Automatic Generation of Maximally Localized Wannier Functions via Optimized Projection Functions and Self-projections

We present an automatized approach towards maximally localized Wannier functions (MLWFs) applicable to both occupied and unoccupied states. We overcome limitations of the standard optimized projection function (OPF) method and its approximations by providing an exact expression for the gradient of the Wannier spread functional with respect to a single semi-unitary OPF matrix. Moreover, we demonstrate that the localization of the resulting Wannier functions (WFs) can be further improved by including projections on reasonably localized WFs, so-called self-projections.