Molecules in Electromagnetic Fields From Ultracold Physics to Controlled Chemistry

A tutorial for calculating the response of molecules to electric and magnetic fields with examples from research in ultracold physics, controlled chemistry, and molecular collisions in fields

Molecules in Electromagnetic Fields is intended to serve as a tutorial for students beginning research, theoretical or experimental, in an area related to molecular physics. The author—a noted expert in the field—offers a systematic discussion of the effects of static and dynamic electric and magnetic fields on the rotational, fine, and hyperfine structure of molecules. The book illustrates how the concepts developed in ultracold physics research have led to what may be the beginning of controlled chemistry in the fully quantum regime.  Offering a glimpse of the current state of the art research, this book suggests future research avenues for ultracold chemistry. 

The text describes theories needed to understand recent exciting developments in the research on trapping molecules, guiding molecular beams, laser control of molecular rotations, and external field control of microscopic intermolecular interactions. In addition, the author presents the description of scattering theory for molecules in electromagnetic fields and offers practical advice for students working on various aspects of molecular interactions. 

This important text:

• Offers information on theeffects of electromagnetic fields on the structure of molecular energy levels
• Includes thorough descriptions of the most useful theories for ultracold molecule researchers
• Presents a wealth of illustrative examples from recent experimental and theoretical work
• Contains helpful exercises that help to reinforce concepts presented throughout text

Written for senior undergraduate and graduate students, professors, researchers, physicists, physical chemists, and chemical physicists, Molecules in Electromagnetic Fields is an interdisciplinary text describing theories and examples from the core of contemporary molecular physics. 

ROMAN V. KREMS is a professor of theoretical chemistry at the University of British Columbia in Vancouver, Canada. His current research focuses on understanding the effects of electromagnetic fields on dynamics of few- and many-body molecular systems, the interaction properties of molecules at extremely low temperatures, and applications of machine learning to molecular physics.

List of Figures xiii

List of Tables xxv

Preface xxvii

Acknowledgments xxxi

1 Introduction to Rotational, Fine, and Hyperfine Structure of Molecular Radicals 1

1.1 Why Molecules are Complex 1

1.2 Separation of Scales 3

1.2.1 Electronic Energy 5

1.2.2 Vibrational Energy 10

1.2.3 Rotational and Fine Structure 14

1.3 Rotation of a Molecule 17

1.4 Hund’s Cases 21

1.4.1 Hund’s Coupling Case (a) 21

1.4.2 Hund’s Coupling Case (b) 22

1.4.3 Hund’s Coupling Case (c) 23

1.5 Parity of Molecular States 23

1.6 General Notation for Molecular States 27

1.7 Hyperfine Structure of Molecules 28

1.7.1 Magnetic Interactions with Nuclei 28

1.7.2 Fermi Contact Interaction 29

1.7.3 Long-Range Magnetic Dipole Interaction 30

1.7.4 Electric Quadrupole Hyperfine Interaction 31

Exercises 31

2 DCStarkEffect 35

2.1 Electric Field Perturbations 35

2.2 Electric Dipole Moment 37

2.3 Linear and Quadratic Stark Shifts 40

2.4 Stark Shifts of Rotational Levels 42

2.4.1 Molecules in a 1Σ Electronic State 42

2.4.2 Molecules in a 2Σ Electronic State 46

2.4.3 Molecules in a 3Σ Electronic State 48

2.4.4 Molecules in a 1Π Electronic State – Λ-Doubling 51

2.4.5 Molecules in a 2Π Electronic State 54

Exercises 56

3 Zeeman Effect 59

3.1 The Electron Spin 59

3.1.1 The Dirac Equation 60

3.2 Zeeman Energy of a Moving Electron 63

3.3 Magnetic Dipole Moment 64

3.4 Zeeman Operator in the Molecule-Fixed Frame 66

3.5 Zeeman Shifts of Rotational Levels 67

3.5.1 Molecules in a 2Σ State 67

3.5.2 Molecules in a 2Π Electronic State 71

3.5.3 Isolated Σ States 74

3.6 Nuclear Zeeman Effect 75

3.6.1 Zeeman Effect in a 1Σ Molecule 76

Exercises 78

4 ACStarkEffect 81

4.1 Periodic Hamiltonians 82

4.2 The Floquet Theory 84

4.2.1 Floquet Matrix 88

4.2.2 Time Evolution Operator 89

4.2.3 Brief Summary of Floquet Theory Results 90

4.3 Two-Mode Floquet Theory 92

4.4 RotatingWave Approximation 94

4.5 Dynamic Dipole Polarizability 96

4.5.1 Polarizability Tensor 97

4.5.2 Dipole Polarizability of a DiatomicMolecule 99

4.5.3 Rotational vs Vibrational vs Electronic Polarizability 101

4.6 Molecules in an Off-Resonant Laser Field 104

4.7 Molecules in a Microwave Field 107

4.8 Molecules in a Quantized Field 109

4.8.1 Field Quantization 109

4.8.2 Interaction of Molecules with Quantized Field 116

4.8.3 Quantized Field vs Floquet Theory 117

Exercises 118

5 Molecular Rotations Under Control 121

5.1 Orientation and Alignment 122

5.1.1 OrientingMolecular Axis in Laboratory Frame 123

5.1.2 Quantum Pendulum 126

5.1.3 Pendular States of Molecules 129

5.1.4 Alignment of Molecules by Intense Laser Fields 131

5.2 Molecular Centrifuge 136

5.3 OrientingMolecules Matters –Which Side Chemistry 140

5.4 Conclusion 142

Exercises 142

6 External Field Traps 145

6.1 Deflection and Focusing of Molecular Beams 146

6.2 Electric (and Magnetic) Slowing of Molecular Beams 151

6.3 Earnshaw’sTheorem 155

6.4 Electric Traps 158

6.5 Magnetic Traps 162

6.6 Optical Dipole Trap 165

6.7 Microwave Trap 167

6.8 Optical Lattices 168

6.9 Some Applications of External Field Traps 171

Exercises 173

7 Molecules in Superimposed Fields 175

7.1 Effects of Combined DC Electric andMagnetic Fields 175

7.1.1 Linear Stark Effect at Low Fields 175

7.1.2 Imaging of Radio-Frequency Fields 178

7.2 Effects of Combined DC and AC Electric Fields 181

7.2.1 Enhancement of Orientation by Laser Fields 181

7.2.2 Tug ofWar Between DC and Microwave Fields 182

8 Molecular Collisions in External Fields 187

8.1 Coupled-ChannelTheory of Molecular Collisions 188

8.1.1 A Very General Formulation 188

8.1.2 Boundary Conditions 191

8.1.3 Scattering Amplitude 194

8.1.4 Scattering Cross Section 197

8.1.5 Scattering of Identical Molecules 200

8.1.6 Numerical Integration of Coupled-Channel Equations 204

8.2 Interactions with External Fields 208

8.2.1 Coupled-Channel Equations in Arbitrary Basis 208

8.2.2 External Field Couplings 209

8.3 The Arthurs–Dalgarno Representation 211

8.4 Scattering Rates 214

9 Matrix Elements of Collision Hamiltonians 217

9.1 Wigner–EckartTheorem 218

9.2 Spherical Tensor Contraction 220

9.3 Collisions in a Magnetic Field 221

9.3.1 Collisions of 1S-Atoms with 2Σ-Molecules 221

9.3.2 Collisions of 1S-Atoms with 3Σ-Molecules 225

9.4 Collisions in an Electric Field 229

9.4.1 Collisions of 2Π Molecules with 1S Atoms 229

9.5 Atom–Molecule Collisions in a Microwave Field 232

9.6 Total Angular Momentum Representation for Collisions in Fields 234

10 Field-Induced Scattering Resonances 239

10.1 Feshbach vs Shape Resonances 239

10.2 The Green’s Operator in Scattering Theory 242

10.3 Feshbach Projection Operators 243

10.4 Resonant Scattering 246

10.5 Calculation of Resonance Locations andWidths 249

10.5.1 Single Open Channel 249

10.5.2 Multiple Open Channels 249

10.6 Locating Field-Induced Resonances 252

11 Field Control of Molecular Collisions 257

11.1 Why to Control Molecular Collisions 257

11.2 Molecular Collisions are Difficult to Control 259

11.3 General Mechanisms for External Field Control 261

11.4 Resonant Scattering 261

11.5 Zeeman and Stark Relaxation at Zero Collision Energy 264

11.6 Effect of Parity Breaking in Combined Fields 269

11.7 Differential Scattering in Electromagnetic Fields 271

11.8 Collisions in Restricted Geometries 272

11.8.1 Threshold Scattering of Molecules in Two Dimensions 276

11.8.2 Collisions in a Quasi-Two-Dimensional Geometry 280

12 Ultracold Controlled Chemistry 283

12.1 Can Chemistry Happen at Zero Kelvin? 284

12.2 Ultracold Stereodynamics 287

12.3 Molecular Beams Under Control 289

12.4 Reactions in Magnetic Traps 289

12.5 Ultracold Chemistry – The Why and What’s Next? 291

12.5.1 Practical Importance of Ultracold Chemistry? 291

12.5.2 Fundamental Importance of Ultracold Controlled Chemistry 293

12.5.3 A Brief Outlook 294

A Unit Conversion Factors 297

B Addition of AngularMomenta 299

B.1 The Clebsch–Gordan Coefficients 301

B.2 TheWigner 3j-Symbols 303

B.3 The Raising and Lowering Operators 304

C Direction Cosine Matrix 307

D Wigner D-Functions 309

D.1 Matrix elements involving D-functions 311

E Spherical tensors 315

E.1 Scalar and Vector Products of Vectors in Spherical Basis 317

E.2 Scalar and Tensor Products of Spherical Tensors 318

References 321

Index 347

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