  Stochastic density functional theory (sDFT) is becoming a valuable tool for
studying ground state properties of extended materials. The computational
complexity of describing the Kohn-Sham orbitals is replaced by introducing a
set of random (stochastic) orbitals leading to linear and often sub-linear
scaling of certain ground-state observable at the account of introducing a
statistical error. Schemes to reduce the noise are essential, for example, for
determining the structure using the forces obtained from sDFT. Recently we have
introduced two embedding schemes to mitigate the statistical fluctuations in
the electron density and resultant forces on the nuclei. Both techniques were
based on fragmenting the system either in real-space or slicing the occupied
space into energy windows, allowing for a significant reduction of the
statistical fluctuations. For chemical accuracy further reduction of the noise
is required, which could be achieved by increasing the number of stochastic
orbitals. However, the convergence is relatively slow as the statistical error
scales as $1/\sqrt{N_\chi}$ according to the central limit theorem, where
$N_\chi$ is the number of random orbitals. In this paper we combined the
aforementioned embedding schemes and introduced a new approach that builds on
overlapped fragments and energy windows. The new approach significantly lowers
the noise for ground state properties such as the electron density, total
energy, and forces on the nuclei, as demonstrated for a G-center in bulk
silicon.
