Extracting effective spin-orbit coupling from first principles calculations via the Wannier representation

The construction of Wannier Hamiltonians (WHs) from density-functional calculations can play an important role in theoretical investigations of condensed-matter systems by combining the virtues of an intuitive tight-binding representation with underlying parameter-free first-principles calculations. Such WHs are in widespread use for Wannier interpolation and extraction of topological properties of band structures. Here, instead, we focus on useful local chemical and physical information that can be extracted from the WH. Site energies and interatomic hoppings are easily extracted from the matrix elements of the WH, but more complicated terms such as spin-orbit coupling (SOC) and crystal field splitting require more care. Here we propose a systematic approach for extracting the strength of such complex operators from a WH, and illustrate its power by applying it to several example systems. These include Fe₂S₂, in which the effective SOC was recently reported [1] to exhibit a giant dependence on the Hubbard on-site U. We quantitatively confirm that this is largely an artifact of an instability to the formation of orbital magnetic order at large U values, and that the dependence of the true SOC strength on U is more modest, though still significant.