 
Summary: COMMUN. MATH. SCI. c 2007 International Press
Vol. 5, No. 4, pp. 9991026
A SUBLINEAR SCALING ALGORITHM FOR COMPUTING THE
ELECTRONIC STRUCTURE OF MATERIALS
CARLOS J. GARC´IACERVERA, JIANFENG LU, AND WEINAN E§
Abstract. We introduce a class of sublinear 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 nonsmooth region. For orbitalfree 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
nonsmooth region. This kind of partition is not used for KohnSham 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.
Key words. sublinear scaling algorithms, asymptotics, DFTcontinuum approximation, density
functional theory
AMS subject classifications. 35Q40, 74Q05, 34E05
