Structure of multielectron atoms

Preface......Page 7
Contents......Page 10
Notation, Used Symbols......Page 17
Acronyms......Page 19
Constants......Page 20
1 The One-Electron Atom......Page 23
 1.1 Formulation of the Schrödinger equation for theHydrogen Atom......Page 24
 1.2 Solutions to the Radial Equation......Page 26
  1.2.1 Eigenstates......Page 27
  1.2.2 Radial Probability Distribution......Page 28
 1.3 Solution of the Angular Equation......Page 31
 1.4 The Total Hydrogenic Wave Function......Page 33
  1.4.1 Energy Levels......Page 35
  1.4.2 Spin......Page 36
  1.4.3 Parity......Page 37
 1.5 Spectroscopic Notation......Page 38
 1.6 Central-Field Wave Functions in Multielectron Atoms......Page 41
 References......Page 42
 2.1 The Multielectron Atom Schrödinger equation......Page 43
  2.1.1 The Three-Body Hamiltonian......Page 44
 2.2 Exchange Symmetries......Page 46
  2.2.1 The Compound Spin Wave Function......Page 47
  2.2.3 The Spatial Wave Function......Page 49
  2.3.1 Two-Electron Antisymmetric Wave Functions......Page 50
  2.3.2 N-Electron Atom Antisymmetric Functions......Page 52
  2.3.4 Matrix Elements for Slater Determinants......Page 53
 2.4 Symmetry in Atomic Structure......Page 56
 References......Page 57
 3.1 Perturbative Treatment of the Exchange Term......Page 58
  3.1.1 The Energy of the Helium Ground State......Page 59
  3.1.2 Energies of the Lowest Excited States in Helium......Page 61
  3.1.3 The Lithium Ground State......Page 64
  3.2.1 Basic Atomic Variational Analyses......Page 66
  3.2.2 High Precision Variational Computations......Page 68
  3.2.3 Lithium and Larger Atoms......Page 69
  3.3.1 The Thomas-Fermi Method in Atomic Structure......Page 71
  3.3.2 Results and Extensions......Page 73
 3.4 Other Approaches — The Need for Approximations in Atomic Structure Theory......Page 75
 References......Page 76
4 The Spin–Orbit Interaction......Page 77
 4.1 Spin–Orbit Interaction for One-Electron Atoms......Page 78
  4.1.1 Magnetic Moments and the Effective Magnetic Field......Page 79
  4.1.2 Fine-Structure in Hydrogen......Page 81
 4.2 Spin–Orbit Interaction for Multielectron Atoms......Page 83
 References......Page 85
5 The Central-Field Approximation......Page 86
 5.1 The Principle of the Central-Field Approximation......Page 87
 5.2 Electron Configurations......Page 89
  5.2.1 Ground State Configurations......Page 90
  5.3.1 Horizontal View of the Periodic System — The Gradual Evolution of Electronic Configurations......Page 95
   5.3.2.1 The Alkalis......Page 97
   5.3.2.5 Halogens......Page 98
   5.3.3.1 Atomic Radius......Page 99
   5.3.3.3 Electron Affinity/Negativity......Page 100
 5.4 The Alkalis — Quantum Defect......Page 101
  5.4.1 The Quantum Defect......Page 103
 References......Page 106
6 Coupling of Angular Momenta – The Vector Model......Page 108
 6.1 The Concept of the Vector Model......Page 109
 6.2 Closed Orbitals......Page 113
  6.2.1 Electron–Electron Repulsion InvolvingClosed Orbitals......Page 114
 6.3 LS-coupling......Page 120
  6.3.1 Atomic Terms......Page 121
  6.3.2 Two Non-Equivalent Electrons......Page 122
  6.3.3 Two Equivalent Electrons......Page 124
  6.3.4 More Than Two Valence Electrons......Page 126
  6.3.5 Fine-Structure......Page 128
 6.4 jj-coupling......Page 131
  6.4.1 jj-coupling Terms......Page 132
  6.4.2 Equivalent Electrons......Page 133
 6.5 LS- and jj-coupling......Page 135
 References......Page 138
7 LS-Coupling......Page 140
  7.1.1 Allowed LS-coupling Terms......Page 141
  7.1.2 Genealogy of Atomic Terms......Page 144
 7.2 Energies of LS-coupling Terms......Page 146
  7.2.1 Reduction of the Problem with theDiagonal-Sum Rule......Page 147
  7.2.2 Energy Contributions from One- and Two-ElectronIntegrals......Page 152
  7.2.3 Slater Integrals......Page 154
  7.2.4 Examples of Term Energies......Page 157
 7.3 Fine-Structure......Page 161
  7.3.1 Multiplet Intervals......Page 162
  7.3.2 Calculation of Fine-Structure Energy Intervals......Page 163
 7.4 Wave Functions......Page 167
  7.4.1 Eigenfunctions of Atomic Terms......Page 168
  7.4.2 Radial Functions for Light Atoms......Page 175
 7.5 Energy Levels in the LS-coupling Scheme......Page 176
 References......Page 179
 8.1 Allowed jj-coupling Terms......Page 180
 8.2 Theoretical Energies of jj-Coupled States......Page 183
  8.2.1 The Spin–Orbit Energies......Page 184
  8.2.2 Energy Contribution from the Electron–ElectronInteraction......Page 185
 8.3 Applicability of jj-coupling......Page 188
 References......Page 190
9 Other Coupling Schemes — Intermediate Cases......Page 191
 9.1 Other Interactions Between Electronic Angular Momenta......Page 192
  9.1.1 Spin–Spin Interaction......Page 193
  9.1.2 Spin-Other-Orbit Interaction......Page 194
  9.1.4 An Example of Magnetic Electron–ElectronInteractions......Page 195
 9.2 J1K-Coupling......Page 196
 9.3 LK-Coupling......Page 199
 9.4 Intermediate Coupling......Page 200
  9.4.1 Transformations Between Coupling Schemes......Page 202
  9.4.2 Examples of Intermediate Coupling for Two-ElectronAtoms......Page 204
 9.5 Configuration Interaction......Page 209
  9.5.1 Analysing the Non-Diagonal Matrix Elements......Page 210
  9.5.2 An Example of Configuration Interaction......Page 212
 9.6 The Difficulties with Assigning Pure State Designations......Page 215
 References......Page 217
 10.1 Isotope Shift......Page 218
  10.1.1 The Reduced Mass Effect......Page 219
  10.1.3 The Volume Effect......Page 220
 10.2 Hyperfine Structure......Page 222
  10.2.1 The Magnetic Dipole Interaction......Page 223
  10.2.2 The Magnetic Dipole hfs Splitting and Shift......Page 225
  10.2.3 The hfs a-Constant......Page 227
  10.2.5 The Electric Quadrupole Interaction......Page 231
  10.2.6 The Nuclear Electric Quadrupole Moment......Page 234
  10.2.7 The Electric Quadrupole Hamiltonian......Page 237
  10.2.8 Higher Order Multipole Moments......Page 239
  10.2.9 Hyperfine Structure in Multielectron Systems......Page 240
  10.2.10 Some Concrete Examples of Atomic HyperfineStructure......Page 241
 10.3 The Finite Size and Finite Mass Nucleus......Page 243
 References......Page 244
11 The Zeeman Effect......Page 246
 11.1 Weak Magnetic Field — The Case with No Hyperfine Structure......Page 247
  11.1.1 The Zeeman Effect Hamiltonian......Page 248
  11.1.2 The Paramagnetic Zeeman Shift......Page 250
  11.2.1 The Zeeman Effect Hamiltonian in the Presence of Hyperfine Structure......Page 255
 11.3 The Paschen-Back Effect — Strong Fields......Page 257
  11.3.1 Perturbation Approach in the Absence of HyperfineStructure......Page 258
  11.3.2 The Case with Hyperfine Structure and a Strong Field......Page 260
 11.4 Intermediate Fields......Page 263
  11.4.1 Diagonalising the Intermediate Field Hamiltonian......Page 264
  11.4.2 Intermediate Field and Hyperfine Structure......Page 265
  11.4.3 The Breit–Rabi Formula......Page 266
 11.5 The Diamagnetic Zeeman Effect and Very Strong Fields......Page 268
  11.5.1 The Landau Region......Page 269
  11.5.3 The Diamagnetic Zeeman Effect as aWeak Perturbation......Page 270
 11.6 Magnetic Interactions......Page 271
 References......Page 272
12 The Stark Effect......Page 273
 12.1 The Linear Stark Effect......Page 274
  12.1.1 The Stark Effect Hamiltonian......Page 275
  12.1.2 Perturbative Treatment of the Linear Stark Effect for a One-Electron Atom......Page 276
  12.1.3 The Linear Stark Effect, for a One-Electron Atom,with Parabolic Coordinates......Page 278
 12.2 Quadratic Stark Effect......Page 281
  12.2.1 The Stark Shift in the Hydrogen Ground State......Page 282
  12.2.2 Stark Shifts for Multielectron Atoms......Page 284
 12.3 Atoms in Static Electric Fields......Page 286
 References......Page 288
13 Complex and Exotic Excitations......Page 289
 13.1 Rydberg Atoms......Page 291
 13.2 Continuum States......Page 293
 13.3 Inner Orbital Excitations and Highly Ionised Atoms......Page 294
  13.3.2 Highly Charged Ions......Page 295
 References......Page 296
 14.1 Self-Consistent Field Methods......Page 297
  14.1.1 The Hartree Equation......Page 298
 14.2 Correlation Methods......Page 301
 14.3 The Starting Point for Further Studies of Atomic Structure......Page 302
 References......Page 303
 A.1 Constants in Atomic Units......Page 305
  A.1.1 The Fine-Structure Constant......Page 306
 References......Page 307
 B.1 The Radial Solutions to the Hydrogenic Schrödinger Equation......Page 308
  B.1.1 Wave Functions......Page 310
 B.2 The One-Electron Schrödinger equation in Parabolic Coordinates......Page 311
 B.3 Radial Integrals and Radial Functions......Page 314
 References......Page 315
 C.1 General Angular Momentum......Page 317
  C.1.1 Eigenvalues......Page 318
 C.2 Orbital Angular Momentum......Page 321
 C.3 Spherical Harmonics......Page 323
  C.3.1 The Addition Theorem for Spherical Harmonics......Page 326
  C.4.1 3j-Symbols......Page 327
  C.4.3 Higher Order nj-Symbols......Page 329
 C.5 Addition of Angular Momenta and Vector CouplingCoefficients......Page 330
  C.5.1 Addition of Quantum Mechanical Angular Momenta......Page 331
   C.5.1.1 Matrix Elements of J J J J2......Page 332
  C.5.2 Transformation Between Representations......Page 333
   C.5.2.1 Derivation of Clebsch-Gordan Coefficients......Page 334
  C.5.3 Coupling of Three Angular Momenta......Page 337
  C.6.1 Scalar, Vector, and Tensor Operators......Page 339
  C.6.2 Statement of the Theorem......Page 340
 References......Page 342
 D.1 Analytical Integration of the Coulomb Repulsion Potential......Page 343
  D.1.1 The Angular Integrals......Page 344
  D.1.2 The Radial Integrals......Page 348
 D.2 Solutions for Hydrogenic Wave Functions......Page 349
 D.3 Slater Integrals......Page 350
 D.4 Coulomb Interaction Energies in jj-coupling......Page 351
 References......Page 356
 E.1 Classical Magnetic Moments......Page 357
  E.1.1 Magnetic Moment Due to the Orbital AngularMomentum......Page 359
  E.1.2 The Electron Spin and Its Associated MagneticMoment......Page 360
  E.2.1 The Dirac Equation......Page 362
  E.2.2 A Relativistic Electron in an Electromagnetic Field......Page 363
 References......Page 365
 F.1 General Formalism......Page 366
 F.2 The Terms of the Hamiltonian......Page 370
  F.2.1 The Zeeman Effect......Page 371
  F.2.2 Interelectronic Magnetic Interactions......Page 372
  F.2.3 The Intrinsic Spin–Orbit Interaction......Page 373
  F.2.4 Hyperfine Structure......Page 374
Index......Page 375