Ultra Long-Range Ab Initio Calculations

We propose a generalization of the Bloch state which involves an additional sum over
a finer grid in reciprocal space around each k-point. This allows for ab-initio
calculations of ultra long-range modulations in the density which may involve millions
of unit cells but with an efficiency rivaling that of single unit cell.
Physical effects on the micron length scale, which nevertheless
depend on details of the electronic structure on nanometer length scales, can be
computed exactly within density functional theory. As a specific example, we apply our method
to a spatially varying field extending far beyond the length scale of a single unit cell.