Atomic DHF calculations

The best-known and widely-quoted tabulation of atomic Dirac-Hartree-Fock energies was published by Desclaux [11], covered elements in the range Z=1 to Z=120 using finite difference methods. A number of computer packages are available to perform MCDHF calculations [19]. Published DHF and Dirac-Fock-Slater (DFS) calculations for atoms are now too numerous to construct a comprehensive catalogue. It is, however, possible to sort the purposes for which these calculations have been performed into general classes. 

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. 

Heavy neutral species In order to model the electron density of heavy elements for subsequent use in the construction of pseudopotential representations of the inner-shell regions near heavy nuclei, atomic DHF calculations are often employed as a starting point [158]. These pseudopotential approximations are used in order to reduce the computational cost of molecular or solid-state calculations of extended systems containing heavy elements. Similarly, atomic DHF calculations are used in the design of atom-centred basis sets, either by the direct 

Scattering MCDHF wavefunctions have recently been used as target states in electron-atom and electron-ion scattering calculations, based mainly on the relativistic R-matrix approach [187-190]. 

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Explicit inclusion of relativistic effects in valence-only calculations has been by far less frequently attempted. Datta, Ewig and van Wazer [135] used a Phillips-Kleinman PP in a study of PbO, whereas Ishikawa and Malli [136] tested PPs of semilocal form in four-component atomic DHF finite difference calculations. This work was extended by Dolg [137] to four-component molecular DHF calculations with a subsequent correlation treatment. In addition a complicated form of Vcv based on the Foldy-Wouthuysen transformation [138] was derived by Pyper [139] and applied in atomic calculations [140]. For all these approaches the computational effort is significantly higher than for the implicit treatment of relativity, and the gain of computational accuracy is not obvious at all. 

From the standpoint of chemical reactivity, the 5s and 5p shells of lanthanides can be considered to be core electrons. Indeed arguments to this effect could be made for Ln 4f orbitals, given their extremely contracted nature. Shown in Figure 3 is a plot from a DHF calculation of a 4f spinor for Gd(III). Note the maximum in the wavefunction at 0.57 A that is, a value comparable to a hydrogen Is orbital Dolg et al. - examined various lanthanide core sizes and found essentially no difference in state splittings of the Ce atom between all-electron calculations and those in which a 28-electron core ([Ar]3d °) is used, Satisfactory results are also obtained for a 46-electron core ([Kr]4di°). Inclusion of 5s and 5p into the core (i.e, a 54-electron [Xe] core)... 

It has been well known for a long time that relativity becomes increasingly important as one descends to heavier elements in the periodic table. What has been less well known is the magnitude of relativistic effects on chemical properties. Pyykkb states that pseudopotentials have been more widely used than any other computational method to probe relativistic effects. This is not surprising because ECPs are designed to facilitate calculations on heavier elements (i.e., those for which relativistic effects are most apparent). Additionally, the way in which ECPs are derived can be used to shed further light on relativistic effects in chemical bonding. One can choose to have the ECP model the core of an atom or atomic ion as determined by a relativistic or nonrelativistic calculation.A standard HF calculation can be used instead of a relativistic DHF calculation as the basis for ECP derivation hence differences in calculated properties can be ascribed to relativity. 

When it comes to contracted basis sets, kinetic balance strictly applied to the contracted large component can lead to problems. While it would be possible to apply the kinetic balance relation to derive a small-component basis from a set of large-component contracted basis functions, this procedure has been shown to be unsuitable in practice (Visscher et al. 1991). The best approach for generating contracted basis sets for relativistic four-component calculations has been to start with an uncontracted large-component basis, and to construct a small-component basis from this basis using kinetic balance. This set is then used in an uncontracted DHF calculation for the atom in question, yielding large- and small-component atomic functions that are kinetically... 

The experience gained in these studies has been invaluable for the development of the BERTHA molecular code much of the material of the present article was first presented in [29] and [30], together with applications to the study of magnetic and hyperfine interactions in atoms and small molecules, NMR shielding constants for H2O and NH3, and P-odd interactions in chiral molecules such as CHBrClF. A detailed study of the water molecule [31] examined the convergence of the DHF and DHFB calculations with a series of uncontracted correlation consistent basis sets due... 

Whilst this demonstrates that calculations using the methods of this paper may prove very useful in studies of molecules containing only low-Z atoms, a major objective has been to study systems containing heavier atoms. So far, only a limited number of molecular calculations have been carried out with BERTHA at the DHF level, mainly in connection with studies of hyperfine and PT-odd effects in heavy polar molecules such as YbF [33] and TIF [13]. The reader is referred to the literature for an assessment of these calculations and for technical details on the construction of basis sets which must not only describe molecular bonding properly but also give a good representation of spinors close to the heavy nuclei to handle the short-range electron-nuclear electroweak interactions. 

Energies of the decomposition reactions MX4 —> MX2 + X2 and MX2 — M + X2 (X = H, F and Cl) for group-14 elements were calculated at the PP CCSD(T) and DHF levels [96,52], The results show a decreasing trend in the stability of the 4+ oxidation state in the group in agreement with the predictions of [150] based on atomic calculations and simple models of bonding. The instability was shown to be a relativistic effect. The neutral state was found to be more stable for element 114 than that for Pb. Thus, element 114 is expected to be less reactive than Pb, and as reactive as Hg. The possibility of the existence of 114F62 is considered [96]. 

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