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We explore the derivation of interatomic exchange interactions inferromagnets within density-functional theory (DFT) and the mapping of DFTresults onto a spin Hamiltonian. We delve into the problem of systemscomprising atoms with strong spontaneous moments together with atoms with weakinduced moments. All moments are considered as degrees of freedom, with thestrong moments thermally fluctuating only in angle and the weak momentsthermally fluctuating in angle and magnitude. We argue that a quadraticdependence of the energy on the weak local moments magnitude, which is a goodapproximation in many cases, allows for an elimination of the weak-momentdegrees of freedom from the thermodynamic expressions in favor of arenormalization of the Heisenberg interactions among the strong moments. Weshow that the renormalization is valid at all temperatures accounting for thethermal fluctuations and resulting in temperature-independent renormalizedinteractions. These are shown to be the ones directly derived from total-energyDFT calculations by constraining the strong-moment directions, as is done e.g.in spin-spiral methods. We furthermore prove that within this framework thethermodynamics of the weak-moment subsystem, and in particular all correlationfunctions, can be derived as polynomials of the correlation functions of thestrong-moment subsystem with coefficients that depend on the spinsusceptibility and that can be calculated within DFT. These conclusions arerigorous under certain physical assumptions on the measure in the magneticphase space. We implement the scheme in the full-potential linearized augmentedplane wave method using the concept of spin-spiral states, consideringapplicable symmetry relations and the use of the magnetic force theorem. Ouranalytical results are corroborated by numerical calculations employing DFT anda Monte Carlo method.

physical review. b, condensed matter and materials physics

Exchange interactions and local moment fluctuation corrections in  ferromagnets at finite temperatures based on non-collinear density-functional  calculations