Low Born effective Charges, High Covalency and Strong Optical Activity in $X_3^{2+}$Bi$^{3-}$N$^{3-}$ ($X$=Ca,Sr,Ba) $inverse$-perovskites

We compute for the first time a complete charge analysis (Bader and Born effective) on $X_3^{2+}$Bi$^{3-}$N$^{3-}$ ($X$=Ca,Sr,Ba). The crystals show a great electron sharing with little possibility of ferroelectricity. $Inverse$ perovskites have been a center of attraction in the recent years and not much is known on the systems under this study. This research addressed some key missing components and decomponents like the hardness and optical spectrum in $X_3^{2+}$Bi$^{3-}$N$^{3-}$ ($X$=Ca,Sr,Ba). The computed lattices slightly deviated from the parent perovskites indicating a future interfacing under a proper substrate. We also found out that all the crystals under this study were semiconducting with direct band gaps but plastic in nature due to strong covalency. The optical spectrum revealed very strong activity in these crystals in the ultraviolet regime. The information herein will definitely guide the experimentalist in fabrication of these materials for novel functionalities.