$\mathrm{B}{\mathrm{C}}_2\mathrm{N}$ is an important class of experimentally synthesizable superhard materials, typically explained by the distribution of B/C/N within a diamondlike framework, which provides a good explanation for their x-ray diffraction (XRD) patterns and hardness. However, there is still a lack of systematic theoretical investigations on the 1:2:1 distribution of B, C, and N atoms in the diamond framework. In this work, based on a 2 × 2 × 2 diamond supercell, all possible nonequivalent 1:2:1 distributions of B/C/N across all subgroups of this supercell were exhaustively explored. There are 465 possible configurations primarily concentrated in the 195, 198, 215, and 216 space groups, with 120, 142, 202, and 1 nonequivalent configurations, respectively. The ground state appears in space group 215 with 10 nonequivalent positions, consistent with previous work [Phys. Rev. B 107, 134101 (2023)]. However, in the previous study, only space group 215 was considered with 10 nonequivalent positions, and only 108 possible configurations were identified, while in our study, we reveal 131 in this case. Additionally, in our systematic and comprehensive work, we found that the systems prefer distributions with C and BN as separated as possible, and their energies decrease as the number of C–C or B–N bonds increases. Finally, we find that all these low-energy structures are superhard insulators, consistent with experimental results, and their XRD patterns are also in agreement with the experimental data, while their Raman spectra can serve as an effective criterion for distinguishing between different distributions.

• Alloys
• Carbon-based materials
• Complex materials
• Crystal structures
• Density functional theory
• First-principles calculations