A series of nitrogen-rich molecules N6XHm, N8XHm, and N10XHm (X = B, Al, Ga, m = 1 and X = C, Si, Ge, m = 2) consisting of N-3 and N-5 radicals, are systematically investigated by using B3LYP and B3PW91 DFT methods. It is found that for the nitrogen-rich molecules, the structures with N-3-chains (N-5-ring) are more stable than those containing a N-3-ring (N-5-chain). This result could be well-explained by the intrinsic stability of the N-3 and N-5 radicals and their charge distribution in nitrogen-rich molecules. The dissociation energies further indicate that the B-doped and C-doped structures are the most stable among the molecules with three elements of group 13 and 14, respectively. Energy decomposition analysis shows the bond of boron-nitrogen is stronger than that of carbon-nitrogen. Detailed bonding analysis demonstrates that the B-N bond is determined by sigma and pi interactions between the B and N atoms, whereas C-N bonds by only sigma interactions. These results imply that the boron atom is more suitable than the carbon atom for building the nitrogen-rich molecules studied in this article.