Local-density Approximation - Spin Polarization

Spin Polarization

The extension of density functionals to spin-polarized systems is straightforward for exchange, where the exact spin-scaling is known, but for correlation further approximations must be employed. A spin polarized system in DFT employs two spin-densities, ρ_α and ρ_β with ρ = ρ_α + ρ_β, and the form of the local-spin-density approximation (LSDA) is

For the exchange energy, the exact result (not just for local density approximations) is known in terms of the spin-unpolarized functional:

The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:

corresponds to the paramagnetic spin-unpolarized situation with equal and spin densities whereas corresponds to the ferromagnetic situation where one spin density vanishes. The spin correlation energy density for a given values of the total density and relative polarization, ε_c(ρ,ς), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.