Slater approximation

The main difficulty in solving the Hartree-Fock equation is caused by the non-local character of the potential in which an electron is orbiting. This causes, in turn, a complicated dependence of the potential, particularly of its exchange part, on the wave functions of electronic shells. There have been a number of attempts to replace it by a local potential, often having an analytical expression (e.g. universal Gaspar potential, Slater approximation for its exchange part, etc.). These forms of potential are usually employed to find wave functions when the requirements for their accuracy are not high or when they serve as the initial functions. 

The three terms represent nuclear, coulomb and exchange-correlation potentials respectively. The third, problematic, term is written as a functional of the density. The same problem which occured in the Hartree-Fock simulation of atomic structure was overcome by defining the one-electron exchange potential with the Slater approximation for a uniform electron gas ... 

The wave eigenfunctions and energy eigenvalues were obtained by real space electronic state calculations, which were performed by the use of the program code SCAT of the DV-Xx molecular orbital (MO) cluster method with the Hartree-Fock-Slater approximation [8,9]. In the method, the exchange-correlation term Vxc in the one-electron Hamiltonian was expressed in terms of the statistical local potential (1),... 

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

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. 

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... 

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... 

This is called a Fermi hole and is the first example we encounter of a particle being dressed (i.e. having its properties modified) by many-body forces. Strictly speaking, the Fermi hole differs for each electron, but the interaction can be made local by averaging it over different orbitals, and this is referred to as the Hartree-Slater approximation. ... 

We solve the dme-dependent relativistic Dirac-equation for complicated ion-atom systems iti the Diiac-Fock-Slater approximation including the many-electron effects by means of an inclusive probability description. Various gas-gas and gas-solid target collision systems are discussed. We find the dominant effect in the observed excitation probability of inner shells to be a reladvisdc dynamic coupling of various levels which in a non-reladvisdc descripdon is zero. The effect of excitation and transfer during the passage within the solid allows us to understand the gas-solid target systems. An ab-inido calculation of K MO X-rays is presented for the system Cl" on Ar. 

Using for the exchange part of the potential the usual energy independent Slater approximation (Xoc potential) we must expand the energy scale by a factor of about... 

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]. 

To construct radial functions and radial distribution functions, based on the Slater approximation, only straightforward modifications to the designs of previous spreadsheets are required. The spreadsheet file, figl-ll.xls, and Figure 1.11, based on lithium 2s data, are based on the following instructions. 

Formally, this example is similar to the Clementi douhle-zeta formulation within the Slater approximation. In order to use the (4s/2s) linear combination we need to know what values to use for the coefficient, ci, in equation 1.26. To model the hydrogen Is orbital, (p r) needs to be normalized. Figures 1.29 and 1.30 show two conditions of the application of such a condition. In Figure 1.29, the normalized linear combination is a poor approximation to the Is orbital. In Figure 1.30 close agreement with the Is function is returned for the appropriate choice of coefficients in equation 1.26. This procedure of minimizing differences is applied extensively in basis set theory. 

This matter is returned to later, at the end of Chapter 3, when matching possibilities to the actual radial functions, rather than Slater approximations to these functions, is discussed. 

The pre-exponential factors in the equation 1.12 normalize the Slater approximations to the radial components of atomic orbitals. Normality is not an inherent property of linear combinations of Slater orbitals, for example, as in Table 1.3, and it is important to check any published coefficients to determine whether normalization is included. In addition, the Slater orbitals for a set of atomic orbitals in an atom are not mutually orthogonal. The results of atomic structure calculations using Slater orbitals, either as single functions or in linear combinations, as in double-zeta sets, of course, are mutually orthogonal, since this property of the eigenfunctions, is mirrored in the final linear combinations returned by the calculations for the eigenvalues. 

Figure 3.1 The spreadsheet detail for orthonormality checking in this example of the Slater approximations to the lithium Is and 2s radial orbitals.

As you see, the calculated 2s orbital energy is very unsatisfactory. But, note, that the kinetic and kinetic energy terms are the same as are returned in the calculation for the 2p orbital, when this is based on the exact function, which has the same form as the Slater approximation. 

To complete Hartree s calculation, we need to evaluate the integrals set out in equations 5.14 and 5.20. For the wave function choice, equation 5.22. the integral in equation 5.14 includes the kinetic energy and nuclear-electron potential energy terms calculated in Chapter 4 for the Slater approximation to the 2s eigenstate in hydrogen. 

I started this manuscript with a quotation from Professor Hartree s famous book on numerical integration. With the development of the methodology of the Hartree-Fock-Slater approximation on a spreadsheet, perhaps to finish, I can be allowed the temerity to quote a well-known comment by one of the giants of quantum mechanics. Professor P.A.M. Dirac (69),... 

Data from an electron diffraction measurement of integrated intensities of Debye reflections for polycrystalline sodium bromide films are used to construct curves of the electron atomic scattering factors of Na and Br. An approximation is given of experimental f curves by analytical expressions obtained in the Slater approximation. The parameters of the approximation are used to calculate the charge on the atoms and also to determine the potential distribution in the sodium bromide unit cell. 

The determination of the charge on the atoms and the electron density and potential distribution in the NaBr lattice was carried out by approximation of the experimental f curves, using analytical expressions for the form factor of electron scattering in the Slater approximation, which are given in [10]. 

The data which we obtained in the approximation can be used to find the potential distribution in the NaBr lattice. The equations from which the potential of the atom can be calculated in the Slater approximation are given in [13]. In our case, if we accept that the charge on the atoms in the NaBr crystal is 0.8, the equations for the ion potentials have the form... 

Figure 1. Exchange potential in the HF approximation (in units e kffTr). Dashed line Slater approximation ... 

The macroscopic dielectric function of Silicon in the Hartree and in the Slater approximation is given in... 

In figure 2 the macroscopic dielectric function in the Hartree and Slater approximations is given for Germanium. For a comparison of our inverse dielectric function of Germanium with other calculations see the paper by K. Kune in these proceedings. 

Another commonly used xc kernel was derived by Petersilka et al. in 1996, and is nowadays referred to as the PGG kernel [22]. Its derivation starts from a simple anal3dic approximation to the EXX potential. This approximation, called the Slater approximation in the context of Hartree-Fock theory, only retains the leading term in the expression for EXX, which reads... 

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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