  Electronic structure of one of the nickelates (LaNiO$_{2}$) and one of the
cuprates (CaCuO$_{2}$) is studied with three self consistent GW-based methods:
scGW, sc(GW+Vertex), and quasiparticle self-consistent GW. Low energy features
obtained in our study are in many respects similar to the features reported in
previous DFT+DMFT studies. Consistent with the DFT+DMFT conclusion, we find
LaNiO$_{2}$ as more correlated than CaCuO$_{2}$. However, correlation effects
included in our study change the DFT Fermi surface near the $\Gamma$ point
differently than it was reported in the DMFT studies. Features which are a few
electron-volts away from the Fermi level are broader in our calculations than
in the DFT+DMFT which reflects the differences between the DFT and the GW
methods. Our results are in qualitative agreement with previous G0W0 results,
but the self-consistency brings in the quantitative differences. Generally,
correlation effects are found to be sufficiently weak in both materials which
allows one to use totally ab-initio diagrammatic approaches like sc(GW+Vertex)
and to avoid the methods with adjustable parameters (DFT+U or DFT+DMFT).
However, the possibility of some strong correlations at low energy which cannot
be captured by perturbative methods cannot be completely excluded. For
instance, differences in the Fermi surface should be resolved: experimental
studies are necessary.
