The spin stiffness (exchange stiffness, spin-wave stiffness) belongs to the most important ground-state properties of itinerant ferromagnets. Its value controls, e.g., the temperature dependence of magnetization at low temperatures, the magnon dispersion law for long wavelengths, or the width of magnetic domain walls. In this contribution, we sketch an ab initio theory of the spin-wave stiffness tensor for itinerant ferromagnets with pair exchange interactions derived from the magnetic force theorem . The resulting formula  involves one-particle Green's functions and effective velocity operators appearing in a recent theory of electron transport . Application of this approach to ideal metal crystals allows one to overcome the problem of ill-convergent lattice summations, as documented by results for bcc Fe, hcp Co, and fcc Ni. Application to random alloys within the coherent potential approximation, illustrated by results for fcc Ni-Fe and bcc Fe-Al systems, enables one to include the disorder-induced vertex corrections, often neglected in evaluation of the exchange interactions.
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