  We critically reexamine the problem of interatomic exchange interactions,
which describe the total energy change caused by infinitesimal rotations of
spins near some equilibrium state. For the small variations, such interactions
can be always related to the response function. However, the form of this
relation can depend on additional approximations. Particularly, the commonly
used magnetic force theorem (MFT) prescribes the linear relation between the
exchange interactions and the response function, while the exact theory
requires this dependence to be inverse. We explore the origin and consequences
of these differences in the definition for the wide class of materials:
ferromagnetic Ni, antiferromagnetic NiO, half-metallic CrO2, multiferroic
HoMnO3, and layered magnets CrCl3 and CrI3. While in most of these cases, MFT
produces quite reasonable results and can be rigorously justifies in the long
wavelength and strong-coupling limits, the exact formulation appears to be more
consistent, especially in dealing with two important issues, which typically
arise in the theory of exchange interactions: (i) the treatment of the ligand
states, and (ii) the choice of the suitable variable for the description of
infinitesimal rotations of spins. Both issues can be efficiently resolved by
employing the ideas of adiabatic spin dynamics supplemented with the exact
expression for the exchange interactions. Particularly, we propose a simple
"downfolding" procedure for the elimination of the ligand spins by transferring
their effect to the interaction parameters between the localized spins.
Furthermore, we argue that the rotations of spin moments are more suitable for
the description of low-energy excitations, while the rotations of the whole
magnetization matrix cause much stronger perturbation in the system of spins.
