dipole Transitions......Page 720
2.2. Matrix Elements......Page 727
2.3. Methods of Calculation......Page 730
3. Interpretation of Nonadiabatic Spectra......Page 731
3.1. Level-by-Level Analysis......Page 732
3.2. Statistical Approach......Page 735
4.1.1. NO2......Page 739
4.1.2. O3......Page 741
4.1.3. Larger Polyatomic Molecules......Page 742
4.2. Photoelectron Spectra......Page 743
4.2.1. Furan, Pyrrole and Thiophene......Page 744
4.2.2. Allene......Page 745
4.3. Emission Spectra......Page 747
4.3.1. NO2......Page 748
4.3.2. Pyrazine......Page 750
5. Conclusions......Page 752
References......Page 755
17. Femtosecond Time-Resolved Spectroscopy of the Dynamics at Conical Intersections G. Stock and W. Domcke......Page 758
1. Introduction......Page 759
2. Preparation and Detection of Nonstationary States......Page 761
2.1. Transient Transmittance......Page 763
2.2. Time-Resolved Fluorescence......Page 767
2.3. Time-Resolved Ionization and Fragment Detection......Page 769
3.1. Absorption and Emission of Photons......Page 771
3.2. Photoelectron Spectra......Page 777
3.3. Computational Aspects......Page 778
4.1. Absorption and Emission of Photons......Page 784
4.2. Photoelectron Spectra......Page 786
5.1. The S1–S2 Conical Intersection in Pyrazine......Page 789
5.1.1. Transmittance Spectra......Page 790
5.1.2. Photoelectron Spectra......Page 796
5.2. Time-Resolved Fluorescence for the X–A Conical Intersection in NO2......Page 802
5.3. A Simple Model of the Photoisomerization of Retinal in Rhodopsin......Page 804
5.3.1. Transmittance Spectra......Page 806
5.3.2. Fluorescence Spectra......Page 808
6. Conclusions......Page 813
References......Page 815
1. Introduction......Page 822
2. Laser-Molecule Interaction and Optimal Control Theory......Page 824
3. Choice of Dynamical Coordinates and Reduced Hamiltonian......Page 826
4. Reaction Surfaces......Page 828
5. Quantum Dynamics on the S1 Surface......Page 829
6. Nonadiabatic Coupling Elements......Page 830
7. Conical Intersections and Geometric Phases......Page 836
8. Wavepacket Dynamics on the Coupled S1/S0 Potential Surfaces......Page 839
10. Conclusion......Page 842
References......Page 844
Index......Page 848