  We introduce a tempering approach with stochastic density functional theory
(sDFT), labeled t-sDFT, which reduces the statistical errors in the estimates
of observable expectation values. This is achieved by rewriting the electronic
density as a sum of a "warm" component complemented by "colder" correction(s).
Since the "warm" component is larger in magnitude but faster to evaluate, we
use many more stochastic orbitals for its evaluation than for the smaller-sized
colder correction(s). This results in a significant reduction of the
statistical fluctuations and the bias compared to sDFT for the same
computational effort. We the method's performance on large hydrogen-passivated
silicon nanocrystals (NCs), finding a reduction in the systematic error in the
energy by more than an order of magnitude, while the systematic errors in the
forces are also quenched. Similarly, the statistical fluctuations are reduced
by factors of around 4-5 for the total energy and around 1.5-2 for the forces
on the atoms. Since the embedding in t-sDFT is fully stochastic, it is possible
to combine t-sDFT with other variants of sDFT such as energy-window sDFT and
embedded-fragmented sDFT.
