Usually, the adiabatic magnetic exchange-energy hypersurface is parametrized in terms of the bilinear Heisenberg interactions in pairs of atoms. For general magnetic configurations, this model is not complete even if it includes pairs with up to infinite interatomic distances. In contrast, the modeling by an in principle infinite spin-cluster expansion is complete for all conceivable magnetic configurations. In the present Letter, it is shown for bcc and fcc iron that a very accurate representation can be reached with a finite expansion with 20 terms which include biquadratic or multispin non-Heisenberg interactions.

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In the conventional s-d model of magnetization dynamics, it is assumed that for an adiabatic situation the s magnetization is ferromagnetically aligned to the d magnetization. For configurations with strong noncollinearity in the d magnetization, this no longer holds true for all cases. In the present paper it is shown that, as a consequence of this noncollinearity between the adiabatic s and d magnetizations, there arise several additional torques in the micromagnetic equation of motion for the d magnetization. The equation is solved for a Néel wall driven by an external field and/or a spin-polarized current, yielding a correction term to the result of the conventional theory. By calculations with the ab initio density-functional electron theory, the magnitude of the correction term is estimated. It is concluded that for 3d metals the effects of the additional torques are very small except possibly for atomic-scale noncollinearities or for the long-term trajectories of complicated magnetization configurations.The dynamics of noncollinear magnetization configurations driven by external magnetic fields or spin-polarized currents is often investigated by micromagnetic simulations with the Landau-Lifshitz-Gilbert ͑LLG͒ equation of motion. 1 The influence of a spin-polarized transport current thereby is treated by including two spin-torque terms ͑an adiabatic and a nonadiabatic spin torque͒ into the LLG equation. These two terms which originally have been introduced in a phenomenological manner 2 have been justified within the s-d model of Zhang and Li. 3 In the s-d model the electronic states are subdivided into "s" states and "d" states. The "s" states represent those electronic states ͑conduction electrons states͒ which dominate the electrical conductivity. For 3d ͑4f͒ metals these are the 4s and 4p ͑5d, 6s, and 6p͒ states which are strongly delocalized, i.e., itinerant. The magnetization related to these states is denoted by m͑r , t͒. The symbol "d" stands for the more localized states which are the 3d ͑4f͒ states and which contribute a magnetization denoted as M d ͑r , t͒ so that the total magnetization is given by M͑r,t͒ = M d ͑r,t͒ + m͑r,t͒. ͑1͒In the s-d model the dynamics of M d ͑r , t͒ is calculated in two steps. In the first step the response of m͑r , t͒ to a given M d ͑r , t͒ is calculated. As soon as m͑r , t͒ is not completely aligned to M d ͑r , t͒, a torque density will be exerted on M d ͑r , t͒ via the s-d exchange interaction, 3In Eq. ͑3͒ S is the effective spin quantum number which is related to the localized atomic moments, and J ex denotes the s-d exchange coupling. In the second step the dynamics of M d is determined from a LLG-type equation of motion,where ␥ is the gyromagnetic ratio, H eff,d is the effective field acting on M d , and ␣ denotes the Gilbert damping constant.In the conventional s-d model 3 m͑r , t͒ is not completely aligned to M d ͑r , t͒ because the conduction electrons experience spin-flip scattering processes ͓characterized by a spinflip scattering time sf , see Eq. ͑7͔͒ and t...