Elasticity of the superconducting metals V, Nb, Ta, Mo, and W at high pressure

First-principles calculations have been performed for V, Nb, Ta, Mo, and W. The recently discovered bcc -> rhombohedral transition for vanadium [Phys. Rev. Lett. 98, 085502 (2007)] was confirmed as the mechanical instability of c(44) was found at P=80 GPa. Furthermore, the c(11), c(12), and c(44) constants for the group-V elements showed erratic behaviors whereas the constants for the group-VI elements were monotonically increasing with pressure. The metals were analyzed with Fermi surface calculations, showing shrinking nesting vectors with pressure for V, Nb, and Ta but were not seen for Mo and W. From electronic topological transition contributions, a critical energy closely situated to the Fermi level for vanadium could be the reason why the elastic constants of V and Nb were difficult to reproduce at ambient pressure.