Beyond Born-Oppenheimer : electronic non-adiabatic coupling terms and conical intersections / by Michael Baer.

Format:

Book

Author/Creator:

Baer, M. (Michael)

Contributor:

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Molecular dynamics--Mathematics.

Molecular dynamics.

Born-Oppenheimer approximation.

Adiabatic invariants.

Physical Description:

xvii, 234 pages : illustrations ; 24 cm

Place of Publication:

Hoboken, N.J. : Wiley, [2006]

Summary:

The Born-Oppenheimer approximation has been fundamental to calculations in molecular spectroscopy and molecular dynamics since the early days of quantum mechanics. This is despite the well-established fact that it is often not valid due to conical intersections that give rise to strong nonadiabatic effects caused by singular nonadiabatic coupling terms (NACTs). In Beyond Born-Oppenheimer, Michael Baer, a leading authority on molecular scattering theory and electronic nonadiabatic processes, addresses this deficiency and introduces a rigorous approach-diabatization-for eliminating troublesome NACTs and deriving well-converged equations to treat the interactions within and between molecules.

Concentrating on both the practical and theoretical aspects of electronic nonadiabatic transitions in molecules, Professor Baer uses a simple mathematical language to rigorously eliminate the singular NACTs and enable reliable calculations of spectroscopic and dynamical cross sections. He presents models of varying complexity to illustrate the validity of the theory and explores the significance of the study of NACTs and the relationship between molecular physics and other fields in physics, particularly electrodynamics. Beyond Born-Oppenheimer is required reading for physicists, physical chemists, and all researchers involved in the quantum mechanical study of molecular systems.

Contents:

1 Mathematical Introduction 1

1.1 Hilbert Space 1

1.1.1 Eigenfunction and Electronic Nonadiabatic Coupling Term 1

1.1.2 Abelian and Non-Abelian Curl Equations 3

1.1.3 Abelian and Non-Abelian Divergence Equations 6

1.2 Hilbert Subspace 8

1.3 Vectorial First-Order Differential Equation and Line Integral 11

1.3.1 Vectorial First-Order Differential Equation 12

1.3.1.1 Study of Abelian Case 12

1.3.1.2 Study of Non-Abelian Case 13

1.3.1.3 Orthogonality 14

1.3.2 Integral Equation 14

1.3.2.1 Integral Equation along an Open Contour 15

1.3.2.2 Integral Equation along a Closed Contour 16

1.3.3 Solution of Differential Vector Equation 20

2 Born-Oppenheimer Approach: Diabatization and Topological Matrix 26

2.1 Time-Independent Treatment 26

2.1.1 Adiabatic Representation 26

2.1.2 Diabatic Representation 28

2.1.3 Adiabatic-to-Diabatic Transformation 30

2.1.3.1 Transformation for Electronic Basis Sets 30

2.1.3.2 Transformation for Nuclear Wavefunctions 33

2.1.3.3 Implications Due to Adiabatic-to-Diabatic Transformation 34

2.2 Application of Complex Eigenfunctions 39

2.2.1 Introducing Time-Independent Phase Factors 39

2.2.1.1 Adiabatic Schrodinger Equation 39

2.2.1.2 Adiabatic-to-Diabatic Transformation 40

2.2.2 Introducing Time-Dependent Phase Factors 41

2.3 Time-Dependent Treatment 43

2.3.1 Time-Dependent Perturbative Approach 43

2.3.2 Time-Dependent Nonperturbative Approach 45

2.3.2.1 Adiabatic Time-Dependent Electronic Basis Set 45

2.3.2.2 Adiabatic Time-Dependent Nuclear Schrodinger Equation 46

2.3.2.3 Time-Dependent Adiabatic-to-Diabatic Transformation 47

2A.1 Dressed Nonadiabatic Coupling Matrix [tau] 51

2A.2 Analyticity of Adiabatic-to-Diabatic Transformation Matrix A in Spacetime Configuration 52

3 Model Studies 58

3.1 Treatment of Analytical Models 58

3.1.1 Two-State Systems 59

3.1.1.1 Adiabatic-to-Diabatic Transformation Matrix 59

3.1.1.2 Topological (D) Matrix 60

3.1.1.3 The Diabatic Potential Matrix 61

3.1.2 Three-State Systems 62

3.1.2.1 Adiabatic-to-Diabatic Transformation Matrix 62

3.1.2.2 Topological Matrix 63

3.1.3 Four-State Systems 64

3.1.3.1 Adiabatic-to-Diabatic Transformation Matrix 64

3.1.3.2 Topological Matrix 65

3.1.4 Comments Related to General Case 66

3.2 Study of 2 x 2 Diabatic Potential Matrix and Related Topics 67

3.2.1 Treatment of General Case 67

3.2.2 The Jahn-Teller Model 70

3.2.3 Elliptic Jahn-Teller Model 72

3.2.4 Distribution of Conical Intersections and Diabatic Potential Matrix 73

3.3 Adiabatic-to-Diabatic Transformation Matrix and Wigner Rotation Matrix 75

3.3.1 Wigner Rotation Matrices 76

3.3.2 Adiabatic-to-Diabatic Transformation Matrix and Wigner d[superscript j] Matrix 77

4 Studies of Molecular Systems 84

4.2 Theoretical Background 85

4.3 Quantization of Nonadiabatic Coupling Matrix: Study of Ab Initio Molecular Systems 87

4.3.1 Two-State Quasiquantization 87

4.3.1.1 {H[subscript 2],H} System 87

4.3.1.2 {H[subscript 2],O} System 91

4.3.1.3 {C[subscript 2],H[subscript 2]} System 92

4.3.2 Multistate Quasiquantization 96

4.3.2.1 {H[subscript 2],H} System 96

4.3.2.2 {C[subscript 2],H} System 99

5 Degeneracy Points and Born-Oppenheimer Coupling Terms as Poles 105

5.1 Relation between Born-Oppenheimer Coupling Terms and Degeneracy Points 105

5.2 Construction of Hilbert Subspace 108

5.3 Sign Flips of Electronic Eigenfunctions 109

5.3.1 Two-State Hilbert Subspace 109

5.3.2 Three-State Hilbert Subspace 110

5.3.3 General Hilbert Subspace 114

5.3.4 Multidegeneracy Point 120

5.3.4.1 General Approach 120

5.3.4.2 Model Studies 121

5.4 Topological Spin 122

5.5 Extended Euler Matrix as a Model for Adiabatic-to-Diabatic Transformation Matrix 125

5.5.2 Two-Dimensional Case 126

5.5.3 Three-Dimensional Case 127

5.5.4 Multidimensional Case 130

5.6 Quantization of [tau] Matrix and its Relation to Size of Configuration Space: Mathieu Equation as a Case of Study 131

5.6.1 Mathieu Equation and Its Eigenfunctions 131

5.6.2 Nonadiabatic Coupling Matrix ([tau]) and Topological Matrix (D) 133

6 Molecular Field 139

6.1 Solenoid as a Model for the Seam 139

6.2 Two-State (Abelian) System 141

6.2.1 Nonadiabatic Coupling Term as a Vector Potential 141

6.2.2 Two-State Curl Equation 143

6.2.3 (Extended) Stokes Theorem 144

6.2.4 Application of Stokes Theorem for Several Two-State Conical Intersections 146

6.2.5 Application of Vector Algebra to Calculate the Field of a Two-State Hilbert Space 147

6.2.6 A Numerical Example: Study of {H[subscript 2],Na} System 149

6.2.7 A Short Summary 151

6.3 Multistate Hilbert Subspace 151

6.3.1 Non-Abelian Stokes Theorem 151

6.3.2 The Curl-Divergence Equations 154

6.3.2.1 Three-State Hilbert Subspace 155

6.3.2.2 Derivation of Poisson Equations 157

6.3.2.3 Strange Matrix Element and Gauge Transformation 159

6.4 A Numerical Study of {H[subscript 2],H} System 160

6.4.2 Introducing Fourier Expansion 160

6.4.3 Introducing Boundary Conditions 161

6.4.4 Numerical Results 162

6.5 Multistate Hilbert Subspace: Breakup of Nonadiabatic Coupling Matrix 162

6.6 Pseudomagnetic Field 167

6.6.1 Quantization of Pseudomagnetic Field along the Seam 168

6.6.2 Non-Abelian Magnetic Fields 168

7 Open Phase and Berry Phase for Molecular Systems 175

7.1 Studies of Ab Initio Systems 175

7.1.2 Open Phase and Berry Phase for Singlevalued Eigenfunctions: Berry's Approach 176

7.1.3 Open Phase and Berry Phase for Multivalued Eigenfunctions: Present Approach 177

7.1.3.1 Derivation of Time-Dependent Equation 177

7.1.3.2 Treatment of Adiabatic Case 179

7.1.3.3 Treatment of Nonadiabatic (General) Case 181

7.1.4 {H[subscript 2],H} System as a Case Study 183

7.2 Phase-Modulus Relations for an External Cyclic Time-Dependent Field 187

7.2.1 Derivation of Reciprocal Relations 187

7.2.2 Mathieu Equation 189

7.2.2.1 Time-Dependent Schrodinger Equations 189

7.2.2.2 Numerical Study of Topological Phase 191

7.2.3 Short Summary 194

8 Extended Born-Oppenheimer Approximations 197

8.2 Born-Oppenheimer Approximation as Applied to a Multistate Model System 199

8.2.1 Extended Approximate Born-Oppenheimer Equation 199

8.2.2 Gauge Invariance Condition for Approximate Born-Oppenheimer Equations 201

8.3 Multistate Born-Oppenheimer Approximation: Energy Considerations to Reduce Dimensions of Diabatic Potential Matrix 201

8.3.2 Derivation of Reduced Diabatic Potential Matrix 202

8.3.3 Application of Extended Euler Matrix: Introducing the N-State Adiabatic-to-Diabatic Transformation Angle 206

8.3.3.2 Derivation of Adiabatic-to-Diabatic Transformation Angle 206

8.3.4 Two-State Diabatic Potential Energy Matrix 210

8.3.4.1 Derivation of Diabatic Potential Matrix 210

8.3.4.2 A Numerical Study of [Delta] W Matrix Elements 211

8.3.4.3 A Different Approach: Helmholtz Decomposition 214

8.4 A Numerical Study of a Reactive (Exchange) Scattering Two-Coordinate Model 214

8.4.1 Basic Equations 214

8.4.2 A Two-Coordinate Reactive (Exchange) Model 216

8.4.3 Numerical Results and Discussion 217.

Notes:

Includes index.

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0471778915

9780471778912

OCLC:

61162263

Online:

Publisher description