Gapped-filtering for efficient Chebyshev expansion of the density projection operator

We develop the gapped-filtering method, whereby a short Chebyshev expansion accurately represents density-matrix operator. The method optimizes coefficients to give correct density matrix at all energies except within gapped region where there are no eigenstates. Gapped filtering reduces number of required terms in compared traditional methods, as long one knows or can determine efficiently HOMO and LUMO positions. reduction is especially noticeable (factors 2-3) when high accuracy sought. To exemplify we use increase efficiency stochastic-GW calculations.