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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.
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.
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.
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.
Über den Autor
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.
Inhaltsverzeichnis
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 MusiaB
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...
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 MusiaB
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...
Details
| Erscheinungsjahr: | 2021 |
|---|---|
| Fachbereich: | Physikalische Chemie |
| Genre: | Chemie |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Buch |
| Seiten: | 688 |
| Inhalt: | 688 S. |
| ISBN-13: | 9781119417750 |
| ISBN-10: | 1119417759 |
| Sprache: | Englisch |
| Einband: | Gebunden |
| Redaktion: |
González, Leticia
Lindh, Roland |
| Herausgeber: | Leticia González/Roland Lindh |
| Hersteller: | Wiley |
| Maße: | 257 x 180 x 38 mm |
| Von/Mit: | Leticia González (u. a.) |
| Erscheinungsdatum: | 01.02.2021 |
| Gewicht: | 1,588 kg |
Über den Autor
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.
Inhaltsverzeichnis
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 MusiaB
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...
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 MusiaB
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...
Details
| Erscheinungsjahr: | 2021 |
|---|---|
| Fachbereich: | Physikalische Chemie |
| Genre: | Chemie |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Buch |
| Seiten: | 688 |
| Inhalt: | 688 S. |
| ISBN-13: | 9781119417750 |
| ISBN-10: | 1119417759 |
| Sprache: | Englisch |
| Einband: | Gebunden |
| Redaktion: |
González, Leticia
Lindh, Roland |
| Herausgeber: | Leticia González/Roland Lindh |
| Hersteller: | Wiley |
| Maße: | 257 x 180 x 38 mm |
| Von/Mit: | Leticia González (u. a.) |
| Erscheinungsdatum: | 01.02.2021 |
| Gewicht: | 1,588 kg |
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