Relativistic Quantum Theory of Atoms and Molecules (eBook)

Professor Grant first became aware of the need to develop a relativistic theory of atomic and molecular structure some 50 years ago in connection with X-ray absorption by heavy metals. In a 1961 paper, he showed that the Dirac-Hartree-Fock equations for atoms could be written in a simple form which has been used in all subsequent atomic calculations. This early work was generalized to permit more accurate multi-configurational calculations in the next two decades, implemented in the widely used GRASP code for relativistic modeling of electronic wavefunctions, energy levels and radiative transition probabilities of spectral lines. The DARC code, an extension of GRASP which is designed to calculate cross sections for atom/ion collisions with low-energy electrons or photons, was developed mainly in the 1980s and is now becoming more relevant for applications involving target atoms of higher atomic number. The BERTHA code is the first relativistic molecular structure code designed to take advantage of the internal structure of Dirac four-component spinors; its speed and accuracy are now beginning to be appreciated and utilized effectively by quantum chemists. The book is designed for all those who would like to know more about the mathematics and physics of relativistic atomic and molecular theory and who wish to use the computational machinery now available to solve problems in atomic and molecular physics and their applications. Professor Grant was elected a Fellow of the Royal Society of London in 1992.

Preface 7
Contents 10
Part I Relativity in atomic and molecular physics 23
1 Relativity in atomic and molecular physics 24
1.1 Elementary ideas 24
1.2 The one-electron atom 28
1.3 Many-electron atoms 40
1.4 Applications to atomic physics 61
1.5 Relativistic molecular structure 66
References 75
Part II Foundations 81
2 Relativistic wave equations for free particles 82
2.1 The special theory of relativity 82
2.2 The Lorentz group 85
2.3 The Poincar ´ e group 92
2.4 The Klein-Gordon equation 100
2.5 The Dirac equation 105
2.6 Maxwell’s equations 115
2.7 Symmetries and local conservation laws 122
2.8 Global conservation laws 126
2.9 Green’s functions 127
References 138
3 The Dirac Equation 140
3.1 Free particles 140
3.2 Spherical symmetry 151
3.3 Hydrogenic atoms 162
3.4 Scattering by a centre of force 171
3.5 Relativistic quantum defect theory 180
3.6 Green’s functions 185
3.7 The nonrelativistic limit: the Pauli approximation 192
3.8 Other aspects of Dirac theory 197
References 197
4 Quantum electrodynamics 199
4.1 Second quantization 199
4.2 Quantization of the electron-positron .eld 207
4.3 Quantization of the Maxwell .eld 214
4.4 Interaction of photons and electrons 218
4.5 Wick’s theorems 224
4.6 Propagators 226
4.7 The S-matrix 235
4.8 Bound states 236
4.9 E.ective interactions 240
4.10 Off-shell potentials 246
4.11 Many-body perturbation theory 250
4.12 MBPT for atoms and molecules 254
4.13 Relativistic approaches to atomic and molecular structure 256
4.14 A strategy for atomic and molecular calculations 261
4.15 Density functional theories 263
References 271
Part III Computational atomic and molecular structure 275
5 Analysis and approximation of Dirac Hamiltonians 276
5.1 Self-adjointness of free particle Hamiltonians 277
5.2 Self-adjointness of Hamiltonians with a local potential 279
5.3 The radial Dirac di.erential operator 282
5.4 The radial Dirac equation for atoms 287
5.5 Variational methods in quantum mechanics 291
5.6 The Rayleigh-Ritz method in relativistic quantum mechanics 302
5.7 Spinor basis sets 307
5.8 L-spinors 310
5.9 S-spinors 320
5.10 G-spinors 322
5.11 Finite di.erence methods 324
5.12 Finite element methods 332
References 339
6 Complex atoms 342
6.1 Dirac-Hartree-Fock theory 342
6.2 One-electron matrix elements of tensor operators 344
6.3 Angular reduction of the Dirac Hamiltonian for a central potential 348
6.4 Matrix elements of 2-body operators 350
6.5 Interaction strengths for the magnetic interactions 357
6.6 Closed shells and con.guration averages 363
6.7 DHF integro-di.erential equations 369
6.8 Con.gurations with incomplete subshells 378
6.9 Atoms with complex con.gurations 393
6.10 CI and MCDHF problems with large CSF sets 401
References 408
7 Computation of atomic structures 410
7.1 Atomic structure calculations with GRASP 410
7.2 GRASP modules 411
7.3 MCDHF integro-di.erential equations 415
7.4 Solving the integro-di.erential equations 418
7.5 Starting the calculation 420
7.6 An EAL calculation 424
7.7 Diagonal and o.-diagonal energy parameters 425
7.8 Koopmans’ theorem and Brillouin’s theorem 428
7.9 Control of MCSCF iterations 433
7.10 Corrections to the Coulomb interaction: Breit and other approximations 435
7.11 QED corrections 436
7.12 Towards higher quality atomic models 440
7.13 X-ray transition energies 445
References 448
8 Computation of atomic properties 450
8.1 Relativistic radiative transition theory 450
8.2 Emission and absorption by one-electron atoms 454
8.3 Radiative transitions in many-electron atoms 460
8.4 Orbital relaxation 461
8.5 Application to atomic transition calculations 465
8.6 Relativistic atomic photo-ionization theory 469
8.7 Hyper.ne interactions 476
8.8 Isotope shifts 481
References 484
9 Continuum processes in many-electron atoms 487
9.1 Relativistic elastic electron-atom scattering 487
9.2 Electron-atom scattering: the close-coupling method 493
9.3 The relativistic R-matrix method 496
9.4 The Buttle correction 508
9.5 R-matrix theory of photo-ionization 509
9.6 The DARC relativistic R-matrix package 510
9.7 Truncation of the close-coupling expansion. The nonrelativistic CCC method 512
9.8 The R-matrix method at intermediate energies 516
9.9 Electron scattering from heavy atoms and ions 520
9.10 The relativistic random phase approximation 529
9.11 RRPA rates for photo-excitation and photo- ionization 536
9.12 Comparison with experiment 539
References 545
10 Molecular structure methods 549
10.1 Molecular and atomic structure methods 549
10.2 Dirac-Hartree-Fock-Breit equations for closed shell atoms 551
10.3 One-centre interaction integrals 555
10.4 Numerical examples 557
10.5 The DHFB method for closed shell molecules 559
10.6 G-spinor basis functions 560
10.7 The charge-current density 561
10.8 Two-centre overlaps 562
10.9 Multi-centre interaction integrals 565
10.10 Fock matrix in terms of G-spinors 574
10.11 Electromagnetic .eld energy 578
10.12 Relativistic density functional calculations 584
10.13 Computational strategies 590
10.14 Multicon.gurational Dirac-Hartree-Fock theory 594
References 600
11 Relativistic calculation of molecular properties 602
11.1 Molecular symmetry 602
11.2 Relativistic e.ects in light molecules 609
11.3 Electromagnetic properties of atoms and molecules 616
11.4 The Zeeman e.ect 618
11.5 Hyper.ne interactions 621
11.6 NMR shielding in small molecules 624
11.7 Molecules with high-Z constituents 628
References 639
A Frequently used formulae and data 642
A.1 Relativistic notation 642
A.2 Dirac matrices 643
A.3 Special functions 646
A.4 Central .eld Dirac spinors and their interactions 651
A.5 Open shells in jj-coupling 666
A.6 Exponents for atomic and molecular G-spinors 669
A.7 Software for relativistic molecular calculations 677
References 679
B Supplementary mathematics 680
B.1 Linear operators on Hilbert space 680
B.2 Lie groups and Lie algebras 688
B.3 Quantum mechanical angular momentum theory 698
B.4 Relativistic symmetry orbitals for double point groups 732
B.5 Basis sets in atomic and molecular physics 737
B.6 Finite di.erence methods for Dirac equations 748
B.7 Eigenfunction expansions for the radially reduced Dirac equation 772
B.8 Iterative processes in nonlinear systems of equations 784
B.9 Lagrangian and Hamiltonian methods 787
B.10 Construction of E coefficients 792
References 798
Index 801