We provide a new variational definition for the spread of an orbital under periodic boundary conditions (PBCs) that is continuous with respect to the gauge, consistent in the thermodynamic limit, well-suited to diffuse orbitals, and systematically adaptable to schemes computing localized Wannier functions. Existing definitions do not satisfy all these desiderata, partly because they depend on an "orbital center"-an ill-defined concept under PBCs. Based on this theoretical development, we showcase a robust and efficient (10x-70x fewer iterations) localization scheme across a range of materials.

翻译：我们提出了一种在周期性边界条件下轨道扩展度的新变分定义，该定义相对于规范是连续的，在热力学极限下具有一致性，适用于弥散轨道，并且能够系统性地适配于计算局域Wannier函数的方案。现有定义无法满足所有这些要求，部分原因在于它们依赖于“轨道中心”——这一概念在周期性边界条件下缺乏明确定义。基于这一理论进展，我们展示了一种在一系列材料中既稳健又高效（迭代次数减少10倍至70倍）的局域化方案。