The search for the ground state energies, ground state structures and related mechanical and dynamic properties was carried out within DFT with VASP (Vienna Ab initio Simulation Package (VASP) using the projector augmented wave (PAW) method for atomic potentials with all valence states (especially, in regard of such light element as carbon) was used. The exchange-correlation (XC) effects inherent to the DFT were considered with the generalized gradient approximation (GGA) following Perdew, Burke and Ernzerhof (PBE) . Relaxing the atoms onto the ground state structures was carried out with the conjugategradient algorithm . The tetrahedron method was applied for both geometry relaxation and total energy calculations using Blöchl et al. corrections and a special scheme . Brillouin-zone (BZ) integrals were approximated using a special k-point sampling . The optimization of the structural parameters was performed until the forces on the atoms were less than 0.02 eV/Å, and all stress components below 0.003 eV/Å 3 . The calculations were converged at an energy cut-off of 400 eV for the plane-wave basis set regarding the k-point integration in the reciprocal space from k x (6)k y (6)k z (6) up to k x k y k z for the final convergence and relaxation to zero strains. In the post-treatment process of the ground state electronic structures, the charge density projections were operated onto the carbon atomic sites. The elastic constants C ij and the phonon band structures were calculated to assess the mechanical and the dynamic stabilities. The phonon and electronic band structures were also obtained and discussed.