Dirac-Fock-Slater

Figure4.7 Relativistic bond contractions A re for Au2 calculated in the years from 1989 to 2001 using different quantum chemical methods. Electron correlation effects Acte = te(corn) — /"e(HF) at the relativistic level are shown on the right hand side of each bar if available. From the left to the right in chronological order Hartree-Fock-Slater results from Ziegler et al. [147] AIMP coupled pair functional results from Stbmberg and Wahlgren [148] EC-ARPP results from Schwerdtfeger [5] EDA results from Haberlen and Rdsch [149] Dirac-Fock-Slater...

Bastug, T, Fleinemann, D., Sepp, W.-D., Kolb, D. and Fricke, B. (1993) All-electron Dirac-Fock-Slater SCF calculations of the Au2 molecule. Chemical Physics Letters, 211, 119-124. 

Finally, a comment regarding relativistic effects and the calculation of one-electron energies and monopole relaxation shifts. The most convenient way to obtain relativistic zlSCF one-electron energies is to use the Dirac-Fock-Slater (DFS) zlSCF values tabulated by Huang et al.82). These are very close (a few tenths of an eV) to DF JSCF, and the relativistic monopole relaxation shift is the given by... 

On the other hand, a large number of relativistic Slater calculations [23] (Dirac-Fock-Slater DFS), in which the RKS-equations are used with the nonrelativistic x-only LDA, can be found in the literature (see e.g. [8,9]). However, no attempt is made to review this extensive body of literature here. 

If this functional is varied with respect to the single particle functions one obtains the Dirac-Fock-Slater equations... 

The experimental XANES spectrum has been reported by Li et al. [13] as shown in Fig 7, where the horizontal scale was calibrated by the data obtained by O Brien et al. [28]. As in the case of a-A Oa, the splitting of peak A is attributable to the spin orbit splitting. The calculated splitting of 2p orbital for a free Si atom using the Dirac-Fock-Slater method is 0.66 eV, which agrees well with the experimental splitting (0.6 eV). 

These very complicated inhomogeneous coupled differential equations can again be simplified by using Slater s approximation. This method is therefore called the relativistic Hartree-Fock-Slater or Dirac-Fock-Slater (DFS) 52—53) calculations, and they have also been done by several authors for the superheavy elements 54-56). 

In all the calculations for the electronic and geometric structures of the system, the density functional method (6-9) was used. The total energy, E, in the Dirac-Fock-Slater approximation is expressed as a functional of charge density... 

KEYWORDS Dirac-Fock-Slater, DV-DFS, uranyl nitrate, neptunyl nitrate, plutonyl nitrate, stability, bonding nature... 

The aim of the present work is to perform a detailed theoretical study of the electronic structures of actinyl nitrates. Relativistic effects are remarkable in the electronic structure and chemical bonding of heavy atoms such as actinide elements[6j. In our previous study, we applied the relativistic discrete variational Dirac-Fock-Slater(DV-DFS) method to study of the electronic structure of uranyl nitrate dihydrate[7]. The accuracy of the DV-DFS method was demonstrate by its ability to reproduce the uranyl nitrate dihydrate experimental X-ray photoelectron spectrum. 

The DV-DFS molecular orbital(MO) method is based on the Dirac-Fock-Slater approximation. This method provides a powerful tool for the study of the electronic structures of molecules containing heavy elements such as uranium[7,8,9,10]. The one-electron molecular Hamiltonian in the Dirac-Fock-Slater MO method is written as... 

From a very general point of view every ion-atom collision system has to be treated as a correlated many-body time-dependent quantum system. To solve this from an ab initio point of view is still impossible. So, one has to rely on various approximations. Nowadays the best method which can be applied to realistic collision systems (which we discuss here) is on the level of the non-selfconsistent time-dependent Hartree-Fock-Slater or, in the relativistic case, the Dirac-Fock-Slater method. Up-to-now no correlation beyond this approximation can be taken into account in the case of 3 or more electrons. (This is in accordance with the definition of correlation given by Lowdin [1] in 1956) In addition no QED contributions, i.e. no correction to the 1/r Coulomb interaction between the electrons, ever have been taken into account, although in very heavy collision systems this effect may become important. This will be discussed in section 5. A short survey of the theory used is followed by our results on impact parameter dependent electron transfer and excitation calculations of ion-atom and ion-solid collisions as well as first results of an ab initio calculation of MO X-rays in such complicated many particle scattering systems. 

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. 

The number of reported molecular DHF calculations is sufficiently small that a fairly complete account is possible. The cases which have been studied in the DHF model all involve small molecules, or molecules which exhibit high spatial symmetry. Larger molecules have been studied using more approximate schemes, ranging from semi-empirical and pseudopotential methods to Dirac-Fock-Slater and density functional methods. These are discussed elsewhere in this book. 

Finite difference methods Benchmark calculations have been performed for a number of diatomic species containing heavy elements, for one-electron systems [195-197], and in the Dirac-Fock-Slater approximation [198,199]. 

In Fig. 7 the purely repulsive Moliere potential is compared with the (partly) attractive Morse potential. It is to be noted from this comparison that the Morse (like the Lennard-Jones) potential, while providing a realistic description of the attractive part of the interaction, becomes insufficient at low intemuclear separations where the purely repulsive potentials are more adequate. While more sophisticated potentials calculated, for example, by employing the Dirac-Fock-Slater (DFS) method (Eckstein etai, 1992), have recently become available, such potentials are generally more complicated and are available only in numerical form, and are thus not very handy for the calculations of interest here. For Si-Si collisions, the interaction potential, being repulsive for small and attractive for... 

Fig. 8. A comparison of the Si-Si screening function calculated within the Dirac-Fock-Slater model (Eckstein et al., 1992) and for the Molifere potential.

Figure 8 compares the Dirac-Fock-Slater potential for Si-Si collisions with the corresponding Moliere potential. While for low intemuclear separations the two potentials are almost indistinguishable from each other, significant differences occur for medium and large intemuclear separations, where both potentials are weak. Nevertheless, trajectories calculated with these two potentials differ significantly at large impact parameters, where the DFS potential becomes attractive (Fig. 9). 

C. Diisterhoft, D. Heinemaim, D. Kolb. Dirac-Fock-Slater calculations for diatomic molecules with a finite element defect correction method (FEM-DKM). Chem. Phys. Lett., 296 (1998) 77-83. 

W.-D. Sepp, D. Kolb, W. Sengjer, H. Har-tung, B. Fricke. Relativistic Dirac-Fock-Slater program to calculate potential-energy curves for diatomic molecules. Phys. Rev. A, 33(6) (1986) 3679-3687. 

DF=Dirac-Fock DFS = Dirac-Fock-Slater DS = Dirac-Slater DVM = discrete variational method EC = electron capture MCDF = multi-configuration DF NR = nonrelativis-... 

Relativistic wave equation for a particle with spin 1/2. Dirac-Fock-Slater method... 

More approximate four-component schemes of solution of the relativistic electronic structure problem have been used to obtain insight in chemical properties connected with relativistic effects. This comprises semiempirical methods such as the Relativistic Extended Hiickel (REX) method as well as the Dirac-Fock-Slater (DFS) method, the relativistic analogue of the Hartree-Fock-Slater (HFS) approach. 

The ionization potentials and low excitation energies calculated for El 22 are shown in Table 2.7. More values may be found in [60]. Intermediate Hamiltonian values for E122 and its monocation were calculated by the Dirac-Coulomb and Dirac-Coulomb-Breit schemes, to obtain the effect of the Breit interaction (2.2). The Breit term contribution is small (0.01-0.04 eV) for transitions not involving/ electrons but increases to 0.07-0.1 eV when/ orbital occupancies are affected, as observed above (Section 2.3.1). The ground state is predicted to be 8s 8p7d, in agreement with early Dirac-Fock(-Slater) calculations [55-57], and not the 8s 8p configuration obtained by density functional theory [58]. The separation of the... 

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