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Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, 2nd Edition

ISBN: 978-3-527-33415-5
750 pages
January 2015
Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, 2nd Edition (3527334157) cover image

Description

Einstein proposed his theory of special relativity in 1905. For a long time it was believed that this theory has no significant impact on chemistry. This view changed in the 1970s when it was realized that (nonrelativistic) Schrödinger quantum mechanics yields results on molecular properties that depart significantly from experimental results. Especially when heavy elements are involved, these quantitative deviations can be so large that qualitative chemical reasoning and understanding is affected. For this to grasp the appropriate many-electron theory has rapidly evolved. Nowadays relativistic approaches are routinely implemented and applied in standard quantum chemical software packages. As it is essential for chemists and physicists to understand relativistic effects in molecules, the first edition of "Relativistic Quantum Chemistry - The fundamental Theory of Molecular Science" had set out to provide a concise, comprehensive, and complete presentation of this theory.

This second edition expands on some of the latest developments in this fascinating field. The text retains its clear and consistent style, allowing for a readily accessible overview of the complex topic. It is also self-contained, building on the fundamental equations and providing the mathematical background necessary. While some parts of the text have been restructured for the sake of clarity a significant amount of new content has also been added. This includes, for example, an in-depth discussion of the Brown-Ravenhall disease, of spin in current-density functional theory, and of exact two-component methods and its local variants.
A strength of the first edition of this textbook was its list of almost 1000 references to the original research literature, which has made it a valuable reference also for experts in the field. In the second edition, more than 100 additional key references have been added - most of them considering the recent developments in the field.
Thus, the book is a must-have for everyone entering the field, as well as for experienced researchers searching for a consistent review.
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Table of Contents

Preface XXI

1 Introduction 1

1.1 Philosophy of this Book 1

1.2 Short Reader’s Guide 4

1.3 Notational Conventions and Choice of Units 6

Part I — Fundamentals 9

2 Elements of Classical Mechanics and Electrodynamics 11

2.1 Elementary Newtonian Mechanics 11

2.1.1 Newton’s Laws of Motion 11

2.1.2 Galilean Transformations 14

2.1.3 Conservation Laws for One Particle in Three Dimensions 20

2.1.4 Collection of N Particles 21

2.2 Lagrangian Formulation 22

2.2.1 Generalized Coordinates and Constraints 22

2.2.2 Hamiltonian Principle and Euler–Lagrange Equations 24

2.2.3 Symmetries and Conservation Laws 28

2.3 Hamiltonian Mechanics 31

2.3.1 Hamiltonian Principle and Canonical Equations 31

2.3.2 Poisson Brackets and Conservation Laws 33

2.3.3 Canonical Transformations 34

2.4 Elementary Electrodynamics 35

2.4.1 Maxwell’s Equations 36

2.4.2 Energy and Momentum of the Electromagnetic Field 38

2.4.3 Plane Electromagnetic Waves in Vacuum 40

2.4.4 Potentials and Gauge Symmetry 42

2.4.5 Survey of Electro– and Magnetostatics 45

2.4.6 One Classical Particle Subject to Electromagnetic Fields 47

2.4.7 Interaction of Two Moving Charged Particles 50

3 Concepts of Special Relativity 53

3.1 Einstein’s Relativity Principle and Lorentz Transformations 53

3.1.1 Deficiencies of Newtonian Mechanics 53

3.1.2 Relativity Principle of Einstein 55

3.1.3 Lorentz Transformations 58

3.1.4 Scalars, Vectors, and Tensors in Minkowski Space 62

3.2 Kinematic Effects in Special Relativity 67

3.2.1 Explicit Form of Special Lorentz Transformations 67

3.2.2 Length Contraction, Time Dilation, and Proper Time 72

3.2.3 Addition of Velocities 75

3.3 Relativistic Dynamics 78

3.3.1 Elementary Relativistic Dynamics 79

3.3.2 Equation of Motion 83

3.3.3 Lagrangian and Hamiltonian Formulation 86

3.4 Covariant Electrodynamics 90

3.4.1 Ingredients 91

3.4.2 Transformation of Electromagnetic Fields 95

3.4.3 Lagrangian Formulation and Equations of Motion 96

3.5 Interaction of Two Moving Charged Particles 101

3.5.1 Scalar and Vector Potentials of a Charge at Rest 102

3.5.2 Retardation from Lorentz Transformation 104

3.5.3 General Expression for the Interaction Energy 105

3.5.4 Interaction Energy at One Instant of Time 105

3.5.5 Symmetrized Darwin Interaction Energy 112

4 Basics of Quantum Mechanics 117

4.1 The Quantum Mechanical State 118

4.1.1 Bracket Notation 118

4.1.2 Expansion in a Complete Basis Set 119

4.1.3 Born Interpretation 119

4.1.4 State Vectors in Hilbert Space 121

4.2 The Equation of Motion 122

4.2.1 Restrictions on the Fundamental Quantum Mechanical Equation 122

4.2.2 Time Evolution and Probabilistic Character 123

4.2.3 Stationary States 123

4.3 Observables 124

4.3.1 Expectation Values 124

4.3.2 Hermitean Operators 125

4.3.3 Unitary Transformations 126

4.3.4 Heisenberg Equation of Motion 127

4.3.5 Hamiltonian in Nonrelativistic Quantum Theory 129

4.3.6 Commutation Relations for Position and Momentum Operators 131

4.3.7 The Schrödinger Velocity Operator 132

4.3.8 Ehrenfest and Hellmann–Feynman Theorems 133

4.3.9 Current Density and Continuity Equation 135

4.4 Angular Momentum and Rotations 139

4.4.1 Classical Angular Momentum 139

4.4.2 Orbital Angular Momentum 140

4.4.3 Coupling of Angular Momenta 145

4.4.4 Spin 147

4.4.5 Coupling of Orbital and Spin Angular Momenta 149

4.5 Pauli Antisymmetry Principle 155

Part II — Dirac’s Theory of the Electron 159

5 Relativistic Theory of the Electron 161

5.1 Correspondence Principle and Klein–Gordon Equation 161

5.1.1 Classical Energy Expression and First Hints from the Correspondence Principle 161

5.1.2 Solutions of the Klein–Gordon Equation 163

5.1.3 The Klein–Gordon Density Distribution 164

5.2 Derivation of the Dirac Equation for a Freely Moving Electron 166

5.2.1 Relation to the Klein–Gordon Equation 166

5.2.2 Explicit Expressions for the Dirac Parameters 167

5.2.3 Continuity Equation and Definition of the 4-Current 169

5.2.4 Lorentz Covariance of the Field-Free Dirac Equation 170

5.3 Solution of the Free-Electron Dirac Equation 173

5.3.1 Particle at Rest 173

5.3.2 Freely Moving Particle 175

5.3.3 The Dirac Velocity Operator 179

5.4 Dirac Electron in External Electromagnetic Potentials 181

5.4.1 Kinematic Momentum 184

5.4.2 Electromagnetic Interaction Energy Operator 184

5.4.3 Nonrelativistic Limit and Pauli Equation 185

5.5 Interpretation of Negative-Energy States: Dirac’s Hole Theory 187

6 The Dirac Hydrogen Atom 193

6.1 Separation of Electronic Motion in a Nuclear Central Field 193

6.2 Schrödinger Hydrogen Atom 197

6.3 Total Angular Momentum 199

6.4 Separation of Angular Coordinates in the Dirac Hamiltonian 200

6.4.1 Spin–Orbit Coupling 200

6.4.2 Relativistic Azimuthal Quantum Number Analog 201

6.4.3 Four-Dimensional Generalization 203

6.4.4 Ansatz for the Spinor 204

6.5 Radial Dirac Equation for Hydrogen-Like Atoms 204

6.5.1 Radial Functions and Orthonormality 205

6.5.2 Radial Eigenvalue Equations 206

6.5.3 Solution of the Coupled Dirac Radial Equations 207

6.5.4 Energy Eigenvalue, Quantization and the Principal Quantum Number 213

6.5.5 The Four-Component Ground State Wave Function 215

6.6 The Nonrelativistic Limit 216

6.7 Choice of the Energy Reference and Matching Energy Scales 218

6.8 Wave Functions and Energy Eigenvalues in the Coulomb Potential 219

6.8.1 Features of Dirac Radial Functions 219

6.8.2 Spectrum of Dirac Hydrogen-like Atoms with Coulombic Potential 221

6.8.3 Radial Density and Expectation Values 223

6.9 Finite Nuclear Size Effects 225

6.9.1 Consequences of the Nuclear Charge Distribution 227

6.9.2 Spinors in External Scalar Potentials of Varying Depth 229

6.10 Momentum Space Representation 233

Part III — Four-Component Many-Electron Theory 235

7 Quantum Electrodynamics 237

7.1 Elementary Quantities and Notation 237

7.1.1 Lagrangian for Electromagnetic Interactions 237

7.1.2 Lorentz and Gauge Symmetry and Equations of Motion 238

7.2 Classical Hamiltonian Description 240

7.2.1 Exact Hamiltonian 240

7.2.2 The Electron–Electron Interaction 241

7.3 Second-Quantized Field-Theoretical Formulation 243

7.4 Implications for the Description of Atoms and Molecules 246

8 First-Quantized Dirac-Based Many-Electron Theory 249

8.1 Two-Electron Systems and the Breit Equation 250

8.1.1 Dirac Equation Generalized for Two Bound-State Electrons 251

8.1.2 The Gaunt Operator for Unretarded Interactions 253

8.1.3 The Breit Operator for Retarded Interactions 256

8.1.4 Exact Retarded Electromagnetic Interaction Energy 260

8.1.5 Breit Interaction from Quantum Electrodynamics 266

8.2 Quasi-Relativistic Many-Particle Hamiltonians 270

8.2.1 Nonrelativistic Hamiltonian for a Molecular System 270

8.2.2 First-Quantized Relativistic Many-Particle Hamiltonian 272

8.2.3 Pathologies of the First-Quantized Formulation 274

8.2.4 Local Model Potentials for One-Particle QED Corrections 278

8.3 Born–Oppenheimer Approximation 279

8.4 Tensor Structure of the Many-Electron Hamiltonian andWave Function 283

8.5 Approximations to the Many-Electron Wave Function 285

8.5.1 The Independent-Particle Model 286

8.5.2 Configuration Interaction 287

8.5.3 Detour: Explicitly Correlated Wave Functions 291

8.5.4 Orthonormality Constraints and Total Energy Expressions 292

8.6 Second Quantization for the Many-Electron Hamiltonian 296

8.6.1 Creation and Annihilation Operators 296

8.6.2 Reduction of Determinantal Matrix Elements to Matrix Elements Over Spinors 297

8.6.3 Many-Electron Hamiltonian and Energy 299

8.6.4 Fock Space and Occupation Number Vectors 300

8.6.5 Fermions and Bosons 301

8.7 Derivation of Effective One-Particle Equations 301

8.7.1 Avoiding Variational Collapse: The Minimax Principle 302

8.7.2 Variation of the Energy Expression 304

8.7.3 Self-Consistent Field Equations 306

8.7.4 Dirac–Hartree–Fock Equations 309

8.7.5 The Relativistic Self-Consistent Field 312

8.8 Relativistic Density Functional Theory 313

8.8.1 Electronic Charge and Current Densities for Many Electrons 314

8.8.2 Current-Density Functional Theory 317

8.8.3 The Four-Component Kohn–Sham Model 318

8.8.4 Electron Density and Spin Density in Relativistic DFT 320

8.8.5 Relativistic Spin-DFT 322

8.8.6 Noncollinear Approaches and Collinear Approximations 323

8.8.7 Relation to the Spin Density 324

8.9 Completion: The Coupled-Cluster Expansion 325

9 Many-Electron Atoms 333

9.1 Transformation of the Many-Electron Hamiltonian to Polar Coordinates 335

9.1.1 Comment on Units 336

9.1.2 Coulomb Interaction in Polar Coordinates 336

9.1.3 Breit Interaction in Polar Coordinates 337

9.1.4 Atomic Many-Electron Hamiltonian 341

9.2 Atomic Many-Electron Wave Function and jj-Coupling 341

9.3 One- and Two-Electron Integrals in Spherical Symmetry 344

9.3.1 One-Electron Integrals 344

9.3.2 Electron–Electron Coulomb Interaction 345

9.3.3 Electron–Electron Frequency-Independent Breit Interaction 349

9.3.4 Calculation of Potential Functions 351

9.4 Total Expectation Values 354

9.4.1 General Expression for the Electronic Energy 354

9.4.2 Breit Contribution to the Total Energy 356

9.4.3 Dirac–Hartree–Fock Total Energy of Closed-Shell Atoms 357

9.5 General Self-Consistent-Field Equations and Atomic Spinors 358

9.5.1 Dirac–Hartree–Fock Equations 360

9.5.2 Comparison of Atomic Hartree–Fock and Dirac–Hartree–Fock Theories 361

9.5.3 Relativistic and Nonrelativistic Electron Densities 364

9.6 Analysis of Radial Functions and Potentials at Short and Long Distances 366

9.6.1 Short-Range Behavior of Atomic Spinors 367

9.6.2 Origin Behavior of Interaction Potentials 370

9.6.3 Short-Range Electron–Electron Coulomb Interaction 371

9.6.4 Exchange Interaction at the Origin 372

9.6.5 Total Electron–Electron Interaction at the Nucleus 376

9.6.6 Asymptotic Behavior of the Interaction Potentials 378

9.7 Numerical Discretization and Solution Techniques 379

9.7.1 Variable Transformations 381

9.7.2 Explicit Transformation Functions 382

9.7.3 Transformed Equations 383

9.7.4 Numerical Solution of Matrix Equations 386

9.7.5 Discretization and Solution of the SCF equations 388

9.7.6 Discretization and Solution of the Poisson Equations 391

9.7.7 Extrapolation Techniques and Other Technical Issues 393

9.8 Results for Total Energies and Radial Functions 395

9.8.1 Electronic Configurations and the Aufbau Principle 397

9.8.2 Radial Functions 397

9.8.3 Effect of the Breit Interaction on Energies and Spinors 399

9.8.4 Effect of the Nuclear Charge Distribution on Total Energies 400

10 General Molecules and Molecular Aggregates 403

10.1 Basis Set Expansion of Molecular Spinors 405

10.1.1 Kinetic Balance 408

10.1.2 Special Choices of Basis Functions 409

10.2 Dirac–Hartree–Fock Electronic Energy in Basis Set Representation 413

10.3 Molecular One- and Two-Electron Integrals 419

10.4 Dirac–Hartree–Fock–Roothaan Matrix Equations 419

10.4.1 Two Possible Routes for the Derivation 420

10.4.2 Treatment of Negative-Energy States 421

10.4.3 Four-Component DFT 422

10.4.4 Symmetry 423

10.4.5 Kramers’ Time Reversal Symmetry 423

10.4.6 Double Groups 424

10.5 Analytic Gradients 425

10.6 Post-Hartree–Fock Methods 428

Part IV — Two-Component Hamiltonians 433

11 Decoupling the Negative-Energy States 435

11.1 Relation of Large and Small Components in One-Electron Equations 435

11.1.1 Restriction on the Potential Energy Operator 436

11.1.2 The X-Operator Formalism 436

11.1.3 Free-Particle Solutions 439

11.2 Closed-Form Unitary Transformation of the Dirac Hamiltonian 440

11.3 The Free-Particle Foldy–Wouthuysen Transformation 443

11.4 General Parametrization of Unitary Transformations 447

11.4.1 Closed-Form Parametrizations 448

11.4.2 Exactly Unitary Series Expansions 449

11.4.3 Approximate Unitary and Truncated Optimum Transformations 451

11.5 Foldy–Wouthuysen Expansion in Powers of 1/c 454

11.5.1 The Lowest-Order Foldy–Wouthuysen Transformation 454

11.5.2 Second-Order Foldy–Wouthuysen Operator: Pauli Hamiltonian 458

11.5.3 Higher-Order Foldy–Wouthuysen Transformations and Their Pathologies 459

11.6 The Infinite-Order Two-Component Two-Step Protocol 462

11.7 Toward Well-Defined Analytic Block-Diagonal Hamiltonians 465

12 Douglas–Kroll–Hess Theory 469

12.1 Sequential Unitary Decoupling Transformations 469

12.2 Explicit Form of the DKH Hamiltonians 471

12.2.1 First Unitary Transformation 471

12.2.2 Second Unitary Transformation 472

12.2.3 Third Unitary Transformation 475

12.3 Infinite-Order DKH Hamiltonians and the Arbitrary-Order DKH Method 476

12.3.1 Convergence of DKH Energies and Variational Stability 477

12.3.2 Infinite-Order Protocol 479

12.3.3 Coefficient Dependence 481

12.3.4 Explicit Expressions of the Positive-Energy Hamiltonians 483

12.3.5 Additional Peculiarities of DKH Theory 485

12.4 Many-Electron DKH Hamiltonians 488

12.4.1 DKH Transformation of One-Electron Terms 488

12.4.2 DKH Transformation of Two-Electron Terms 489

12.5 Computational Aspects of DKH Calculations 492

12.5.1 Exploiting a Resolution of the Identity 494

12.5.2 Advantages of Scalar-Relativistic DKH Hamiltonians 496

12.5.3 Approximations for Complicated Terms 498

12.5.4 DKH Gradients 500

13 Elimination Techniques 503

13.1 Naive Reduction: Pauli Elimination 503

13.2 Breit–Pauli Theory 507

13.2.1 Foldy–Wouthuysen Transformation of the Breit Equation 508

13.2.2 Transformation of the Two-Electron Interaction 509

13.2.3 The Breit–Pauli Hamiltonian 518

13.3 The Cowan–Griffin andWood–Boring Approaches 522

13.4 Elimination for Different Representations of Dirac Matrices 523

13.5 Regular Approximations 524

Part V — Chemistry with Relativistic Hamiltonians 527

14 Special Computational Techniques 529

14.1 From the Modified Dirac Equation to Exact-Two-Component Methods 530

14.1.1 Normalized Elimination of the Small Component 531

14.1.2 Exact-Decoupling Methods 533

14.1.3 Approximations in Many-Electron Calculations 546

14.1.4 Numerical Comparison 548

14.2 Locality of Relativistic Contributions 551

14.3 Local Exact Decoupling 553

14.3.1 Atomic Unitary Transformation 554

14.3.2 Local Decomposition of the X-Operator 555

14.3.3 Local Approximations to the Exact-Decoupling Transformation 556

14.3.4 Numerical Comparison 559

14.4 Efficient Calculation of Spin–Orbit Coupling Effects 561

14.5 Relativistic Effective Core Potentials 564

15 External Electromagnetic Fields and Molecular Properties 567

15.1 Four-Component Perturbation and Response Theory 569

15.1.1 Variational Treatment 570

15.1.2 Perturbation Theory 570

15.1.3 The Dirac-Like One-Electron Picture 573

15.1.4 Two Types of Properties 575

15.2 Reduction to Two-Component Form and Picture Change Artifacts 576

15.2.1 Origin of Picture Change Errors 577

15.2.2 Picture-Change-Free Transformed Properties 580

15.2.3 Foldy–Wouthuysen Transformation of Properties 580

15.2.4 Breit–Pauli Hamiltonian with Electromagnetic Fields 581

15.3 Douglas–Kroll–Hess Property Transformation 582

15.3.1 The Variational DKH Scheme for Perturbing Potentials 583

15.3.2 Most General Electromagnetic Property 584

15.3.3 Perturbative Approach 587

15.3.4 Automated Generation of DKH Property Operators 592

15.3.5 Consequences for the Electron Density Distribution 593

15.3.6 DKH Perturbation Theory with Magnetic Fields 595

15.4 Magnetic Fields in Resonance Spectroscopies 595

15.4.1 The Notorious Diamagnetic Term 595

15.4.2 Gauge Origin and London Orbitals 596

15.4.3 Explicit Form of Perturbation Operators 597

15.4.4 Spin Hamiltonian 598

15.5 Electric Field Gradient and Nuclear Quadrupole Moment 599

15.6 Parity Violation and Electro-Weak Chemistry 602

16 Relativistic Effects in Chemistry 605

16.1 Effects in Atoms with Consequences for Chemical Bonding 608

16.2 Is Spin a Relativistic Effect? 612

16.3 Z-Dependence of Relativistic Effects: Perturbation Theory 613

16.4 Potential Energy Surfaces and Spectroscopic Parameters 614

16.4.1 Dihydrogen 616

16.4.2 Thallium Hydride 617

16.4.3 The Gold Dimer 619

16.4.4 Tin Oxide and Cesium Hydride 622

16.5 Lanthanides and Actinides 622

16.5.1 Lanthanide and Actinide Contraction 623

16.5.2 Electronic Spectra of Actinide Compounds 623

16.6 Electron Density of Transition Metal Complexes 625

16.7 Relativistic Quantum Chemical Calculations in Practice 629

Appendix 631

A Vector and Tensor Calculus 633

A.1 Three-Dimensional Expressions 633

A.1.1 Algebraic Vector and Tensor Operations 633

A.1.2 Differential Vector Operations 634

A.1.3 Integral Theorems and Distributions 635

A.1.4 Total Differentials and Time Derivatives 637

A.2 Four-Dimensional Expressions 638

A.2.1 Algebraic Vector and Tensor Operations 638

A.2.2 Differential Vector Operations 638

B Kinetic Energy in Generalized Coordinates 641

C Technical Proofs for Special Relativity 643

C.1 Invariance of Space-Time Interval 643

C.2 Uniqueness of Lorentz Transformations 644

C.3 Useful Trigonometric and Hyperbolic Formulae for Lorentz Transformations 646

D Relations for Pauli and Dirac Matrices 649

D.1 Pauli Spin Matrices 649

D.2 Dirac’s Relation 650

D.2.1 Momenta and Vector Fields 651

D.2.2 Four-Dimensional Generalization 652

E Fourier Transformations 653

E.1 Definition and General Properties 653

E.2 Fourier Transformation of the Coulomb Potential 654

F Gordon Decomposition 657

F.1 One-Electron Case 657

F.2 Many-Electron Case 659

G Discretization and Quadrature Schemes 661

G.1 Numerov Approach toward Second-Order Differential Equations 661

G.2 Numerov Approach for First-Order Differential Equations 663

G.3 Simpson’s Quadrature Formula 665

G.4 Bickley’s Central-Difference Formulae 665

H List of Abbreviations and Acronyms 667

I List of Symbols 669

J Units and Dimensions 673

References 675

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Author Information

Markus Reiher obtained his PhD in Theoretical Chemistry in 1998, working in the group of Juergen Hinze at the University of Bielefeld on relativistic atomic structure theory. He completed his habilitation on transition-metal catalysis and vibrational spectroscopy at the University of Erlangen in the group of Bernd Artur Hess in 2002. During that time he had the opportunity to return to relativistic theories when working with Bernd Hess and Alex Wolf. From 2003 to 2005, Markus Reiher was Privatdozent at the University of Bonn and then moved to the University of Jena as Professor for Physical Chemistry in 2005. Since the beginning of 2006 he has been Professor for Theoretical Chemistry at ETH Zurich. Markus Reiher's research interests in molecular physics and chemistry are broad and diverse.

Alexander Wolf studied physics at the University of Erlangen and at Imperial College, London. In 2004, he completed his PhD in Theoretical Chemistry in the group of Bernd Artur Hess in Erlangen. His thesis elaborated on the generalized Douglas-Kroll-Hess transformation and efficient decoupling schemes for the Dirac Hamiltonian. As a postdoc he continued to work on these topics in the group of Markus Reiher at the universities of Bonn (2004) and Jena (2005). Since 2006 he has been engaged in financial risk management for various consultancies and is currently working in the area of structuring and modeling of life insurance products. On a regular basis he has been using his spare time to delve into his old passion, relativistic quantum mechanics and quantum chemistry.

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New to This Edition

The new edition is updated to include the latest developments in the field. Some content of the chapters has been restructured for the sake of clarity.

A significant amount of completely new content including:

- extension of the discussion of continuum dissolution

- extended discussion of spin in relativistic density functional theory

- exact-decoupling procedures with a focus on X2C

- local approximations to exact-decoupling methods

- more than 100 new key references

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