Overview

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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Author:   Leticia González ,  Roland Lindh
Publisher:   John Wiley & Sons Inc
Imprint:   John Wiley & Sons Inc
Dimensions:   Width: 17.80cm , Height: 4.10cm , Length: 25.20cm
Weight:   1.588kg
ISBN:  

9781119417750

ISBN 10:   1119417759
Pages:   688
Publication Date:   10 December 2020
Audience:   Professional and scholarly ,  College/higher education ,  Professional & Vocational ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

List of Contributors xix Preface xxiii 1 Motivation and Basic Concepts 1 Sandra Gómez, Ignacio Fdez. Galván, Roland Lindh, and Leticia Gonzalez 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 Ferre, 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-ClusterWave 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 RASWave 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 andWeyl 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. Galván 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 Gonzalez 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 ConditionalWave Function Approach 579 18.5.1.1 Hermitian ConditionalWave Function Approach 581 18.5.2 The Interacting ConditionalWave 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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Professor Leticia González teaches at the Department of Chemistry at the University of Vienna, Austria. She is a theoretical chemist world-known for her work on molecular excited states and ultrafast dynamics simulations. Besides publishing over 250 papers and several reviews on excited states and dynamics, she has developed the SHARC program package to simulate non-adiabatic dynamics. Professor Roland Lindh currently teaches at Uppsala University, Sweden. He is a member of the editorial board of International Journal of Quantum Chemistry and the MOLCAS quantum chemistry program project. He co-authored the book ""Multiconfigurational Quantum Chemistry"" and is an expert on method development for multiconfigurational wave function theory.

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