Noncollinear Approaches and Collinear Approximations

In between full current-density functional theory and relativistic density-only KS-DFT, different intermediate formulations of relativistic KS-DFT are possible. For a relativistic system of noninteracting electrons, whose wave function is exactly described by a single Slater determinant, Eq. (8.219) yields for the magnetization, after application of the Slater-Condon rules for the evaluation of the integral, a contribution of 

ipi r) denotes one of the N 4-spinors of Eq. (8.213). The noninteracting reference system can then be set up in such a way that, in addition to the electron density, some parts of the magnetization agree with those of the fully interacting system. 

These different choices for the noninteracting reference system imply dif-ferent definitions of the noninteracting kinetic energy functionals, Tg p, mz] and m ], and accordingly also of the exchange-correlation energies, 

E l)[p,Mz] and E p, m ] [394], For these choices, for which in addition to the electron density one further quantity is to be reproduced by the KS system, the resulting exchange-correlation potential has two components. This is in close analogy to the case of nonrelativistic unrestricted KS-DFT formulated for a- and /5-spin densities, j0 and pp. For this reason, approximate exchange-correlation functionals developed for nonrelativistic DFT are simply employed in actual relativistic spin-DFT calculations. However, the exchange-correlation potential is defined differently in nonrelativistic unrestricted KS-DFT and in the relativistic collinear and noncollinear cases. Consequently, different exact conditions also apply to the exchange-correlation functional [394]. 

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