The density functional theory calculations were done with the Vienna ab initio simulation package (VASP) using the projector augmented wave (PAW) method. The electron exchange-correlation functional was described by the generalized gradient approximation (GGA) in the form proposed by Perdew, Burke, and Ernzerhof (PBE). The Ti 1-x-y Al x Ta y N solid solutions were modeled by 2×2×2 rocksalt cubic supercells containing 64 atoms in which the metal sublattice was occupied by randomly distributed Ti, Al, and Ta atoms. As an example of such supercells, the structure containing 50% of Ti, 25% of Al and 25% of Ta atoms on the the metal sublattice is shown in Fig. . The structure relaxation considering both the atomic positions and lattice vectors was performed by the conjugate gradient scheme until the maximum force on each atom was less than 0.001 eV/Å, and the total energy was