“…The generalized gradient approximation (GGA), formulated by Perdew, Burke, and Ernzerhof (PBE), 35 was employed for the exchange and correlation energy terms because the GGA describes spin-polarized transition metals and compounds better than the local-density approximation (LDA). 36,37 The cutoff energy of the wave functions was 550 eV, and the cutoff energies of the augmentation functions were 700 eV for carbides and nitrides. The electronic wave functions were sampled on -centered grids, 38 16 × 16 × 24, 16 × 12 × 16, and 12 × 12 × 16 in the first Brillouin zone (BZ) of θ -Ni 3 X, of 3R-Ni 3 X, and of hcp-Ni 3 X y (0 y 1), respectively.…”