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A Kubo-Greenwood-like equation for the Gilbert damping parameter $\alpha$ ispresented that is based on the linear response formalism. Its implementationusing the fully relativistic Korringa-Kohn-Rostoker (KKR) band structure methodin combination with Coherent Potential Approximation (CPA) alloy theory allowsit to be applied to a wide range of situations. This is demonstrated withresults obtained for the bcc alloy system Fe$_x$Co$_{1-x}$ as well as for aseries of alloys of permalloy with 5d transition metals.  To account for the thermal displacements of atoms as a scattering mechanism,an alloy-analogy model is introduced. The corresponding calculations for Nicorrectly describe the rapid change of $\alpha$ when small amounts ofsubstitutional Cu are introduced.

Ab-initio calculation of the Gilbert damping parameter via linear response formalism