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Quantum Chemistry and Dynamics of Excited States. Methods and Applications. Edition No. 1

  • ID: 5186243
  • Book
  • December 2020
  • 688 Pages
  • John Wiley and Sons Ltd
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An introduction to the rapidly evolving methodology of electronic excited states

For academic researchers, postdocs, graduate and undergraduate students, Quantum Chemistry and Dynamics of Excited States: Methods and Applications reports the most updated and accurate theoretical techniques to treat electronic excited states. From methods to deal with stationary calculations through time-dependent simulations of molecular systems, this book serves as a guide for beginners in the field and knowledge seekers alike. Taking into account the most recent theory developments and representative applications, it also covers the often-overlooked gap between theoretical and computational chemistry.

An excellent reference for both researchers and students, Excited States provides essential knowledge on quantum chemistry, an in-depth overview of the latest developments, and theoretical techniques around the properties and nonadiabatic dynamics of chemical systems.

Readers will learn:

●      Essential theoretical techniques to describe the properties and dynamics of chemical systems

●      Electronic Structure methods for stationary calculations

●      Methods for electronic excited states from both a quantum chemical and time-dependent point of view

●      A breakdown of the most recent developments in the past 30 years

For those searching for a better understanding of excited states as they relate to chemistry, biochemistry, industrial chemistry, and beyond, Quantum Chemistry and Dynamics of Excited States provides a solid education in the necessary foundations and important theories of excited states in photochemistry and ultrafast phenomena.

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List of Contributors xix

Preface xxiii

1 Motivation and Basic Concepts 1
Sandra Gómez, Ignacio Fdez. Galván, Roland Lindh, and Leticia González

1.1 Mission and Motivation 1

1.2 Atomic Units 4

1.3 The Molecular Hamiltonian 5

1.4 Dirac or Bra-Ket Notation 6

1.5 Index Definitions 7

1.6 Second Quantization Formalism 7

1.7 Born–Oppenheimer Approximation and Potential Energy Surfaces 9

1.8 Adiabatic Versus Diabatic Representations 10

1.9 Conical Intersections 11

1.10 Further Reading 12

1.11 Acknowledgments 12

Part I Quantum Chemistry 13

2 Time-Dependent Density Functional Theory 15
Miquel Huix-Rotllant, Nicolas Ferré, and Mario Barbatti

2.1 Introduction 15

2.2 TDDFT Fundamentals 16

2.2.1 The Runge–Gross Theorems 16

2.2.2 The Time-Dependent Kohn–Sham Approach 18

2.2.3 Solutions of Time-Dependent Kohn–Sham Equations 19

2.2.3.1 Real-Time TDDFT 19

2.2.3.2 Linear-Response TDDFT 20

2.3 Linear-Response TDDFT in Action 22

2.3.1 Vertical Excitations and Energy Surfaces 22

2.3.1.1 Vertical Excitations: How Good are They? 23

2.3.1.2 Reconstructed Energy Surfaces: How Good are They? 25

2.3.2 Conical Intersections 28

2.3.3 Coupling Terms and Auxiliary Wave Functions 30

2.3.3.1 The Casida Ansatz 30

2.3.3.2 Time-Derivative Non-Adiabatic Couplings 31

2.3.4 Non-Adiabatic Dynamics 32

2.4 Excited States and Dynamics with TDDFT Variants and Beyond 34

2.5 Conclusions 35

Acknowledgments 36

References 36

3 Multi-Configurational Density Functional Theory: Progress and Challenges 47
Erik Donovan Hedegård

3.1 Introduction 47

3.2 Wave Function Theory 50

3.3 Kohn–Sham Density Functional Theory 50

3.3.1 Density Functional Approximations 53

3.3.2 Density Functional Theory for Excited States 54

3.3.2.1 Issues Within the Time-Dependent Density Functional Theory Ansatz 55

3.3.2.2 Self-Interaction Error 55

3.3.2.3 Degeneracies, Near-Degeneracies and the Symmetry Dilemma 56

3.4 Multi-Configurational Density Functional Theory 57

3.4.1 Semi-Empirical Multi-Configurational Density Functional Theory 57

3.4.2 Multi-Configurational Density Functional Theory Based the On-Top Pair Density 58

3.4.2.1 Density Matrices and the On-Top Pair Density 59

3.4.2.2 Energy Functional and Excited States with the On-Top Pair Density 60

3.4.3 Multi-Configurational Density Functional Theory Based on Range-Separation 61

3.4.3.1 Energy Functional and Excited States in Range-Separated Methods 62

3.4.3.2 The Range-Separation Parameter in Excited State Calculations 62

3.5 Illustrative Examples 64

3.5.1 Excited States of Organic Molecules 64

3.5.2 Excited States for a Transition Metal Complex 65

3.6 Outlook 66

Acknowledgments 67

References 67

4 Equation-of-Motion Coupled-Cluster Models 77
Monika Musiał

4.1 Introduction 77

4.2 Theoretical Background 79

4.2.1 Coupled-Cluster Wave Function 79

4.2.2 The Equation-of-Motion Approach 80

4.2.3 Similarity-Transformed Hamiltonian 81

4.2.4 Davidson Diagonalization Algorithm 82

4.3 Excited States: EE-EOM-CC 84

4.3.1 EE-EOM-CCSD Model 84

4.3.2 EE-EOM-CCSDT Model 86

4.3.3 EE-EOM-CC Results 87

4.4 Ionized States: IP-EOM-CC 89

4.4.1 IP-EOM-CCSD Model 89

4.4.2 IP-EOM-CCSDT Model 89

4.4.3 IP-EOM-CC Results 90

4.5 Electron-Attached States: EA-EOM-CC 91

4.5.1 EA-EOM-CCSD Model 92

4.5.2 EA-EOM-CCSDT Model 92

4.5.3 EA-EOM-CC Results 92

4.6 Doubly-Ionized States: DIP-EOM-CC 94

4.6.1 DIP-EOM-CCSD Model 95

4.6.2 DIP-EOM-CCSDT Model 95

4.6.3 DIP-EOM-CC Results 96

4.7 Doubly Electron-Attached States: DEA-EOM-CC 97

4.7.1 DEA-EOM-CCSD Model 98

4.7.2 DEA-EOM-CCSDT Model 98

4.7.3 DEA-EOM-CC Results 98

4.8 Size-Extensivity Issue in the EOM-CC Theory 100

4.9 Final Remarks 102

References 103

5 The Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator 109
Andreas Dreuw

5.1 Original Derivation via Green’s Functions 110

5.2 The Intermediate State Representation 112

5.3 Calculation of Excited State Properties and Analysis 114

5.3.1 Excited State Properties 114

5.3.2 Excited-State Wave Function and Density Analyses 116

5.4 Properties and Limitations of ADC 117

5.5 Variants of EE-ADC 119

5.5.1 Extended ADC(2) 119

5.5.2 Unrestricted EE-ADC Schemes 120

5.5.3 Spin-Flip EE-ADC Schemes 121

5.5.4 Spin-Opposite-Scaled ADC Schemes 122

5.5.5 Core-Valence Separated (CVS) EE-ADC 123

5.6 Describing Molecular Photochemistry with ADC Methods 125

5.6.1 Potential Energy Surfaces 125

5.6.2 Environment Models within ADC 126

5.7 Brief Summary and Perspective 126

Bibliography 127

6 Foundation of Multi-Configurational Quantum Chemistry 133
Giovanni Li Manni, Kai Guther, Dongxia Ma, and Werner Dobrautz

6.1 Scaling Problem in FCI, CAS and RAS Wave Functions 136

6.2 Factorization and Coupling of Slater Determinants 138

6.2.1 Slater Condon Rules 140

6.3 Configuration State Functions 141

6.3.1 The Unitary Group Approach (UGA) 142

6.3.1.1 Analogy between CSFs and Spherical Harmonics 143

6.3.1.2 Gel’fand-Tsetlin Basis 143

6.3.1.3 Paldus and Weyl Tables 145

6.3.1.4 The Step-Vector 148

6.3.2 The Graphical Unitary Group Approach (GUGA) 148

6.3.3 Evaluation of Non-Vanishing Hamiltonian Matrix Elements 153

6.3.3.1 One-Body Coupling Coefficients 154

6.3.3.2 Two-Body Matrix Elements 157

6.4 Configuration Interaction Eigenvalue Problem 158

6.4.1 Iterative Methods 159

6.4.1.1 Lanczos Algorithm 159

6.4.1.2 Davidson Algorithm 160

6.4.2 Direct-CI Algorithm 162

6.5 The CASSCF Method 165

6.5.1 The MCSCF Parameterization 167

6.5.2 The MCSCF Gradient and Hessian 169

6.5.3 One-Step and Two-Step Procedures 170

6.5.4 Augmented Hessian Method 171

6.5.5 Matrix form of the First and Second Derivatives in MCSCF 171

6.5.6 Quadratically Converging Method with Optimal Convergence 175

6.5.7 Orbital-CI Coupling Terms 178

6.5.8 Super-CI for the Orbital Optimization 179

6.5.9 Redundancy of Active Orbital Rotations 181

6.6 Restricted and Generalized Active Space Wave Functions 182

6.6.1 GUGA Applied to CAS, RAS and GAS Wave Functions 184

6.6.2 Redundancies in GASSCF Orbital Rotations 186

6.6.3 MCSCF Molecular Orbitals 187

6.6.4 GASSCF Applied to the Gd2 Molecule 188

6.7 Excited States 189

6.7.1 Multi-State CI Solver 190

6.7.2 State-Specific and State-Averaged MCSCF 191

6.8 Stochastic Multiconfigurational Approaches 191

6.8.1 FCIQMC Working Equation 192

6.8.2 Multi-Wave Function Approach for Excited States 196

6.8.3 Sampling Reduced Density Matrices 196

Bibliography 198

7 The Density Matrix Renormalization Group for Strong Correlation in Ground and Excited States 205
Leon Freitag and Markus Reiher

7.1 Introduction 205

7.2 DMRG Theory 207

7.2.1 Renormalization Group Formulation 207

7.2.2 Matrix Product States and Matrix Product Operators 210

7.2.3 MPS-MPO Formulation of DMRG 214

7.2.4 Connection between the Renormalization Group and the MPS-MPO Formulation of DMRG 217

7.2.5 Developments to Enhance DMRG Convergence and Performance 218

7.3 DMRG and Orbital Entanglement 218

7.4 DMRG in Practice 220

7.4.1 Calculating Excited States with DMRG 220

7.4.2 Factors Affecting the DMRG Convergence and Accuracy 220

7.4.3 Post-DMRG Methods for Dynamic Correlation and Environment Effects 221

7.4.4 Analytical Energy Gradients and Non-Adiabatic Coupling Matrix Elements 222

7.4.5 Tensor Network States 224

7.5 Applications in Quantum Chemistry 225

7.6 Conclusions 230

Acknowledgment 231

References 231

8 Excited-State Calculations with Quantum Monte Carlo 247
Jonas Feldt and Claudia Filippi

8.1 Introduction 247

8.2 Variational Monte Carlo 249

8.3 Diffusion Monte Carlo 252

8.4 Wave Functions and their Optimization 256

8.4.1 Stochastic Reconfiguration Method 258

8.4.2 Linear Method 259

8.5 Excited States 261

8.5.1 Energy-Based Methods 261

8.5.2 Time-Dependent Linear-Response VMC 263

8.5.3 Variance-Based Methods 264

8.6 Applications to Excited States of Molecular Systems 265

8.7 Alternatives to Diffusion Monte Carlo 269

Bibliography 270

9 Multi-Reference Configuration Interaction 277
Felix Plasser and Hans Lischka

9.1 Introduction 277

9.2 Basics 278

9.2.1 Configuration Interaction and the Variational Principle 278

9.2.2 The Size-Extensivity Problem of Truncated CI 280

9.2.3 Multi-Reference Configuration Spaces 282

9.2.4 Many-Electron Basis Functions: Determinants and CSFs 286

9.2.5 Workflow 287

9.3 Types of MRCI 289

9.3.1 Uncontracted and Contracted MRCI 289

9.3.2 MRCI with Extensivity Corrections 291

9.3.3 Types of Selection Schemes 293

9.3.4 Construction of Orbitals 293

9.4 Popular Implementations 294

9.5 Conclusions 295

References 295

10 Multi-Configurational Reference Perturbation Theory with a CASSCF Reference Function 299
Roland Lindh and Ignacio Fdez. Galvan

10.1 Rayleigh–Schrödinger Perturbation Theory 300

10.1.1 The Single-State Theory 300

10.1.1.1 The Conventional Projectional Derivation 300

10.1.1.2 The Bi-Variational Approach 304

10.1.2 Convergence Properties and Intruder States 308

10.1.2.1 Real and Imaginary Shift Techniques 310

10.2 Møller–Plesset Perturbation Theory 313

10.2.1 The Reference Function 314

10.2.2 The Partitioning of the Hamiltonian 315

10.2.3 The First-Order Interacting Space and Second-Order Energy Correction 316

10.3 State-Specific Multi-Configurational Reference Perturbation Methods 320

10.3.1 The Generation of the Reference Hamiltonian 321

10.3.2 CAS-MP2 Theory 322

10.3.3 CASPT2 Theory 323

10.3.3.1 The Partitioning of the Hamiltonian 324

10.3.3.2 The First-Order Interacting Space 325

10.3.3.3 Other Active Space References 328

10.3.3.4 Benchmark Results 329

10.3.3.5 IPEA Shift 330

10.3.4 MRMP2 Theory 331

10.3.4.1 The Partitioning of the Hamiltonian 331

10.3.4.2 The First-Order Interacting Space 332

10.3.5 NEVPT2 Theory 333

10.3.5.1 The Partitioning of the Hamiltonian 333

10.3.5.2 The First-Order Interacting Space 335

10.3.6 Performance Improvements 336

10.4 Quasi-Degenerate Perturbation Theory 338

10.5 Multi-State Multi-Configurational Reference Perturbation Methods 341

10.5.1 Multi-State CASPT2 Theory 341

10.5.2 Extended MS-CASPT2 Theory 342

10.6 Summary and Outlook 343

Acknowledgments 345

References 345

Appendix 350

Part II Nuclear Dynamics 355

11 Exact Quantum Dynamics (Wave Packets) in Reduced Dimensionality 357
Sebastian Reiter, Daniel Keefer, and Regina de Vivie-Riedle

11.1 Introduction 357

11.2 Fundamentals of Molecular Quantum Dynamics 358

11.2.1 Wave Packet Dynamics 358

11.2.2 Time-Propagator Schemes 360

11.2.3 Excited State Wave Packet Dynamics 362

11.2.4 Surfaces and Coupling Elements in Reactive Coordinates 362

11.3 Choice of Dynamical Coordinates and Hamiltonian in Reduced Dimensionality 364

11.3.1 Manual Selection by Chemical Intuition 364

11.3.2 The G-Matrix Formalism 365

11.3.2.1 General Setup 366

11.3.2.2 Practical Computation of the G-Matrix Elements 367

11.3.2.3 Photorelaxation of Uracil in Linear Reactive Coordinates 367

11.3.3 Automatic Generation of Linear Coordinates 369

11.3.3.1 IRC Based Approach 369

11.3.3.2 Trajectory-Based Approach 371

11.3.3.3 Comparison of Both Techniques for Linear Subspaces 372

11.3.4 Automatic Generation of Non-Linear Coordinates 374

11.4 Summary and Further Remarks 378

References 379

12 Multi-Configuration Time-Dependent Hartree Methods: From Quantum to Semiclassical and Quantum-Classical 383
M. Bonfanti, G. A. Worth, and I. Burghardt

12.1 Introduction 383

12.2 Time-Dependent Variational Principle and MCTDH 385

12.2.1 Variational Principle and Tangent Space Projections 385

12.2.2 MCTDH: Variational Multi-Configurational Wave Functions 386

12.2.2.1 MCTDH Wave Function Ansatz 386

12.2.2.2 MCTDH Equations of Motion 388

12.2.3 ML-MCTDH: Hierarchical Representations 389

12.3 Gaussian-Based MCTDH 390

12.3.1 G-MCTDH and vMCG 390

12.3.1.1 G-MCTDH Wave Function Ansatz 391

12.3.1.2 G-MCTDH Equations of Motion 392

12.3.1.3 vMCG Equations of Motion 393

12.3.2 2L-GMCTDH 394

12.3.2.1 Wave Function Ansatz 394

12.3.2.2 Equations of Motion 395

12.4 Quantum-Classical Multi-Configurational Approaches 396

12.4.1 Quantum-Classical Limit of G-MCTDH 396

12.4.2 Quantum-Classical Scheme with Finite-Width Wave Packets 398

12.4.3 Related Approaches 399

12.5 How to use MCTDH & Co 399

12.6 Synopsis and Application to Donor–Acceptor Complex 400

12.6.1 Hamiltonian, Spectral Densities, and Potential Surfaces 400

12.6.2 Ultrafast Coherent Charge Transfer Dynamics 402

12.6.3 Comparison of Methods 403

12.7 Conclusions and Outlook 405

Acknowledgments 406

References 406

13 Gaussian Wave Packets and the DD-vMCG Approach 413
Graham A. Worth and Benjamin Lasorne

13.1 Historical Background 413

13.2 Basic Theory 415

13.2.1 Gaussian Wave Packets 415

13.2.2 General Equations of Motion 418

13.2.2.1 Coefficients and Parameters 418

13.2.2.2 CX-Formalism 419

13.2.2.3 Nuclear and Electronic Degrees of Freedom 420

13.2.3 Variational Multi-Configurational Gaussian Approach 422

13.3 Example Calculations 424

13.4 Tunneling Dynamics: Salicylaldimine 425

13.5 Non-Adiabatic Dynamics: The Butatriene Cation 426

13.6 Direct Non-Adiabatic Dynamics: Formamide 428

13.7 Summary 431

13.8 Practical Implementation 431

Acknowledgments 431

References 431

14 Full and Ab Initio Multiple Spawning 435
Basile F. E. Curchod

14.1 Introduction 435

14.2 Time-Dependent Molecular Schrödinger Equation in a Gaussian Basis 436

14.2.1 Central Equations of Motion 436

14.2.2 Dynamics of the Trajectory Basis Functions 439

14.3 Full Multiple Spawning 440

14.3.1 Full Multiple Spawning Equations 440

14.3.2 Spawning Algorithm 442

14.4 Extending Full Multiple Spawning 443

14.4.1 External Field in Full Multiple Spawning 444

14.4.2 Spin-Orbit Coupling in Full Multiple Spawning 445

14.5 Ab Initio Multiple Spawning 447

14.5.1 From Full- to Ab Initio Multiple Spawning 447

14.5.2 Testing the Approximations of Ab Initio Multiple Spawning 449

14.5.3 On-the-Fly Ab Initio Multiple Spawning 450

14.5.4 Ab Initio Multiple Spawning versus Trajectory Surface Hopping 451

14.6 Dissecting an Ab Initio Multiple Spawning Dynamics 454

14.6.1 The Different Steps of an Ab Initio Multiple Spawning Dynamics 454

14.6.2 Example of Ab Initio Multiple Spawning Dynamics – the Photo-Chemistry of Cyclohexadiene 455

14.7 In Silico Photo-Chemistry with Ab Initio Multiple Spawning 459

14.8 Summary 462

References 463

15 Ehrenfest Methods for Electron and Nuclear Dynamics 469
Adam Kirrander and Morgane Vacher

15.1 Introduction 469

15.2 Theory of the (Simple) Ehrenfest Method 470

15.2.1 Wave Function Ansatz 471

15.2.2 Equations of Motion 472

15.3 Theory of the Multi-Configurational Ehrenfest Method 474

15.3.1 Wave Function Ansatz 474

15.3.2 Equations of Motion 476

15.3.3 Computational Aspects 479

15.4 Applications 480

15.4.1 Coupled Electron and Nuclear Dynamics Upon Sudden Ionization 481

15.4.2 Ultrafast Scattering as a Probe of Nuclear Dynamics 485

15.5 Conclusion 490

References 491

16 Surface Hopping Molecular Dynamics 499
Sebastian Mai, Philipp Marquetand, and Leticia González

16.1 Introduction 499

16.2 Basics of Surface Hopping 500

16.2.1 Advantages and Disadvantages 500

16.2.2 General Algorithm 501

16.3 Surface Hopping Ingredients 503

16.3.1 Nuclear Motion 503

16.3.2 Wave Function Propagation 504

16.3.3 Decoherence 505

16.3.4 Surface Hopping Algorithm 507

16.3.5 Kinetic Energy Adjustment and Frustrated Hops 509

16.3.6 Coupling Terms and Representations 511

16.4 Practical Remarks 513

16.4.1 Choice of the Electronic Structure Method 513

16.4.2 Initial Conditions 516

16.4.3 Example Application and Trajectory Analysis 518

16.5 Popular Implementations 521

16.6 Conclusion and Outlook 522

Acknowledgments 522

References 522

17 Exact Factorization of the Electron–Nuclear Wave Function: Theory and Applications 531
Federica Agostini and E. K. U. Gross

17.1 Introduction 531

17.2 The Time-Dependent Molecular Problem in the Exact-Factorization Formulation 533

17.2.1 Wave Function Ansatz 533

17.2.2 Equations of Motion 535

17.3 The Born–Oppenheimer Framework and the Exact Factorization 536

17.3.1 One-Dimensional Case: Time-Dependent Potential Energy Surface 538

17.3.2 Two-Dimensional Case: Time-Dependent Potential Energy Surface and Time-Dependent Vector Potential 542

17.4 Trajectory-Based Solution of the Exact-Factorization Equations 545

17.4.1 CT-MQC: The Approximations 546

17.4.2 CT-MQC: Photo-Induced Ring Opening in Oxirane 549

17.4.3 CT-MQC: The Algorithm 551

17.5 The Molecular Berry Phase 553

17.6 Conclusions 556

Acknowledgments 556

References 556

18 Bohmian Approaches to Non-Adiabatic Molecular Dynamics 563

Guillermo Albareda and Ivano Tavernelli

18.1 Introduction 563

18.2 A Practical Overview of Bohmian Mechanics 565

18.2.1 The Postulates 565

18.2.2 Computation of Bohmian Trajectories 566

18.2.2.1 Trajectories from the Schrödinger Equation 566

18.2.2.2 Trajectories from the Hamilton–Jacobi Equation 567

18.2.2.3 Trajectories from a Complex Action 568

18.2.3 Computation of Expectation Values 569

18.3 The Born–Huang Picture of Molecular Dynamics 569

18.3.1 The Molecular Schrödinger Equation in Position Space 569

18.3.2 Schrödinger Equation in the Born–Huang Basis 570

18.3.2.1 The Born–Oppenheimer Approximation: The Adiabatic Case 571

18.3.2.2 Non-Adiabatic Dynamics 572

18.4 BH-Based Approaches 573

18.4.1 The Non-Adiabatic Bohmian Dynamics Equations (NABDY) 573

18.4.2 Implementation in Molecular Dynamics: The Adiabatic Case 575

18.4.3 The Approximate Quantum Potential Approach 577

18.5 Non-BH Approaches 579

18.5.1 The Conditional Wave Function Approach 579

18.5.1.1 Hermitian Conditional Wave Function Approach 581

18.5.2 The Interacting Conditional Wave Function Approach 582

18.5.3 Time-Dependent Quantum Monte Carlo 585

18.6 Conclusions 588

References 589

19 Semiclassical Molecular Dynamics for Spectroscopic Calculations 595
Riccardo Conte and Michele Ceotto

19.1 Introduction 595

19.2 From Feynman’s Path Integral to van Vleck’s Semiclassical Propagator 598

19.3 The Semiclassical Initial Value Representation and the Heller–Herman–Kluk–Kay Formulation 601

19.4 A Derivation of the Heller–Herman–Kluk–Kay Propagator 603

19.5 The Time-Averaging Filter 604

19.6 The Multiple Coherent States SCIVR 606

19.7 The “Divide-and-Conquer” SCIVR 610

19.8 Mixed SCIVR Dynamics: Towards Semiclassical Spectroscopy in Condensed Phase 615

19.9 Semiclassical Spectroscopy Workflow 618

19.10 A Taste of Semiclassical Spectroscopy 619

19.11 Summary and Conclusions 622

Acknowledgments 624

Bibliography 624

20 Path-Integral Approaches to Non-Adiabatic Dynamics 629
Maximilian A. C. Saller, Johan E. Runeson, and Jeremy O. Richardson

20.1 Introduction 629

20.2 Semiclassical Theory 631

20.2.1 Mapping Approach 631

20.2.2 Linearized Semiclassical Dynamics 632

20.3 Non-Equilibrium Dynamics 633

20.3.1 Spin-Boson Systems 634

20.3.2 Non-Equilibrium Correlation Functions 636

20.4 Non-Adiabatic Path-Integral Theory 640

20.4.1 Mean-Field Path-Integral Sampling 640

20.4.2 Non-Adiabatic Ring-Polymer Molecular Dynamics 641

20.4.3 Alleviation of the Negative Sign 644

20.4.4 Practical Implementation of Monte Carlo Sampling 644

20.5 Equilibrium Correlation Functions 646

20.6 Conclusions 648

Acknowledgments 649

References 649

Index 655

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Roland Lindh
Leticia González
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