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In 1987, Liechtenstein et al. came up with the idea to formulate the problemof interatomic exchange interactions, which would describe the energy changecaused by the infinitesimal rotations of spins, in terms of the magneticsusceptibility. The formulation appears to be very generic and, for isotropicsystems, expresses the energy change in the form of the Heisenberg model,irrespectively on which microscopic mechanism stands behind the interactionparameters. Moreover, this approach establishes the relationship between theexchange interactions and the electronic structure obtained, for instance, inthe first-principles calculations based on the density functional theory. Thepurpose of this review is to elaborate basic ideas of the linear responsetheories for the exchange interactions as well as more recent developments. Thespecial attention is paid to the approximations underlying the original methodof Liechtenstein et al. in comparison with its more recent and more rigorousextensions, the roles of the on-site Coulomb interactions and the ligandstates, and calculations of antisymmetric Dzyaloshinskii-Moriya interactions,which can be performed alongside with the isotropic exchange, within onecomputational scheme. The abilities of the linear response theories as well asmany theoretical nuances, which may arise in the analysis of interatomicexchange interactions, are illustrated on magnetic van der Walls materialsCr$X_3$ ($X$$=$ Cl, I), half-metallic ferromagnet CrO$_2$, ferromagnetic Weylsemimetal Co$_3$Sn$_2$S$_2$, and orthorhombic manganites $A$MnO$_3$ ($A$$=$ La,Ho), known for the peculiar interplay of the lattice distortion, spin, andorbital ordering.

Linear response theories for interatomic exchange interactions