  We develop an efficient approach to evaluate range-separated exact exchange
for grid or plane-wave based representations within the Generalized Kohn-Sham
DFT (GKS-DFT) framework. The Coulomb kernel is fragmented in reciprocal space,
and we employ a mixed deterministic-stochastic representation, retaining long
wavelength (low-$k$) contributions deterministically and using a sparse
("fragmented") stochastic basis for the high-$k$ part. Coupled with a
projection of the Hamiltonian onto a subspace of valence and conduction states
from a prior local-DFT calculation, this method allows for the calculation of
long-range exchange of large molecular systems with hundreds and potentially
thousands of coupled valence states delocalized over millions of grid points.
We find that even a small number of valence and conduction states is sufficient
for converging the HOMO and LUMO energies of the GKS-DFT. Excellent tuning of
long-range separated hybrids (RSH) is easily obtained in the method for very
large systems, as exemplified here for the chlorophyll hexamer of Photosystem
II with 1,320 electrons.
