  Recently, two nonempirical hybrid functionals, dielectric-dependent
range-separated hybrid functional based on the Coulomb-attenuating method
(DD-RSH-CAM) and doubly screened hybrid functional (DSH), have been suggested
by [Chen et al, Phys. Rev. Mater. 2, 073803 (2018)] and [Cui et al, J. Phys.
Chem. Lett. 9, 2338 (2018)], respectively. These two hybrid functionals are
both based on a common model dielectric function approach, but differ in the
way how to non-empirically obtain the range-separation parameter. By retaining
the full short-range Fock exchange and a fraction of the long-range Fock
exchange that equals the inverse of the dielectric constant, both DD-RSH-CAM
and DSH turn out to perform very well in predicting the band gaps for a large
variety of semiconductors and insulators. Here, we assess how these two hybrid
functionals perform on challenging antiferromagnetic transition-metal monoxides
MnO, FeO, CoO, and NiO by comparing them to other conventional hybrid
functionals and the $GW$ method. We find that single-shot DD0-RSH-CAM and DSH0
improve the band gaps towards experiments as compared to conventional hybrid
functionals. The magnetic moments are slightly increased, but the predicted
dielectric constants are decreased. The valence band density of states (DOS)
predicted by DD0-RSH-CAM and DSH0 are as satisfactory as HSE03 in comparison to
experimental spectra, however, the conduction band DOS are shifted to higher
energies by about 2 eV compared to HSE03. Self-consistent DD-RSH-CAM and DSH
deteriorate the results with a significant overestimation of band gaps.
