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.
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
