Spin-density-functional theory

Density-functional formalism may be extended to a spin-density formalism, 

Useful estimates of exc and the corresponding potentials vxc have been given by von Barth and Hedin [7.11], and by Gunnarsson and Lundqv ist [7.12]. 

Since we want to be able to do calculations on magnetic materials, we must expand the formalism to allow for spin-dependent external potentials. This is quite easily done with the use of the Levy constrained search formulation. 

As can be proven in exactly the same way as above, two different non-degenerate ground states will always lead to different four-vectors (n(r), m(r)) [22, 23], where n and rhz can be related to the density of spin-up electrons n+ and spin-down electrons n respectively as  

This means that we can write the energy functional 

What has not yet been proven is that a given ground state corresponds to a unique vector of external fields (v(r),B(r)). The fact is that one can construct [22] external fields such that they yield common eigenstates of the corresponding non-interacting Hamiltonians, but it has not been shown that these eigenstates are actually ground states [24]. 

The difficulty of proving this is that each spin component only determines the potential up to a constant, and that we only can get rid of one of the constants by changing energy scales in our potential [25]. This is fortunately enough not a big problem when doing actual calculations, since we then have v(r) and B(r) given [26]. It has been shown that the Levy constrained search formulation can be extended without problems to the spin-dependent case [27]. 

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Highest occupied molecular orbital Intermediate neglect of differential overlap Linear combination of atomic orbitals Local density approximation Local spin density functional theory Lowest unoccupied molecular orbital Many-body perturbation theory Modified INDO version 3 Modified neglect of diatomic overlap Molecular orbital Moller-Plesset... 

Local spin density functional theory (LSDFT) is an extension of regular DFT in the same way that restricted and unrestricted Hartree-Fock extensions were developed to deal with systems containing unpaired electrons. In this theory both the electron density and the spin density are fundamental quantities with the net spin density being the difference between the density of up-spin and down-spin electrons ... 

It is clear that an ah initio calculation of the ground state of AF Cr, based on actual experimental data on the magnetic structure, would be at the moment absolutely unfeasible. That is why most calculations are performed for a vector Q = 2ir/a (1,0,0). In this case Cr has a CsCl unit cell. The local magnetic moments at different atoms are equal in magnitude but opposite in direction. Such an approach is used, in particular, in papers [2, 3, 4], in which the electronic structure of Cr is calculated within the framework of spin density functional theory. Our paper [6] is devoted to the study of the influence of relativistic effects on the electronic structure of chromium. The results of calculations demonstrate that the relativistic effects completely change the structure of the Or electron spectrum, which leads to its anisotropy for the directions being identical in the non-relativistic approach. 

Vignale, G., and Rasolt, M., 1988, Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields , Phys. Rev. B 37 10685. 

Perdew, J. P., Ernzerhof, M., Burke, K., Savin, A., 1997, On-Top Pair-Density Interpretation of Spin Density Functional Theory, With Applications to Magnetism , Int. J. Quant. Chem., 61, 197. 

SPIN-DENSITY FUNCTIONAL THEORY General Density Functional Theory... 

Evidently, the LSD and GGA approximations are working, but not in the way the standard spin-density functional theory would lead us to expect. In Ref [36], a nearly-exact alternative theory, to which LSD and GGA are also approximations, is constructed, which yields an alternative physical interpretation in the absence of a strong external magnetic field. In this theory, Hf(r) and rti(r) are not the physical spin densities, but are only intermediate objects (like the Kohn-Sham orbitals or Fermi surface) used to construct two physical predictions the total electron density n(r) from... 

Perdew JP, Ernzerhof M, Burke K, Savin A. On-top pair-density interpretation of spin-density functional theory, with applications to magnetism to appear in Int. J. Quantum Chem. 

A. Kafafi and E. R. H. El-Gharkawy,/. Phys. Chem. A, 102,3202 (1998). A Simple Coupling Scheme Between Hartree-Fock and Local Spin-Density Functional Theories. 

M. A. Whitehead and S. Manoli, Phys. Rev. A, 38,630 (1988). Generalized-Exchange-Local-Spin-Density-Functional Theory Self-Interaction Correction. 

A method to calculate J]j, based on the local approximation to spin density functional theory has been developed by Liechtenstein et al. [51, 52]. Using spherical charge and spin densities and a local force theorem, expression for Jjj is... 

Gritsenko O, Baerends EJ (2002) The analog of Koopmans theorem in spin-density functional theory, J Chem Phys, 117 9154-9159... 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

Eqs. (101)-(103) constitute the KS scheme of time-dependent spin-density functional theory. With the xc action functional [ t. i] defined in analogy to Eq. (43), the spin-dependent xc potentials can be represented as functional derivatives ... 

Equations (4.34)-(4.37) are immediately identified as the relativistic extension of the standard form of nonrelativistic spin-density functional theory. 

While the concentration dependence of the experimental fields are reproduced rather well by the theoretical fields (a phase transition to the BCC structure occurs around 65% Fe), the later ones are obviously too small. This finding has been ascribed in the past to a shortcoming of plain spin density functional theory in dealing with the core polarization mechanism (Ebert et al. 1988a). Recent work done on the basis of the optimized potential method (OPM) gave results for the pure elements Fe, Co and Ni in very good agreement with experiment (Akai and Kotani 1999). 

Calculating xw within the framework of plain spin density functional theory (SDFT), there is no modification of the electronic potential due to the induced orbital magnetization. Working instead within the more appropriate current density functional theory, however, there would be a correction to the exchange correlation potential just as in the case of the spin susceptibility giving rise to a Stoner-like enhancement. Alternatively, this effect can be accounted for by adopting Brooks s orbital polarization formalism (Brooks 1985). 

To keep the notation simple, spin labels are either ignored or condensed into a common variable x = (rs) in most of this text. They will only be put back explicitly in discussing spin-density-functional theory, in Sec. 6. 

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