The modern quantum theory of magnetism in solids is getting commonly derived using Green's functions formalism. The popularity draws itself from remarkable opportunities to capture the microscopic landscape of exchange interactions, starting from a tightbinding representation of the electronic structure. Indeed, the conventional method of infinitesimal spin rotations, considered in terms of local force theorem, opens vast prospects of investigations regarding the magnetic environment, as well as pairwise atomic couplings. However, this theoretical concept practically does not devoid of intrinsic inconsistencies. In particular, naturally expected correspondence between single and pairwise infinitesimal spin rotations is being numerically revealed to diverge. In this work, we elaborate this question on the model example and canonical case of bcc iron. Our analytical derivations discovered the principal preference of on-site magnetic precursors if the compositions of individual atomic interactions are in focus. The problem of extremely slow or even absent spatial convergence while considering metallic compounds was solved by suggesting the original technique, based on reciprocal space framework. Using fundamental Fourier transform-inspired interconnection between suggested technique and traditional spatial representation, we shed light on symmetry breaking in bcc Fe on the level of orbitally decomposed total exchange surrounding.

In this work we conduct a numerical search of non-trivial mechanisms, leading to new tendencies towards long-range ferromagnetic ordering in two-dimensional materials. For this purpose we employ an original variant of pairwise infinitesimal spin rotations technique to establish the magnetic transition temperature as the rigid function of basic crystal's parameters. It favored the numerical optimization of this function using modified genetic algorithm, designed to harvest local extrema. It resulted in revealing the moderate metallicity, accompanied by essential orbital anisotropy, as the prime configuration, which provides the most favoring conditions to ferromagnetic ordering, related to double-exchange and superexchange mechanisms.