Panorama

DFT is the workhorse of any modern computational materials, but it is limited by its computational implementation. Even with High-Performance-Computing centers, the ever-growing demands of materials science makes DFT calculations expensive (sometimes even prohibitely expensive).

I work with a technique called Wannierization that can be applied after an inexpensive DFT calculation, albeit not highly accurate, in order to reduce the complexity of any a posteriori calculation that requires higher accuracy (say interpolation, magnetic-force-theorem, etc.).

Wannierization in itself is an interesting problem, but the main goal of using such tools is to discover new materials with interesting magnetic behaviour (high-throughput searches).

Questions we want to answer

Picture the magnetic exchange interaction. Starting from a very simple two-body configuration, it is through localized orbitals that one obtains an exchange of the form

$$ J=J_D + \frac{U}{4}-\sqrt{t^2+\frac{U^2}{16}}, $$

where $U$ and $t$ are integrals related to the on-site Coulomb interaction and the hopping parameter. From the picture below, it

Of course, this is anlogous to how exchange is studied in molecules. Here, a localized atomic orbital basis may suffice (though not always). In solids, however, Wannier functions are a complete description of the electronic structure. In other words, there is a direct way to build these Wannier orbitals starting from the Bloch wavefunctions. In practice, this construction is obstructed by numerous details that fit better in another post. Furthermore, assuming that one gets the localized orbital picture, the methods for obtaining $J$ starting from such basis functions is also its own challenge.

Nonetheless, expressions like this bring a lot of intuition (e.g. an orbital-decomposed $J$), and while their predictions might be off for certain materials, these limits also tell a nice story of what’s going on with the electrons in a material, and that’s what makes them valuable.

More standard applications of Wannier functions are interpolating band-structure (in a Slater-Koster sense), evaluation of transport quantities (from a Berry phase approach).