Self-consistent field relativistic calculations

Ab initio calculations can be performed at the Hartree-Fock level of approximation, equivalent to a self-consistent-field (SCF) calculation, or at a post Hartree-Fock level which includes the effects of correlation — defined to be everything that the Hartree-Fock level of approximation leaves out of a non-relativistic solution to the Schrodinger equation (within the clamped-nuclei Born-Oppenhe-imer approximation). 

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 Pauli Hamiltonian is ideally suited for carrying out relativistic corrections as a first-order perturbation to a non-relativistic Hamiltonian. In recent years, several authors have considered inclusion of the Pauli terms in variational self-consistent field (SCF) calculations. Wadt, Hay and... 

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... 

Although the Pauli Hamiltonian was derived for use in perturbational computations as corrections to nonrelativistic Hamiltonian, these terms have been used in variational self-consistent field (SCF) calculations by Hay and Wadt, These authors have incorporated the relativistic terms into the ECPs, except that the spin-orbit effects are not included in their ECPs, and, thus, in the molecular calculations based on this scheme. 

Model potential methods and their utilization in atomic structure calculations are reviewed in [139], main attention being paid to analytic effective model potentials in the Coulomb and non-Coulomb approximations, to effective model potentials based on the Thomas-Fermi statistical model of the atom, as well as employing a self-consistent field core potential. Relativistic effects in model potential calculations are discussed there, too. Paper [140] has examples of numerous model potential calculations of various atomic spectroscopic properties. 

The two first columns give the non-relativistic and relativistic Xa results (without the resolved spin-orbit coupling shown in the three last columns) by Boring and Wood701 with the major origin (uranium 5 f or fluorine n and a orbitals) of the one-electron functions. These results are calculated in a self-consistent field as one-component (rather than four-... 

The total electron density at the nucleus p(0) depends on the nature of the chemical bonding. Positive contributions to p(0) arise from the atomic 6s populations of the molecular orbitals on the gold ion, while a decrease in p(0) is caused by the atomic bd populations due to their shielding effects on s electrons. The relative magnitudes of the various contributions are known from the results of relativistic free-ion self-consistent field calculations for gold and for other d transition elements. The atomic 6p populations, on the other hand, yield only small contributions both by shielding of s electrons and by their relativistic density at the nucleus. Hence p(0) and consequently the isomer shift will mainly depend on the atomic 6s and 5[Pg.281]

Numerical self-consistent fields have been established (see, for example [9,30,31]) and we shall record below a few of the consequences of the calculations, in particular for the chemical potential p, when we have considered the relativistic generalization of the TF theory (Sect. 6.4 below). 

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). 

Reiher, M. (1998) Development and implementation of numerical algorithms for the solution of multi-configuration self-consistent field equations for relativistic atomic structure calculations. PhD thesis, FakultSt fUr Chemie, University Bielefeld, Germany. 

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. 

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