  In many cases, the relativistic spin-orbit (SO) interaction is regarded to be
small and can be treated using perturbation theory. The major obstacle on this
route comes from the fact that the SO interaction can also polarize the
electron system and produce additional contributions to the perturbation
theory, arising from the electron-electron interactions. In electronic
structure calculations, it may even lead to necessity to abandon the
perturbation theory and return to the self-consistently solution of
Kohn-Sham-like equations with the effective potential $\hat{v}$, incorporating
the effects of the electron-electron interactions and the SO coupling, even
though the latter is small. In this work, we present the theory of
self-consistent linear response (SCLR), which allows us to get rid of numerical
self-consistency and formulate it analytically in the first order of the SO
coupling. This strategy is applied to the Hartree-Fock solution of the
effective Hubbard model, derived from electronic structure calculations in the
Wannier basis. By using $\hat{v}$, obtained from SCLR, one can successfully
reproduce results of ordinary calculations for the orbital magnetization and
other properties in the first order of the SO coupling. Particularly, SCLR
appears to be extremely useful for calculations of antisymmetric
Dzyaloshinskii-Moriya (DM) interactions based on the magnetic force theorem.
Furthermore, due to the powerful 2n+1 theorem, the SCLR theory allows us to
calculate the magnetic anisotropy energy up to the third order of the SO
coupling. The fruitfulness of this approach is illustrated on a number of
example, including the spin canting in YTiO$_3$ and LaMnO$_3$, formation of
spiral magnetic order in BiFeO$_3$, and the magnetic inversion symmetry
breaking in BiMnO$_3$, which gives rise to both ferroelectric activity and DM
interactions, responsible for the ferromagnetism.
