725 Index (2n + 1)-rule 473, 480 anti-electron 188 LS-coupling 148 anti-matter 188 X-operator techniques 462 anticommutation relations 167 ΛS-coupling 148 anticommutator 167 β+-decay 245 antisymmetry 156 ωω-coupling 155 atomic nucleus jj-coupling 155, 342 – moving 195 n-rule 588 – point-charge potential 194 p2-basis 495 atomic units 7 2-electron systems 251 atoms 4-current 169, 170, 238, 261, 266, – hydrogen-like 194 314 – many-electron 333 – many-electron case 314, 315 Au2 619 4-momentum 79 Aufbau principle 397 4-potential 266 aurophilic effect 607 4-spinor 169 4-velocity 79 Baker–Campbell–Hausdorff ex- pansion 328 ab initio theory 1 balance accidental degeneracy 221 – kinetic 185, 408 AMFI 547, 562 – magnetic 185, 408 analytic gradients 425 Barysz–Sadlej–Snijders approach angular momentum 40, 139, 197 462, 534, 542 – commutators 142 basis function – coupling 145, 149, 199 – one-electron 284 – coupling of orbital and spin 149 basis set – operator 140 – Cartesian Gaussians 411 – orbit–orbit coupling 520 – complete 119 – orbital 197 – contractions 407 – spin–spin coupling 520 – expansion 119 – total 149, 199 – finite 119 Relativistic Quantum Chemistry. Markus Reiher and Alexander Wolf © 2015 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2015 by WILEY-VCH Verlag GmbH & Co. KGaA. 726 Index – Gauss-type functions 409 central-difference formulae 665 – one-electron 284 charge density 36, 120, 169, 314 – one-particle basis functions 405 – many-electron case 314, 315 – polarization functions 407 charge-current density 91 – primitives 407 CI 404 – Slater-type functions 412 – coupling coefficients 299 Bethe–Salpeter equation 251 – density matrix 299 Bickley formulae 386, 501, 665 – direct 405 Biot–Savart law 47 – structure factors 299 Bohr magneton 187, 519 CIPSO 563 boost CISD 289 – Galilei 15 clamped-nuclei approximation – Lorentz 60, 68, 102 279 Born interpretation 119, 135 clamped-nucleus approximation Born–Oppenheimer approxima- 194 tion 279 classical mechanics boson 156, 164, 301 – Hamiltonian 31 box normalization 174 – Lagrangian 22 bra-ket notation 118 – Newtonian 11 Breit interaction 242, 337, 521 – relativistic 53 – frequency-dependent operator Clebsch–Gordan 268 – coefficient 146, 151, 343 – in spherical coordinates 337 – summation 151 – long-wavelength limit 268 Clifford algebra 173, 238 – operator 257 closed shells 357 – potential functions 353 cluster amplitudes 326 – radial integrals 349 co-vector 63 – retardation operator 258 collinear 323 – two-component operator 492 commutation relation 131, 141 Breit–Pauli Hamiltonian 259, 507 – equal time 156 Brown–Ravenhall disease 274 commutation symmetry BSS 462, 534, 542 – orbital angular momentum 199 Bulirsch–Stoer method 393 – total angular momentum 199 commutator 128, 129, 199 canonical equations components of the spinor – classical mechanics 31 – large 185 – Hamilton 32 – pseudo-large 531 – Hartree–Fock 310 – small 185 canonical spinors 310 configuration interaction canonical transformations 34 – CISD 289 CASPT2 290, 624 – excitation hierarchy 288 CASSCF 290 Index 727 – multi-reference 289, 624 – nonrelativistic 136 conjugate variables 131 – relativistic 170, 314 conservation laws 20, 29 – transverse part 114 – commutator 129 cusp condition – Poisson brackets 33 – electron–nucleus cusp 368 constraints 22 contact term 507 d’Alembert operator 44, 66 continuity equation 37, 91, 136, Darwin interaction 110, 256 164, 165, 169, 314, 315, 594 Darwin operator 458, 506 continuum dissolution 274 degeneracy 149 coordinates – accidental 221 – generalized 22 delta distribution – spherical 140, 196 – representation 637 correlated spinors 395 – three-dimensional 636 correlation energy 311 density distribution 120 correspondence principle 122, density functional theory 313 129, 161, 184, 254, 271 density matrix 299 Coulomb gauge 44, 241, 243 density matrix renormalization Coulomb operator group 283 – electron–electron interaction density operator 135, 578 255, 336 derivative theory 425 – electron–nucleus interaction 194 DFT 313 – in spherical coordinates 336 – Kohn–Sham 318 – tensor product 284 – spin 322 coupled cluster 325, 404 – time-dependent 573 – amplitudes 326 diagonal local approximation to – CCSD 327 Hamiltonian 554 – connected amplitudes 327 diagonal local approximation to – disconnected amplitudes 327 unitary transformation 556 – Fock space 329, 624 diamagnetic term 187 – size consistency 327 dihydrogen 616 coupled product basis 146 DIIS 312 covariant 15 Dirac Cowan–Griffin operator 522 – 4-current 169, 238, 261, 266 cross product 634 – bra-ket notation 118 CsH 622 – charge density 170, 238, 261, 314 curl 634 – charge distribution 170 curl theorem 635 – continuity equation 169 current density 36, 136 – current density 170, 238, 261, – induced 138 314 – longitudinal part 114 – density distribution 170 728 Index – electromagnetic interaction en- – energy for atoms 357 ergy 184 – energy in basis representation – electron density 170, 238, 261 415 – energy hydrogen-like atoms – equations for atoms 360 213, 214, 217 – Roothaan equation 288 – equation 166, 193 – SCF equations 309, 310 – Hamiltonian 193 direct perturbation theory 613 – hole theory 188, 244 direct product 251, 283, 650 – hydrogen atom 193 direct sum 300 – picture 578 discretization schemes 379 – relation 504, 505, 650 dispersion forces 607 Dirac equation 166 divergence 634 – 4-current 314, 315 divergence theorem 635 – continuity equation 314, 315 DKH picture 578 – derivation 167 DKH transformation 543 – field-free equation 166 – electron density 486 – in external electromagnetic – gradient 500 fields 182 – see Douglas–Kroll–Hess 469 – interpretive aspects 187 DLH 554 – Lorentz covariance 170 DLU 556 – minimal coupling 181 DMRG 283, 621 – modified equation 500, 531 double groups 423 – momentum-space representa- doublet state 149 tion 233 Douglas–Kroll–Hess transforma- – particle at rest 173, 174 tion 454, 457, 469, 543 – relation to Klein–Gordon equa- – coefficient dependence 481 tion 166 – convergence 477 – solutions for free particles 173 – electron density 593 – standard representation 168, – gradient 500 486, 523 – infinite-order 479 – Weyl representation 168, 486, – many-particle extension 488 523 – resolution of the identity 494 Dirac relation 650 – scalar approximation 496 Dirac–Coulomb Hamiltonian 273 – spin-averaged approximation Dirac–Coulomb–Breit Hamilto- 496 nian 274 – spin-free approximation 496 Dirac–Fock–Roothaan calculations DPT 613 288 dual field strength tensor 94 Dirac–Hartree–Fock theory 287, duality transformation 94 295, 299, 361 duplexity problem 147, 166 – energy 413 dynamic electron correlation 290 Index 729 ECP elimination techniques – see effective core potential 564 – normalized elimination 531 effective core potential 564 – regular approximation 524 – energy-consistent 566 energy – shape-consistent 566 – kinetic 21 – small core 565 – potential 21 – spin–orbit 566 – reference level 185, 218 EFG 599 – shift by rest energy 218 Ehrenfest theorem 133, 315 energy conservation 29 eigenvalue equation 124 energy-momentum relation 82 Einstein’s summation convention EPR 57 – see ESR 572 electric field equation of motion – longitudinal component 114, – Euler–Lagrange 26, 48, 49 241 – Hamilton 32 – transverse component 114, 241 – in quantum mechanics 122 electric field gradient 599 – Newton 12, 27 electro-weak theory 603 equivalence restriction 205, 214, electrodynamics 342, 357 – classical 35 ESR 572, 595 – quantum 237 Euclidean space 17 electromagnetic fields 35 Euler formula 646 electromagnetic interaction event 14 – energy 184 exact two-component approach – QED 266 537 electromagnetic waves 40 exact two-component method 462 electron correlation 291, 311 exact-decoupling methods 533 – coupled cluster 325 – local variant 554 – dynamic 290 – local variants 553 – static 290, 397 excitation operator 365 electron density 120, 169, 314, 320, expectation value 124 364, 625 – radial density 223 – many-electron case 314, 315 external potential 195 electron–positron pair creation external-field no-pair projection 189 538 electronic configuration 342, 397 electronic energy 280 FCI 285 – atoms 354 Fermi contact term 598 electronic Schrödinger equation fermion 156, 301 280 Feynman diagrams 278 elementary particles 1 Feynman gauge 268 Feynman slash notation 172, 238 730 Index field strength tensor 92 – transformation 16, 29 fields 32 gauge fine-structure constant 7 – Coulomb 44 fine-structure splitting 221, 396 – Feynman 268 finite differences 500 – Lorenz 44, 92, 268 finite nuclei 225 gauge field 91 finite-element method 335 gauge origin 596 finite-field methods 573 gauge transformation 92 first-principles theory 1 – classical electrodynamics 43 Fitzgerald–Lorentz contraction 73 – classical Lagrangian 28, 112 fixed-nucleus approximation 194 – quantum electrodynamics 238 Fock operator 308, 420 Gaunt Fock space 246, 300 – coefficients 347 Foldy–Wouthuysen transforma- – operator 255, 263 tion Gauss’ theorem 635 – free-particle case 443, 471, 508, generalized coordinates 22 534, 542 geometry gradients 425, 500, 533 – general unitary transformation GIAO 596 454 gold dimer 619 – higher-order 459 Gordon decomposition 314, 315, – kinetic balance 446 321, 657 – molecular properties 580 gradient 634 FORA 524 Grant coefficients 347 force 13 Green’s function 636 four-component Hamiltonian 273 group theory 148, 155 Fourier transformation 233, 653 – double groups 423 – back transformation 653 – irreducible representation 148 frame of reference – point groups 423 – inertial 12 – rotations 19 free-particle solutions 175 gyromagnetic ratio 187, 246, 572