Relativistic Self-Consistent Fields

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

The self-consistent field approach in relativistic quantum chemistry provides one of the most convenient and useful computational tools for the study of the electronic structure and properties of atoms, molecules and solids just as it does in nonrelativistic quantum chemistry. This chapter describes only methods in which the motion of electrons is described by the Dirac operator, namely 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

There are other reasons why methods based on Dirac Hamiltonians have been unpopular with quantum chemists. Dirac theory is relatively unfamiliar, and the field is not well served with textbooks that treat the topic with the needs of quantum chemists in mind. Matrix self-consistent-field equations are usually derived from variational arguments, and as a result of the debates on variational collapse and continuum dissolution , many people believe that such derivations are invalid for relativistic problems. Most implementations of the Dirac formalism have made no attempt to exploit the rich internal structure of Dirac 

4-component spinors resulting in power hungry code requiring huge amounts of memory. Inevitably, this relegates Dirac methods, if one trusts the results, to the unimportant role of providing expensive benchmarks for more approximate computational schemes. 

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This paper outlines the current status of the BERTHA program. This has been named (with her approval) after Lady Jefferies (f902-f999) nee Bertha Swirles), a colleague of D.R. Hartree at Cambridge University and a distinguished applied mathematician who, amongst other achievements, wrote the earliest paper on relativistic self-consistent fields for atoms [f]. 

In the conclusion of this section let us notice that a wealth of data on the applications of the relativistic self-consistent field method to the studies of the hyperfine structure of atomic levels is collected in [149]. Investigations of the hyperfine structure by the methods of perturbation theory are described in monograph [17]. 

The K(3 IKOi x-ray intensity ratio is an easily measurable quantity with relatively high precision and has been studied extensively for /f-x-ray emission by radioactive decay, photoionization, and charged-particle bombardment (1-3). Except for the case of heavy-ion impact where multiple ionization processes are dominant, it is generally accepted that this ratio is a characteristic quantity for each element. The experimental results are usually compared with the theoretical values for a single isolated atom and good agreement is obtained with the relativistic self-consistent-field calculations by Scofield (4). 

Torbohm, Fricke and Rosen [55] used relativistic self-consistent field (RSCF) calculations to express the wavefunctions within the nucleus as a power series expansion, which was used to express the field isotope shift in terms of the moments (r ) of the charge distribution. The effect of the higher moments is commonly accounted for by introducing a parameter Ac = c(Cc), where the correction 1 — k is found to be about 3% for Z 55, 5% for Z 70, and 6% for Z w 80. For hyperfine structure, the higher moments are found to be considerably more important, as discussed below. 

The perturbation theory of relativistic QED, see for example [47,65], is the source of widely used methods of nonrelativistic many-body perturbation theory (MBPT) [66]. We demonstrate how it can also be used to formulate the theory of relativistic self-consistent fields as the first step in a more elaborate theory of MBPT incorporating radiative corrections. 

Highly-ionized atoms DHF calculations on isoelectronic sequences of few-electron ions serve as the starting point of fundamental studies of physical phenomena, though many-body corrections are now applied routinely using relativistic many-body theory. Relativistic self-consistent field studies are used as the basis of investigations of systematic trends in ionization energies [137-144], radiative transition probabilities [145-148], and quantum electrodynamic corrections [149-151] in few-electron systems. Increased experimental precision in these areas has driven the development of many-body methods to model the electron correlation effects, and the inclusion of Breit interaction in the evaluation of both one-body and many-body corrections. 

One feature of relativistic self-consistent field calculations to which attention should be drawn is the fact that the Breit interaction can be easily included in the self-consistent field iterations once the algebraic approximation has been invoked. This should be contrasted with the situation in atomic calculations using numerical methods in which the Breit interaction is treated by first-order perturbation theory. 

Internal conversion coefficients (ICC) were obtained from relativistic self-consistent-field Dirac-Fock calculations by Band et al. (2002). They presented results for E1,...E5, M1,...M5 transitions in the energy range Ey= 1 — 2,000 keV for K, Li, L2, L3 atomic shells of elements Z = 10 — 126. The total ICCs and graphs for ICCs were also published. The Dirac-Fock values are in better agreement with experimental results than the relativistic Hartree-Fock-Slater theoretical ones. 

Recently new ICCs have been obtained from relativistic self-consistent-field Dirac-Fock (DF) calculations for each Zbetween 10 and 126, for K, Li, L2, and L3 atomic shells for nuclear-transition multipolarities E1-E5 and M1-M5, and for nuclear-transition energies from 1 keV above the Lj threshold to 2,000 keV (Band et al. 2002). The total ICC values were calculated from the sum of partial ICC values from all atomic shells. The calculated K and total values are, on average, about 3% lower than the theoretical relativistic Hartree-Fock-Slater values, and agree better with the most accurate experimental ICC values. A selection of total ICCs is plotted inO Figs. 11.2—11.7, for atomic numbers Z = 10, 30, 50, 70, 90, and 110. The full set of tables and graphs can be found in the original publication. 

Malli, G.L., 1983b, Relativistic self-consistent field theory for molecules, in Relativistic Effects in Atoms, Molecules and Solids, NATO ASI Series, Series B Physics, Vol. 87, ed. G.L. Malli (Plenum, New York) p. 183. 

Ca,< ) iDy Dy and Tr, numerous aims of these investigations were hyperfine structure interaction constants, nuclear moments, effects of configuration interaction and core polarization in the electron shell of atoms, molecular data like effective spin rotation constants, etc. The combination of experimental data and advanced theoretical procedures, like relativistic self-consistent field calculations, provided many fruitful results regarding the reliability of electronic wave functions of atoms. 

B. Swirles. The Relativistic Self-Consistent Field. Proc. Roy. Soc. London A, 152 (1935) 625-649. 

P. Grant. Relativistic self-consistent fields. Proc. Phys. See., 86 (1965) 523-527. 

F. C. Smith, W. R. Johnson. Relativistic Self-Consistent Fields with Exchange. 

G. Malli, J. Qreg. Ab initio relativistic self-consistent-field (RSCF) wavefunctions for the diatomics Li2 and Be2. Chem. Phys. Lett., 69(2) (1980) 313-314. 

J. C. Barthelat, M. Pelissier, P. Durand. Analytical relativistic self-consistent-field calculations for atoms. Phys. Rev. A, 21(6) (1980) 1773-1785. 

S. Fraga, Theoret. Chim. Acta, 2,406 (1964). Non-Relativistic Self-Consistent Field Theory. IL... 

O. Matsuoka, ]. Chem. Phys., 97, 2271 (1992). Relativistic Self-Consistent-Field Methods for Molecules. III. All-Electron Calculations on Diatomics Hydrogen Iodide, Hydro-iodine(l ). Hydrogen Astatide and Hydroastatine(l ) (HI, HI+, AtH, and AtH+). 

I. Lindgren, A. Rosen Relativistic self-consistent field calculations with application to hyperfine interaction. Pt.I Relativistic self-consistent fields, Pt.II Relativistic theory of atomic hyperfine interaction. Case Stud. Atom. Phys. 4, 93 (1974), Pt.IlI Comparison between theoretical and experimental hyperfine structure results. Case Stud. Atom. Phys. 4, 197 (1974)... 

The common deviation from standard f electron structure in the light An is other than IV. As substantiated, for example, by relativistic self-consistent field type calculations (Freeman and Koelling 1974, Freeman 1980, Arko et al. 1985) the radial extend of 5f electrons is large and comparable to that of 3d electrons in transition... 

Malli, G.L. Thirty years of relativistic self-consistent field theory for molecules Relativistic and electron correlation effects for atomic and molecular systems of transactinide superheavy elements up to ekaplutonium E126 with g-atomic spinors in the ground state configuration. Theor. Chem. Ace. 118, 473 82 (2007)... 

Relativistic self-consistent field calculation for mercury, D. F. Mayers, Proc. R. Soc. London. Ser. A, Math. Phys. Set (1934-1990), 1957, 241, 93. 

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