  Linear-scaling implementations of density functional theory (DFT) reach their
intended efficiency regime only when applied to systems having a physical size
larger than the range of their Kohn-Sham density matrix (DM). This causes a
problem since many types of large systems of interest have a rather broad DM
range and are therefore not amenable to analysis using DFT methods. For this
reason, the recently proposed stochastic DFT (sDFT), avoiding exhaustive DM
evaluations, is emerging as an attractive alternative linear-scaling approach.
This review develops a general formulation of sDFT in terms of a
(non)orthogonal basis representation and offers an analysis of the statistical
errors (SEs) involved in the calculation. Using a new Gaussian-type basis-set
implementation of sDFT, applied to water clusters and silicon nanocrystals, it
demonstrates and explains how the standard deviation and the bias depend on the
sampling rate and the system size in various types of calculations. We also
develop basis-set embedded-fragments theory, demonstrating its utility for
reducing the SEs for energy, density of states and nuclear force calculations.
Finally, we discuss the algorithmic complexity of sDFT, showing it has CPU
wall-time linear-scaling. The method parallelizes well over distributed
processors with good scalability and therefore may find use in the upcoming
exascale computing architectures.
