Relativistic Hartree-Fock-Slater calculations

Relativistic corrections have also been included in DFT calculations using perturbation theory, first by Herman and Skillman (1963) and later by Snijders and Baerends (Snijders 1979, Snijders and Baerends 1977, 1978). Following a non-relativistic Hartree-Fock-Slater calculation, first-order perturbation theory was used to calculate the relativistic corrections from the Breit-Pauli terms of 0(c ). Herman and Skillman applied this approach to first order only for the energies. 

Fig. 24.5 Relativistic splitting of radii of valence electrons of antimony, bismuth, and element 115 (relativistic Hartree-Fock-Slater calculations).

Desclaux [11] performed true relativistic Dirac-Fock calculations with exchange to obtain orbital binding energies for every atom. Relativistic Hartree-Fock-Slater calculations were made by Huang et al. [12] and later improved by ab initio Dirac-Hartree-Slater wave functions for elements with Z = 70 to 106 in [13]. 

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). 

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. 

Also available are the results of relativistic relaxed-orbital ab initio calculations of L-shell Coster-Kronig transition energies for all possible transitions in berkelium atoms [75], relativistic relaxed-orbital Hartree-Fock-Slater calculations of the neutral-atom electron binding energies in berkelium [76], and... 

These were calculated by the method of Barnes and Smith from spin-orbit splittings in atomic spectra without relativistic correction (open circles). Values calculated from relativistic Hartree-Fock-Slater atomic wave functions are included for comparison (filled circles). 

Carlson, T. A. Lu, C. C. Tucker, T. C. Nestor, C. W. Malik, F. B. Eigenvalues, Radial Expectation Values, and Potentials for Free Atoms from Z = 2 to 126 as calculated from Relativistic Hartree-Fock-Slater Atomic Wave Functions, Oak Ridge National Laboratory, 1970, pp. 1-29. 

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...

We present in Table I results from calculations on bond energies, bond distances and vibrational frequencies for the simple MH hydrides of the coinage triad M=Cu,Ag,and Au as well as the isoelectronic series MH+ with M=Zn,Cd,and Hg.Table I contains experimental data (lH) as well as results from non-relativistic (JLl) and relativistic (4.) Hartree-Fock-Slater (HFS) calculations. Results from a similar set of calculations on the metal-dimers M2 (M=Cu,Ag,and Au) as well as the dications (M=Zn,Cd,and Hg) are presented in Table II. 

The calculations of the photoionization cross section of the atomic subshell have previously been performed using Hartree-Fock-Slater one-electron model by several workers. Table 1 compares the photoionization cross sections of the atomic orbital electrons obtained in the present work with those previously reported by Scofield for some atoms. Scofield has used the relativistic wave functions. The... 

Relativistic molecular orbital calculations have been performed for the study of the atomic-number dependence of the relativistic effects on chemical bonding by examining the hexafluorides XFg (X=S, Se, Mo, Ru, Rh, Te, W, Re, Os, hr, Pt, U, Np, Pu) and diatomic molecules (CuH, AgH, AuH), using the discrete-variational Dirac-Slater and Hartree-Fock-Slater methods. The conclusions obtained in the present work are sununarized. 

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. 

Table 3 presents relativistic effects on several properties calculated as the difference (A) obtained in calculations which included the quasirelativistic correction, and corresponding calculations that excluded the correction, and used Hartree-Fock-Slater core orbitals rather than Dirac-Slater. The method finds significant relativistic Pt-C bond shortening, and little effect on the CO bond. The effect on adsorption energy is dramatic. Eads increases by about 50% when relativity is included. There is also an increase in the Pt-C force constant and frequency. The shortened Pt-C bond results in an increase in CO frequency through a wall effect, a Pauli repulsion effect. Ref. 34 ascribed the anomalously small shift in CO frequency from gas phase to adsorbed on Pt to the relativistic effect. 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

The energy level structure is particularly important to understand the relativistic effects on the valence states of the actinide atoms. O Figure 18.18 shows relativistic effects on the energy level structure of Am calculated by the nonrelativistic Hartree-Fock-Slater and the relativistic... 

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