  In 1987, Liechtenstein et al. came up with the idea to formulate the problem
of interatomic exchange interactions, which would describe the energy change
caused by the infinitesimal rotations of spins, in terms of the magnetic
susceptibility. The formulation appears to be very generic and, for isotropic
systems, expresses the energy change in the form of the Heisenberg model,
irrespectively on which microscopic mechanism stands behind the interaction
parameters. Moreover, this approach establishes the relationship between the
exchange interactions and the electronic structure obtained, for instance, in
the first-principles calculations based on the density functional theory. The
purpose of this review is to elaborate basic ideas of the linear response
theories for the exchange interactions as well as more recent developments. The
special attention is paid to the approximations underlying the original method
of Liechtenstein et al. in comparison with its more recent and more rigorous
extensions, the roles of the on-site Coulomb interactions and the ligand
states, and calculations of antisymmetric Dzyaloshinskii-Moriya interactions,
which can be performed alongside with the isotropic exchange, within one
computational scheme. The abilities of the linear response theories as well as
many theoretical nuances, which may arise in the analysis of interatomic
exchange interactions, are illustrated on magnetic van der Walls materials
Cr$X_3$ ($X$$=$ Cl, I), half-metallic ferromagnet CrO$_2$, ferromagnetic Weyl
semimetal Co$_3$Sn$_2$S$_2$, and orthorhombic manganites $A$MnO$_3$ ($A$$=$ La,
Ho), known for the peculiar interplay of the lattice distortion, spin, and
orbital ordering.
