The Four-Component Kohn-Sham Model

We have already seen in section 8.1 that (i) a Dirac electron with electromagnetic potentials created by all other electrons [cf. Eq. (8.2)] cannot be solved analytically, which is the reason why the total wave function as given in Eq. (8.4) cannot be calculated, and also that (ii) the electromagnetic interactions may be conveniently expressed through the 4-currents of the electrons as given in Eq. (8.31) for the two-electron case. Now, we seek a one-electron Dirac equation, which can be solved exactly so that a Hartree-type product becomes the exact wave function of this system. Such a separation, in order to be exact (after what has been said in section 8.5), requires a Hamiltonian, which is a sum of strictly local operators. The local interaction terms may be extracted from a 4-current based interaction energy such as that in Eq. (8.31). Of course, we need to take into account Pauli exchange effects that were omitted in section 8.1.4, and we also need to take account of electron correlation effects. This leads us to the Kohn-Sham (KS) model of DFT. 

The question now is how the spinors in the Kohn-Sham Slater determinant are obtained in order to compute the densities. With all the precautions discussed in extenso for the energy expectation value calculated from unbounded operators, a functional derivative may be calculated from the energy functional of Eq. (8.211) similarly to the derivation given in section 8.7. In 1973, Rajagopal and Callaway [392] derived in this way the most general relativistic KS equations. 

In addition to the exchange- orrelation energy functional E q, we recognize the familiar interaction integrals of Eq. (8.31). The external potentials, ( ext and Aext, contain the electron-nucleus interaction as usual and the functional derivatives ate defined as 

The notation SExc[j ]/ j f) is a composite representation for the three (spatial) components = 1,2,3. 

In order to fully understand the role of the electron density in relativistic DFT, we follow the original work in Ref. [394] from which we extract the essence relevant to the relativistic formulation of DFT. 

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