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.

中文翻译：

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扩展平衡的 Wannier 函数：稳健且可自动化的轨道定位

我们引入了一种 Wannier 函数 (WFs)，它通过最小化常规扩展函数获得，惩罚项与扩展分布的方差成正比。这种修改后的万尼尔化方案不太容易产生具有一个或几个局部性差的轨道的无效解决方案，使其非常适合复杂系统或高通量应用。此外，我们提出了一个可自动化的协议，用于选择初始猜测并确定定位算法的最佳波段数（或等效的 WF）。该方法的改进性能和稳健性在一系列不同的测试系统中得到了证明，包括金刚石中的氮空位中心、具有原子吸附物的金属板、铁电体的自发极化、30种无机单层材料，包括金属和半导体。这些方法是在 Python 中实现的，作为原子模拟环境的一部分。