Spin-DFT

Exchange Potential for a Pure-State System in the Spin DFT. . 80... 

Here, we should mention that there exists an extensive discussion in the literature on the capabilities of spin-DFT regarding, for instance, the question whether the Kohn-Sham spin density has to be equal to the spin density of the fully interacting system of electrons (and in the case of open-shell singlet broken-symmetry (BS) determinants (see below) for binuclear transition-metal clusters this is certainly not the case see Ref. (33) for a more detailed discussion). But the situation is much more subtle and one may basically set up the variational procedure in a Kohn-Sham framework such that the spin density of the Kohn-Sham system of noninteracting fermions represents the true spin density. However, the frame of this review is not sufficient to present all details on this matter (34,35). 

Fig. 1 Fe-terminated (a) and 0-terminated (b) a-Fe203(0001) surface structures. The interlayer relaxations shown in (a) were measured by XPD and predicted by two independent theoretical methods. Those shown in (b) are predicted by spin-DFT theory, but have not been confirmed experimentally.

Originally the fourth statement was considered to be as sound as the other three. However, it has become clear very recently, as a consequence of work of H. Eschrig and W. Pickett [32] and, independently, of the author with G. Vi-gnale [33, 34], that there are significant exceptions to it. In fact, the fourth substatement holds only when one formulates DFT exclusively in terms of the charge density, as we have done up to this point. It does not hold when one works with spin densities (spin-DFT) or current densities (current-DFT).14 In these (and some other) cases the densities still determine the wave function, but they do not uniquely determine the corresponding potentials. This so-called nonuniqueness problem has been discovered only recently, and its consequences are now beginning to be explored [27, 32, 33, 34, 35, 36, 37, 38]. It is clear, however, that the fourth substatement is, from a practical point of view, the least important of the four, and most applications of DFT do not have to be reconsidered as a consequence of its eventual failure. (But some do see Refs. [33, 34] for examples.)... 

As has been alluded to by mentioning the electronic spin, an extension of density-functional theory to spin-density-functional theory is possible by simply decomposing the total electron density p(r) into the spin densities p r) and p (r) although there are some recent doubts as to the validity of the entire idea [149,150]. In practice, spin-DFT works very well. Spin-DFT is needed whenever the electrons feel an external magnetic field or are subject to spontaneous spin-polarization, such as in magnetic atoms, molecules, and solids (for example, itinerant magnets, see Section 3.5). Spin-DFT also allows different... 

The generalization of density-functional theory that allows different orbitals for electrons with different spins is called spin-density-functional theory (Parr and Yang, Chapter 8). In spin-DFT, one deals separately with the electron density p (r) due to the spin-a electrons and the density p (r) of the spin-/3 electrons, and functionals such as become functionals of these two quantities P ]- One deals with sepa-... 

For species with all electrons paired and molecular geometries in the region of the equilibrium geometry, we can expect that p = p, and spin-DFT will reduce to the ordinary form of DFT. 

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]. 

It is instructive to see how the nonrelativistic unrestricted KS-DFT formalism emerges from relativistic spin-DFT. This is most easily seen for the collinear approach, although the noncollinear one also reduces to the same nonrela-... 

Gunnarsson and Lundqvist did an LSDA spin-DFT calculation of the H2 molecule, expanding the occupied KS orbitals using the s and Is AOs as basis functions. For internuclear separations up to 3.2 bohrs, they found the lowest energy KS orbitals to be = N Sa + Dfc). However, for internuclear separations greater than... 

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