| Author: |
BJÖRN O. ROOS, ROLAND LINDH, PER ÅKE MALMQVIST, VALERA VERYAZOV, PER-OLOF WIDMARK
|
| Published in: | John Wiley & Sons |
| Release Year: | 2016 |
| ISBN: | 978-1119-2-7787-3 |
| Pages: | 230 |
| Edition: | First Edition |
| File Size: | 5 MB |
| File Type: | |
| Language: | English |
Description of Multiconfigrational Quantum Chemistry
The intention of the Multiconfigrational Quantum Chemistry book is to introduce the reader to the multiconfigurational approaches in quantum chemistry. These methods are more difficult to learn to use and there does not exist any textbook in the field that takes the students from the simple Hartree–Fock method to the advanced multireference methods such as multireference configuration interaction (MRCI), or the complete active space self-consistent field (CASSCF) method. The intention is to describe these and other wave function-based methods such that the treatment can be followed by any student with basic knowledge in quantum mechanics and quantum chemistry.
Using many illustrative examples, we shall show how these methods can be applied in various areas of chemistry, such as chemical reactions in the ground and excited states, transition metal, and other heavy element systems. These methods are based on a well-defined wave function with exact spin and symmetry and are therefore well suited for detailed analysis of various bonding situations. A simple example is the oxygen molecule, which has a 3Σ−g ground state.
Already this label tells us much about the wave function and the electronic structure. It is a triplet state (S = 1), it is symmetric around the molecular axis (Σ), it is a grade function, and it is antisymmetric with respect to a mirror plane through the molecular axis. None of these properties are well defined in some methods widely used today. It becomes even worse for the first excited state, 1Δg, which cannot be properly described with single configurational methods due to its multiconfigurational character.
This failure can have severe consequences in studies of oxygen-containing biological systems. It is true that these wave function-based methods cannot yet be applied to as large systems as can, for example, density functional theory (DFT), but the method development is fast and increases the possibilities for every year.
Using many illustrative examples, we shall show how these methods can be applied in various areas of chemistry, such as chemical reactions in the ground and excited states, transition metal, and other heavy element systems. These methods are based on a well-defined wave function with exact spin and symmetry and are therefore well suited for detailed analysis of various bonding situations. A simple example is the oxygen molecule, which has a 3Σ−g ground state.
Already this label tells us much about the wave function and the electronic structure. It is a triplet state (S = 1), it is symmetric around the molecular axis (Σ), it is a grade function, and it is antisymmetric with respect to a mirror plane through the molecular axis. None of these properties are well defined in some methods widely used today. It becomes even worse for the first excited state, 1Δg, which cannot be properly described with single configurational methods due to its multiconfigurational character.
This failure can have severe consequences in studies of oxygen-containing biological systems. It is true that these wave function-based methods cannot yet be applied to as large systems as can, for example, density functional theory (DFT), but the method development is fast and increases the possibilities for every year.
Content of Multiconfigrational Quantum Chemistry
1 Introduction 1
1.1 References, 4
2 Mathematical Background 7
2.1 Introduction, 7
2.2 Convenient Matrix Algebra, 7
2.3 Many-Electron Basis Functions, 11
2.4 Probability Basics, 14
2.5 Density Functions for Particles, 16
2.6 Wave Functions and Density Functions, 17
2.7 Density Matrices, 18
2.8 References, 22
3 Molecular Orbital Theory 23
3.1 Atomic Orbitals, 24
3.1.1 The Hydrogen Atom, 24
3.1.2 The Helium Atom, 26
3.1.3 Many Electron Atoms, 28
3.2 Molecular Orbitals, 29
3.2.1 The Born–Oppenheimer Approximation, 29
3.2.2 The LCAO Method, 30
3.2.3 The Helium Dimer, 34
3.2.4 The Lithium and Beryllium Dimers, 35
3.2.5 The B to Ne Dimers, 35
3.2.6 Heteronuclear Diatomic Molecules, 37
3.2.7 Polyatomic Molecules, 39
3.3 Further Reading, 41
4 Hartree–Fock Theory 43
4.1 The Hartree–Fock Theory, 44
4.1.1 Approximating the Wave Function, 44
4.1.2 The Hartree–Fock Equations, 45
4.2 Restrictions on The Hartree–Fock Wave Function, 49
4.2.1 Spin Properties of Hartree–Fock Wave Functions, 50
4.3 The Roothaan–Hall Equations, 53
4.4 Practical Issues, 55
4.4.1 Dissociation of Hydrogen Molecule, 55
4.4.2 The Hartree-Fock Solution, 56
4.5 Further Reading, 57
4.6 References, 58
5 Relativistic Effects 59
5.1 Relativistic Effects in Chemistry, 59
5.2 Relativistic Quantum Chemistry, 62
5.3 The Douglas–Kroll–Hess Transformation, 64
5.4 Further Reading, 66
5.5 References, 66
6 Basis Sets 69
6.1 General Concepts, 69
6.2 Slater Type Orbitals, STOs, 70
6.3 Gaussian Type Orbitals, GTOs, 71
6.3.1 Shell Structure Organization, 71
6.3.2 Cartesian and Real Spherical Harmonics Angular Momentum
Functions, 72
6.4 Constructing Basis Sets, 72
6.4.1 Obtaining Exponents, 73
6.4.2 Contraction Schemes, 73
6.4.3 Convergence in the Basis Set Size, 77
6.5 Selection of Basis Sets, 79
6.5.1 Effect of the Hamiltonian, 79
6.5.2 Core Correlation, 80
6.5.3 Other Issues, 81
6.6 References, 81
7 Second Quantization and Multiconfigurational Wave Functions 85
7.1 Second Quantization, 85
7.2 Second Quantization Operators, 86
7.3 Spin and Spin-Free Formalisms, 89
7.4 Further Reading, 90
7.5 References, 91
8 Electron Correlation 93
8.1 Dynamical and Nondynamical Correlation, 93
8.2 The Interelectron Cusp, 94
8.3 Broken Bonds. (σ)2→(σ∗)2, 97
8.4 Multiple Bonds, Aromatic Rings, 99
8.5 Other Correlation Issues, 100
8.6 Further Reading, 102
8.7 References, 102
9 Multiconfigurational SCF Theory 103
9.1 Multiconfigurational SCF Theory, 103
9.1.1 The H2 Molecule, 104
9.1.2 Multiple Bonds, 107
9.1.3 Molecules with Competing Valence Structures, 108
9.1.4 Transition States on Energy Surfaces, 109
9.1.5 Other Cases of Near-Degeneracy Effects, 110
9.1.6 Static and Dynamic Correlation, 111
9.2 Determination of the MCSCF Wave Function, 114
9.2.1 Exponential Operators and Orbital Transformations, 115
9.2.2 Slater Determinants and Spin-Adapted State Functions, 117
9.2.3 The MCSCF Gradient and Hessian, 119
9.3 Complete and Restricted Active Spaces, the CASSCF and RASSCF
Methods, 121
9.3.1 State Average MCSCF, 125
9.3.2 Novel MCSCF Methods, 125
9.4 Choosing Active Space, 126
9.4.1 Atoms and Atomic Ions, 126
9.4.2 Molecules Built from Main Group Atoms, 128
9.5 References, 130
10 The RAS State-Interaction Method 131
10.1 The Biorthogonal Transformation, 131
10.2 Common One-Electron Properties, 133
10.3 Wigner–Eckart Coefficients for Spin-Orbit Interaction, 134
10.4 Unconventional Usage of RASSI, 135
10.5 Further Reading, 136
10.6 References, 136
11 The Multireference CI Method 137
11.1 Single-Reference CI. Nonextensivity, 137
11.2 Multireference CI, 139
11.3 Further Reading, 140
11.4 References, 140
12 Multiconfigurational Reference Perturbation Theory 143
12.1 CASPT2 theory, 143
12.1.1 Introduction, 143
12.1.2 Quasi-Degenerate Rayleigh–Schrödinger Perturbation
Theory, 144
12.1.3 The First-Order Interacting Space, 145
12.1.4 Multiconfigurational Root States, 146
12.1.5 The CASPT2 Equations, 148
12.1.6 IPEA, RASPT2, and MS-CASPT2, 154
12.2 References, 155
13 CASPT2/CASSCF Applications 157
13.1 Orbital Representations, 158
13.1.1 Starting Orbitals: Atomic Orbitals, 162
13.1.2 Starting Orbitals: Molecular Orbitals, 164
13.2 Specific Applications, 167
13.2.1 Ground State Reactions, 167
13.2.2 Excited States–Vertical Excitation Energies, 171
13.2.3 Photochemistry and Photophysics, 184
13.2.4 Transition Metal Chemistry, 194
13.2.5 Spin-Orbit Chemistry, 202
13.2.6 Lanthanide Chemistry, 207
13.2.7 Actinide Chemistry, 209
13.2.8 RASSCF/RASPT2 Applications, 212
13.3 References, 216
Summary and Conclusion 219
Index 221
1.1 References, 4
2 Mathematical Background 7
2.1 Introduction, 7
2.2 Convenient Matrix Algebra, 7
2.3 Many-Electron Basis Functions, 11
2.4 Probability Basics, 14
2.5 Density Functions for Particles, 16
2.6 Wave Functions and Density Functions, 17
2.7 Density Matrices, 18
2.8 References, 22
3 Molecular Orbital Theory 23
3.1 Atomic Orbitals, 24
3.1.1 The Hydrogen Atom, 24
3.1.2 The Helium Atom, 26
3.1.3 Many Electron Atoms, 28
3.2 Molecular Orbitals, 29
3.2.1 The Born–Oppenheimer Approximation, 29
3.2.2 The LCAO Method, 30
3.2.3 The Helium Dimer, 34
3.2.4 The Lithium and Beryllium Dimers, 35
3.2.5 The B to Ne Dimers, 35
3.2.6 Heteronuclear Diatomic Molecules, 37
3.2.7 Polyatomic Molecules, 39
3.3 Further Reading, 41
4 Hartree–Fock Theory 43
4.1 The Hartree–Fock Theory, 44
4.1.1 Approximating the Wave Function, 44
4.1.2 The Hartree–Fock Equations, 45
4.2 Restrictions on The Hartree–Fock Wave Function, 49
4.2.1 Spin Properties of Hartree–Fock Wave Functions, 50
4.3 The Roothaan–Hall Equations, 53
4.4 Practical Issues, 55
4.4.1 Dissociation of Hydrogen Molecule, 55
4.4.2 The Hartree-Fock Solution, 56
4.5 Further Reading, 57
4.6 References, 58
5 Relativistic Effects 59
5.1 Relativistic Effects in Chemistry, 59
5.2 Relativistic Quantum Chemistry, 62
5.3 The Douglas–Kroll–Hess Transformation, 64
5.4 Further Reading, 66
5.5 References, 66
6 Basis Sets 69
6.1 General Concepts, 69
6.2 Slater Type Orbitals, STOs, 70
6.3 Gaussian Type Orbitals, GTOs, 71
6.3.1 Shell Structure Organization, 71
6.3.2 Cartesian and Real Spherical Harmonics Angular Momentum
Functions, 72
6.4 Constructing Basis Sets, 72
6.4.1 Obtaining Exponents, 73
6.4.2 Contraction Schemes, 73
6.4.3 Convergence in the Basis Set Size, 77
6.5 Selection of Basis Sets, 79
6.5.1 Effect of the Hamiltonian, 79
6.5.2 Core Correlation, 80
6.5.3 Other Issues, 81
6.6 References, 81
7 Second Quantization and Multiconfigurational Wave Functions 85
7.1 Second Quantization, 85
7.2 Second Quantization Operators, 86
7.3 Spin and Spin-Free Formalisms, 89
7.4 Further Reading, 90
7.5 References, 91
8 Electron Correlation 93
8.1 Dynamical and Nondynamical Correlation, 93
8.2 The Interelectron Cusp, 94
8.3 Broken Bonds. (σ)2→(σ∗)2, 97
8.4 Multiple Bonds, Aromatic Rings, 99
8.5 Other Correlation Issues, 100
8.6 Further Reading, 102
8.7 References, 102
9 Multiconfigurational SCF Theory 103
9.1 Multiconfigurational SCF Theory, 103
9.1.1 The H2 Molecule, 104
9.1.2 Multiple Bonds, 107
9.1.3 Molecules with Competing Valence Structures, 108
9.1.4 Transition States on Energy Surfaces, 109
9.1.5 Other Cases of Near-Degeneracy Effects, 110
9.1.6 Static and Dynamic Correlation, 111
9.2 Determination of the MCSCF Wave Function, 114
9.2.1 Exponential Operators and Orbital Transformations, 115
9.2.2 Slater Determinants and Spin-Adapted State Functions, 117
9.2.3 The MCSCF Gradient and Hessian, 119
9.3 Complete and Restricted Active Spaces, the CASSCF and RASSCF
Methods, 121
9.3.1 State Average MCSCF, 125
9.3.2 Novel MCSCF Methods, 125
9.4 Choosing Active Space, 126
9.4.1 Atoms and Atomic Ions, 126
9.4.2 Molecules Built from Main Group Atoms, 128
9.5 References, 130
10 The RAS State-Interaction Method 131
10.1 The Biorthogonal Transformation, 131
10.2 Common One-Electron Properties, 133
10.3 Wigner–Eckart Coefficients for Spin-Orbit Interaction, 134
10.4 Unconventional Usage of RASSI, 135
10.5 Further Reading, 136
10.6 References, 136
11 The Multireference CI Method 137
11.1 Single-Reference CI. Nonextensivity, 137
11.2 Multireference CI, 139
11.3 Further Reading, 140
11.4 References, 140
12 Multiconfigurational Reference Perturbation Theory 143
12.1 CASPT2 theory, 143
12.1.1 Introduction, 143
12.1.2 Quasi-Degenerate Rayleigh–Schrödinger Perturbation
Theory, 144
12.1.3 The First-Order Interacting Space, 145
12.1.4 Multiconfigurational Root States, 146
12.1.5 The CASPT2 Equations, 148
12.1.6 IPEA, RASPT2, and MS-CASPT2, 154
12.2 References, 155
13 CASPT2/CASSCF Applications 157
13.1 Orbital Representations, 158
13.1.1 Starting Orbitals: Atomic Orbitals, 162
13.1.2 Starting Orbitals: Molecular Orbitals, 164
13.2 Specific Applications, 167
13.2.1 Ground State Reactions, 167
13.2.2 Excited States–Vertical Excitation Energies, 171
13.2.3 Photochemistry and Photophysics, 184
13.2.4 Transition Metal Chemistry, 194
13.2.5 Spin-Orbit Chemistry, 202
13.2.6 Lanthanide Chemistry, 207
13.2.7 Actinide Chemistry, 209
13.2.8 RASSCF/RASPT2 Applications, 212
13.3 References, 216
Summary and Conclusion 219
Index 221
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