Here, we derive rigorously a description of magnetic inertia in the extended Landau-Lifshitz-Gilbert (LLG) equation starting from a fundamental and relativistic Dirac-Kohn-Sham framework. We use a unitary transformation, the so called called Foldy-Wouthuysen transformation, up to the order of 1/c4. In this way, the particle and anti-particle in fully relativistic description become decoupled and a Hamiltonian describing only the particles is derived. This Hamiltonian involves the nonrelativistic Schrodinger-Pauli Hamiltonian together with the relativistic corrections of the order 1/c2 and 1/c4.
With the thus-derived Hamiltonian, we calculate the corresponding spin dynamics leading to the LLG equation of motion. Our result exemplify that the relativistic correction terms of 1/c2 are responsible for the Gilbert damping, however, the relativistic correction terms of 1/c4 are responsible for magnetic inertial dynamics. Therefore, we predict that the intrinsic magnetic inertia is a higher-order relativistic spin-orbit coupling effect and is expected to be prominent only on ultrashort timescales (subpicoseconds). We also show that the corresponding Gilbert damping and magnetic inertia parameters are related to one another through the imaginary and real parts of the magnetic susceptibility tensor, respectively.