Spread-balanced Wannier functions: Robust and automatable orbital localization

We introduce a type of Wannier functions (WFs) obtained by minimizing the conventional spread functional with a penalty term proportional to the variance of the spread distribution. This modified Wannierization scheme is less prone to produce ineffective solutions featuring one or several poorly localized orbitals, making it well suited for complex systems or high-throughput applications. Furthermore, we propose an automatable protocol for selecting the initial guess and determine the optimal number of bands (or equivalently, WFs) for the localization algorithm. The improved performance and robustness of the approach is demonstrated for a diverse set of test systems including the nitrogen-vacancy center in diamond, metal slabs with atomic adsorbates, spontaneous polarization of ferroelectrics, and 30 inorganic monolayer materials comprising both metals and semiconductors. The methods are implemented in Python as part of the Atomic Simulation Environment.