Hartree—Fock—Slater values

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. 

What does this mean We have replaced the non-local and therefore fairly complicated exchange term of Hartree-Fock theory as given in equation (3-3) by a simple approximate expression which depends only on the local values of the electron density. Thus, this expression represents a density functional for the exchange energy. As noted above, this formula was originally explicitly derived as an approximation to the HF scheme, without any reference to density functional theory. To improve the quality of this approximation an adjustable, semiempirical parameter a was introduced into the pre-factor Cx which leads to the Xa or Hartree-Fock-Slater (HFS) method which enjoyed a significant amount of popularity among physicists, but never had much impact in chemistry,... 

The extended Hiickel theory calculations, used in this work and discussed below, are based on the approaches of Hoffmann Although VSIP values given by Cusachs, Reynolds and Barnard were explored for use as the Coulomb integrals, the VSIP values obtained from a Hartree-Fock-Slater approximation by Herman and Skillman were consistently used in the present EHT calculations by this author. Both the geometric mean formula due to Mulliken and Cusachs formula ) were considered for the Hamiltonian construction, but the Mulliken-Wolfsberg-Helmholtz arithmetic mean formula was chosen for use. 

Accurate distance determinations depend critically on the accurate determination of phase shifts. There are two general approaches to this problem theoretical and empirical determination. The main approaches to the theoretical calculation of phase shifts are based on the Hartree-Fock (HF) and Hartree-Fock-Slater (HFS) methods. However, both of these are too involved for general use. Teo and Lee used the theoretical approach of Lee and Beni to calculate and tabulate theoretical phase shifts for the majority of elements. Use of these theoretical phase shifts requires the use of an adjustable q in the data analysis (vide supra). Most recently, McKale and co-workers performed ab initio calculations of amplitude and phase functions using a curved wave formalism for the range of k values 2 < k < 20. 

As with ligand-field absorption spectra, the Nephelauxetic effect [41, 42] will also impact L-edge XAS data. For L-edge spectra, the effect of a reduction in interelectron repulsion is also to reduce the ffee-ion state splitting and make the spectra somewhat more orbital like. This is demonstrated by the sequence of simulated low-spin Fe(III) L-edge X-ray absorption spectra shown in Fig. 13, where the e - e repulsion is reduced systematically from [i = 100% to 60%. This starts from 80% of the Hartree-Fock calculated values of the Slater Condon Shortley parameters for e - e repulsion. 

Figure 2 gives the theoretical values of the atomic scattering factors of indium and phosphorus calculated by the Hartree-Fock-Slater method [9], as well as the experimental values of these factors [1]. 

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. 

First, we already mentioned that the Hartree-Fock Slater integrals F do not match the fitted ones the ab initio values are 1.1 to 1.5 timesthe phenomenological ones. This difference can be taken into account by an expansion of the rare earth radial wavefunction which, in counterpart, gives higher radial integrals (r ). If A is the mean ratio between the phenomenological F and the Hartree-Fock ones, then may replace the simple... 

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. 

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