Atomic-Structure Calculations

The electronic structure of atoms has been studied for many decades on the basis of four-component methods (Grant 1994 Kim 1993a,b Reiher and Hess 2000 Sapir-stein 1993, 1998). Nevertheless, significant improvements have been achieved in recent years. Even one-electron atoms still give us new insight into four-component electronic structure theory (Andrae 1997 Autschbach and Schwarz 2000 Chen and Goldman 1993 Chen etal. 1994 Pyykko and Seth 1997). In this section we review methodological improvements as well as new implementations and typical applications. 

In general, the many-electron wave function is expressed in terms of antisymmetrized products of one-electron functions and the clamped-nucleus approximation as well as the central-field and equivalence restriction for the orbitals is used. Thus the one-electron spinor takes the form 

Total energies Ej (in a.u.) and orbital energies (in a.u.) from both a numerical nonrela-tivistic Hartree-Fock (HF) calculation for the ground configuration [1] and an analytical nonre-lativistic HF calculation using Slater-type orbitals (STO) for the ground state [2], are  

The problem of the exchange term in Hartree-Fock equations has been treated in different ways. The HF-Slater (HFS) method was used in [15]. Numerical SCF calculations of ground-state total energies in relativistic and nonrelativistic approximations are compared in [16, 17]. HFS wavefunctions served as zeroth-order eigenfunctions to compute the relativistic Hamiltonian. In [18], seven contributions to the total energy (including magnetic interaction, retardation, and vacuum polarization terms) are detailed. 

In the Xa theory, the exchange potential depends on an adjustable a parameter which is evaluated throughout the Periodic Table [19]. In an extension of this method, the a para- 

Analytical representations of the Thomas-Fermi (TF) or TF-Dirac (TFD) potentials were used in [25]. The TFD-Weizsacker (TFDW) variational equations are solved numerically to obtain total energies of the ground term in [26]. These energies are also calculated in an explicit parameter-free approximation [27]. 

The pseudopotential approach was used in [28] and was also applied in the density-functional formalism [29]. For chemical problems, ab initio effective core potentials including some relativistic effects may be of practical use [30]. 

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Herman, F. and Skillman, S. (1963) Atomic Structure Calculations, Prentice Hall, Englewood Cliffs, NJ. 

Parpia, F.A., Froese-Fischer, C. and Grant, l.P. (1996) GRASP92 A package for large-scale relativistic atomic structure calculations. Computer Physics Communications, 94, 249—271. 

Fig. 7.78 Linear relation of the quadmpole splitting A q = ( jl)eqQ (1 + j /3)l/2 and the isomer shift b for aurous (a) and auric (b) compounds. Also included is a correlation with the relative change in electron density at the gold nucleus, Ali/r(o)P, as derived from Dirac-Fock atomic structure calculations for several electron configurations of gold. An approximate scale of the EFG (in the principal axes system) is given on the right-hand ordinate (from [341])...

J. B. Mann, Atomic Structure Calculations, Los Alamos Scientific Laboratory, Univ. California, Los Alamos, NM, Part I Hartree—Fock Energy Results for the Elements Hydrogen to Lawrencium, 1967 Part II Hartree-Fock Wave Functions and Radial Expectation Values, 1968. 

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. 

J. Migdalek. Model-Potential Methods in Atomic Structure Calculations, Zeszyty Naukove Universytetu Jagiellonskiego, MXV, Prace Fizyczne, Zeszyt 32, Krakow, 1990. 

I. Martinson (ed.). Frontiers in Atomic Structure Calculations and Spectroscopy (Proceedings of a workshop held in Lund, Sweden, May 24-25, 1988), Physica Scripta, RS15 (1989). 

G. Gaigalas, Z. Rudzikas, Ch. Froese Fischer, An efficient approach for spin-angular integrations in atomic structure calculations, J. Phys. B At. Mol. Opt. Phys., 30, 3747-3771 (1997). 

On the other hand, ab initio (meaning from the beginning in Latin) methods use a correct Hamiltonian operator, which includes kinetic energy of the electrons, attractions between electrons and nuclei, and repulsions between electrons and those between nuclei, to calculate all integrals without making use of any experimental data other than the values of the fundamental constants. An example of these methods is the self-consistent field (SCF) method first introduced by D. R. Hartree and V. Fock in the 1920s. This method was briefly described in Chapter 2, in connection with the atomic structure calculations. Before proceeding further, it should be mentioned that ab initio does not mean exact or totally correct. This is because, as we have seen in the SCF treatment, approximations are still made in ab initio methods. 

Fig. 17.9. Comparisons of the results of three recent advanced atomic structure calculations [15-17], The ratios of the oscillator strengths (/-values) of [15] and [16], as well as [17] and [16] are plotted on a logarithmic scale against the MCHF [16] oscillator strengths...

This spectrum presents an instructive case for the large uncertainties in atomic transition probabilities, obtained even with sophisticated multiconfiguration calculations. For this ion, three extensive and detailed atomic structure calculations have been undertaken in the last ten years [15-17]. 

F. Herman and S. Skillman Atomic Structure Calculations (Englewood Cliffs, Prentice-Hall, N.J., 1963). 

Herman, F. Skillman, S. "Atomic Structure Calculations" Englewood Cliffs, N.J., Prentice-Hall, 1963. 

Herman, F., Skillman, S. Atomic structure calculations. Englewood Cliffs, N. J. Prentice-Hall 1963,... 

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