This review addresses the method of explicit calculations of interatomic exchange interactions of magnetic materials. This involves exchange mechanisms normally referred to as a Heisenberg exchange, a Dzyaloshinskii-Moriya interaction, and an anisotropic symmetric exchange. The connection between microscopic theories of the electronic structure, such as density functional theory and dynamical mean-field theory, and interatomic exchange is examined. The different aspects of extracting information for an effective spin Hamiltonian that involves thousands of atoms, from electronic structure calculations considering significantly fewer atoms (1–50), is highlighted. Examples of exchange interactions of a large group of materials is presented, which involves heavy elements of the $3d$ period, alloys between transition metals, Heusler compounds, multilayer systems as well as overlayers and adatoms on a substrate, transition metal oxides, $4f$ elements, magnetic materials in two dimensions, and molecular magnets. Where possible, a comparison to experimental data is made that becomes focused on the magnon dispersion. The influence of relativity is reviewed in a few cases, as is the importance of dynamical correlations. Development to theories that handle out-of-equilibrium conditions is also described here. The review ends with a description of extensions of the theories behind explicit calculations of interatomic exchange to nonmagnetic situations, such as those that describe chemical (charge) order and superconductivity.

中文翻译：

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固体中磁相互作用的定量理论

本综述讨论了磁性材料原子间交换相互作用的显式计算方法。这涉及通常称为海森堡交换、Dzyaloshinskii-Moriya 相互作用和各向异性对称交换的交换机制。研究了电子结构的微观理论（例如密度泛函理论和动态平均场理论）与原子间交换之间的联系。强调了从电子结构计算中提取涉及数千个原子的有效自旋哈密顿量的信息的不同方面，考虑到更少的原子（1-50）。给出了一大群材料的交换相互作用的例子，其中涉及重元素$3d$时期，过渡金属之间的合金，赫斯勒化合物，多层系统以及基底上的覆盖层和吸附原子，过渡金属氧化物，$4F$元素、二维磁性材料和分子磁体。在可能的情况下，与实验数据进行比较，重点关注磁振子色散。在一些案例中回顾了相对论的影响，以及动态相关性的重要性。这里还描述了处理非平衡条件的理论的发展。综述最后描述了原子间交换到非磁性情况的显式计算背后的理论扩展，例如那些描述化学（电荷）秩序和超导性的理论。