Relativistic quantum chemistry: the fundamental theory of molecular science
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Weinheim
Wiley-VCH
2009
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| Online-Zugang: | http://d-nb.info/991676920/04 http://deposit.dnb.de/cgi-bin/dokserv?id=3191974&prov=M&dok_var=1&dok_ext=htm Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XIX, 669 S. graph. Darst. |
| ISBN: | 9783527312924 |
Internformat
MARC
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| 100 | 1 | |a Reiher, Markus |d 1971- |e Verfasser |0 (DE-588)1043004467 |4 aut | |
| 245 | 1 | 0 | |a Relativistic quantum chemistry |b the fundamental theory of molecular science |c Markus Reiher and Alexander Wolf |
| 264 | 1 | |a Weinheim |b Wiley-VCH |c 2009 | |
| 300 | |a XIX, 669 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Quantenchemie - Lehrbuch | |
| 650 | 4 | |a Quantum chemistry | |
| 650 | 4 | |a Relativistic quantum theory | |
| 650 | 0 | 7 | |a Quantenchemie |0 (DE-588)4047979-1 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
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| 700 | 1 | |a Wolf, Alexander |d 1975- |e Verfasser |0 (DE-588)12906744X |4 aut | |
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Datensatz im Suchindex
| _version_ | 1805088361298264064 |
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| adam_text |
Contents
Preface XVII
1 Introduction 1
1.1 Philosophy of this Book 2
1.2 Short Reader's Guide 4
1.3 Notational Conventions and Choice of Units 6
Parti — Fundamentals 9
2 Elements of Classical Mechanics and Electrodynamics 11
2.1 Elementary Newtonian Mechanics 11
2.1.1 Newton's Laws of Motion 12
2.1.2 Galilean Transformations 14
2.1.2.1 Relativity Principle of Galilei 14
2.1.2.2 General Galilean Transformations and Boosts 25
2.1.2.3 Galilei Covariance of Newton's Laws 16
2.1.2.4 Scalars, Vectors, Tensors in 3-Dimensional Space 17
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 23
2.2.2.1 Discrete System of Point Particles 23
2.2.2.2 Explicit Example: Planar Pendulum 26
2.2.2.3 Continuous Systems of Fields 26
2.2.3 Symmetries and Conservation Laws 28
2.2.3.1 Gauge Transformations of the Lagrangian 28
2.2.3.2 Energy and Momentum Conservation 28
2.2.3.3 General Space-Time Symmetries 29
2.3 Hamiltonian Mechanics 30
2.3.1 Hamiltonian Principle and Canonical Equations 30
VI Contents
2.3.1.1 System of Point Particles 30
2.3.1.2 Continuous System of Fields 32
2.3.2 Poisson Brackets and Conservation Laws 32
2.3.3 Canonical Transformations 34
2.4 Elementary Electrodynamics 35
2 A.I Maxwell's Equations 35
2.4.2 Energy and Momentum of the Electromagnetic Field 37
2.4.2.1 Energy and Poynting's Theorem 37
2.4.2.2 Momentum and Maxwell's Stress Tensor 38
2.4.2.3 Angular Momentum 40
2.4.3 Plane Electromagnetic Waves in Vacuum 40
2.4.4 Potentials and Gauge Symmetry 41
2.4.4.1 Lorentz Gauge 43
2.4.4.2 Coulomb Gauge 44
2.4.4.3 Retarded Potentials 44
2.4.5 Survey of Electro- and Magnetostatics 45
2.4.5.1 Electrostatics 45
2.4.5.2 Magnetostatics 46
2.4.6 One Classical Particle Subject to Electromagnetic Fields 47
2.4.7 Interaction of Two Moving Charged Particles 49
Concepts of Special Relativity 51
3.1 Einstein's Relativity Principle and Lorentz Transformations 51
3.1.1 Deficiencies of Newtonian Mechanics 51
3.1.2 Relativity Principle of Einstein 53
3.1.3 Lorentz Transformations 56
3.1.3.1 Definition of General Lorentz Transformations 56
3.1.3.2 Classification of Lorentz Transformations 57
3.1.3.3 Inverse Lorentz Transformation 58
3.1 A Scalars, Vectors, and Tensors in Minkowski Space 60
3.1.4.1 Contra-and Covariant Components 60
3.1.4.2 Properties of Scalars, Vectors, and Tensors 61
3.2 Kinematical Effects in Special Relativity 65
3.2.1 Explicit Form of Special Lorentz Transformations 65
3.2.1.1 Lorentz Boost in One Direction 65
3.2.1.2 General Lorentz Boost 68
3.2.2 Length Contraction, Time Dilation, and Proper Time 70
3.2.2.1 Length Contraction 70
3.2.2.2 Time Dilation 71
3.2.2.3 Proper Time 72
3.2.3 Addition of Velocities 73
3.2.3.1 Parallel Velocities 73
Contents VII
3.2.3.2 General Velocities 75
3.3 Relativistic Dynamics 76
3.3.1 Elementary Relativistic Dynamics 77
3.3.1.1 Trajectories and Relativistic Velocity 77
3.3.1.2 Relativistic Momentum and Energy 77
3.3.1.3 Energy-Momentum Relation 79
3.3.2 Equation of Motion 81
3.3.2.1 Minkowski Force 81
3.3.2.2 Lorentz Force 82
3.3.3 Lagrangian and Hamiltonian Formulation 84
3.3.3.1 Relativistic Free Particle 84
3.3.3.2 Particle in Electromagnetic Fields 86
3.4 Covariant Electrodynamics 88
3.4.1 Ingredients 88
3.4.1.1 Charge-Current Density 88
3A.I.2 Gauge Field 89
3.4.1.3 Field Strength Tensor 90
3.4.2 Transformation of Electromagnetic Fields 92
3.4.3 Lagrangian Formulation and Equations of Motion 93
3.4.3.1 Lagrangian for the Electrodynamic Field 93
3.4.3.2 Minimal Coupling 95
3.4.3.3 Euler-Lagrange Equations 96
3.5 Interaction of Two Moving Charged Particles 98
3.5.1 Scalar and Vector Potentials of a Charge at Rest 99
3.5.2 Retardation from Lorentz Transformation 101
3.5.3 General Expression for the Interaction Energy 102
3.5.4 Interaction Energy at One Instant of Time 103
3.5.4.1 Taylor Expansion of Potential and Energy 104
3.5.4.2 Variables of Charge Two at Time of Charge One 105
3.5.4.3 Final Expansion of the Interaction Energy 106
3.5.4.4 Expansion of the Retardation Time 707
3.5.4.5 General Darwin Interaction Energy 707
3.5.5 Symmetrized Darwin Interaction Energy 709
Basics of Quantum Mechanics 113
4.1 The Quantum Mechanical State 7 74
4.1.1 Bracket Notation 7 74
4.1.2 Expansion in a Complete Basis Set 775
4.1.3 Born Interpretation 775
4.1.4 State Vectors in Hilbert Space 776
4.2 The Equation of Motion 778
VIII Contents
4.2.1 Restrictions on the Fundamental Quantum Mechanical
Equation 118
4.2.2 Time Evolution and Probabilistic Character 118
4.2.3 Stationary States 119
4.3 Observables 120
4.3.1 Expectation Values 120
4.3.2 Hermitean Operators 121
4.3.3 Unitary Transformations 121
4.3.4 Heisenberg Equation of Motion 122
4.3.5 Hamiltonian in Nonrelativistic Quantum Theory 125
4.3.6 Commutation Relations for Position and Momentum
Operators 127
4.3.7 The Schrodinger Velocity Operator 228
4.3.8 Ehrenfest and Hellmann-Feynman Theorems 229
4.3.9 Current Density and Continuity Equation 130
4.4 Angular Momentum and Rotations 232
4.4.1 Orbital Angular Momentum 133
4.4.2 Coupling of Angular Momenta 238
4.4.3 Spin 240
4.4.4 Coupling of Orbital and Spin Angular Momenta 243
4.5 Pauli Antisymmetry Principle 248
Part II — Dirac's Theory of the Electron 151
5 Relativistic Theory of the Electron 153
5.1 Correspondence Principle and Klein-Gordon Equation 253
5.1.1 Classical Energy Expression and First Hints from the
Correspondence Principle 253
5.1.2 Solutions of the Klein-Gordon Equation 255
5.1.3 The Klein-Gordon Density Distribution 256
5.2 Derivation of the Dirac Equation for a Freely Moving Electron 258
5.2.1 Relation to the Klein-Gordon Equation 158
5.2.2 Explicit Expressions for the Dirac Parameters 259
5.2.3 Continuity Equation and Definition of the 4-Current 262
5.2.4 Lorentz Covariance of the Field-Free Dirac Equation 262
5.2.4.1 Covariant Form 262
5.2.4.2 Lorentz Transformation of the Dirac Spinor 263
5.2.4.3 Higher Level of Abstraction and Clifford Algebra 264
5.3 Solution of the Free-Electron Dirac Equation 265
5.3.1 Particle at Rest 265
5.3.2 Freely Moving Particle 267
5.3.3 The Dirac Velocity Operator 272
Contents IX
5.4 Dirac Electron in External Electromagnetic Potentials 173
5.4.1 Kinematic Momentum 174
5.4.2 Electromagnetic Interaction Energy Operator 175
5.4.3 Nonrelativistic Limit and Pauli Equation 175
5.5 Interpretation of Negative-Energy States: Dirac's Hole Theory 178
6 The Dirac Hydrogen Atom 183
6.1 Separation of Electronic Motion in a Nuclear Central Field 183
6.2 Schrodinger Hydrogen Atom 186
6.3 Total Angular Momentum 189
6.4 Separation of Angular Coordinates in the Dirac Hamiltonian 190
6.4.1 Spin-Orbit Coupling 190
6.4.2 Relativistic Azimuthai Quantum Number Analog 191
6A3 Four-Dimensional Generalization 192
6.4.4 Ansatz for the Spinor 193
6.5 Radial Dirac Equation for Hydrogen-Like Atoms 194
6.5.1 Radial Functions and Orthonormality 195
6.5.2 Radial Eigenvalue Equations 196
6.5.3 Solution of the Coupled Dirac Radial Equations 197
6.5.4 Energy Eigenvalue, Quantization and the Principal Quantum
Number 202
6.5.5 The Four-Component Ground State Wave Function 205
6.6 The Nonrelativistic Limit 205
6.7 Choice of the Energy Reference and Matching Energy Scales 207
6.8 Wave Functions and Energy Eigenvalues in the Coulomb
Potential 209
6.8.1 Features of Dirac Radial Functions 209
6.8.2 Spectrum of Dirac Hydrogen-like Atoms with Coulombic
Potential 210
6.8.3 Radial Density and Expectation Values 212
6.9 Finite Nuclear Size Effects 224
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 222
Part III — Four-Component Many-Electron Theory 225
7 Quantum Electrodynamics 227
7.1 Elementary Quantities and Notation 227
7.1.1 Lagrangian for Electromagnetic Interactions 227
7.1.2 Lorentz and Gauge Symmetry and Equations of Motion 228
7.2 Classical Hamiltonian Description 230
X Contents
7.2.1 Exact Hamiltonian 230
7.2.2 The Electron-Electron Interaction 231
7.3 Second-Quantized Field-Theoretical Formulation 233
7A Implications for the Description of Atoms and Molecules 236
8 First-Quantized Dirac-Based Many-Electron Theory 239
8.1 Two-Electron Systems and the Breit Equation 240
8.1.1 Dirac Equation Generalized for Two Bound-State Electrons 241
8.1.2 The Gaunt Operator for Unretarded Interactions 243
8.1-3 The Breit Operator for Retarded Interactions 246
8.1.4 Exact Retarded Electromagnetic Interaction Energy 252
8.1.5 Breit Interaction from Quantum Electrodynamics 256
8.2 Quasi-Relativistic Many-Particle Hamiltonians 260
8.2.1 Nonrelativistic Hamiltonian for a Molecular System 260
8.2.2 First-Quantized Relativistic Many-Particle Hamiltonian 262
8.2.3 Pathologies of the First-Quantized Formulation 264
8.2.3.1 Boundedness and Variational Collapse 264
8.2.3.2 Continuum Dissolution 265
8.2.4 Local Model Potentials for One-Particle QED Corrections 266
8.3 Born-Oppenheimer Approximation 267
8.4 Tensor Structure of the Many-Electron Hamiltonian and Wave
Function 271
8.5 Approximations to the Many-Electron Wave Function 274
8.5.1 The Independent-Particle Model 274
8.5.2 Configuration Interaction 275
8.5.3 Detour: Explicitly Correlated Wave Functions 279
8.5.4 Orthonormality Constraints and Total Energy Expressions 281
8.6 Second Quantization for the Many-Electron Hamiltonian 284
8.6.1 Creation and Annihilation Operators 285
8.6.2 Reduction of Determinantal Matrix Elements to Matrix Elements
Over Spinors 286
8.6.3 Many-Electron Hamiltonian and Energy 2S7
8.6.4 Fock Space and Occupation Number Vectors 288
8.6.5 Fermions and Bosons 289
8.7 Derivation of Effective One-Particle Equations 290
8.7.1 The Minimax Principle 290
8.7.2 Variation of the Energy Expression 292
8.7.2.1 Variational Conditions 292
8.7.2.2 The CI Eigenvalue Problem 292
8.7.3 Self-Consistent Field Equations 294
8.7.4 Dirac-Hartree-Fock Equations 297
8.7.5 The Relativistic Self-Consistent Field 299
Contents XI
8.8 Relativistic Density Functional Theory 301
8.8.1 Electronic Charge and Current Densities for Many Electrons 302
8.8.2 Current-Density Functional Theory 305
8.8.3 The Four-Component Kohn-Sham Model 306
8.8.4 Noncollinear Approaches and Collinear Approximations 308
8.9 Completion: The Coupled-Cluster Expansion 308
Many-Electron Atoms 315
9.1 Transformation of the Many-Electron Hamiltonian to Polar
Coordinates 317
9.1.1 Comment on Units 318
9.1.2 Coulomb Interaction in Polar Coordinates 318
9.1.3 Breit Interaction in Polar Coordinates 319
9.1.4 Atomic Many-Electron Hamiltonian 322
9.2 Atomic Many-Electron Wave Function and //'-Coupling 323
9.3 One- and Two-Electron Integrals in Spherical Symmetry 326
9.3.1 One-Electron Integrals 326
9.3.2 Electron-Electron Coulomb Interaction 327
9.3.3 Electron-Electron Frequency-Independent Breit Interaction 330
9.3.4 Calculation of Potential Functions 333
9.3.4.1 First-Order Differential Equations 334
9.3.4.2 Derivation of the Radial Poisson Equation 334
9.3.4.3 Breit Potential Functions 335
9.4 Total Expectation Values 336
9 A.I General Expression for the Electronic Energy 336
9.4.2 Breit Contribution to the Total Energy 337
9 A3 Dirac-Hartree-Fock Total Energy of Closed-Shell Atoms 339
9.5 General Self-Consistent-Field Equations and Atomic Spinors 340
9.5.1 Dirac-Hartree-Fock Equations 342
9.5.2 Comparison of Atomic Hartree-Fock and Dirac-Hartree-Fock
Theories 342
9.5.3 Relativistic and Nonrelativistic Electron Densities 346
9.6 Analysis of Radial Functions and Potentials at Short and Long
Distances 348
9.6.1 Short-Range Behavior of Atomic Spinors 349
9.6.1.1 Cusp-Analogous Condition at the Nucleus 350
9.6.1.2 Coulomb Potential Functions 350
9.6.2 Origin Behavior of Interaction Potentials 351
9.6.3 Short-Range Electron-Electron Coulomb Interaction 353
9.6A Exchange Interaction at the Origin 353
9.6.5 Total Electron-Electron Interaction at the Nucleus 357
9.6.6 Asymptotic Behavior of the Interaction Potentials 360
XII Contents
9.7 Numerical Discretization and Solution Techniques 361
9.7 A Variable Transformations 362
9.7.2 Explicit Transformation Functions 363
9.7.2.1 The Logarithmic Grid 364
9.7.2.2 The Rational Grid 364
9.7.3 Transformed Equations 364
9.7.3.1 SCF Equations 365
9.7.3.2 Regular Solution Functions for Point-Nucleus Case 365
9.7.3.3 Poisson Equations 366
9.7A Numerical Solution of Matrix Equations 367
9.7.5 Discretization and Solution of the SCF equations 369
9.7.6 Discretization and Solution of the Poisson Equations 372
9.7.7 Extrapolation Techniques and Other Technical Issues 374
9.8 Results for Total Energies and Radial Functions 376
9.8.1 Electronic Configurations and the Aufbau Principle 378
9.8.2 Radial Functions 378
9.8.3 Effect of the Breit Interaction on Energies and Spinors 380
9.8.4 Effect of the Nuclear Charge Distribution on Total Energies 381
10 General Molecules and Molecular Aggregates 385
10.1 Basis Set Expansion of Molecular Spinors 387
10.1.1 Kinetic Balance 390
10.1.2 Special Choices of Basis Functions 391
10.2 Dirac-Hartree-Fock Electronic Energy in Basis Set
Representation 394
10.3 Molecular One- and Two-Electron Integrals 400
10.4 Dirac-Hartree-Fock-Roothaan Matrix Equations 401
10.4.1 Two Possible Routes for the Derivation 402
10.4.2 Treatment of Negative-Energy States 403
10.4.3 Four-Component DFT 404
10.4.4 Symmetry 404
10.4.5 Kramers' Time Reversal Symmetry 405
10.4.6 Double Groups 406
10.5 Analytic Gradients 406
10.6 Post-Hartree-Fock Methods 409
Part IV — Two-Component Hamiltonians 413
11 Decoupling the Negative-Energy States 415
11.1 Relation of Large and Small Components in One-Electron
Equations 435
11.1.1 Restriction on the Potential Energy Operator 416
Contents XIII
11.1.2 The X-Operator Formalism 416
11.1.3 Free-Particle Solutions 419
11.2 Closed-Form Unitary Transformation of the Dirac Hamiltonian 420
11.3 The Free-Particle Foldy-Wouthuysen Transformation 423
11A General Parametrization of Unitary Transformations 427
11.4.1 Closed-Form Parametrizations 428
11.4.2 Exactly Unitary Series Expansions 429
11.4.3 Approximate Unitary and Truncated Optimum
Transformations 431
11.5 Foldy-Wouthuysen Expansion in Powers of 1/c 434
11.5.1 The Lowest-Order Foldy-Wouthuysen Transformation 434
11.5.2 Second-Order Foldy-Wouthuysen Operator: Pauli
Hamiltonian 438
11.5.3 Higher-Order Foldy-Wouthuysen Transformations and Their
Pathologies 439
11.6 The Infinite-Order Two-Component One-Step Protocol 442
11.7 Toward Weil-Defined Analytic Block-Diagonal Hamiltonians 445
12 Douglas-Kroll-Hess Theory 447
12.1 Sequential Unitary Decoupling Transformations 447
12.2 Explicit Form of the DKH Hamiltonians 449
12.2.1 First Unitary Transformation 449
12.2.2 Second Unitary Transformation 450
12.2.3 Third Unitary Transformation 453
12.3 Infinite-Order DKH Hamiltonians and the Arbitrary-Order DKH
Method 454
12.3.1 Convergence of DKH Energies and Variational Stability 455
12.3.2 Infinite-Order Protocol 457
12.3.3 Coefficient Dependence 459
12.3.4 Explicit Expressions of the Positive-Energy Hamiltonians 461
12.3.5 Additional Peculiarities of DKH Theory 463
12.3.5.1 Two-Component Electron Density Distribution 464
12.3.5.2 Off-Diagonal Potential Operators 464
12.3.5.3 Nonrelativistic Limit 465
12.3.5.4 Rigorous Analytic Results 465
12.4 Many-Electron DKH Hamiltonians 465
12.4.1 DKH Transformation of One-Electron Terms 466
12.4.2 DKH Transformation of Two-Electron Terms 467
12.5 Computational Aspects of DKH Calculations 470
12.5.1 Exploiting a Resolution of the Identity 477
12.5.2 Advantages of Scalar-Relativistic DKH Hamiltonians 473
12.5.3 Approximations for Complicated Terms 475
XIV Contents
12.5.3.1 Spin-Orbit Operators 476
12.5.3.2 Two-Electron Terms 476
12.5.3.3 Large One-Electron Basis Sets 476
12.5 A DKH Gradients 477
13 Elimination Techniques 479
13.1 Naive Reduction: Pauli Elimination 479
13.2 Breit-Pauli Theory 483
13.2.1 Foldy-Wouthuysen Transformation of the Breit Equation 484
13.2.2 Transformation of the Two-Electron Interaction 485
13.2.2.1 All-Even Operators 486
13.2.2.2 Transformed Coulomb Contribution 488
13.2.2.3 Transformed Breit Contribution 489
13.2.3 The Breit-Pauli Hamiltonian 494
13.3 The Cowan-Griffin and Wood-Boring Approach 498
13.4 Elimination for Different Representations of Dirac Matrices 499
13.5 Regular Approximations 500
Part V — Chemistry with Relativistic Hamiltonians 503
14 Special Computational Techniques 505
14.1 The Modified Dirac Equation 506
14.2 Efficient Calculation of Spin-Orbit Coupling Effects 509
14.3 Locality in Four-Component Methods 522
14.4 Relativistic Effective Core Potentials 513
15 External Electromagnetic Fields and Molecular Properties 517
15.1 Four-Component Perturbation and Response Theory 529
15.1.1 Variational Treatment 520
15.1.2 Perturbation Theory 520
15.1.3 The Dirac-Like One-Electron Picture 523
15.1.4 Two Types of Properties 525
15.2 Reduction to Two-Component Form and Picture Change
Artifacts 526
15.2.1 Origin of Picture Change Errors 527
15.2.2 Picture-Change-Free Transformed Properties 530
15.2.3 Foldy-Wouthuysen Transformation of Properties 530
15.2 A Breit-Pauli Hamiltonian with Electromagnetic Fields 531
15.3 Douglas-Kroll-Hess Property Transformation 532
15.3.1 The Variational DKH Scheme for Perturbing Potentials 533
15.3.2 Most General Electromagnetic Property 534
15.3.3 Perturbative Approach 537
Contents XV
15.3.3.1 Direct DKH Transformation of First-Order Energy 537
15.3.3.2 Expressions of 3rd Order in Unperturbed Potential 539
15.3.3.3 Alternative Transformation for First-Order Energy 540
15.3.4 Automated Generation of DKH Property Operators 542
15.3.5 Consequences for the Electron Density Distribution 543
15.3.6 DKH Perturbation Theory with Magnetic Fields 544
15.4 Magnetic Fields in Resonance Spectroscopies 545
15.4.1 The Notorious Diamagnetic Term 545
15.4.2 Gauge Origin and London Orbitals 546
15.4.3 Explicit Form of Perturbation Operators 547
15.4.4 Spin Hamiltonian 547
15.5 Electric Field Gradient and Nuclear Quadrupole Moment 549
15.6 Parity Violation and Electro-Weak Chemistry 552
16 Relativistic Effects in Chemistry 555
16.1 Effects in Atoms with Consequences for Chemical Bonding 558
16.2 Is Spin a Relativistic Effect? 562
16.3 Z-Dependence of Relativistic Effects: Perturbation Theory 563
16.4 Potential Energy Surfaces and Spectroscopic Parameters 564
16.4.1 Thallium Hydride 565
16.4.2 The Gold Dimer 567
16.4.3 Tin Oxide and Cesium Hydride 570
16.5 Lanthanides and Actinides 570
16.5.1 Lanthanide and Actinide Contraction 571
16.5.2 Electronic Spectra of Actinide Compounds 572
16.6 Electron Density of Transition Metal Complexes 573
16.7 Relativistic Quantum Chemical Calculations in Practice 577
Appendix 580
A Vector and Tensor Calculus 581
A.I Three-Dimensional Expressions 581
A.1.1 Algebraic Vector and Tensor Operations 581
A. 1.2 Differential Vector Operations 582
A.1.3 Integral Theorems and Distributions 583
A.1.4 Total Differentials and Time Derivatives 585
A.2 Four-Dimensional Expressions 586
A.2.1 Algebraic Vector and Tensor Operations 586
A.2.2 Differential Vector Operations 586
B Kinetic Energy in Generalized Coordinates 589
XVI Contents
C Technical Proofs for Special Relativity 591
C.I Invariance of Space-Time Interval 591
C.2 Uniqueness of Lorentz Transformations 592
C.3 Useful Trigonometric and Hyperbolic Formulae for Lorentz
Transformations 594
D Relations for Pauli and Dirac Matrices 597
D.I Pauli Spin Matrices 597
D.2 Dirac's Relation 598
D.2.1 Momenta and Vector Fields 599
D.2.2 Four-Dimensional Generalization 600
E Fourier Transformations 601
E.I Definition and General Properties 601
E.2 Fourier Transformation of the Coulomb Potential 602
F Discretization and Quadrature Schemes 605
F.I Numerov Approach toward Second-Order Differential
Equations 605
F.2 Numerov Approach for First-Order Differential Equations 607
F.3 Simpson's Quadrature Formula 609
F.4 Bickley's Central-Difference Formulae 609
G List of Abbreviations and Acronyms 611
H List of Symbols 613
References 615 |
| adam_txt |
Contents
Preface XVII
1 Introduction 1
1.1 Philosophy of this Book 2
1.2 Short Reader's Guide 4
1.3 Notational Conventions and Choice of Units 6
Parti — Fundamentals 9
2 Elements of Classical Mechanics and Electrodynamics 11
2.1 Elementary Newtonian Mechanics 11
2.1.1 Newton's Laws of Motion 12
2.1.2 Galilean Transformations 14
2.1.2.1 Relativity Principle of Galilei 14
2.1.2.2 General Galilean Transformations and Boosts 25
2.1.2.3 Galilei Covariance of Newton's Laws 16
2.1.2.4 Scalars, Vectors, Tensors in 3-Dimensional Space 17
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 23
2.2.2.1 Discrete System of Point Particles 23
2.2.2.2 Explicit Example: Planar Pendulum 26
2.2.2.3 Continuous Systems of Fields 26
2.2.3 Symmetries and Conservation Laws 28
2.2.3.1 Gauge Transformations of the Lagrangian 28
2.2.3.2 Energy and Momentum Conservation 28
2.2.3.3 General Space-Time Symmetries 29
2.3 Hamiltonian Mechanics 30
2.3.1 Hamiltonian Principle and Canonical Equations 30
VI Contents
2.3.1.1 System of Point Particles 30
2.3.1.2 Continuous System of Fields 32
2.3.2 Poisson Brackets and Conservation Laws 32
2.3.3 Canonical Transformations 34
2.4 Elementary Electrodynamics 35
2 A.I Maxwell's Equations 35
2.4.2 Energy and Momentum of the Electromagnetic Field 37
2.4.2.1 Energy and Poynting's Theorem 37
2.4.2.2 Momentum and Maxwell's Stress Tensor 38
2.4.2.3 Angular Momentum 40
2.4.3 Plane Electromagnetic Waves in Vacuum 40
2.4.4 Potentials and Gauge Symmetry 41
2.4.4.1 Lorentz Gauge 43
2.4.4.2 Coulomb Gauge 44
2.4.4.3 Retarded Potentials 44
2.4.5 Survey of Electro- and Magnetostatics 45
2.4.5.1 Electrostatics 45
2.4.5.2 Magnetostatics 46
2.4.6 One Classical Particle Subject to Electromagnetic Fields 47
2.4.7 Interaction of Two Moving Charged Particles 49
Concepts of Special Relativity 51
3.1 Einstein's Relativity Principle and Lorentz Transformations 51
3.1.1 Deficiencies of Newtonian Mechanics 51
3.1.2 Relativity Principle of Einstein 53
3.1.3 Lorentz Transformations 56
3.1.3.1 Definition of General Lorentz Transformations 56
3.1.3.2 Classification of Lorentz Transformations 57
3.1.3.3 Inverse Lorentz Transformation 58
3.1 A Scalars, Vectors, and Tensors in Minkowski Space 60
3.1.4.1 Contra-and Covariant Components 60
3.1.4.2 Properties of Scalars, Vectors, and Tensors 61
3.2 Kinematical Effects in Special Relativity 65
3.2.1 Explicit Form of Special Lorentz Transformations 65
3.2.1.1 Lorentz Boost in One Direction 65
3.2.1.2 General Lorentz Boost 68
3.2.2 Length Contraction, Time Dilation, and Proper Time 70
3.2.2.1 Length Contraction 70
3.2.2.2 Time Dilation 71
3.2.2.3 Proper Time 72
3.2.3 Addition of Velocities 73
3.2.3.1 Parallel Velocities 73
Contents VII
3.2.3.2 General Velocities 75
3.3 Relativistic Dynamics 76
3.3.1 Elementary Relativistic Dynamics 77
3.3.1.1 Trajectories and Relativistic Velocity 77
3.3.1.2 Relativistic Momentum and Energy 77
3.3.1.3 Energy-Momentum Relation 79
3.3.2 Equation of Motion 81
3.3.2.1 Minkowski Force 81
3.3.2.2 Lorentz Force 82
3.3.3 Lagrangian and Hamiltonian Formulation 84
3.3.3.1 Relativistic Free Particle 84
3.3.3.2 Particle in Electromagnetic Fields 86
3.4 Covariant Electrodynamics 88
3.4.1 Ingredients 88
3.4.1.1 Charge-Current Density 88
3A.I.2 Gauge Field 89
3.4.1.3 Field Strength Tensor 90
3.4.2 Transformation of Electromagnetic Fields 92
3.4.3 Lagrangian Formulation and Equations of Motion 93
3.4.3.1 Lagrangian for the Electrodynamic Field 93
3.4.3.2 Minimal Coupling 95
3.4.3.3 Euler-Lagrange Equations 96
3.5 Interaction of Two Moving Charged Particles 98
3.5.1 Scalar and Vector Potentials of a Charge at Rest 99
3.5.2 Retardation from Lorentz Transformation 101
3.5.3 General Expression for the Interaction Energy 102
3.5.4 Interaction Energy at One Instant of Time 103
3.5.4.1 Taylor Expansion of Potential and Energy 104
3.5.4.2 Variables of Charge Two at Time of Charge One 105
3.5.4.3 Final Expansion of the Interaction Energy 106
3.5.4.4 Expansion of the Retardation Time 707
3.5.4.5 General Darwin Interaction Energy 707
3.5.5 Symmetrized Darwin Interaction Energy 709
Basics of Quantum Mechanics 113
4.1 The Quantum Mechanical State 7 74
4.1.1 Bracket Notation 7 74
4.1.2 Expansion in a Complete Basis Set 775
4.1.3 Born Interpretation 775
4.1.4 State Vectors in Hilbert Space 776
4.2 The Equation of Motion 778
VIII Contents
4.2.1 Restrictions on the Fundamental Quantum Mechanical
Equation 118
4.2.2 Time Evolution and Probabilistic Character 118
4.2.3 Stationary States 119
4.3 Observables 120
4.3.1 Expectation Values 120
4.3.2 Hermitean Operators 121
4.3.3 Unitary Transformations 121
4.3.4 Heisenberg Equation of Motion 122
4.3.5 Hamiltonian in Nonrelativistic Quantum Theory 125
4.3.6 Commutation Relations for Position and Momentum
Operators 127
4.3.7 The Schrodinger Velocity Operator 228
4.3.8 Ehrenfest and Hellmann-Feynman Theorems 229
4.3.9 Current Density and Continuity Equation 130
4.4 Angular Momentum and Rotations 232
4.4.1 Orbital Angular Momentum 133
4.4.2 Coupling of Angular Momenta 238
4.4.3 Spin 240
4.4.4 Coupling of Orbital and Spin Angular Momenta 243
4.5 Pauli Antisymmetry Principle 248
Part II — Dirac's Theory of the Electron 151
5 Relativistic Theory of the Electron 153
5.1 Correspondence Principle and Klein-Gordon Equation 253
5.1.1 Classical Energy Expression and First Hints from the
Correspondence Principle 253
5.1.2 Solutions of the Klein-Gordon Equation 255
5.1.3 The Klein-Gordon Density Distribution 256
5.2 Derivation of the Dirac Equation for a Freely Moving Electron 258
5.2.1 Relation to the Klein-Gordon Equation 158
5.2.2 Explicit Expressions for the Dirac Parameters 259
5.2.3 Continuity Equation and Definition of the 4-Current 262
5.2.4 Lorentz Covariance of the Field-Free Dirac Equation 262
5.2.4.1 Covariant Form 262
5.2.4.2 Lorentz Transformation of the Dirac Spinor 263
5.2.4.3 Higher Level of Abstraction and Clifford Algebra 264
5.3 Solution of the Free-Electron Dirac Equation 265
5.3.1 Particle at Rest 265
5.3.2 Freely Moving Particle 267
5.3.3 The Dirac Velocity Operator 272
Contents IX
5.4 Dirac Electron in External Electromagnetic Potentials 173
5.4.1 Kinematic Momentum 174
5.4.2 Electromagnetic Interaction Energy Operator 175
5.4.3 Nonrelativistic Limit and Pauli Equation 175
5.5 Interpretation of Negative-Energy States: Dirac's Hole Theory 178
6 The Dirac Hydrogen Atom 183
6.1 Separation of Electronic Motion in a Nuclear Central Field 183
6.2 Schrodinger Hydrogen Atom 186
6.3 Total Angular Momentum 189
6.4 Separation of Angular Coordinates in the Dirac Hamiltonian 190
6.4.1 Spin-Orbit Coupling 190
6.4.2 Relativistic Azimuthai Quantum Number Analog 191
6A3 Four-Dimensional Generalization 192
6.4.4 Ansatz for the Spinor 193
6.5 Radial Dirac Equation for Hydrogen-Like Atoms 194
6.5.1 Radial Functions and Orthonormality 195
6.5.2 Radial Eigenvalue Equations 196
6.5.3 Solution of the Coupled Dirac Radial Equations 197
6.5.4 Energy Eigenvalue, Quantization and the Principal Quantum
Number 202
6.5.5 The Four-Component Ground State Wave Function 205
6.6 The Nonrelativistic Limit 205
6.7 Choice of the Energy Reference and Matching Energy Scales 207
6.8 Wave Functions and Energy Eigenvalues in the Coulomb
Potential 209
6.8.1 Features of Dirac Radial Functions 209
6.8.2 Spectrum of Dirac Hydrogen-like Atoms with Coulombic
Potential 210
6.8.3 Radial Density and Expectation Values 212
6.9 Finite Nuclear Size Effects 224
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 222
Part III — Four-Component Many-Electron Theory 225
7 Quantum Electrodynamics 227
7.1 Elementary Quantities and Notation 227
7.1.1 Lagrangian for Electromagnetic Interactions 227
7.1.2 Lorentz and Gauge Symmetry and Equations of Motion 228
7.2 Classical Hamiltonian Description 230
X Contents
7.2.1 Exact Hamiltonian 230
7.2.2 The Electron-Electron Interaction 231
7.3 Second-Quantized Field-Theoretical Formulation 233
7A Implications for the Description of Atoms and Molecules 236
8 First-Quantized Dirac-Based Many-Electron Theory 239
8.1 Two-Electron Systems and the Breit Equation 240
8.1.1 Dirac Equation Generalized for Two Bound-State Electrons 241
8.1.2 The Gaunt Operator for Unretarded Interactions 243
8.1-3 The Breit Operator for Retarded Interactions 246
8.1.4 Exact Retarded Electromagnetic Interaction Energy 252
8.1.5 Breit Interaction from Quantum Electrodynamics 256
8.2 Quasi-Relativistic Many-Particle Hamiltonians 260
8.2.1 Nonrelativistic Hamiltonian for a Molecular System 260
8.2.2 First-Quantized Relativistic Many-Particle Hamiltonian 262
8.2.3 Pathologies of the First-Quantized Formulation 264
8.2.3.1 Boundedness and Variational Collapse 264
8.2.3.2 Continuum Dissolution 265
8.2.4 Local Model Potentials for One-Particle QED Corrections 266
8.3 Born-Oppenheimer Approximation 267
8.4 Tensor Structure of the Many-Electron Hamiltonian and Wave
Function 271
8.5 Approximations to the Many-Electron Wave Function 274
8.5.1 The Independent-Particle Model 274
8.5.2 Configuration Interaction 275
8.5.3 Detour: Explicitly Correlated Wave Functions 279
8.5.4 Orthonormality Constraints and Total Energy Expressions 281
8.6 Second Quantization for the Many-Electron Hamiltonian 284
8.6.1 Creation and Annihilation Operators 285
8.6.2 Reduction of Determinantal Matrix Elements to Matrix Elements
Over Spinors 286
8.6.3 Many-Electron Hamiltonian and Energy 2S7
8.6.4 Fock Space and Occupation Number Vectors 288
8.6.5 Fermions and Bosons 289
8.7 Derivation of Effective One-Particle Equations 290
8.7.1 The Minimax Principle 290
8.7.2 Variation of the Energy Expression 292
8.7.2.1 Variational Conditions 292
8.7.2.2 The CI Eigenvalue Problem 292
8.7.3 Self-Consistent Field Equations 294
8.7.4 Dirac-Hartree-Fock Equations 297
8.7.5 The Relativistic Self-Consistent Field 299
Contents XI
8.8 Relativistic Density Functional Theory 301
8.8.1 Electronic Charge and Current Densities for Many Electrons 302
8.8.2 Current-Density Functional Theory 305
8.8.3 The Four-Component Kohn-Sham Model 306
8.8.4 Noncollinear Approaches and Collinear Approximations 308
8.9 Completion: The Coupled-Cluster Expansion 308
Many-Electron Atoms 315
9.1 Transformation of the Many-Electron Hamiltonian to Polar
Coordinates 317
9.1.1 Comment on Units 318
9.1.2 Coulomb Interaction in Polar Coordinates 318
9.1.3 Breit Interaction in Polar Coordinates 319
9.1.4 Atomic Many-Electron Hamiltonian 322
9.2 Atomic Many-Electron Wave Function and //'-Coupling 323
9.3 One- and Two-Electron Integrals in Spherical Symmetry 326
9.3.1 One-Electron Integrals 326
9.3.2 Electron-Electron Coulomb Interaction 327
9.3.3 Electron-Electron Frequency-Independent Breit Interaction 330
9.3.4 Calculation of Potential Functions 333
9.3.4.1 First-Order Differential Equations 334
9.3.4.2 Derivation of the Radial Poisson Equation 334
9.3.4.3 Breit Potential Functions 335
9.4 Total Expectation Values 336
9 A.I General Expression for the Electronic Energy 336
9.4.2 Breit Contribution to the Total Energy 337
9 A3 Dirac-Hartree-Fock Total Energy of Closed-Shell Atoms 339
9.5 General Self-Consistent-Field Equations and Atomic Spinors 340
9.5.1 Dirac-Hartree-Fock Equations 342
9.5.2 Comparison of Atomic Hartree-Fock and Dirac-Hartree-Fock
Theories 342
9.5.3 Relativistic and Nonrelativistic Electron Densities 346
9.6 Analysis of Radial Functions and Potentials at Short and Long
Distances 348
9.6.1 Short-Range Behavior of Atomic Spinors 349
9.6.1.1 Cusp-Analogous Condition at the Nucleus 350
9.6.1.2 Coulomb Potential Functions 350
9.6.2 Origin Behavior of Interaction Potentials 351
9.6.3 Short-Range Electron-Electron Coulomb Interaction 353
9.6A Exchange Interaction at the Origin 353
9.6.5 Total Electron-Electron Interaction at the Nucleus 357
9.6.6 Asymptotic Behavior of the Interaction Potentials 360
XII Contents
9.7 Numerical Discretization and Solution Techniques 361
9.7 A Variable Transformations 362
9.7.2 Explicit Transformation Functions 363
9.7.2.1 The Logarithmic Grid 364
9.7.2.2 The Rational Grid 364
9.7.3 Transformed Equations 364
9.7.3.1 SCF Equations 365
9.7.3.2 Regular Solution Functions for Point-Nucleus Case 365
9.7.3.3 Poisson Equations 366
9.7A Numerical Solution of Matrix Equations 367
9.7.5 Discretization and Solution of the SCF equations 369
9.7.6 Discretization and Solution of the Poisson Equations 372
9.7.7 Extrapolation Techniques and Other Technical Issues 374
9.8 Results for Total Energies and Radial Functions 376
9.8.1 Electronic Configurations and the Aufbau Principle 378
9.8.2 Radial Functions 378
9.8.3 Effect of the Breit Interaction on Energies and Spinors 380
9.8.4 Effect of the Nuclear Charge Distribution on Total Energies 381
10 General Molecules and Molecular Aggregates 385
10.1 Basis Set Expansion of Molecular Spinors 387
10.1.1 Kinetic Balance 390
10.1.2 Special Choices of Basis Functions 391
10.2 Dirac-Hartree-Fock Electronic Energy in Basis Set
Representation 394
10.3 Molecular One- and Two-Electron Integrals 400
10.4 Dirac-Hartree-Fock-Roothaan Matrix Equations 401
10.4.1 Two Possible Routes for the Derivation 402
10.4.2 Treatment of Negative-Energy States 403
10.4.3 Four-Component DFT 404
10.4.4 Symmetry 404
10.4.5 Kramers' Time Reversal Symmetry 405
10.4.6 Double Groups 406
10.5 Analytic Gradients 406
10.6 Post-Hartree-Fock Methods 409
Part IV — Two-Component Hamiltonians 413
11 Decoupling the Negative-Energy States 415
11.1 Relation of Large and Small Components in One-Electron
Equations 435
11.1.1 Restriction on the Potential Energy Operator 416
Contents XIII
11.1.2 The X-Operator Formalism 416
11.1.3 Free-Particle Solutions 419
11.2 Closed-Form Unitary Transformation of the Dirac Hamiltonian 420
11.3 The Free-Particle Foldy-Wouthuysen Transformation 423
11A General Parametrization of Unitary Transformations 427
11.4.1 Closed-Form Parametrizations 428
11.4.2 Exactly Unitary Series Expansions 429
11.4.3 Approximate Unitary and Truncated Optimum
Transformations 431
11.5 Foldy-Wouthuysen Expansion in Powers of 1/c 434
11.5.1 The Lowest-Order Foldy-Wouthuysen Transformation 434
11.5.2 Second-Order Foldy-Wouthuysen Operator: Pauli
Hamiltonian 438
11.5.3 Higher-Order Foldy-Wouthuysen Transformations and Their
Pathologies 439
11.6 The Infinite-Order Two-Component One-Step Protocol 442
11.7 Toward Weil-Defined Analytic Block-Diagonal Hamiltonians 445
12 Douglas-Kroll-Hess Theory 447
12.1 Sequential Unitary Decoupling Transformations 447
12.2 Explicit Form of the DKH Hamiltonians 449
12.2.1 First Unitary Transformation 449
12.2.2 Second Unitary Transformation 450
12.2.3 Third Unitary Transformation 453
12.3 Infinite-Order DKH Hamiltonians and the Arbitrary-Order DKH
Method 454
12.3.1 Convergence of DKH Energies and Variational Stability 455
12.3.2 Infinite-Order Protocol 457
12.3.3 Coefficient Dependence 459
12.3.4 Explicit Expressions of the Positive-Energy Hamiltonians 461
12.3.5 Additional Peculiarities of DKH Theory 463
12.3.5.1 Two-Component Electron Density Distribution 464
12.3.5.2 Off-Diagonal Potential Operators 464
12.3.5.3 Nonrelativistic Limit 465
12.3.5.4 Rigorous Analytic Results 465
12.4 Many-Electron DKH Hamiltonians 465
12.4.1 DKH Transformation of One-Electron Terms 466
12.4.2 DKH Transformation of Two-Electron Terms 467
12.5 Computational Aspects of DKH Calculations 470
12.5.1 Exploiting a Resolution of the Identity 477
12.5.2 Advantages of Scalar-Relativistic DKH Hamiltonians 473
12.5.3 Approximations for Complicated Terms 475
XIV Contents
12.5.3.1 Spin-Orbit Operators 476
12.5.3.2 Two-Electron Terms 476
12.5.3.3 Large One-Electron Basis Sets 476
12.5 A DKH Gradients 477
13 Elimination Techniques 479
13.1 Naive Reduction: Pauli Elimination 479
13.2 Breit-Pauli Theory 483
13.2.1 Foldy-Wouthuysen Transformation of the Breit Equation 484
13.2.2 Transformation of the Two-Electron Interaction 485
13.2.2.1 All-Even Operators 486
13.2.2.2 Transformed Coulomb Contribution 488
13.2.2.3 Transformed Breit Contribution 489
13.2.3 The Breit-Pauli Hamiltonian 494
13.3 The Cowan-Griffin and Wood-Boring Approach 498
13.4 Elimination for Different Representations of Dirac Matrices 499
13.5 Regular Approximations 500
Part V — Chemistry with Relativistic Hamiltonians 503
14 Special Computational Techniques 505
14.1 The Modified Dirac Equation 506
14.2 Efficient Calculation of Spin-Orbit Coupling Effects 509
14.3 Locality in Four-Component Methods 522
14.4 Relativistic Effective Core Potentials 513
15 External Electromagnetic Fields and Molecular Properties 517
15.1 Four-Component Perturbation and Response Theory 529
15.1.1 Variational Treatment 520
15.1.2 Perturbation Theory 520
15.1.3 The Dirac-Like One-Electron Picture 523
15.1.4 Two Types of Properties 525
15.2 Reduction to Two-Component Form and Picture Change
Artifacts 526
15.2.1 Origin of Picture Change Errors 527
15.2.2 Picture-Change-Free Transformed Properties 530
15.2.3 Foldy-Wouthuysen Transformation of Properties 530
15.2 A Breit-Pauli Hamiltonian with Electromagnetic Fields 531
15.3 Douglas-Kroll-Hess Property Transformation 532
15.3.1 The Variational DKH Scheme for Perturbing Potentials 533
15.3.2 Most General Electromagnetic Property 534
15.3.3 Perturbative Approach 537
Contents XV
15.3.3.1 Direct DKH Transformation of First-Order Energy 537
15.3.3.2 Expressions of 3rd Order in Unperturbed Potential 539
15.3.3.3 Alternative Transformation for First-Order Energy 540
15.3.4 Automated Generation of DKH Property Operators 542
15.3.5 Consequences for the Electron Density Distribution 543
15.3.6 DKH Perturbation Theory with Magnetic Fields 544
15.4 Magnetic Fields in Resonance Spectroscopies 545
15.4.1 The Notorious Diamagnetic Term 545
15.4.2 Gauge Origin and London Orbitals 546
15.4.3 Explicit Form of Perturbation Operators 547
15.4.4 Spin Hamiltonian 547
15.5 Electric Field Gradient and Nuclear Quadrupole Moment 549
15.6 Parity Violation and Electro-Weak Chemistry 552
16 Relativistic Effects in Chemistry 555
16.1 Effects in Atoms with Consequences for Chemical Bonding 558
16.2 Is Spin a Relativistic Effect? 562
16.3 Z-Dependence of Relativistic Effects: Perturbation Theory 563
16.4 Potential Energy Surfaces and Spectroscopic Parameters 564
16.4.1 Thallium Hydride 565
16.4.2 The Gold Dimer 567
16.4.3 Tin Oxide and Cesium Hydride 570
16.5 Lanthanides and Actinides 570
16.5.1 Lanthanide and Actinide Contraction 571
16.5.2 Electronic Spectra of Actinide Compounds 572
16.6 Electron Density of Transition Metal Complexes 573
16.7 Relativistic Quantum Chemical Calculations in Practice 577
Appendix 580
A Vector and Tensor Calculus 581
A.I Three-Dimensional Expressions 581
A.1.1 Algebraic Vector and Tensor Operations 581
A. 1.2 Differential Vector Operations 582
A.1.3 Integral Theorems and Distributions 583
A.1.4 Total Differentials and Time Derivatives 585
A.2 Four-Dimensional Expressions 586
A.2.1 Algebraic Vector and Tensor Operations 586
A.2.2 Differential Vector Operations 586
B Kinetic Energy in Generalized Coordinates 589
XVI Contents
C Technical Proofs for Special Relativity 591
C.I Invariance of Space-Time Interval 591
C.2 Uniqueness of Lorentz Transformations 592
C.3 Useful Trigonometric and Hyperbolic Formulae for Lorentz
Transformations 594
D Relations for Pauli and Dirac Matrices 597
D.I Pauli Spin Matrices 597
D.2 Dirac's Relation 598
D.2.1 Momenta and Vector Fields 599
D.2.2 Four-Dimensional Generalization 600
E Fourier Transformations 601
E.I Definition and General Properties 601
E.2 Fourier Transformation of the Coulomb Potential 602
F Discretization and Quadrature Schemes 605
F.I Numerov Approach toward Second-Order Differential
Equations 605
F.2 Numerov Approach for First-Order Differential Equations 607
F.3 Simpson's Quadrature Formula 609
F.4 Bickley's Central-Difference Formulae 609
G List of Abbreviations and Acronyms 611
H List of Symbols 613
References 615 |
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| author | Reiher, Markus 1971- Wolf, Alexander 1975- |
| author_GND | (DE-588)1043004467 (DE-588)12906744X |
| author_facet | Reiher, Markus 1971- Wolf, Alexander 1975- |
| author_role | aut aut |
| author_sort | Reiher, Markus 1971- |
| author_variant | m r mr a w aw |
| building | Verbundindex |
| bvnumber | BV021327049 |
| callnumber-first | Q - Science |
| callnumber-label | QD462 |
| callnumber-raw | QD462.6.R42 |
| callnumber-search | QD462.6.R42 |
| callnumber-sort | QD 3462.6 R42 |
| callnumber-subject | QD - Chemistry |
| classification_rvk | VE 5650 |
| classification_tum | CHE 150f |
| ctrlnum | (OCoLC)305077917 (DE-599)BVBBV021327049 |
| dewey-full | 541.28 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 541 - Physical chemistry |
| dewey-raw | 541.28 |
| dewey-search | 541.28 |
| dewey-sort | 3541.28 |
| dewey-tens | 540 - Chemistry and allied sciences |
| discipline | Chemie / Pharmazie Physik Chemie |
| discipline_str_mv | Chemie / Pharmazie Physik Chemie |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV021327049 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:00:38Z |
| indexdate | 2024-07-20T09:06:47Z |
| institution | BVB |
| isbn | 9783527312924 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014647380 |
| oclc_num | 305077917 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-29T DE-19 DE-BY-UBM DE-634 DE-11 DE-703 DE-83 DE-384 |
| owner_facet | DE-91G DE-BY-TUM DE-29T DE-19 DE-BY-UBM DE-634 DE-11 DE-703 DE-83 DE-384 |
| physical | XIX, 669 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Wiley-VCH |
| record_format | marc |
| spelling | Reiher, Markus 1971- Verfasser (DE-588)1043004467 aut Relativistic quantum chemistry the fundamental theory of molecular science Markus Reiher and Alexander Wolf Weinheim Wiley-VCH 2009 XIX, 669 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Quantenchemie - Lehrbuch Quantum chemistry Relativistic quantum theory Quantenchemie (DE-588)4047979-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Quantenchemie (DE-588)4047979-1 s DE-604 Wolf, Alexander 1975- Verfasser (DE-588)12906744X aut http://d-nb.info/991676920/04 http://deposit.dnb.de/cgi-bin/dokserv?id=3191974&prov=M&dok_var=1&dok_ext=htm HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014647380&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Reiher, Markus 1971- Wolf, Alexander 1975- Relativistic quantum chemistry the fundamental theory of molecular science Quantenchemie - Lehrbuch Quantum chemistry Relativistic quantum theory Quantenchemie (DE-588)4047979-1 gnd |
| subject_GND | (DE-588)4047979-1 (DE-588)4123623-3 |
| title | Relativistic quantum chemistry the fundamental theory of molecular science |
| title_auth | Relativistic quantum chemistry the fundamental theory of molecular science |
| title_exact_search | Relativistic quantum chemistry the fundamental theory of molecular science |
| title_exact_search_txtP | Relativistic quantum chemistry the fundamental theory of molecular science |
| title_full | Relativistic quantum chemistry the fundamental theory of molecular science Markus Reiher and Alexander Wolf |
| title_fullStr | Relativistic quantum chemistry the fundamental theory of molecular science Markus Reiher and Alexander Wolf |
| title_full_unstemmed | Relativistic quantum chemistry the fundamental theory of molecular science Markus Reiher and Alexander Wolf |
| title_short | Relativistic quantum chemistry |
| title_sort | relativistic quantum chemistry the fundamental theory of molecular science |
| title_sub | the fundamental theory of molecular science |
| topic | Quantenchemie - Lehrbuch Quantum chemistry Relativistic quantum theory Quantenchemie (DE-588)4047979-1 gnd |
| topic_facet | Quantenchemie - Lehrbuch Quantum chemistry Relativistic quantum theory Quantenchemie Lehrbuch |
| url | http://d-nb.info/991676920/04 http://deposit.dnb.de/cgi-bin/dokserv?id=3191974&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014647380&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT reihermarkus relativisticquantumchemistrythefundamentaltheoryofmolecularscience AT wolfalexander relativisticquantumchemistrythefundamentaltheoryofmolecularscience |