An approach to compute exchange parameters of the Heisenberg model in plane-wave-based methods is presented. This calculation scheme is based on the Green's function method and Wannier function projection technique. It was implemented in the framework of the pseudopotential method and tested on such materials as NiO, FeO, ${\mathrm{Li}}_2{\mathrm{MnO}}_3$, and ${\mathrm{KCuF}}_3$. The obtained exchange constants are in a good agreement with both the total energy calculations and experimental estimations for NiO and ${\mathrm{KCuF}}_3$. In the case of FeO our calculations explain the pressure dependence of the Néel temperature. ${\mathrm{Li}}_2{\mathrm{MnO}}_3$ turns out to be a Slater insulator with antiferromagnetic nearest-neighbor exchange defined by the spin splitting. The proposed approach provides a unique way to analyze magnetic interactions, since it allows one to calculate orbital contributions to the total exchange coupling and study the mechanism of the exchange coupling.

The problem of weak ferromagnetism in antiferromagnets due to canting of magnetic moments is treated using Green’s function technique. At first the eigenvalues and eigenfunctions of the electronic Hamiltonian corresponding to collinear magnetic configuration are calculated which are then used to determine the first and the second variations of the total energy as a function of the magnetic moments canting angle. Spin-orbit coupling is taken into account via perturbation theory. The results of calculations are used to determine an effective spin Hamiltonian. This Hamiltonian can be mapped on the conventional spin Hamiltonian that allows one to determine parameters of isotropic and anisotropic (Dzyaloshinskii-Moriya) exchange interactions. The method is applied to the typical antiferromagnets with weak ferromagnetism $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ and ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$. The obtained values of the magnetic moments canting angles are in good agreement with previous theoretical results and are in reasonable agreement with experimental data.

A distinct approach for calculations of Dzyaloshinskii-Moriya interactions in molecules and crystals is proposed. It is based on the exact perturbation expansion of the total energy of weak ferromagnets in the canting angle with the only assumption of local Hubbard-type interactions. This scheme leads to a simple and transparent analytical expression for the Dzyaloshinskii-Moriya vector with a natural separation into spin and orbital contributions. The main problem was transferred to calculations of effective tight-binding parameters in the properly chosen basis including the spin-orbit coupling. Test calculations for ${\text{La}}_2{\text{CuO}}_4$ give the value of the canting angle in agreement with experimental data.

Significance: Structural complexity of patterns, systems, and processes is a very basic and intuitively clear concept in human’s perception of reality that is very difficult to describe quantitatively. A demand in a mathematical notion that properly reflects complexity of hierarchical nonrandom structures exists in many areas of science, from geology to social sciences. Here, we propose an easy to compute, robust, and universal definition of complexity based on interscale dissimilarity of patterns. Using classical magnetic patterns as an example, we demonstrate that our approach leads to maximization of complexity for the most visually nontrivial patterns and can be used to detect phase transitions with high accuracy, making it a promising tool for studying pattern formation in a variety of systems.