  The electronic structure and magnetic properties of CaCr$\mathrm{O}_3$ have
been calculated by two methods, including hybrid-exchange density-function
theory and density-functional theory + $U$. The computed densities of states
from both of these methods are in a qualitative agreement with the previous
x-ray spectroscopy. On the other hand, the opening of the band gap separates
them apart. hybrid-exchange density-functional theory always gives a finite
band gap, down to $\sim 1.2$ eV from HSE06 functional, whereas by tuning the
Hubbard-$U$ parameter down to $0.5$ eV, a conducting state with AFM-C (defined
in the text) spin configuration can be achieved. From hybrid density-functional
theory, the computed nearest-neighbouring exchange interaction along the
$c$-axis and in the $ab$-plane are $\sim 4$ meV and $\sim 6$ meV
(anti-ferromagnetic), respectively, which are qualitatively in agreement with
the previous magnetic measurements. These anti-ferromagnetic exchange
interaction, together with the in-plane anti-ferro-orbital ordering will induce
a spin-orbital frustration, which could play a role for the abnormal electronic
properties in CaCrO$_3$. In hybrid-exchange density-functional theory, an
abrupt reduction ($\sim 0.2$ eV) of the majority-spin band gap of the
ferromagnetic state between 60 K and 100 K has been found as lowering
temperature, which shows a strong link to the previous optical conductivity
measurements in [A. C. Komarek, et. al., Phys. Rev. B \textbf{84}, 125114
(2011)]. In sharp contrast, the density-functional theory + $U$ methods
predicted AFM-C state as the lowest AFM state for the crystal structure
measured below 90 K, above which AFM-A is however the lowest. The closely
related concepts including electron-hole liquid and surface-plasmon-mediating
spin-spin interactions have been discussed as well.
