  We investigate abilities of various linear response based techniques for
extracting parameters of antisymmetric Dzyaloshinskii-Moriya (DM) interactions
from the first-principles electronic structure calculations. For these
purposes, we further elaborate the idea of Sandratskii [Phys. Rev. B 96, 024450
(2017)], which states that $z$ component of the DM vector can be computed by
retaining only spin-diagonal part of the spin-orbit (SO) coupling. We start our
analysis with the magnetic force theorem (MFT), which relies on additional
approximations resulting in the linear dependence of the exchange interactions
on the response tensor, and compare it with the exact approach formulated in
terms of the inverse response. We propose the downfolding procedure
transferring the effect of these ligand spins into parameters of effective
interactions between the localized spins. These techniques are applied for the
series of CrCl$_3$ and CrI$_3$ based materials, including bulk, monolayer,
bilayer, and three-layer systems. Particularly, we discuss how the DM
interactions are induced by the inversion symmetry breaking at the surface or
by the electric field. As long as the SO interaction of the heavy ligand atoms
is taken into account in the calculations of the response tensor between the Cr
$3d$ states, the MFT appears to be a good approximation for the DM
interactions, being in contrast with the isotropic exchange, for which MFT and
the exact method provide quite a different description. Finally, we discuss the
relevance of our approach to other techniques ever proposed for calculations of
the DM interactions. Particularly, we argue that the spin-current model for the
DM interactions can be derived from the MFT based expression and is the
relativistic counterpart of the double exchange, occurring in metallic systems
in the limit of infinite exchange splitting.
