Many-body methods for atoms and molecules / Rajat Kumar Chaudhuri, Sudip Kumar Chattopadhyay.

Format:

Book

Author/Creator:

Chaudhuri, Rajat Kumar, author.

Language:

English

Subjects (All):

Electron configuration.

Physical Description:

xvii, 219 pages ; 24 cm

Place of Publication:

Boca Raton, FL : CRC Press, [2017]

Summary:

Brings Readers from the Threshold to the Frontier of Modern Research, Many-Body Methods for Atoms and Molecules addresses two major classes of theories of electron correlation: the many-body perturbation theory and coupled cluster methods. It discusses the issues related to the formal development and consequent numerical implementation of the methods from the standpoint of a practicing theoretician. The book will enable readers to understand the future development of state-of-the-art multi-reference coupled cluster methods as well as their perturbative counterparts. The book begins with an introduction to the issues relevant to the development of correlated methods in general. It next gives a formally rigorous treatment of aspects that build the foundation toward the theoretical development of methods capable of tackling problems of electronic correlation. The authors proceed to cover perturbation theory first in a fundamental way and then in the multi-reference context. They also describe the idea of state-specific theories, Fock space-based multi-reference coupled cluster methods, and basic issues of the single-reference coupled cluster method. The book concludes with state-of-the-art methods of modern electronic structure. Features, Covers basic issues of electron correlation problems, Explores the frontier areas of electron correlation methods of modern electronic structure, Examines advances in the multi-reference approaches in the contexts of perturbation and coupled cluster theories, Emphasizes the essential theoretical details and related algebraic and diagrammatic formulations, including the logical simplification, by adopting a diagrammatic approach Book jacket.

Contents:

1 Introduction 1

1.1 Background 1

1.2 Born-Oppenheimer approximation 3

1.3 Approximate methods 6

1.3.1 Variational method: Linear variation principle 6

1.4 Independent particle model 8

1.4.1 Hartree product 8

1.4.2 Slater determinant 9

1.4.3 Slater's rule 10

1.4.4 Hartree-Fock method 11

1.4.5 Hartree-Fock-Roothan method 13

1.4.6 Brillouin's theorem 14

1.4.7 Koopmans' theorem 14

1.5 Configuration interaction 15

1.6 Electron correlation 19

1.7 Size extensivity and consistency 20

2 Occupation Number Representation 23

2.1 Background 23

2.2 Creation and annihilation operators 24

2.3 Occupation number representation of operators 28

2.4 Evaluation of matrix elements 29

2.4.1 Number operators 29

2.4.2 Overlap matrix elements 29

2.4.3 Matrix elements between vacuum states 30

2.5 Normal order product of ordinary operators 31

2.5.1 Wick's theorem for ordinary operators 32

2.6 Hole-particle formalism and Fermi vacuum 34

2.7 Evaluation of Hamiltonian elements between reference states 36

2.8 Normal order product for a Fermi vacuum 37

2.9 Normal product form of quantum mechanical operators 40

2.9.1 Evaluation of matrix elements between states 43

2.9.2 Vacuum expectation value of composite operators 44

2.10 Graphical representation of normal product operators 44

3 Perturbation Theory 49

3.1 Background 49

3.2 Rayleigh-Schrödinger perturbation theory: Traditional approach 50

3.2.1 Systematic derivation of order by order perturbation expansion 50

3.2.2 Wigner's (2n + 1) rule 53

3.2.3 Hylleraas variation principle 54

3.3 Projection operator based formulation of perturbation theory 56

3.4 Brillouin-Wigner perturbation theory 58

3.5 Rayleigh-Schrodinger perturbation theory 59

3.6 Wave operator based formulation of Rayleigh- Schrödinger perturbation theory 61

3.7 Choice of zero-order Hamiltonian H₀ 64

3.8 Intruder state problems in Rayleigh-Schrödinger perturbation theory 65

3.9 Comparison of Brillouin-Wigner and Rayleigh-Schrödinger perturbation theories 69

4 Multireference Perturbation Theory 73

4.1 Introduction 73

4.2 Choice of Fermi vacuum and the hole-particle states 75

4.3 Multiconfiguration self-consistent field method 78

4.4 Improved virtual orbital complete active space configuration method 81

4.4.1 Closed-shell ground state 81

4.4.2 Restricted open-shell doublet state 82

4.5 Classification of perturbative methods 83

4.6 Formal multireference perturbation theory for complete model space 84

4.6.1 Order by order expansion of the wave operator 87

4.6.2 Linked cluster theorem 91

4.7 Multireference perturbation theory for incomplete model space 92

4.8 Intermediate Hamiltonian methods 97

4.8.1 Generalized degenerate perturbation theory 99

4.9 Effective valence shell Hamiltonian method 100

5 State-Specific Perturbation Theory 107

5.1 Background 107

5.2 Multireference Möller-Plesset second-order perturbation theory 109

5.3 Multiconfiguration quasi-degenerate perturbation theory 113

5.4 Complete active space second-order perturbation theory 114

5.4.1 General formulation 114

5.4.2 Connection with effective Hamiltonian based method 117

5.5 Multistate complete active space second-order perturbation theory 118

6 Coupled Cluster Method 121

6.1 Introduction 121

6.2 Single-reference coupled cluster method 122

6.3 Separability 124

6.4 Relation with full configuration interaction method 125

6.5 Coupled cluster equation for doubles and singles-doubles approximations 127

6.5.1 Coupled cluster doubles method 127

6.5.2 Coupled cluster singles-doubles method 131

6.6 Evaluation of the matrix elements for coupled cluster doubles equations 133

6.6.1 Evaluation of correlation energy matrix elements 139

6.7 Diagrammatic representation of coupled cluster doubles matrix elements 140

6.7.1 Generation of coupled cluster diagrams 140

6.7.2 Diagram rules and evaluation of matrix elements 145

6.8 Emergence of many-body perturbation theory from CC method 147

6.9 Other variants of CC theory 149

7 Fock Space Multireference Coupled Cluster Method 151

7.1 Background 151

7.2 Choice of wave operator for multireference systems 153

7.3 Connectivity of the effective Hamiltonian 155

7.4 Fock space coupled cluster theory for energy difference 156

7.4.1 Hierarchical generation of coupled cluster equations 160

7.4.2 Quadratic: nature of Fock space coupled cluster equations 162

7.5 Systematic generation of cluster equations for various valence sectors 163

7.5.1 Coupled cluster equations for the (1,0) valence sector 163

7.5.2 Coupled cluster equations for the (0,1) valence sector 165

7.5.3 Coupled cluster equations for the (1,1) valence sector 167

7.6 Equation of motion coupled cluster method 172

7.7 Relationship between FSMRCC and EOMCC 175

7.8 Numerical examples 178

7.8.1 Ionization potential of Be I and Na I isoelectronic sequence 178

7.8.2 Ionization potential of Yb I 179

7.9 Intermediate Hamiltonian based multireference coupled cluster theory 180

7.9.1 Similarity transformation based formulation 182

7.9.2 Eigenvalue independent partitioning based coupled cluster formulation 184

8 Hilbert Space Coupled Cluster Theory 187

8.1 Introduction 187

8.2 State universal multireference Preference coupled cluster theory 188

8.2.1 State universal multireference perturbation theory 190

8.3 Development of state-specific theories 192

8.3.1 State-specific Brillouin-Wigner muftireference coupled cluster theory 193

8.3.2 State specific MkMRCC theory 195

8.3.3 State-specific multireference perturbation theory 197.

ISBN:

9781482211900

1482211904

OCLC:

964298209