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We propose a systematic and sequential expansion of theLandau-Lifshitz-Gilbert equation utilizing the dependence of the Gilbertdamping tensor on the angle between magnetic moments, which arises frommulti-body scattering processes. The tensor consists of a damping-like term anda correction to the gyromagnetic ratio. Based on electronic structure theory,both terms are shown to depend on e.g. the scalar, anisotropic, vector-chiraland scalar-chiral products of magnetic moments: $\vec{e}_i\cdot\vec{e}_j$,$(\vec{n}_{ij}\cdot\vec{e}_i)(\vec{n}_{ij}\cdot\vec{e}_j)$,$\vec{n}_{ij}\cdot(\vec{e}_i\times\vec{e}_j)$, $(\vec{e}_i\cdot\vec{e}_j)^2$,$\vec{e}_i\cdot(\vec{e}_j\times\vec{e}_k)$..., where some terms are subjectedto the spin-orbit field $\vec{n}_{ij}$ in first and second order. We explorethe magnitude of the different contributions using both the Alexander-Andersonmodel and time-dependent density functional theory in magnetic adatoms anddimers deposited on Au(111) surface.

Generalization of the Landau-Lifshitz-Gilbert equation by multi-body contributions to Gilbert damping for non-collinear magnets