Problems with exchange-correlation potential

We have to consider the calculation of the fourth term, the problem term, in the KS operator of Eq. 7.23, the exchange-correlation potential vXc(r). This is defined as the functional derivative [36, 37] of the exchange-correlation energy functional, fsxc[p(r)], with respect to the electron density functional (Eq. 7.23). The exchange-correlation energy UX( lp(r)], a functional of the electron density function p(r), is a quantity which depends on the function p(r ) and on just what mathematical form the... 

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 Vignale-Rasolt CDFT formalism can be obtained as the weakly relativistic limit of the fully relativistic Kohn-Sham-Dirac equation (5.1). This property has been exploited to set up a computational scheme that works in the framework of nonrelativistic CDFT and accounts for the spin-orbit coupling at the same time (Ebert et al. 1997a). This hybrid scheme deals with the kinematic part of the problem in a fully relativistic way, whereas the exchange-correlation potential terms are treated consistently to first order in 1 /c. In particular, the corresponding modified Dirac equation... 

Another problem is that for fee Fe, the Generalized Gradient Approximation (GGA) is assumed to be the most accurate approximation for the exchange-correlation potential, but most of the studies on the FeNi alloys have been made using the Local Spin Density Approximation (LSDA). Therefore, it was of some importance to perform an exhaustive study of FeNi alloys using the GGA, and this was done in Paper VIII, where we have investigated the FeNi alloys from pure Fe to alloys with a Ni concentration of 50%. 

We are immediately confronted with the problem of how to find the unknown exchange-correlaticm energy Exc- which is replaced by an imknown exchange-correlation potential in the form of a functional derivalive... 

The DFT models can be tested when applied to exactly solvable problems with electronic correlation (like the harmonium, as discussed in Chapter 4). It turns out that despite the exchange and correlation DFT potentials deviating from the exact ones, the total energy is quite accurate. 

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