Springer Atomic, Optical, and Plasma Physics

Theory and Computation

By: Ian P Grant

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Hardcover | 28 November 2006

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Relativistic quantum electrodynamics, which describes the electromagneticinteractions of electrons and atomic nuclei, provides the basis for modeling the electronic structure of atoms, molecules and solids and of their interactions with photons and other projectiles. The theory underlying the widely used GRASP relativistic atomic structure program, the DARC electron-atom scattering code and the new BERTHA relativistic molecular structure program is presented in depth, together with computational aspects relevant to practical calculations. Along with an understanding of the physics and mathematics, the reader will gain some idea of how to use these programs to predict energy levels, ionization energies, electron affinities, transition probabilities, hyperfine effects and other properties of atoms and molecules.

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From the reviews: "Relativistic Quantum Theory of Atoms and Molecules is a book that should be on the desk of every atomic-physics student and research scientist. Written by a master of the field, it contains a wealth of information about relativistic atomic and molecular structures." (Walter Johnson, Physics Today, January, 2008)

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Non-Fiction Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Nuclear Physics Atomic & Molecular Physics Chemistry Physical Chemistry Relativity Physics

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Published: 23rd November 2010

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Relativity in atomic and molecular physics
Relativity in atomic and molecular physics	p. 3
Elementary ideas	p. 3
The one-electron atom	p. 7
Classical Kepler orbits	p. 7
The Bohr atom	p. 8
X-ray spectra and Moseley's Law	p. 9
Transition to quantum mechanics	p. 9
Sommerfeld's relativistic orbits and Dirac's wave equation	p. 10
Dirac and Schrodinger charge distributions	p. 13
The Dirac hydrogenic spectrum at high Z	p. 17
Many-electron atoms	p. 19
Central field models of the atom	p. 20
Closed and open shells	p. 21
Mean field potentials	p. 23
Comparison of Hartree-Fock and Dirac-Hartree-Fock models for ground states	p. 24
The mechanism of shell filling	p. 35
Other approaches	p. 38
Applications to atomic physics	p. 40
X-ray spectra	p. 41
Applications to astrophysics and plasma physics	p. 43
Modelling atomic processes in plasmas	p. 44
Relativistic molecular structure	p. 45
Relativistic interpretations of chemical anomalies	p. 46
Relativistic effective core potentials and other approximations	p. 48
Dirac four-component methods for molecules	p. 51
Parity violation and hyperfine interactions	p. 52
High-precision spectroscopy of small molecules containing light elements	p. 54
References	p. 54
Foundations
Relativistic wave equations for free particles	p. 63
The special theory of relativity	p. 63
The Lorentz group	p. 66
* Spinor representation of Lorentz transformations	p. 67
* Infinitesimal Lorentz transformations and their generators	p. 69
* Representations of the Lorentz group	p. 70
The Poincare group	p. 73
* Representations of the Poincare group	p. 75
* Space and time reflections	p. 77
The Klein-Gordon equation	p. 81
The Dirac equation	p. 86
[gamma]-Matrices and covariant form of Dirac's equation	p. 87
* Lagrangian formulation of Dirac's equation	p. 88
Foldy canonical form and the Foldy-Wouthuysen transformation	p. 90
* Position operators in Dirac theory	p. 92
Dirac particles in electromagnetic fields	p. 94
* Negative energy states	p. 95
Maxwell's equations	p. 96
Covariant form of Maxwell's equations	p. 96
* Lagrangian formulation	p. 99
Gauge invariance	p. 100
* Motion of a test charge	p. 102
* Symmetries and local conservation laws	p. 103
* Global conservation laws	p. 107
* Green's functions	p. 108
Nonrelativistic Green's functions	p. 109
Klein-Gordon operator	p. 112
Maxwell's equations: the zero-mass case	p. 116
Free-particle Dirac equation	p. 118
References	p. 119
The Dirac Equation	p. 121
Free particles	p. 121
Properties of Dirac matrices	p. 122
Covariance properties	p. 123
Bilinear covariants	p. 125
Plane wave solutions	p. 126
Energy and spin projectors	p. 128
Charge conjugation	p. 130
Spherical symmetry	p. 132
Angular structure	p. 133
The operator [sigma subscript r]	p. 134
The operator c[sigma] . p	p. 136
Separation of radial and spin-angular parts	p. 136
Angular density distributions	p. 137
Radial solutions for the free particle	p. 141
Partial wave normalization	p. 142
Hydrogenic atoms	p. 143
Solution of the radial equations	p. 146
The bound state solutions	p. 148
Charge distributions and energy levels in hydrogenic atoms	p. 149
* The continuum solutions	p. 150
Scattering by a centre of force	p. 152
Nonrelativistic potential scattering	p. 152
* Relativistic Coulomb scattering	p. 155
* Polarization effects in Coulomb scattering	p. 158
Historical note	p. 160
* Relativistic quantum defect theory	p. 161
Green's functions	p. 166
* Partial wave Green's functions	p. 167
The partial wave Green's function for the free Dirac particle	p. 169
Summation over partial waves in the free electron case	p. 170
* Green's function for hydrogenic ions	p. 171
The nonrelativistic limit: the Pauli approximation	p. 173
The Pauli approximation	p. 173
The Foldy-Wouthuysen and related transformations	p. 175
Other aspects of Dirac theory	p. 178
References	p. 178
Quantum electrodynamics	p. 181
Second quantization	p. 181
Quantization of the Schrodinger equation	p. 181
Identical particles: the symmetric case	p. 184
Identical particles: the antisymmetric case	p. 188
Quantization of the electron-positron field	p. 189
The Furry picture	p. 189
The free electron case	p. 192
Quantization of the Maxwell field	p. 196
Interaction of photons and electrons	p. 200
The equations of motion	p. 200
The Furry picture	p. 202
The interaction picture	p. 203
Wick's theorems	p. 206
Propagators	p. 208
Photon propagators	p. 208
Electron-positron propagators	p. 209
Feynman diagrams	p. 210
Second order interaction: U[superscript (2)] (t, t[subscript 0])	p. 212
Feynman rules	p. 216
The S-matrix	p. 217
Bound states	p. 218
A perturbation expansion	p. 218
Gell-Mann, Low, Sucher energy shift	p. 219
Effective interactions	p. 222
One-photon exchange: Feynman gauge	p. 222
One-photon exchange: Coulomb gauge	p. 225
* Off-shell potentials: heuristic argument	p. 227
One-photon exchange: the first order energy shift	p. 227
* Off-shell potentials	p. 228
Many-body perturbation theory	p. 232
Nonrelativistic many-body theory	p. 233
MBPT for atoms and molecules	p. 236
Particle-hole formalism	p. 236
Computational methods	p. 238
Relativistic approaches to atomic and molecular structure	p. 238
The no-virtual-pair approximation (NVPA)	p. 238
The NVPA as an antidote to "continuum dissolution"	p. 239
The NVPA and "variational collapse"	p. 242
Semirelativistic approaches	p. 243
A strategy for atomic and molecular calculations	p. 243
Density functional theories	p. 245
Basic ideas of RDFT	p. 246
The relativistic Hohenberg-Kohn theorem	p. 247
The relativistic Kohn-Sham equations	p. 248
Exchange and correlation functionals	p. 249
The optimized potential method	p. 252
References	p. 253
Computational atomic and molecular structure
Analysis and approximation of Dirac Hamiltonians	p. 259
Self-adjointness of free particle Hamiltonians	p. 260
Free particles: the Schrodinger case	p. 260
Free particles: the Dirac case	p. 261
Self-adjointness of Hamiltonians with a local potential	p. 262
The Schrodinger case	p. 263
The Dirac case	p. 264
The radial Dirac differential operator	p. 265
The boundary condition at a singular endpoint	p. 266
The Dirac radial operator with one singular endpoint	p. 267
The radial Dirac equation for atoms	p. 270
Power series solutions near r = 0	p. 272
Power series solutions in the nonrelativistic limit	p. 273
The boundary condition at the origin	p. 274
Variational methods in quantum mechanics	p. 274
Min-max theorems and the Ritz method	p. 276
Convergence of the Rayleigh-Ritz eigenvalues in nonrelativistic quantum mechanics	p. 279
Convergence of the Rayleigh-Ritz method in nonrelativistic quantum mechanics	p. 283
The Rayleigh-Ritz method in relativistic quantum mechanics	p. 285
The finite matrix method for the Dirac equation	p. 285
Convergence of Rayleigh-Ritz methods for Dirac Hamiltonians	p. 287
Spinor basis sets	p. 290
L-spinors	p. 293
Kinetic matching and the nonrelativistic limit	p. 295
Orthogonality properties	p. 295
Linear independence of L-spinors	p. 296
Completeness of L-spinors	p. 297
Charge conjugation and L-spinors	p. 297
Construction of [Pi superscript Beta Beta prime], S[superscript Beta Beta prime], and U[superscript Beta Beta prime] matrices for hydrogenic atoms	p. 298
Numerical study of L-spinor performance in hydrogenic atoms	p. 299
S-spinors	p. 303
Construction of [Pi superscript Beta Beta prime], S[superscript Beta Beta prime], and U[superscript Beta Beta prime] for hydrogenic atoms	p. 304
G-spinors	p. 305
Finite difference methods	p. 307
Methods of solution	p. 309
Acceptable solutions	p. 312
Finite element methods	p. 315
B-splines	p. 316
Variational formulation of finite element schemes	p. 316
Schrodinger equations	p. 318
Dirac equations	p. 319
References	p. 322
Complex atoms	p. 325
Dirac-Hartree-Fock theory	p. 325
One-electron matrix elements of tensor operators	p. 327
2-spinor matrix elements of even operators	p. 328
2-spinor matrix elements of odd operators	p. 330
Angular reduction of the Dirac Hamiltonian for a central potential	p. 331
Matrix elements of 2-body operators	p. 333
The Coulomb interaction	p. 333
Relativistic corrections to the Coulomb interaction	p. 334
The Gaunt interaction	p. 335
The Moller interaction	p. 336
The transverse photon interaction in Coulomb gauge	p. 337
The Breit interaction	p. 339
Interaction strengths for the magnetic interactions	p. 340
The transverse photon interaction	p. 340
The Breit interaction	p. 344
Closed shells and configuration averages	p. 346
The Dirac-Hartree-Fock model	p. 346
Inclusion of magnetic interactions	p. 349
Average of configuration models	p. 350
DHF integro-differential equations	p. 352
Construction of electrostatic potentials	p. 356
Construction of magnetic potentials	p. 356
Algorithms for potentials and Slater integrals	p. 359
Configurations with incomplete subshells	p. 361
Atomic states with incomplete subshells	p. 361
Partially filled subshells in jj-coupling	p. 363
Creation and annihilation operators as irreducible tensor operators. Quasispin	p. 364
Double tensor operators	p. 366
Parentage	p. 367
Coefficients of fractional parentage in the seniority scheme	p. 370
Equivalent electrons in LS-coupling	p. 374
Atoms with complex configurations	p. 376
Recoupling coefficients	p. 379
Matrix elements between open shell states	p. 379
Matrix elements of two-electron operators of type G	p. 382
CI and MCDHF problems with large CSF sets	p. 384
Decoupling active electrons	p. 385
One-electron matrix elements	p. 388
Two-electron matrix elements	p. 388
References	p. 391
Computation of atomic structures	p. 393
Atomic structure calculations with GRASP	p. 393
GRASP modules	p. 394
MCDHF integro-differential equations	p. 398
Solving the integro-differential equations	p. 401
Starting the calculation	p. 403
The radial grid	p. 403
The nuclear mass	p. 405
The nuclear size	p. 405
Initial estimates for radial wavefunctions	p. 406
An EAL calculation	p. 407
Diagonal and off-diagonal energy parameters	p. 408
Koopmans' theorem and Brillouin's theorem	p. 411
Froese Fischer's analysis	p. 412
Control of MCSCF iterations	p. 416
Corrections to the Coulomb interaction: Breit and other approximations	p. 418
QED corrections	p. 419
Towards higher quality atomic models	p. 423
CSF sets for electron correlation: active space methods	p. 424
Example: intercombination transitions in Be-like ions	p. 426
X-ray transition energies	p. 428
References	p. 431
Computation of atomic properties	p. 433
Relativistic radiative transition theory	p. 433
Line transitions	p. 433
Multipole expansion of the radiation field	p. 434
Emission and absorption by one-electron atoms	p. 437
Evaluation of one-electron transition amplitudes	p. 439
The nonrelativistic limit: Pauli approximation	p. 440
Radiative transitions in many-electron atoms	p. 443
Transitions in highly ionized atoms: Fe XXIII	p. 443
Orbital relaxation	p. 444
Application to atomic transition calculations	p. 448
Large-scale calculations of energies and transition rates	p. 451
Relativistic atomic photo-ionization theory	p. 452
The differential cross-section for photo-ionization	p. 453
Low energies: the electric dipole case	p. 455
Angular distributions and polarization parameters for a single channel	p. 457
Other aspects of photo-ionization	p. 458
Hyperfine interactions	p. 459
Hyperfine interactions in the many-electron atom	p. 461
Isotope shifts	p. 464
Nuclear motion	p. 464
Nuclear volume effect	p. 466
References	p. 467
Continuum processes in many-electron atoms	p. 471
Relativistic elastic electron-atom scattering	p. 471
Model potentials	p. 473
Computational issues	p. 473
Other approaches	p. 474
Determination of phase-shifts	p. 475
Summation of the partial wave expansion	p. 477
Electron-atom scattering: the close-coupling method	p. 477
Low-energy elastic and inelastic collisions	p. 477
The distorted wave approximation	p. 480
The relativistic R-matrix method	p. 480
The radial Dirac equation on a finite interval	p. 481
Bloch operators	p. 483
The inner region, r [Less than Equal] a	p. 485
The outer region, r [Greater than] a	p. 489
Matching inner and outer solutions	p. 489
The Buttle correction	p. 492
R-matrix theory of photo-ionization	p. 493
The DARC relativistic R-matrix package	p. 494
Truncation of the close-coupling expansion. The nonrelativistic CCC method	p. 496
The R-matrix method at intermediate energies	p. 500
Electron scattering from heavy atoms and ions	p. 504
Early work	p. 504
Electron scattering from the mercury atom	p. 505
Scattering of polarized electrons from polarized atoms	p. 509
The relativistic random phase approximation	p. 513
The RRPA equations	p. 513
Radial equations	p. 516
Multipole transition amplitudes	p. 518
RRPA rates for photo-excitation and photo-ionization	p. 520
Photo-excitation	p. 520
Photo-ionization	p. 520
Comparison with experiment	p. 523
Photo-ionization of outer atomic subshells at high Z	p. 523
Beyond RRPA	p. 526
References	p. 529
Molecular structure methods	p. 533
Molecular and atomic structure methods	p. 533
Dirac-Hartree-Fock-Breit equations for closed shell atoms	p. 535
DHFB energy of a closed shell atom	p. 535
Spinor basis function representation	p. 535
Matrix of the radial Dirac operator	p. 536
Coulomb Slater integrals	p. 536
Breit integrals for closed shells	p. 537
The DHFB Fock matrix	p. 538
One-centre interaction integrals	p. 539
Numerical examples	p. 541
The DHFB method for closed shell molecules	p. 543
G-spinor basis functions	p. 544
The charge-current density	p. 545
Two-centre overlaps	p. 546
Relativistic expansion coefficients	p. 547
Symmetry properties of E[subscript q] coefficients	p. 548
Multi-centre interaction integrals	p. 549
Auxiliary integrals involving HGTFs	p. 550
Multi-centre one-electron integrals	p. 551
Multi-centre two-electron integrals	p. 556
Fock matrix in terms of G-spinors	p. 558
The BERTHA integral package	p. 560
Electromagnetic field energy	p. 562
Interaction energy in terms of internal fields	p. 562
The nonrelativistic Fock matrix	p. 565
The relativistic Fock matrix	p. 565
Implementation of the field formulation	p. 566
Relativistic density functional calculations	p. 568
Computational strategies	p. 574
The Roothaan bound	p. 575
Integral-direct Fock matrix evaluation	p. 575
Symmetry properties of interaction matrix elements	p. 576
Stepwise refinement	p. 576
Level-shifting	p. 577
Multiconfigurational Dirac-Hartree-Fock theory	p. 578
Orbital optimization	p. 578
References	p. 584
Relativistic calculation of molecular properties	p. 587
Molecular symmetry	p. 587
Diatomic molecules	p. 589
Polyatomic molecules	p. 591
Relativistic effects in light molecules	p. 594
Nonrelativistic Breit-Pauli model	p. 594
DHF and DHFB calculations for water using BERTHA	p. 596
Second-order many-body corrections	p. 598
Relativistic study of the potential energy surface and vibration-rotation levels of water	p. 599
Electromagnetic properties of atoms and molecules	p. 601
Gauge transformations in electromagnetic processes	p. 601
B-spinors	p. 602
The Zeeman effect	p. 603
Hyperfine interactions	p. 606
NMR shielding in small molecules	p. 609
NMR shielding constants for [superscript 17]O in water	p. 611
NMR shielding constants for [superscript 15]N in ammonia	p. 612
Molecules with high-Z constituents	p. 613
Electronic structure of TlF	p. 614
Electronic structure of YbF	p. 617
DHF+CI study of uranium hexafluoride	p. 620
References	p. 624
Frequently used formulae and data	p. 627
Relativistic notation	p. 627
Dirac matrices	p. 628
Special functions	p. 631
Spherical Bessel functions	p. 631
Confluent hypergeometric functions	p. 632
Generalized Laguerre polynomials	p. 633
Hermite polynomials	p. 634
Incomplete gamma functions	p. 635
Incomplete Beta functions	p. 635
Continued fraction evaluation	p. 636
Central field Dirac spinors and their interactions	p. 636
Central field Dirac spinors	p. 636
Matrix elements of simple ITOs	p. 639
Magnetic interactions	p. 642
Effective interaction strengths for two-body operators	p. 644
Open shells in jj-coupling	p. 651
Exponents for atomic and molecular G-spinors	p. 654
Software for relativistic molecular calculations	p. 662
BERTHA	p. 662
DIRAC	p. 662
MOLFDIR	p. 663
References	p. 664
Supplementary mathematics	p. 665
Linear operators on Hilbert space	p. 665
Hilbert spaces	p. 665
Linear operators	p. 665
Spectrum and resolvent of linear operators	p. 666
Self-adjoint operators	p. 666
Observables and self-adjoint operators	p. 669
Commuting operators	p. 671
Unitary and anti-unitary operators	p. 671
Lie groups and Lie algebras	p. 673
Lie groups	p. 673
Lie algebras	p. 674
Representations of Lie groups and Lie algebras	p. 676
The Cartan-Weyl classification	p. 677
Casimir operators	p. 678
Kronecker products of group representations	p. 679
Tensor operators and the Wigner-Eckart theorem	p. 681
Quantum mechanical angular momentum theory	p. 683
The rotation group	p. 683
Abstract angular momentum	p. 684
Orbital angular momentum	p. 685
Representation functions	p. 687
Kronecker products of irreducible representations	p. 691
Coupling of three or more angular momenta	p. 693
The 3j-symbol	p. 694
The 6j-symbol	p. 694
The 9j-symbols	p. 696
Graphical treatment of angular momentum algebra	p. 697
Diagrammatic treatment of Clebsch-Gordan coefficients	p. 699
Diagrammatic treatment of 3jm-symbols	p. 701
Generalized angular momentum couphng schemes	p. 703
GCG and njm coefficients	p. 706
Manipulating angular momentum diagrams	p. 708
Tensor operators and the Wigner-Eckart theorem	p. 709
Composite tensor operators	p. 711
Diagrammatic representation of tensor operators	p. 713
Relativistic symmetry orbitals for double point groups	p. 717
Construction of symmetry orbitals	p. 717
Linear independence of molecular symmetry orbitals	p. 719
Reduction of operator matrices	p. 719
Time reversal	p. 720
The TSYM software package	p. 722
Basis sets in atomic and molecular physics	p. 722
The Coulomb Sturmian functions	p. 727
Completeness and linear independence of Coulomb Sturmians	p. 729
Basis sets of exponential-type functions	p. 730
Finite difference methods for Dirac equations	p. 733
An existence theorem	p. 733
Initial value methods	p. 734
Linear multistep methods	p. 736
The nodal structure of Dirac radial wavefunctions	p. 738
Discretization of two-point boundary value problems	p. 741
Two-point boundary value problems: the deferred correction method	p. 744
Construction of difference corrections	p. 746
Single stepping algorithms	p. 748
Stepping outwards from the origin	p. 749
Algorithm for the outer region	p. 751
The boundary condition at T = t[subscript N]	p. 753
The boundary condition at the origin	p. 754
Improving a trial solution	p. 755
Eigenfunction expansions for the radially reduced Dirac equation	p. 757
The fundamental lemma	p. 757
Boundary conditions: the two-point boundary value problem	p. 758
Boundary conditions at the nucleus	p. 759
Pauli approximation at R[subscript 2]	p. 759
The MIT bag model at R[subscript 2]	p. 759
The eigenvalue spectrum	p. 760
The inhomogeneous boundary value problem	p. 761
Eigenfunction expansions	p. 766
Iterative processes in nonlinear systems of equations	p. 769
Lagrangian and Hamiltonian methods	p. 772
Lagrange's equations	p. 772
Hamilton's equations	p. 773
Symmetries and conservation laws	p. 775
Construction of E coefficients	p. 777
E-coefficients through Cartesian intermediates	p. 777
Recurrence relations for E-coefficients	p. 779
Implementation issues	p. 780
References	p. 783
Index	p. 787
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Springer Atomic, Optical, and Plasma Physics

Theory and Computation

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