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The electronic structure and magnetic properties of CaCr$\mathrm{O}_3$ havebeen calculated by two methods, including hybrid-exchange density-functiontheory and density-functional theory + $U$. The computed densities of statesfrom both of these methods are in a qualitative agreement with the previousx-ray spectroscopy. On the other hand, the opening of the band gap separatesthem apart. hybrid-exchange density-functional theory always gives a finiteband gap, down to $\sim 1.2$ eV from HSE06 functional, whereas by tuning theHubbard-$U$ parameter down to $0.5$ eV, a conducting state with AFM-C (definedin the text) spin configuration can be achieved. From hybrid density-functionaltheory, 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 withthe previous magnetic measurements. These anti-ferromagnetic exchangeinteraction, together with the in-plane anti-ferro-orbital ordering will inducea spin-orbital frustration, which could play a role for the abnormal electronicproperties in CaCrO$_3$. In hybrid-exchange density-functional theory, anabrupt reduction ($\sim 0.2$ eV) of the majority-spin band gap of theferromagnetic state between 60 K and 100 K has been found as loweringtemperature, which shows a strong link to the previous optical conductivitymeasurements in [A. C. Komarek, et. al., Phys. Rev. B \textbf{84}, 125114(2011)]. In sharp contrast, the density-functional theory + $U$ methodspredicted AFM-C state as the lowest AFM state for the crystal structuremeasured below 90 K, above which AFM-A is however the lowest. The closelyrelated concepts including electron-hole liquid and surface-plasmon-mediatingspin-spin interactions have been discussed as well.

Electronic structure and magnetic properties of $\mathrm{CaCr}\mathrm{O}_3$: The interplay between spin- and orbital-orderings