We introduce a class of sub-linear scaling algorithms for analyzing the electronic structure of crystalline solids with isolated defects. We divide the localized orbitals of the electrons into two sets: one set associated with the atoms in the region where the deformation of the material is smooth (smooth region), and the other set associated with the atoms around the defects (non- smooth region). The orbitals associated with atoms in the smooth region can be approximated accurately using asymptotic analysis. The results can then be used in the original formulation to find the orbitals in the non-smooth region. For orbital-free density functional theory, one can simply partition the electron density into a sum of the density in the smooth region and a density in the non-smooth region. This kind of partition is not used for Kohn-Sham density functional theory and one uses instead the partition of the set of orbitals. As a byproduct, we develop the necessary real space formulations and we present a formulation of the electronic structure problem for a subsystem, when the electronic structure for the remaining part is known.
"A sub-linear scaling algorithm for computing the electronic structure of materials." Commun. Math. Sci. 5 (4) 999 - 1026, December 2007.