Principles and Practices of Molecular Properties

Patrick Norman is Professor and Head of Theoretical Chemistry and Biology at KTH Royal Institute of Technology, Stockholm, Sweden. His research interests include response theory for non-resonant and resonant external fields in the UV/vis and X-ray regions. He is a co-author of the Dalton program. Kenneth Ruud is Professor of Theoretical Chemistry at the University of Tromsø – The Arctic University of Norway. His research interests include linear and nonlinear response theory for mixed electric and magnetic fields as well as vibrational and medium effects. He is a co-author of the Dalton program. Trond Saue is a directeur de recherché of the French National Center for Scientific Research (CNRS) working at Université Toulouse III-Paul Sabatier in France. His research focuses on relativistic methods in theoretical chemistry. He is a principal author of the DIRAC program.

Preface xi

1 Introduction 1

2 Quantum Mechanics 11

2.1 Fundamentals 11

2.1.1 Postulates of Quantum Mechanics 11

2.1.2 Lagrangian and Hamiltonian Formalisms 11

2.1.3 Wave Functions and Operators 18

2.2 Time Evolution of Wave Functions 22

2.3 Time Evolution of Expectation Values 25

2.4 Variational Principle 27

Further Reading 29

3 Particles and Fields 31

3.1 Microscopic Maxwell’s Equations 32

3.1.1 General Considerations 32

3.1.2 The Stationary Case 34

3.1.3 The General Case 38

3.1.4 Electromagnetic Potentials and Gauge Freedom 39

3.1.5 Electromagnetic Waves and Polarization 41

3.1.6 Electrodynamics: Relativistic and Nonrelativistic Formulations 45

3.2 Particles in Electromagnetic Fields 48

3.2.1 The Classical Mechanical Hamiltonian 48

3.2.2 The Quantum-Mechanical Hamiltonian 52

3.3 Electric and Magnetic Multipoles 57

3.3.1 Multipolar Gauge 57

3.3.2 Multipole Expansions 59

3.3.3 The Electric Dipole Approximation and Beyond 63

3.3.4 Origin Dependence of Electric and Magnetic Multipoles 64

3.3.5 Electric Multipoles 65

3.3.5.1 General Versus Traceless Forms 65

3.3.5.2 What We Can Learn from Symmetry 68

3.3.6 Magnetic Multipoles 69

3.3.7 Electric Dipole Radiation 70

3.4 Macroscopic Maxwell’s Equations 72

3.4.1 Spatial Averaging 72

3.4.2 Polarization and Magnetization 73

3.4.3 Maxwell’s Equations in Matter 77

3.4.4 Constitutive Relations 79

3.5 Linear Media 81

3.5.1 Boundary Conditions 82

3.5.2 Polarization in Linear Media 86

3.5.3 Electromagnetic Waves in a Linear Medium 92

3.5.4 Frequency Dependence of the Permittivity 96

3.5.4.1 Kramers–Kronig Relations 97

3.5.4.2 Relaxation in the Debye Model 98

3.5.4.3 Resonances in the Lorentz Model 101

3.5.4.4 Refraction and Absorption 105

3.5.5 Rotational Averages 107

3.5.6 A Note About Dimensions, Units, and Magnitudes 110

Further Reading 111

4 Symmetry 113

4.1 Fundamentals 113

4.1.1 Symmetry Operations and Groups 113

4.1.2 Group Representation 117

4.2 Time Symmetries 120

4.3 Spatial Symmetries 125

4.3.1 Spatial Inversion 125

4.3.2 Rotations 127

Further Reading 134

5 Exact-State Response Theory 135

5.1 Responses in Two-Level System 135

5.2 Molecular Electric Properties 145

5.3 Reference-State Parameterizations 151

5.4 Equations of Motion 156

5.4.1 Time Evolution of Projection Amplitudes 157

5.4.2 Time Evolution of Rotation Amplitudes 159

5.5 Response Functions 163

5.5.1 First-Order Properties 166

5.5.2 Second-Order Properties 166

5.5.3 Third-Order Properties 169

5.5.4 Fourth-Order Properties 174

5.5.5 Higher-Order Properties 179

5.6 Dispersion 179

5.7 Oscillator Strength and Sum Rules 183

5.8 Absorption 185

5.9 Residue Analysis 190

5.10 Relaxation 194

5.10.1 Density Operator 195

5.10.2 Liouville Equation 196

5.10.3 Density Matrix from Perturbation Theory 200

5.10.4 Linear Response Functions from the Density Matrix 201

5.10.5 Nonlinear Response Functions from the Density Matrix 204

5.10.6 Relaxation in Wave Function Theory 204

5.10.7 Absorption Cross Section 207

5.10.8 Einstein Coefficients 210

Further Reading 211

6 Electronic and Nuclear Contributions to Molecular Properties 213

6.1 Born–Oppenheimer Approximation 213

6.2 Separation of Response Functions 216

6.3 Molecular Vibrations and Normal Coordinates 221

6.4 Perturbation Theory for Vibrational Wave Functions 225

6.5 Zero-Point Vibrational Contributions to Properties 227

6.5.1 First-Order Anharmonic Contributions 227

6.5.2 Importance of Zero-Point Vibrational Corrections 231

6.5.3 Temperature Effects 234

6.6 Pure Vibrational Contributions to Properties 235

6.6.1 Perturbation Theory Approach 235

6.6.2 Pure Vibrational Effects from an Analysis of the Electric-Field Dependence of the Molecular Geometry 238

6.7 Adiabatic Vibronic Theory for Electronic Excitation Processes 244

6.7.1 Franck–Condon Integrals 248

6.7.2 Vibronic Effects in a Diatomic System 250

6.7.3 Linear Coupling Model 252

6.7.4 Herzberg–Teller Corrections and Vibronically Induced Transitions 252

Further Reading 253

7 Approximate Electronic State Response Theory 255

7.1 Reference State Parameterizations 255

7.1.1 Single Determinant 255

7.1.2 Configuration Interaction 263

7.1.3 Multiconfiguration Self-Consistent Field 266

7.1.4 Coupled Cluster 268

7.2 Equations of Motion 271

7.2.1 Ehrenfest Theorem 271

7.2.2 Quasi-Energy Derivatives 275

7.3 Response Functions 276

7.3.1 Single Determinant Approaches 276

7.3.2 Configuration Interaction 281

7.3.3 Multiconfiguration Self-Consistent Field 281

7.3.4 Matrix Structure in the SCF, CI, and MCSCF Approximations 281

7.3.5 Coupled Cluster 285

7.4 Residue Analysis 288

7.5 Relaxation 291

Further Reading 293

8 Response Functions and Spectroscopies 295

8.1 Nuclear Interactions 296

8.1.1 Nuclear Charge Distribution 296

8.1.2 Hyperfine Structure 301

8.1.2.1 Nuclear Magnetic Dipole Moment 301

8.1.2.2 Nuclear Electric Quadrupole Moment 305

8.2 Zeeman Interaction and Electron Paramagnetic Resonance 310

8.3 Polarizabilities 317

8.3.1 Linear Polarizability 317

8.3.1.1 Weak Intermolecular Forces 321

8.3.2 Nonlinear Polarizabilities 325

8.4 Magnetizability 326

8.4.1 The Origin Dependence of the Magnetizability 328

8.4.2 Magnetizabilities from Magnetically Induced Currents 331

8.4.3 Isotropic Magnetizabilities and Pascal’s Rule 332

8.5 Electronic Absorption and Emission Spectroscopies 335

8.5.1 Visible and Ultraviolet Absorption 338

8.5.2 Fluorescence Spectroscopy 343

8.5.3 Phosphorescence 344

8.5.4 Multiphoton Absorption 347

8.5.4.1 Multiphoton Absorption Cross Sections 348

8.5.4.2 Few-State Models for Two-Photon Absorption Cross Section 350

8.5.4.3 General Multiphoton Absorption Processes 351

8.5.5 X-ray Absorption 354

8.5.5.1 Core-Excited States 355

8.5.5.2 Field Polarization 358

8.5.5.3 Static Exchange Approximation 360

8.5.5.4 Complex or Damped Response Theory 362

8.6 Birefringences and Dichroisms 364

8.6.1 Natural Optical Activity 366

8.6.2 Electronic Circular Dichroism 372

8.6.3 Nonlinear Birefringences 375

8.6.3.1 Magnetic Circular Dichroism 376

8.6.3.2 Electric Field Gradient-Induced Birefringence 379

8.7 Vibrational Spectroscopies 381

8.7.1 Infrared Absorption 381

8.7.1.1 Double-Harmonic Approximation 381

8.7.1.2 Anharmonic Corrections 383

8.7.2 Vibrational Circular Dichroism 384

8.7.3 Raman Scattering 388

8.7.3.1 Raman Scattering from a Classical Point of View 388

8.7.3.2 Raman Scattering from a Quantum Mechanical Point of View 392

8.7.4 Vibrational Raman Optical Activity 402

8.8 Nuclear Magnetic Resonance 408

8.8.1 The NMR Experiment 408

8.8.2 NMR Parameters 413

Further Reading 417

Appendicies

A Abbreviations 419

B Units 421

C Second Quantization 423

C.1 Creation and Annihilation Operators 423

C.2 Fock Space 425

C.3 The Number Operator 426

C.4 The Electronic Hamiltonian on Second-Quantized Form 427

C.5 Spin in Second Quantization 429

D Fourier Transforms 431

E Operator Algebra 435

F Spin Matrix Algebra 439

G Angular Momentum Algebra 441

H Variational Perturbation Theory 445

I Two-Level Atom 451

I.1 Rabi Oscillations 452

I.2 Time-Dependent Perturbation Theory 454

I.3 The Quasi-energy Approach 455

Index 457