Atomistic spin dynamic method with both damping and moment of inertia effects included from first principles

We consider spin dynamics for implementation in an atomistic framework and we address the feasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In the spirit of an {\it s-d} -like interaction between the magnetization and electron spin, we derive a generalized equation of motion for the magnetization dynamics in the semi-classical limit, which is non-local in both space and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation, also including the moment of inertia, and demonstrate how the exchange interaction, damping, and moment of inertia, all can be calculated from first principles.