  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.
