  Various methods going beyond density-functional theory (DFT), such as DFT+U,
hybrid functionals, meta-GGAs, GW, and DFT-embedded dynamical mean field theory
(eDMFT), have been developed to describe the electronic structure of correlated
materials, but it is unclear how accurate these methods can be expected to be
when applied to a given strongly correlated solid. It is thus of pressing
interest to compare their accuracy as they apply to different categories of
materials. Here, we introduce a novel paradigm in which a chosen set of
beyond-DFT methods is systematically and uniformly tested on a chosen class of
materials. For a first application, we choose the target materials to be the
binary transition-metal oxides FeO, CoO, MnO, and NiO in their
antiferromagnetic phase and present a head-to-head comparison of spectral
properties as computed using the various methods. We also compare with
available experimental angle-resolved photoemission spectroscopy (ARPES),
inverse-photoemission spectroscopy, and with optical absorption. We find both
B3LYP and eDMFT can reproduce the experiments quite well, with eDMFT doing best
in particular when comparing with the ARPES data.
