InNCo$_{3}$ and InNNi$_{3}$ play dominant roles in the total dnesity of states of these compounds. 

In order to understand the bonding nature among the ions in InNCo$_{3}$ and InNNi$_{3}$, we analyzed the charge density contours of InNCo$_{3}$ and InNNi$_{3}$ in the(110) plane, as shown in Fig. 4. From Fig. 4, it is found that a certain amount of charges are accumulated in the intermediate region between N and Co atoms in InNCo$_{3}$, and slightly more charges are accumulated intermediate region between N and Ni atoms in InNNi$_{3}$. This gives an evidence for the strong hybridization between N and transition metal (Co/Ni) atoms, indicating that the N-Co and N-Ni bondings exhibit strong covalent characteristics and the latter is slightly stronger than the former. The similar bonding characteristics for Ni-N atoms or Ni-C atoms were also found in other Ni-based ternary nitrides or carbides AXNi$_{3}$ $^{10,14,37,38}$. Therefore, our results suggest that the magnetic properties of InNNi$_{3}$ reported in experiment 15 are very likely due to the non-stoichiometry effect, which was also found in the cases of AlCNi$_{3}$ and GaCNi$_{3}$ $^{12,16–19}$. 

## IV. CONCLUSIONS 

In summary, we performed the first-principles calculations to study the elastic and elec- tronic properties of cubic antiperovskites InNCo$_{3}$ and InNNi$_{3}$. Based on the Voigt, Reuss and Hill bounds, the shear, Young’s moduli and Poisson’s ratio have also been estimated for the InNCo$_{3}$ and InNNi$_{3}$ polycrystals. The theoretically predicted equilibrium lattice parameters are in good agreement with the available experimental data. Our calculations show that the 3 d states of transition metal atoms in InNCo$_{3}$ and InNNi$_{3}$ play dominant roles near the Fermi levels. InNCo$_{3}$ energetically prefers to the ferromagnetic state. The mag- netic ground state of InNNi$_{3}$, which is same to other Ni-based ternary nitrides or carbides with a cubic anti-perovskite structure, is a stable paramagnetic (non-magnetic) state. This could be understood from that the hybridization between Ni-3 d and N-2 p states in InNNi$_{3}$ is slightly stronger than the one between Co-3 d and N-2 p states in InNCo$_{3}$ because of the more 3 d electrons in Ni. 

