| Preface |
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v | |
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1 | (22) |
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1 | (1) |
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2 | (5) |
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7 | (3) |
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Defining an effective Hamiltonian |
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7 | (2) |
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Electron-electron interaction |
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9 | (1) |
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Relativity and Atomic Structure |
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10 | (4) |
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14 | (6) |
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Dirac-Fock one-centre method |
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17 | (2) |
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Relativistic quantum chemistry |
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19 | (1) |
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20 | (3) |
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23 | (84) |
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23 | (3) |
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Introducing the Dirac Equation |
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26 | (5) |
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26 | (2) |
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Dirac equation in an external field |
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28 | (1) |
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Why the Dirac matrices are four dimensional |
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29 | (2) |
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State Space and Interpretation |
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31 | (3) |
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A Hilbert space for the Dirace equation |
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31 | (1) |
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Tentative one-particle interpretation |
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32 | (2) |
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Solving the Dirac Equation |
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34 | (10) |
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The stationary Dirac equation |
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36 | (1) |
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Fourier transform of the free Dirac equation |
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37 | (2) |
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Momentum space eigenvectors of the Dirac operator |
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39 | (1) |
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40 | (1) |
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41 | (1) |
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42 | (2) |
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44 | (2) |
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Positive and negative energies |
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44 | (1) |
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45 | (1) |
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46 | (3) |
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Standard position and velocity |
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46 | (1) |
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Classical velocity and zitterbewegung |
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47 | (2) |
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Electron-Positron Interpretation |
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49 | (5) |
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54 | (7) |
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54 | (1) |
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Convariance of the Dirac equation |
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55 | (2) |
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57 | (2) |
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59 | (2) |
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Discrete Lorentz transformations |
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61 | (1) |
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Classification of External Fields |
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61 | (4) |
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Poincare transformations of external fields |
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61 | (1) |
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62 | (1) |
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Electromagnetic vector potential |
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63 | (1) |
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Anomalous magnetic moment |
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63 | (1) |
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Anomalous electric moment |
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64 | (1) |
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64 | (1) |
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64 | (1) |
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Properties of Dirac Operators |
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65 | (4) |
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Short Description of the Nonrelativistic Limit |
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69 | (8) |
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71 | (4) |
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g-factor and Thomas precession |
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75 | (2) |
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77 | (11) |
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Which potentials are spherically symmetric? |
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77 | (2) |
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Introducing polar coordinates |
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79 | (2) |
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Operators commuting with the Dirac operator |
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81 | (1) |
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Angular momentum eigenfunctions |
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82 | (2) |
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The angular momentum eigenspaces |
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84 | (2) |
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The partial-wave subspaces |
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86 | (2) |
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88 | (16) |
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89 | (1) |
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Systematics of eigenstates |
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90 | (2) |
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Fall to the center, self-adjointness, and all that |
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92 | (2) |
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Influence of anomalous magnetic moment |
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94 | (1) |
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A useful similarity transformation |
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94 | (2) |
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96 | (2) |
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98 | (2) |
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100 | (3) |
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103 | (1) |
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104 | (3) |
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Relativistic Self-Consistent Fields |
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107 | (96) |
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107 | (5) |
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112 | (25) |
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Special relativistic notation: Minkowski space-time. Lorentz transformation |
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112 | (5) |
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117 | (1) |
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The Dirac equation for free particles |
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118 | (2) |
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Dirac equation in external electromagnetic fields |
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120 | (2) |
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122 | (1) |
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Quantization of the Dirac field in the Furry picture |
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123 | (3) |
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Quantization of the Maxwell field |
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126 | (3) |
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Relativistic Hamiltonian for many-electron systems |
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129 | (2) |
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The electron-electron interaction energy |
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131 | (4) |
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The Dirac-Hartree-Fock-Breit model |
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135 | (2) |
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Finite Matrix Methods for Dirac Hamiltonians |
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137 | (20) |
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Dirac central field wavefunctions |
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139 | (3) |
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142 | (1) |
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143 | (2) |
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Group theoretical properties |
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145 | (1) |
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146 | (1) |
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146 | (2) |
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The Rayleigh-Ritz method for Dirac Hamiltonians |
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148 | (2) |
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150 | (3) |
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153 | (2) |
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155 | (2) |
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157 | (11) |
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158 | (1) |
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159 | (1) |
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160 | (1) |
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Atoms: elimination of angular integration |
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161 | (3) |
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One-centre integrals for S-spinors and G-spinors |
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164 | (2) |
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166 | (2) |
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DHFB Theory for Molecules |
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168 | (10) |
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G-spinors, SGTF, CGTF and HGTF |
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169 | (3) |
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172 | (2) |
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Integrals over products of G-spinors |
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174 | (2) |
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The generalized McMurchie-Davidson algorithm |
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176 | (2) |
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Implementation: the Bertha Code |
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178 | (8) |
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Choice of basis set exponents |
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178 | (1) |
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Computational cost of integral generation |
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179 | (1) |
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Economization using spinor symmetry |
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180 | (1) |
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Integral-direct Fock matrix evaluation |
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180 | (1) |
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Elimination of small terms |
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181 | (1) |
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181 | (1) |
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182 | (1) |
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Relativistic methods for large systems |
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183 | (3) |
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186 | (5) |
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Optimiation with respect to the spinor basis |
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188 | (1) |
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The approach to self-consistency |
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189 | (2) |
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Survey of Relativistic Mean Field Calculations |
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191 | (3) |
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191 | (1) |
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Molecular DHF calculations |
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192 | (2) |
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194 | (9) |
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Nuclear Charge Density Distributions in Quantum Chemistry |
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203 | (56) |
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203 | (2) |
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205 | (6) |
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205 | (1) |
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206 | (4) |
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Electric and magnetic fields generated by the nucleus |
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210 | (1) |
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Nuclear Charge Density Distributions: Their Potential and Other Properties |
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211 | (10) |
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212 | (2) |
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Spherical nuclear charge density distributions |
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214 | (5) |
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Standardization of charge density distributions and relation to experimental data |
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219 | (2) |
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Nuclear Charge Density Distribution Models |
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221 | (13) |
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Point-like charge density distribution |
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222 | (1) |
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`Spherical shell' charge density distribution |
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223 | (1) |
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`Homogeneous' charge density distribution |
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224 | (1) |
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Gauss-type charge density distribution |
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225 | (1) |
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Fermi-type charge density distribution |
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226 | (2) |
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Fourier-Bessel charge density distribution |
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228 | (3) |
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Sum-of-Gaussians charge density distribution |
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231 | (3) |
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Nuclear Models in Quantum Chemistry |
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234 | (12) |
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The nuclear electrostatic potential |
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234 | (3) |
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Electronic structure calculations - the numerical approach |
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237 | (2) |
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239 | (3) |
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242 | (3) |
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Electronic structure calculations - the algebraic approach |
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245 | (1) |
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Other Properties Depending on the Nuclear Charge Distribution |
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246 | (4) |
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The electron-nucleus contact term |
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246 | (1) |
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Higher quantum electrodynamic effects |
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247 | (1) |
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248 | (2) |
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Higher nuclear electric multipole moments |
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250 | (1) |
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250 | (9) |
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Basis Sets for Relativistic Calculations |
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259 | (32) |
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259 | (2) |
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The Dirac Equation for the Hydrogen Atom |
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261 | (2) |
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263 | (3) |
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The Kinetic Balance Requirement |
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266 | (4) |
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The Optimization of Basis Sets |
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270 | (3) |
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Describing the Small R Region |
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273 | (2) |
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Basis Set Shell Structure |
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275 | (2) |
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277 | (1) |
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278 | (3) |
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Large-Small Component Balance |
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281 | (1) |
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Examples of 4-Component Basis Sets in Applications |
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282 | (6) |
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288 | (3) |
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Post Dirac-Fock-Methods - Electron Correlation |
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291 | (41) |
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291 | (2) |
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The Dirac-Coulomb-Briet Hamiltonian |
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293 | (10) |
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The Hamiltonian in second quantization |
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295 | (7) |
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Symmetry relations between the matrix elements |
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302 | (1) |
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303 | (4) |
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Neglect of spin-orbit coupling |
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304 | (2) |
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Approximations to the Coulomb-Breit matrix elements |
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306 | (1) |
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Many-Body Perturbation Theory |
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307 | (2) |
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Configuration Interaction |
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309 | (12) |
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Kramers-unrestricted configuration interaction |
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311 | (2) |
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Use of group chains to utilize molecular symmetry |
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313 | (2) |
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Coupling coefficients in non-symmetry adapted graphs |
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315 | (3) |
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Coupling coefficients in symmetry adapted graphs |
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318 | (1) |
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Kramers-restricted configuration interaction |
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319 | (2) |
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321 | (8) |
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Single reference unrestricted coupled cluster theory |
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321 | (4) |
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Single reference restricted coupled cluster theory |
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325 | (1) |
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Fock space coupled cluster methods |
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326 | (3) |
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329 | (3) |
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Post Dirac-Fock-Methods - Properties |
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332 | (69) |
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332 | (1) |
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Theory of Molecular Properties |
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333 | (15) |
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334 | (5) |
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Rayleigh-Schrodinger perturbation theory |
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339 | (2) |
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Variational perturbation theory |
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341 | (5) |
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346 | (2) |
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Electromagnetic Interactions |
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348 | (21) |
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Classical electrodynamics |
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348 | (4) |
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The interaction of charged particles with electromagnetic fields |
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352 | (4) |
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The Dirac equation in the presence of external electromagnetic fields |
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356 | (6) |
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362 | (1) |
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362 | (2) |
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364 | (3) |
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367 | (2) |
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369 | (10) |
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The zeroth order Hamiltonian |
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369 | (2) |
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General structure of perturbation operators |
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371 | (2) |
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373 | (1) |
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374 | (2) |
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376 | (1) |
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376 | (1) |
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Relativistic and non-relativistic perturbation operators |
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377 | (2) |
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Molecular Properties at the Closed-Shell 4-Component Relativistic Hartree-Fock Level |
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379 | (15) |
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379 | (4) |
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First order response of the wavefunction |
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383 | (6) |
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389 | (1) |
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The linear response function |
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390 | (1) |
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The quadratic response function |
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391 | (3) |
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394 | (7) |
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394 | (1) |
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Double perturbation theory |
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395 | (2) |
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397 | (4) |
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401 | (67) |
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401 | (2) |
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403 | (30) |
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Dirac equation for the electron in an external field |
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403 | (4) |
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Electromagnetic field: classical theory |
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407 | (5) |
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Quantization of electromagnetic field |
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412 | (4) |
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Quantization of electron-positron field |
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416 | (3) |
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Interaction of quantized fields |
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419 | (3) |
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422 | (4) |
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Ultraviolet divergencies: regularization and renormalization |
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426 | (7) |
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QED Theory of the Interelectron Interaction in Atoms |
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433 | (12) |
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Adiabatic S-matrix formalism |
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433 | (2) |
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435 | (3) |
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438 | (3) |
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Dirac-Hartree-Fock approximation |
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441 | (2) |
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Dirac-Columb-Breit Hamiltonian |
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443 | (2) |
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QED Corrections for Light Atoms |
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445 | (8) |
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Radiative corrections in hydrogen: lowest order |
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445 | (4) |
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Radiative corrections in hydrogen: higher orders |
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449 | (2) |
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QED corrections in many-electron atoms |
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451 | (2) |
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QED Corrections in Heavy Atoms |
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453 | (15) |
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Electron self-energy: potential expansion |
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453 | (4) |
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Electron self-energy: partial wave renormalization |
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457 | (2) |
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459 | (2) |
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Radiative corrections for the ns valence electrons for heavy atoms |
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461 | (7) |
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468 | (55) |
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468 | (3) |
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The Weak Interactions and Atomic Physics |
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471 | (4) |
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Heavy Ions as a Laboratory for Many-Body Theory |
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475 | (17) |
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Furry representation and S-matrix theory |
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476 | (3) |
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479 | (1) |
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480 | (4) |
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484 | (6) |
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490 | (1) |
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491 | (1) |
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Parity Nonconservation in Cesium |
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492 | (25) |
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Many body perturbation theory |
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495 | (8) |
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503 | (2) |
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505 | (4) |
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Sum-over-states for PNC amplitude |
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509 | (1) |
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Smaller PNC contributions |
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510 | (1) |
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510 | (1) |
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511 | (1) |
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Nuclear spin-dependent effects |
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512 | (1) |
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513 | (1) |
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Comparison with experiment |
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513 | (1) |
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Prospects for higher theoretical accuracy |
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514 | (1) |
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515 | (1) |
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516 | (1) |
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516 | (1) |
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516 | (1) |
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517 | (6) |
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Relativistic Density Functional Theory: Foundations and Basic Formalism |
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523 | (99) |
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524 | (5) |
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Field Theoretical Background |
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529 | (7) |
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Foundations and Basic Formalism |
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536 | (23) |
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536 | (6) |
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Relativistic Kohn-Sham equations |
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542 | (7) |
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549 | (4) |
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Relativistic optimized potential method: A third generation of density functionals |
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553 | (4) |
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557 | (2) |
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Relativistic Exchange-Correlation Functional: Concepts and Illustrative Results |
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559 | (23) |
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Relativistic implicit functionals |
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561 | (11) |
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Relativistic local density approximation |
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572 | (7) |
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Relativistic generalized gradient approximation |
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579 | (3) |
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582 | (40) |
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Appendix: Quantization of Noninteracting Fermions |
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583 | (8) |
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Appendix: Renormalization Scheme of Vacuum QED |
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591 | (8) |
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Appendix: Relativistic Homogeneous Electron Gas |
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599 | (1) |
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599 | (1) |
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600 | (5) |
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605 | (5) |
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Appendix: Renormalization of Inhomogeneous Electron Gas |
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610 | (3) |
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Appendix: Gradient Corrections to the Relativistic LDA |
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613 | (9) |
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Two-Component Methods and the Generalized Douglas-Kroll Transformation |
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622 | (42) |
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622 | (4) |
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Methods to Decouple the Dirac Equation |
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626 | (15) |
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Relationship between the large and small components |
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626 | (4) |
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Elimination of the small component |
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630 | (3) |
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Transformation to two components |
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633 | (8) |
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641 | (11) |
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Generalized parameterisation of a unitary matrix |
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643 | (2) |
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The generalized Douglas-Kroll transformation |
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645 | (5) |
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Aspects of implementation |
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650 | (2) |
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Numerical Results with DKH3 and DKH4 |
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652 | (4) |
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Transformation of the Wavefunction - Picture Change |
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656 | (3) |
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Conclusions and Perspectives |
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659 | (5) |
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Perturbation Theory of Relativistic Effects |
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664 | (94) |
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Introduction. Why Perturbation Theory? |
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665 | (3) |
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The Non-Relativistic Limit |
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668 | (24) |
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668 | (1) |
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Eigenvalues and eigenfunctions |
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669 | (2) |
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The Levy-Leblond equation |
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671 | (4) |
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The Levy-Leblond equation in a central field |
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675 | (1) |
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The resolvent of the Dirac operator |
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676 | (1) |
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`Second-order Dirac equations' |
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677 | (1) |
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The Foldy-Wouthuysen transformation |
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678 | (3) |
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Electrodynamics in the vacuum, independent of the system of units |
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681 | (3) |
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The non-relativistic limit of electrodynamics |
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684 | (1) |
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The Levy-Leblond equation in a magnetic field |
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685 | (3) |
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Magnetic properties in the nonrelativistic limit |
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688 | (2) |
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Is spin a relativistic effect? |
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690 | (2) |
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Perturbation Theory Based on the Foldy-Wouthuysen Transformation |
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692 | (8) |
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A FW transformation in two steps |
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692 | (2) |
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Perturbation expansion of the operators |
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694 | (1) |
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PT of the eigenfunctions and eigenvalues |
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695 | (1) |
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695 | (1) |
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A non-hermitian variant of the FW transformation |
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696 | (1) |
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Properties of the X operator |
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697 | (1) |
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The Douglas-Kroll-Hess transformation |
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698 | (2) |
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Direct Perturbation Theory |
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700 | (15) |
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The perturbation expansion |
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700 | (2) |
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702 | (1) |
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Formulation in terms of upper and lower components |
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702 | (2) |
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Application to H-like ions |
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704 | (2) |
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706 | (1) |
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Relation to Breit-Pauli form |
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707 | (2) |
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DPT of electric properties |
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709 | (3) |
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DPT of magnetic properties |
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712 | (1) |
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713 | (1) |
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The regular approximation and the method of Moore |
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714 | (1) |
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Stationary Direct Perturbation Theory |
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715 | (6) |
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Stationary conditions and stationary functionals |
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715 | (3) |
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|
718 | (1) |
|
Regularization of the trial function |
|
|
719 | (2) |
|
Quasidegenerate Direct Perturbation Theory |
|
|
721 | (7) |
|
Effective Hamiltonians in a model space |
|
|
721 | (3) |
|
Matrix representation of the effective Hamiltonian in the model space |
|
|
724 | (1) |
|
The FW transformation revisited |
|
|
725 | (1) |
|
Infinite-order quasi-degenerate DPT |
|
|
726 | (2) |
|
|
|
728 | (21) |
|
The Dirac-Coulomb and the Levy-Leblond-Coulomb operator |
|
|
728 | (4) |
|
The Gaunt and the Breit interaction |
|
|
732 | (2) |
|
|
|
734 | (3) |
|
The Brown-Ravenhall disease and related problems |
|
|
737 | (1) |
|
Relativistic Hartree-Fock in terms of DPT |
|
|
738 | (7) |
|
Relativistic MC-SCF in terms of DPT |
|
|
745 | (1) |
|
Density functional theory (DFT) in terms of DPT |
|
|
746 | (1) |
|
Relativistic corrections to explicitly correlated wavefunctions |
|
|
746 | (3) |
|
Direct Perturbation Theory Using Energy Gradients or the Finite Perturbations |
|
|
749 | (2) |
|
Conclusions, Merits and Drawbacks of Direct Perturbation Theory |
|
|
751 | (1) |
|
Appendix: The Concept of Effective Hamiltonians |
|
|
752 | (2) |
|
|
|
754 | (4) |
|
Perturbation Theory Based on Quasi-Relativistic Hamiltonians |
|
|
758 | (35) |
|
|
|
|
|
758 | (2) |
|
|
|
760 | (4) |
|
Elimination of the small component |
|
|
760 | (1) |
|
The Levy-Leblond equation |
|
|
761 | (1) |
|
|
|
761 | (1) |
|
|
|
762 | (1) |
|
|
|
762 | (1) |
|
The transformed Dirac equation |
|
|
763 | (1) |
|
Quasi-Relativistic Hamiltonians |
|
|
764 | (3) |
|
|
|
765 | (1) |
|
|
|
765 | (1) |
|
|
|
766 | (1) |
|
|
|
766 | (1) |
|
|
|
767 | (1) |
|
Perturbation Energy Expansions |
|
|
767 | (4) |
|
|
|
771 | (6) |
|
|
|
771 | (3) |
|
|
|
774 | (1) |
|
|
|
774 | (3) |
|
|
|
777 | (1) |
|
|
|
778 | (10) |
|
|
|
778 | (1) |
|
|
|
778 | (1) |
|
|
|
779 | (4) |
|
|
|
783 | (1) |
|
|
|
783 | (5) |
|
|
|
788 | (5) |
|
Relativistic Effective Core Potentials |
|
|
793 | (70) |
|
|
|
|
|
793 | (102) |
|
|
|
895 | |
|
|
|
801 | (5) |
|
Dirac-Coulomb(-Breit) Hamiltonian |
|
|
802 | (2) |
|
Douglas-Kroll-Hess Hamiltonian |
|
|
804 | (1) |
|
|
|
805 | (1) |
|
|
|
806 | (38) |
|
|
|
807 | (3) |
|
Valence-only model Hamiltonian |
|
|
810 | (2) |
|
|
|
812 | (3) |
|
|
|
815 | (4) |
|
Analytical form of pseudopotentials |
|
|
819 | (2) |
|
Shape-consistent pseudopotentials |
|
|
821 | (3) |
|
Energy-consistent pseudopotentials |
|
|
824 | (5) |
|
Core-polarization potentials |
|
|
829 | (6) |
|
Core-core/nucleus repulsion |
|
|
835 | (1) |
|
|
|
836 | (8) |
|
|
|
844 | (11) |
|
|
|
845 | (2) |
|
|
|
847 | (8) |
|
|
|
855 | (8) |
|
Relativistic Solid State Theory |
|
|
863 | (57) |
|
|
|
|
|
864 | (1) |
|
Effects due to Relativistic Shifts in ε(k) |
|
|
865 | (4) |
|
Electronic States: SO-Coupling and Crystal Symmetry |
|
|
869 | (17) |
|
Spin splitting in semiconductors |
|
|
872 | (14) |
|
Electronic States: SO-Coupling and Spin Polarization |
|
|
886 | (14) |
|
Relativistic band structures |
|
|
887 | (9) |
|
|
|
896 | (4) |
|
Magnetooptical and Magnetoelastic Effects |
|
|
900 | (9) |
|
|
|
901 | (3) |
|
Magnetic anisotropy, magnetostriction |
|
|
904 | (2) |
|
|
|
906 | (3) |
|
|
|
909 | (11) |
| Index |
|
920 | |
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