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.

• Received 3 June 2022

DOI:https://doi.org/10.1103/RevModPhys.95.035004