Monovacancies in body-centered cubic d-transition metals were studied using first-principles calculations. A full-potential linear muffin-tin orbital method was used, together with the local-density approximation and the generalized-gradient approximation, to calculate the volume-relaxed vacancy formation energies. A complementary ab initio pseudopotential method was used to calculate volume-relaxed and structure-relaxed local-density approximation formation energies and formation volumes in V, Nb, Mo, Ta and W. Fully relaxed pseudopotential geometries were also applied to full-potential linear muffin-tin orbital and generalized-gradient approximation calculations. Various clear trends arose from the results. Thus, for a given fully-relaxed geometry, the full-potential linear muffin-tin orbital local-density approximation and pseudopotential local-density approximation formation energies were almost identical. The lowest calculated formation energies were within, or close to, the experimental error bars for all of these body-centered cubic metals; except Cr. The overall agreement with experiment was better for 4d- and 5d-metals than for the 3d-metals. The generalized-gradient approximation and local-density approximation formation energies were very similar for 4d- and 5d-metals. However, for 3d-metals (especially Fe), the generalized-gradient approximation was better. The volume-relaxation and structure-relaxation contributions lowered the calculated formation energy by 0.1 to 0.5eV, and improved the agreement with experiment. The fully-relaxed local-density approximation formation volumes ranged from 0.45 to 0.62 (equilibrium) atomic volumes, and the predominant structural effects were an approximate 5% inward relaxation of the first-nearest neighbour shell for group-V metals, and a corresponding 1% inward relaxation for group-VI metals; with the exception of Mo, where the second-shell atoms also relaxed inwardly by some 1%.

First-Principles Formation Energies of Monovacancies in BCC Transition Metals. P.Söderlind, L.H.Yang, J.A.Moriarty, J.M.Wills: Physical Review B, 2000, 61[4], 2579-86