Exchange potential Slater derivation

This formula was used by Slater [385] to define an effective local exchange potential. The generally unsatisfactory results obtained in calculations with this potential indicate that the locality hypothesis fails for the density functional derivative of the exchange energy Ex [294],... 

For the self-consistent orbitals, we have determined1"4 the exact analytical asymptotic structure in the classically forbidden vacuum region of (i) the Slater potential VJ (r), (u) the functional derivative (exchange potential)... 

Another reason for the choice of the title is the above-mentioned introduction of the Xa-method and the MS-Xa method by Slater and coworkers. There are, however, in particular two other reasons for choosing the title. The first is the formulation of the Density Functional Theory by Hohenberg and Kohn in 1964 [19], which today is probably one of the most quoted papers in electronic structure calculations. This basic work was followed by another important paper in 1965 by Kohn and Sham [20], where they showed how one could use the method for practical calculations and introduced the Kohn-Sham, KS, exchange potential. Exactly the same expression for the exchange potential had previously been derived by Caspar [21], This exchange potential is therefore often known as the Caspar-Kohn-Sham, GKS, potential. Another very important reason for choice of the title is the introduction of the three dimensional numerical integration method by Ellis and Painter in 1968-1970 [22-24]. This... 

Slater (1951) simply and significantly improved upon the Fermi-Amaldi approach by averaging the potential derived from the indirect part of U in the HF method. That potential is the HF exchange potential, and its average is... 

In the DFT, as in the Hartree-Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistency. The many-electron wave function is still written as a Slater determinant. However, the wave functions used to construct the Slater determinant are not the one-electron wave functions of the Hartree-Fock approximation. In the DFT, these wave functions have no individual meaning. They are merely used to construct the total electron-charge density. The difference between the Hartree-Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange correlation potential, Vxc[ ](t), a functional derivative of the exchange correlation energy, Exc, that, in turn, is a functional (a function of a function) of the electron density. In DFT, the Schrodinger equation is expressed as ... 

Instead of relying on experimental data for the ionization potentials, the essential EH energy (H ) and orbital contraction Q parameters can also be deduced from theoretical calculations [115,116]. Recently, a complete set of EH parameters has been derived from atomic Hartree-Fock-Slater calculations (an early form of density-functional theory, see Section 2.12) which were also adjusted to fit some experimental data. The parameter set thus derived [117] includes exchange, some correlation, and also the influences of relativity for convenience, we include these data in Table 2.1. These parameters may be used to study the trends in the periodic table and, also, to perform simple calculations. Other sets of EH parameters, from very different sources, are also available. These then typically include better basis sets (such as double- parameters for d orbitals) although they are less self-consistent for the whole periodic table. 

Exchange-only. The KLI approximation was originally derived for the exchange-only functional for the KLI exchange-only potential y f (r) simply replace with Later it was applied to SIC schemes and for an XC potential based on the self-energy. In the exchange-only case the first term in Eq. (130) is the so called Slater potential... 

The band structure of semiconducting SmSe shown in Fig. 57 is calculated with the augmented plane-wave method (APW). The approximation is used to obtain the muffin-tin potentials with a = 0.67 for the exchange, assuming an intermediate state for Sm with the configuration 4f d instead of pure 4f . The atomic wave functions are derived in the Hartree-Fock-Slater approximation, Farberovich [1]. The band structure model of [1] is qualitatively confirmed by an analysis of the reflection and electroreflection spectra of SmSe single crystals with the minimum direct gap located at the X point. However, the next direct gap is at r and no indications of reflection structures which are attributable to K point excitations are observed, Kurita et al. [2]. Earlier, the band structure for the T-X (i.e. (100)) direction was calculated by... 

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