(2N + 1)-Rule 473, 480 LS-Coupling 148 X-Operator Techniques 462 AS

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(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 expansion 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

– 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 FSCC 329, 624 full configuration interaction 285, H2 616 288 hadron 164 functional derivative 320 Hamiltonian Furry picture 270, 274 – Breit–Pauli 507 – Dirac–Coulomb 273 Galilean transformation 15 – Dirac–Coulomb–Breit 274 Galilei – four-component 273 – boost 15, 17 – no-pair 277 – covariance 17 – operator 122 – group 16 – QED 245 – principle of relativity 14 Hamiltonian function 30, 32 Index 731

Hamiltonian principle 31 interaction Hartree product 286 – instantaneous Coulomb 195 Hartree units 7 – spin–spin 256 Hartree–Fock approximation 295 – two electrons 251 Hartree–Fock theory 287, 299, 361 interaction energy – canonical equations 310 – Darwin 110 – energy 413 – electromagnetic 184 – energy in basis representation invariance 414 – in form 15 – restricted energy 414 – space-time interval 643 – SCF equations 310 IODK 543 – time-dependent 573 IOFW 543 Hausdorff expansion 328 IORA 524, 532 Heisenberg IOTC 462, 534, 539, 543 – equation of motion 128, 180 irreducible representation 148, – picture 127, 180 155 – uncertainty relation 131 helium 251 kinematic momentum 184 Hellmann–Feynman theorem 134, kinetic balance 185, 302, 404, 408, 572 409, 411, 437, 446, 531, 533, 551 hermitean operator 121, 126 – approximate 408, 411, 438 hermiticity 121, 126 – hierarchy 409 – momentum operator 131 kinetic energy – nonrelativistic Hamiltonian 130 – in generalized coordinates 641 Hilbert space 121, 283 Klein–Gordon equation 163 homochirality 603 – density distribution 164 hydrogen atom – derivation 162 – Dirac 193 Kohn–Sham theory 318 – energy eigenvalues 221 Kramers pair 423 – momentum-space representa- Kronecker product 251, 283, 650 tion 233 Kronecker symbol 14, 19 – Schrödinger 197, 211, 216–218 Lagrangian hydrogen-like atoms 194 – classical physics 22 hyperbolic functions 646 – Dirac–Hartree–Fock functional hyperﬁne coupling constants 598 306 independent particle model 285 – gauge transformation 28, 112 inert pair effect 611 – MCSCF functional 306 inertial force 13 – multipliers 306 inertial system 12 – QED 238 inﬁnite-order two-component Lamb shift 223, 278 method 462 Landé factor 187 732 Index

lanthanide contraction 623 many-particle system 21 Laplacian operator 46, 130, 634 mass 13 LCAO 307, 406 – inertial 13 Legendre – rest 13 – operator 142 mass–velocity operator 458, 506 – polynomials 143, 337 matrix element 126 length contraction 72, 73 Maxwell equations 36 Levi-Cività tensor 13, 142 Maxwell stress tensor 39 Liénard–Wiechert potential 111 MCSCF 289, 405, 409 linear combination of atomic or- – coupling coefficients 299 bitals 307, 406 – density matrix 299 local exact decoupling 553 – structure factors 299 logarithmic grid 382 meson 164 London orbitals 596 minimal coupling 97, 181, 254, longitudinal contribution 114, 241 575 Lorentz minimax principle 302 – boost 60, 102 Minkowski – co-vector 63 – force 83 – contraction 73 – space 57 – covariance 122, 238 – tensor 57 – covariant 4-vector 63 model potential 564 – factor 69 modified Dirac equation 500, 531 – force 48 molecular structure 280 Lorentz group 59 moment of force 20 – homogeneous 60 moment of inertia 139 – inhomogeneous 59 momentum 39 Lorentz transformation 59 – angular 13 – improper 60 – canonical 48 – proper 60 – kinematic 13, 49 – uniqueness of 644 – kinematic operator 184 Lorenz gauge 44, 92, 268 – linear 13, 49 lower components 231 – mechanical 13, 49 Lévy-Leblond equation 612, 614 – Poisson brackets 49 – radial 139, 198 magnetic balance 185, 408 momentum conservation 29 magnetic interaction momentum-space representation – nonrelativistic theory 258 132, 233 – relativistic theory 255, 263 MRCI 289, 624 magnetic moment multi-configuration self-consistent – nuclear 572 field 289 magnetic resonance 572, 595 magnetization 321 Index 733 multi-reference configuration in- – finite 225 teraction 289, 624 – point-charge 225 multiplicity 149 numerical grid multipole expansion 337, 599 – logarithmic 382 Møller–Plesset perturbation the- – rational 382 ory 404, 405 numerical methods 379, 409 – differentiation 665 negative-energy solutions 175, – integration 665 219 Numerov method 389, 661 NESC 531, 539 Newton observables – laws of motion 11 – eigenvalue equation 124 – quantum analog 134 – in quantum mechanics 124 Newton–Raphson optimization occupation number vector 300 430 operator NMR 572, 595 – angular momentum 140 no-pair approximation – anticommutator 167 – projectors 274 – Breit–Pauli 259, 507 no-pair Hamiltonian 277 – commutator 129 nodes – Cowan–Griffin 522 – radial 220 – d’Alembert 44, 66 noncollinear 323 – even 445 nonrelativistic limit 311 – hermitean 121, 126 – DKH transformation 487 – Laplacian 46 – for atoms 216 – multiplicative 223 – general 185 – odd 445 normal ordering 244 – one-electron spin–orbit 506 normal product 244, 301 – parity 144, 153 normalization – radial momentum 198 – 4-spinors 206, 292 – velocity 179, 184 – basis functions 119 – Wood–Boring 522 – CSFs 293 orbit–orbit coupling 520 – Pauli spinors 206 orbital angular momentum 197 – radial functions 206, 342 orbital rotation 429 – Slater determinants 292 orthogonalization nuclear charge – 4-spinors 206, 292 – finite distribution 225 – basis functions 119 – point-like nuclei 225 – CSFs 293 nuclear quadrupole moment 601 – Pauli spinors 206 nuclear Schrödinger equation 282 – radial function 342 nuclear spin 195, 572 – radial functions 206 nucleus – Slater determinants 292 734 Index

orthonormality point groups 423 – 4-spinors 292 Poisson brackets 33, 49, 129, 131 – basis functions 119 Poisson equation 241, 507 – CSFs 293 – radial form 353 – Pauli spinors 206 Poisson integral 44 – radial functions 206, 342 position-space representation 132 – Slater determinants 292 positive-energy solutions 175, 219 orthonormality constraints 292 positron 188 orthonormalization potential 21 – 4-spinors 206 – external 195 – Liénard–Wiechert 111 pair creation 189 – retardation 103 pair interaction 50, 101 – scalar 42, 181 paramagnetic term 187 – vector 42, 181 parity Poynting theorem 38 – even 144 principal quantum number 214 – odd 144 probability density 120 – operator 144, 153 product – violation 603 – direct 121, 251, 283, 488, 650 particle density 120, 135 – Kronecker 121, 251, 283, 488, Pauli 650 – antisymmetry principle 155 – normal 244 – approximation 503 – tensor 121, 251, 283, 488, 650 – elimination 503 projection – equation 185, 186 – external-ﬁeld no-pair 538 – exclusion principle 301 – operators 277 – Hamiltonian 506 projection operators 409 – spin matrices 148, 168, 649 proper length 72 – spinor 150, 204 proper time 74 – theory 503 proton 189 Pauli Hamiltonian 458 pseudo potential 564 Pauli principle 286, 301 pseudo-large component 531 periodic boundary conditions 233 perturbation theory 570 QED 237 – Rayleigh–Schrödinger 570 – electromagnetic interaction 266 phosphorescence 561 – Hamiltonian 245 photon 266 – self-energy 278 – polarization of 266 – vacuum polarization 278 picture-change error 577 quadrature 665 – two-electron interaction 497 quantum electrodynamics 190, plane waves 175, 314 237 Poincaré group 59 – gauge ﬁxation 241 Index 735

– gauge invariance 238 relativistic energy 80 – Lorentz covariance 238 relativistic momentum 80 – second quantization 243 relativity principle quantum field theory 155 – of Einstein 55 quantum interference 122 – of Galilei 14 quantum mechanical state 118 resolution of the identity 119 – state function 118 response theory 570 – state vector 118 retarded potentials 45, 103 – uniqueness 126 Richardson extrapolation 386, 393 quantum mechanics rotation 139 – nonrelativistic 129, 197, 218 rotation group 19 – Schrödinger 197, 218 Russell–Saunders coupling 148 quantum number 129 – angular momentum 143 scalar potential 42, 181 – azimuthal 143 scalar product 633 – magnetic 143 – Euclidean 633 – principal 214 – four-dimensional 62 – radial 213 scalar-relativistic approximation quartet state 149 325, 491, 496, 547 quintet state 149 SCF convergence 312 Schrödinger Racah algebra 334, 343 – energy hydrogen-like atoms radial density 223, 364 198, 218 radial momentum 139 – equation 197, 211, 216 radial momentum operator 198 – Hamiltonian 197, 211, 216 radial nodes 220 – hydrogen atom 197, 211, 216– radial Poisson equation 353 218 radial quantum number 213 – picture 127 random phase approximation 573 – quantum mechanics 129, 197, rapidity 70 218 rational grid 382 – velocity operator 132, 137 Rayleigh–Schrödinger perturba- Schrödinger equation tion theory 570 – electronic 280 regular approximations 524 – nuclear 282 relativistic corrections second quantization 190, 296 – Darwin operator 506 – in quantum electrodynamics – mass–velocity operator 506 243 – spin–orbit operator 506 self-adjoint operator 121 relativistic effect 605 self-consistent field 309, 312 – kinematic 325, 496, 547, 626 – multi-configuration 289 – scalar 325, 491, 496, 547 self-energy 278 – spin–orbit 626 shell 214 736 Index

simple Coulombic correction 552 strong nuclear force 164 Simpson formula 389, 665 Sturm’s theorem 663 singlet state 149 superposition 122, 124 size consistency 289, 327 symmetry 423 Slater–Condon rules 295, 296 system of inertia 12 small component 219 SnO 622 TD-DFT 573 SO(3) 19 tensor 17 solid state 233 – dual field strength 94 speed of light 7 – field strength 92 spherical coordinates 140, 196 – Minkowski 57 spherical harmonics 143 tensor product 251, 283, 650 – addition theorem 144, 337 tensor-network states 283 – Cartesian 145 term symbols 148, 155 – completeness 144 time – orthogonality 144 – absolute 16 spherical spinor 150 – dilation 73 spin 147, 166, 168 – in nonrelativistic physics 15 – density 320, 322, 324 – reversal symmetry 423 – Hamiltonian 598 TlH 617 – nuclear 195, 572 torque 20 spin statistics theorem 120, 155 total angular momentum 149, 199 spin Zeeman interaction 322 total energy spin–orbit coupling 200, 221 – atoms 354 – approximate Hamiltonians 562 transformation – operator 458, 506 – canonical 34 – perturbation theory 562 – Galilean 15 spin–orbit splitting 220, 396 – Lorentz 55 spin–spin coupling 256, 520 – passive 15 spin-DFT 322 transformation techniques 440 spinor 169 transverse contribution 114, 241 – spherical 150 transverse gauge 243 split notation 176, 185, 412, 504 trigonometric functions 646 square-root operator 162, 233, triplet state 149 439, 461 two-electron interaction 251 standard representation 168, 486, Uehling potential 278 523 uncertainty relation 131 static electron correlation 290 uncoupled product basis 146 stationary state 123 uniqueness of Lorentz transforma- Stern–Gerlach experiment 147 tions 644 Stokes theorem 21, 635 unitarity condition 126 Index 737 unitary operator 126 vector potential 42, 181 unitary transformation 126, 127, vector product 634 309, 429, 440, 441 velocity 13 – atomic 554 – Dirac operator 179, 184 – Cayley parametrization 448 – operator 132, 137 – closed-form expressions 448 vibronic coupling 281 – exponential parametrization 447 – general parametrization 447 wave equation 41 – local 554 wave function 118 – McWeeny parametrization 448 – explicitly correlated 291 – optimum coefficients 452 wave packet 179 – sequence 459, 469 wave vector 267 – series expansion 449 Weyl representation 168, 486, 523 – square-root parametrization 448 Wichmann–Kroll correction 278 units 7, 336, 380 Wick’s theorem 301 – Gaussian units 7 Wigner 3j-symbol 146, 343, 351 – Hartree atomic units 7 – triangle condition 347 – natural units 7 Wood–Boring operator 522 upper components 231 X2C 462, 533, 537 vacuum polarization 278 XQR 539 variational collapse 274, 277, 302, 354 Zeeman effect 147 variational principle 302, 306 Zeeman interaction 322 vector 17 zero-field splitting 520 – Galilei 19 Zitterbewegung 181 – Lorentz 63 ZORA 524 vector coupling coefficient 146 – scaled variant 525