This volume aims at a comprehensive introduction into the theory of nonadiabatic molecular processes an increasingly relevant and rapidly expanding segment of molecular quantum dynamics. This very active and current field of research deals with molecular interactions involving transitions between electronic states, which occur typically in cases of reactive scattering between molecules, photoexcitation or strong vibronic and rotational coupling between electronic and nuclear degrees of freedom.
The main objective of Electron Dynamics in Molecular Interactions is to provide a synoptic presentation of some very recent theoretical efforts and to contrast them with the more traditional models of nonadiabatic molecular processes. In these presented models derived from their quantum dynamical fundaments, their interrelations are discussed, and their characteristic applications to concrete chemical systems are also outlined. This volume also includes an assessment of the present status of electron dynamics and a report on novel developments to meet the current challenges in the field.
There is a need for a systematic comparative treatise as nonadiabatic theories, which are of considerably higher complexity than the more traditional adiabatic approaches, are steadily gaining in importance. This volume addresses a broad readership ranging from physics or chemistry graduate students to specialists in the field of theoretical quantum dynamics.

About This Book xix

Introduction: Electron Multistate Molecular Dynamics xxv

Part I: Preparations 1

1 Ab Initio Theory of Electronic Structure 3

1.1 Molecular Orbitals 4

      1.1.1 Molecular and atomic orbitals 9

      1.1.2 Expectation values from molecular orbitals: The example of spin-orbit coupling 11

1.2 The Minimal LCAO Model 12

1.3 Hartree-Fock Theory 16

      1.3.1 The Hartree-Fock equations 17

      1.3.2 Koopmans' theorem 21

      1.3.3 The Hartree-Fock Hamiltonian 22

1.4 The Restricted and the Unrestricted Hartree-Fock Formalism 23

      1.4.1 The restricted Hartree-Fock method 24

      1.4.2 The unrestricted Hartree-Fock method 25

      1.4.3 The Roothaan formalism 27

1.5 Post-Hartree-Fock Methods 30

      1.5.1 Configuration interaction 31

      1.5.2 Many-body perturbation theory 34

1.6 Excited Electronic States 36

1.7 Appendix: The Functional Derivative 41

2 The Adiabatic and the Diabatic Representation 45

2.1 The Born-Oppenheimer Approximation 46

2.2 Harmonic Vibrational Modes 50

2.3 Adiabatic and Diabatic Frames 54

      2.3.1 The diabatic approximation 58

2.4 Gauge Theoretical Form of the Nuclear Equation 59

2.5 Avoided Crossings, Degeneracies, Conical Intersections 63

      2.5.1 Avoided crossings 63

      2.5.2 Conical intersections and the Jahn-Teller effect 65

      2.5.3 Jahn-Teller distortion 68

      2.5.4 The molecular Aharonov-Bohm effect 71

      2.5.5 The geometric phase in molecular pseudorotation 75

2.6 Locating the Seam 78

      2.6.1 Intersection-adapted coordinates 79

      2.6.2 Determination of the seam space 82

      2.6.3 Seam subspaces by Lagrangian minimization 84

3 Basic Concepts of Scattering Theory 87

3.1 The Time-Dependent and the Time-Independent View of Scattering Processes 88

3.2 Quantum Mechanical Equations of Motion 89

3.3 The Scattering Matrix 94

      3.3.1 The Moller operators 96

      3.3.2 The Lippmann-Schwinger equations 98

      3.3.3 Unitarity of the S-matrix 100

3.4 Elastic Scattering by a Spherical Potential 104

      3.4.1 The asymptotic scattering solution 104

      3.4.2 T-, S-, and K-matrix boundary conditions 107

      3.4.3 The elastic cross section Ill

3.5 Resonances 113

4 Semiclassical Notions 121

4.1 Path Integrals and the Quantum Propagator 122

      4.1.1 The quantum and the semiclassical propagator 123

      4.1.2 The Van Vleck propagator 128

      4.1.3 The monodromy matrix 133

4.2 The WKB Approximation 136

      4.2.1 The WKB wave function 136

      4.2.2 The Bohr-Sommerfeld quantization rules for bound WKB states 144

4.3 The Wigner Function: A Quantum Mechanical Phase Space Distribution 146

      4.3.1 Defining properties of the Wigner function 146

      4.3.2 Time dependence of the Wigner function 149

      4.3.3 The Moyal formalism 153

4.4 Coherent States 154

      4.4.1 Coherent and particle number states 155

      4.4.2 Coherent states as minimal uncertainty solutions 158

      4.4.3 The nuclear coherent state 159

5 Open Systems: Elements of Rate Theory 161

5.1 Classical Rate Theory 163

5.2 Quantum Transition State Theory 169

      5.2.1 The quantum transition state approximation 171

5.3 The Euclidean Path Integral 174

      5.3.1 Classical polymer isomorphism 178

5.4 Centroid Dynamics 181

5.5 The Path Integral Form of the Golden Rule Rate Constant 187

5.6 Beyond the Golden Rule: Reduced Density Matrix Theory 190

      5.6.1 A two-state problem 195

Part II: Methods 199

6 Time-Independent Theory ofMolecular Collisions I: Multichannel Scattering 201

6.1 The Multichannel Problem 201

6.2 The Lippmann-Schwinger Equation for Inelastic Scattering 206

6.3 The Born Approximation 210

      6.3.1 The distorted-wave Born approximation (DWBA) 212

6.4 Microreversibility 213

6.5 R-matrix and Log Derivative Propagation 217

      6.5.1 The log derivative method 220

6.6 Reactive Scattering I: The Differential Equation Approach 223

      6.6.1 Jacobi coordinates 224

      6.6.2 Hyperspherical coordinates 227

6.7 Space-Fixed and Body-Fixed Frames of Reference 229

      6.7.1 Space-fixed representation 230

      6.7.2 Body-fixed representation 232

6.8 Reactive Scattering II: The Integral Equation Approach 238

7 Time-Independent Theory of Molecular Collisions II: The Electronic Problem 245

7.1 Inclusion of the Electronic System 246

      7.1.1 The triatomic case 247

      7.1.2 The adiabatic case 251

      7.1.3 The diabatic case 253

7.2 Case Study: The F + H2 Reaction 255

7.3 Variational Procedures 263

      7.3.1 The Kohn variational principle 267

      7.3.2 Kohn anomalies 269

7.4 Case Study: Quenching of the Sodium Atom 3p State by Interaction with Hydrogen Molecules 271

      7.4.1 Basis sets 277

      7.4.2 Algebraic realization of the outgoing wave variational principle 279

      7.4.3 Exciplex funnel dynamics 280

7.5 The Landau-Zener-Stückelberg Model ofNonadiabatic Transitions 282

8 The Time-Dependent Self-Consistent Field Theory 291

8.1 Time-Dependent Variational Principles 292

      8.1.1 Time-dependent perturbations 295

      8.1.2 Free and forced oscillations 299

8.2 The Time-Dependent Hartree-Fock Theory: Application to Molecules 300

8.3 Wave-Function-Based Ab Initio Molecular Dynamics 305

      8.3.1 Direct molecular dynamics in the time-dependent Hartree-Fock framework 305

      8.3.2 Classical trajectories within TDHF dynamics 307

      8.3.3 The Hellmann-Feynman theorem 312

      8.3.4 Ehrenfest dynamics 315

      8.3.5 Car-Parrinello dynamics 318

8.4 Time-Dependent Hartree-Fock Dynamics in the Eikonal Approximation 322

      8.4.1 The eikonal approximation 324

      8.4.2 TDHF approach to the electronic problem within the eikonal approximation 327

      8.4.3 The Liouville-von Neumann equation in a traveling orbital basis 329

8.5 Case Study: Light Emission in Slow Proton-Hydrogen Collisions 333

9 Evolution of Coherent Molecular States: Electron Nuclear Dynamics Theory 341

9.1 The Thouless Representation 343

9.2 The END Equations 348

      9.2.1 Derivation of the END equations 349

      9.2.2 Interpretation of the END equations 354

9.3 Two Special Cases: The Boosted Self-Consistent Field and the Linearized END Equations 357

      9.3.1 The boosted electronic system 357

      9.3.2 The linear version of the electronic END equations 360

9.4 Inclusion of Nuclear Quantum Effects 361

      9.4.1 Trajectory interference 362

      9.4.2 Case study: H impact on molecular and atomic targets by END theory 366

      9.4.3 Rovibrational analysis of the nuclear system 372

9.5 Nonadiabatic Effects in Bound Systems by END Theory: The Pseudorotation of H+3 376

10 The Classical Electron Analog 383

10.1 Critique of the Ehrenfest Representation 384

10.2 The Classical Electron Analog 386

      10.2.1 The CEA equations of motion 388

      10.2.2 Adiabatic representation of the Hamilton function 389

      10.2.3 The classical analog of the electronic two-state problem 391

10.3 CEA Theory Applied to a Conical Intersection Problem 395

11 Hopping and Spawning 403

11.1 The Trajectory Surface Hopping Method 404

11.2 The Fewest Switches Algorithm 408

      11.2.1 Three test cases 413

      11.2.2 Complex-valued trajectories 418

11.3 Spawning 419

      11.3.1 Applications to model problems 425

11.4 Case Study: The Dynamics of Na*-Quenching by Collision with Hydrogen Molecules 429

11.5 Comparison with Other Methods 433

12 Semiclassical Propagator Techniques 437

12.1 The Path Integral Approach to Molecular Dynamics 438

12.2 Semiclassical Propagation and Surface Hopping 442

12.3 The Initial Value Representation 447

12.4 The Mapping Approach to Electronic Degrees of Freedom 451

      12.4.1 The Schwinger mapping formalism 452

      12.4.2 Extension to general N-level systems 454

12.5 The Mapping Technique Applied to Nonadiabatic Dynamics 457

      12.5.1 The SC-IVR approach applied to nonadiabatic model cases 460

      12.5.2 Comparison with the Ehrenfest model 464

12.6 Case Study: The S1—S2 Transition in Pyrazine: SC-IVR Treatment of a Conical Intersection Problem 466

12.7 Numerical Procedures for Semiclassical Propagation Methods 471

      12.7.1 Monte Carlo integration 473

      12.7.2 Filinov filtering 476

      12.7.3 The forward-backward initial value representation 479

12.8 Cellular Dynamics 482

13 Quantum Hydrodynamics I: Coupled Trajectories in Bohmian Mechanics 491

13.1 Elements of the Quantum Theory of Motion 492

      13.1.1 Quantum trajectories 495

      13.1.2 The pilot wave and the guided particle 498

13.2 Lagrangian Quantum Hydrodynamics 499

      13.2.1 Assembling the wave function 502

      13.2.2 Technical challenges for quantum trajectory propagation 503

13.3 Nonadiabatic Lagrangian Quantum Hydrodynamics 505

13.4 The Classical Limit of the Quantum Theory of Motion 512

14 Quantum Hydrodynamics II: The Semiclassical Liouville—Von Neumann Equation 517

14.1 The Semiclassical Liouville Formalism for Multistate Problems 518

      14.1.1 Two coupled states: A model problem 524

14.2 Phase Space Trajectory Implementation 528

14.3 Generalized Quantum Hydrodynamics: Mixed States 534

      14.3.1 Pure states 538

      14.3.2 Mixed states 540

14.4 Coupled Electronic States 542

15 Wave Packet Propagation Methods 547

15.1 The Grid Representation 548

      15.1.1 The discrete variable representation (DVR) 553

      15.1.2 The fast Fourier transform (FFT) 555

15.2 Numerical Wave Packet Propagation Techniques 557

      15.2.1 The Crank-Nicolson scheme 557

      15.2.2 Split operator propagation 559

      15.2.3 Propagator expansion techniques 560

15.3 The Multiconfiguration Time-Dependent Hartree (MCTDH) Method 564

      15.3.1 The time-dependent Hartree (TDH) approach 565

      15.3.2 The multiconfiguration time-dependent Hartree (MCTDH) approach 567

      15.3.3 The MCTDH equations 569

15.4 Case Study: Photostability of Biologically Relevant Molecules 574

      15.4.1 Ultrafast deexcitation by passage through conical intersections in nucleic acid bases and base pairs 575

      15.4.2 Dynamics at the 1πσ*-S0 conical intersection of pyrrole 579

16 Density Functional Dynamics 587

16.1 Fundamentals of Density Functional Theory 588

      16.1.1 Exchange-correlation potentials 593

16.2 Excited Electronic States in DFT 595

16.3 Time-Dependent Density Functional Theory 598

      16.3.1 TDDFT in the linear response domain 601

      16.3.2 Time-dependent current density functional theory 603

16.4 Direct Molecular Dynamics Based on DFT 605

      16.4.1 Calculating molecular photoabsorption spectra 606

      16.4.2 Molecular bonding properties analyzed by the electron localization function 608

      16.4.3 Combining TDDFT with standard methods of nonadiabatic dynamics 609

17 Decoherence 613

17.1 The Dissipative Liouville-von Neumann Equation 615

17.2 Evaluating Decoherence Times in a Semiclassical Framework 626

      17.2.1 Ensemble average of the decoherence function 632

17.3 Case Study: The Dynamics of Electron Hydration 636

      17.3.1 Isotope effects in hydrated electron relaxation 638

17.4 Continuous Surface Switching: A Compromise between Mean-Field and Individual Surface Propagation 644

17.5 Decay of Mixing 648

      17.5.1 Decoherence time 657

      17.5.2 Determining the decoherent state 658

Part III: Special Topics 661

18 Ultrafast Optical Spectroscopy 663

18.1 Linear and Nonlinear Polarization 664

      18.1.1 Deriving the pump-probe signal 666

18.2 Theory of Nonlinear Polarization in Femtosecond Molecular Spectroscopy 669

      18.2.1 The perturbative approach 672

      18.2.2 The non-perturbative approach 681

18.3 Polarization Studies of cis-trans Isomerization 682

      18.3.1 Adiabatic formulation 688

18.4 The Density Matrix Approach to Simulating Pump-Probe Signals 690

      18.4.1 The pump-probe signal 698

18.5 Case Study: Ultrafast Spectroscopy on Non-Stoichiometric Alkali-Halide Clusters 700

      18.5.1 Effective single-electron systems of the form NanFn-1 701

      18.5.2 Extension to nonadiabatic dynamics 706

18.6 Appendix: Derivation of the Pump—Probe Signal S(td) 711

19 Optical Control of Electron Multistate Molecular Dynamics 715

19.1 Interaction of a Molecule with a Pulse of Light 716

19.2 The Tannor-Rice Scheme: Optimal Control 719

19.3 The Brumer-Shapiro Scheme: Coherent Control 725

19.4 Case Study: Coherent Control of ICN Photodissociation 730

19.5 Optimal Control in Pump-Probe Spectroscopy 736

      19.5.1 Case study: Application to Na3F2 743

20 Electron Transfer in Condensed Media 749

20.1 The Electronic Hamiltonian 753

20.2 Electronic-Vibronic Coupling: The Spin-Boson Hamiltonian 756

20.3 Adiabatic versus Nonadiabatic Electron Transfer 760

20.4 Thermally Activated Transfer 763

20.5 Inclusion of Nuclear Tunneling 767

20.5.1 The continuous limit of nuclear frequencies 771

20.6 Effects of Polar Solvents on Electron Transfer 774

      20.6.1 The dielectric displacement field 776

      20.6.2 Polarization and polarizability 778

      20.6.3 The free energy functional 783

      20.6.4 The electron transfer rate in a polar environment 786

20.7 Ultrafast Electron Transfer 790

20.8 Case Study: Aqueous Ferrous-Ferric Exchange 793

      20.8.1 Monte Carlo modeling 793

      20.8.2 Euclidean path integral simulations 798

      20.8.3 Recent quantum dynamical extensions 804

20.9 Appendix: Formulae Relevant for Electron Transfer Theory within the Marcus Model 805

      20.9.1 Electron transfer in a vibrational bath: Formal procedures used in the derivation of the rate constant 806

      20.9.2 Derivation of the effective free energy functional Eq.(20.109) 810

      20.9.3 The density of states for electron transfer in a solvent: Calculating the trace Eq.(20.118) 813

21 Electronic Friction in Molecule-Surface Interactions 817

21.1 Langevin Formulation of Ehrenfest Dynamics 820

21.2 An Ab Initio Model for Electronic Friction 824

21.3 Case Study: Nonadiabatic Effects in the Interaction between the Cu(100) Surface and a CO Molecule 828

      21.3.1 Vibrational relaxation of CO on the Cu(100) surface: The impact of electronic friction 828

      21.3.2 Vibrational excitation and hot diffusion 835

21.4 Beyond Langevin Theory 838

Epilogue 841

Bibliography 847

Index 875

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