  Over this past decade, we combined the idea of stochastic resolution of
identity with a variety of electronic structure methods. In our stochastic
Kohn-Sham DFT method, the density is an average over multiple stochastic
samples, with stochastic errors that decrease as the inverse square root of the
number of sampling orbitals. Here we develop a stochastic embedding density
functional theory method (se-DFT) that selectively reduces the stochastic error
(specifically on the forces) for a selected sub-system(s). The motivation,
similar to that of other quantum embedding methods, is that for many systems of
practical interest the properties are often determined by only a small
sub-system. In stochastic embedding DFT two sets of orbitals are used: a
deterministic one associated with the embedded subspace, and the rest which is
described by a stochastic set. The method is exact in the limit of large number
of stochastic samples. We apply se-DFT to study a p-nitroaniline molecule in
water, where the statistical errors in the forces on the system (the
p-nitroaniline molecule) are reduced by an order of magnitude compared with
non-embedding stochastic DFT.
