  We explore the derivation of interatomic exchange interactions in
ferromagnets within density-functional theory (DFT) and the mapping of DFT
results onto a spin Hamiltonian. We delve into the problem of systems
comprising atoms with strong spontaneous moments together with atoms with weak
induced moments. All moments are considered as degrees of freedom, with the
strong moments thermally fluctuating only in angle and the weak moments
thermally fluctuating in angle and magnitude. We argue that a quadratic
dependence of the energy on the weak local moments magnitude, which is a good
approximation in many cases, allows for an elimination of the weak-moment
degrees of freedom from the thermodynamic expressions in favor of a
renormalization of the Heisenberg interactions among the strong moments. We
show that the renormalization is valid at all temperatures accounting for the
thermal fluctuations and resulting in temperature-independent renormalized
interactions. These are shown to be the ones directly derived from total-energy
DFT calculations by constraining the strong-moment directions, as is done e.g.
in spin-spiral methods. We furthermore prove that within this framework the
thermodynamics of the weak-moment subsystem, and in particular all correlation
functions, can be derived as polynomials of the correlation functions of the
strong-moment subsystem with coefficients that depend on the spin
susceptibility and that can be calculated within DFT. These conclusions are
rigorous under certain physical assumptions on the measure in the magnetic
phase space. We implement the scheme in the full-potential linearized augmented
plane wave method using the concept of spin-spiral states, considering
applicable symmetry relations and the use of the magnetic force theorem. Our
analytical results are corroborated by numerical calculations employing DFT and
a Monte Carlo method.
