Relativistic magnetic interactions from nonorthogonal basis sets

We propose a method to determine the magnetic exchange interaction and onsite anisotropy tensors of extended Heisenberg spin models from density functional theory including relativistic effects. The method is based on the Liechtenstein-Katsnelson-Antropov-Gubanov torque formalism, whereby energy variations upon infinitesimal rotations are performed. We assume that the Kohn-Sham Hamiltonian is expanded in a nonorthogonal basis set of pseudoatomic orbitals. We define local operators that are both Hermitian and satisfy relevant sum rules. We demonstrate that in the presence of spin-orbit coupling a correct mapping from the density functional total energy to a spin model that relies on the rotation of the exchange field part of the Hamiltonian can not be accounted for by transforming the full Hamiltonian. We derive a set of sum rules that pose stringent validity tests on any specific calculation. We showcase the flexibility and accuracy of the method by computing the exchange and anisotropy tensors of both well-studied magnetic nanostructures and of recently synthesized two-dimensional magnets. Specifically, we benchmark our approach against the established Korringa-Kohn-Rostoker Green's function method and show that they agree well. Finally, we demonstrate how the application of biaxial strain on the two-dimensional magnet $\mathrm{T}\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$ can trigger a magnetic phase transition.