Exchange potential approximation

Since the relativistic DV-Xa method takes into account fully relativistic effects under Slater s exchange potential approximation, this is very useful for studying the electronic structures and chemical bonding in molecules containing heavy... 

We have used the basis set of the Linear-Muffin-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed the Vosko-Wilk-Nusair parametrization for the exchange-correlation energy density and potential. In conjunction with this we have treated the alloying effects for random and partially ordered phases with a multisublattice generalization of the coherent potential approximation (CPA). 

The presence of the nonlocal exchange potentials in the Hartree-Fock equations greatly complicates their solution and necessitates further approximations. Several of these are discussed in the following subsection. In the evaluation of any calculations, it is important to recognize their common (and imperfect) origin, as well as the seriousness of the particular approximations made in solving the equations. 

Given in Table 7.1 are the results [24] of the total energy of some atoms obtained by solving the Kohn-Sham equation self-consistently with the exchange potential Wx within the central field approximation. The energy is obtained from Equation 7.10... 

The shape function had a role in theoretical chemistry and physics long before it was named by Parr and Bartolotti. For example, in x-ray measurements of the electron density, what one actually measures is the shape function—the relative abundance of electrons at different locations in the molecule. Determining the actual electron density requires calibration to a standard with known electron density. On the theoretical side, the shape function appears early in the history of Thomas-Fermi theory. For example, the Majorana-Fermi-Amaldi approximation to the exchange potential is just [3,4]... 

This method is usually thought as an approach allowing one to find the exact exchange potential. It may be considered [17] as an approximation to the exact GS problem, similar to the HF approximation namely, the solution of the optimized potential (OP) approximation - the energy Egg and the wave function Gs - stems from the following minimization problem... 

Now we adopt the above observation, concerning and in order to split the exact expression for in Eq. (188) into the sum of and UcPP interpreting these terms as approximate exchange potential and approximate correlation potential, respectively. Thus... 

In the remainder of this section, we give a brief overview of some of the functionals that are most widely used in plane-wave DFT calculations by examining each of the different approaches identified in Fig. 10.2 in turn. The simplest approximation to the true Kohn-Sham functional is the local density approximation (LDA). In the LDA, the local exchange-correlation potential in the Kohn-Sham equations [Eq. (1.5)] is defined as the exchange potential for the spatially uniform electron gas with the same density as the local electron density ... 

Fig. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

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