  We develop a stochastic formulation of the optimally-tuned range-separated
hybrid density functional theory which enables significant reduction of the
computational effort and scaling of the non-local exchange operator at the
price of introducing a controllable statistical error. Our method is based on
stochastic representations of the Coulomb convolution integral and of the
generalized Kohn-Sham density matrix. The computational cost of the approach is
similar to that of usual Kohn-Sham density functional theory, yet it provides
much more accurate description of the quasiparticle energies for the frontier
orbitals. This is illustrated for a series of silicon nanocrystals up to sizes
exceeding 3000 electrons. Comparison with the stochastic GW many-body
perturbation technique indicates excellent agreement for the fundamental band
gap energies, good agreement for the band-edge quasiparticle excitations, and
very low statistical errors in the total energy for large systems. The present
approach has a major advantage over one-shot GW by providing a self-consistent
Hamiltonian which is central for additional post-processing, for example in the
stochastic Bethe-Salpeter approach.
