  The stochastic density functional theory (DFT) [Phys. Rev. Lett. 111, 106402
(2013)] is a valuable linear scaling approach to Kohn-Sham DFT that does not
rely on the sparsity of the density matrix. Linear (and often sub-linear)
scaling is achieved by introducing a controlled statistical error in the
density, energy and forces. The statistical error (noise) is proportional to
the inverse square root of the number of stochastic orbitals and thus decreases
slowly, however, by dividing the system to fragments that are embedded
stochastically, the statistical error can be reduced significantly. This has
been shown to provide remarkable results for non-covalently bonded systems,
however, the application to covalently bonded systems had limited success,
particularly for delocalized electrons. Here, we show that the statistical
error in the density correlates with both the density and the density matrix of
the system and propose a new fragmentation scheme that elegantly interpolates
between overlapped fragments. We assess the performance of the approach for
bulk silicon of varying supercell sizes (up to $N_{e}=16384$ electrons) and
show that overlapped fragments reduce significantly the statistical noise even
for systems with a delocalized density matrix.
