Fermi surface nesting and pre-martensitic softening in V-Nb system under high pressure*

First-principles total energy calculation based on the exact muffin-tin orbital and full potential linear muffin-tin orbital methods were used to calculate the equation of state and shear elastic constants of bcc V, Nb, and the V$_{95}$Nb$_{05}$ disordered alloy as a function of pressure up to 6 Mbar. We found a mechanical instability in C$_{44}$ and a corresponding softening in C$^{'}$ at pressures $\sim $ 2 Mbar for V. Both shear elastic constants show softening at pressures $\sim $ 0.5 Mbar for Nb. Substitution of 5 at. {\%} of V with Nb removes the instability of V with respect to trigonal distortions in the vicinity of 2 Mbar pressure, but still leaves the softening of C$_{44}$ in this pressure region. We argue that the pressure induced softening in the shear elastic constants of V and Nb can be attributed simultaneously to three different electronic structure peculiarities, namely to the Fermi Surface nesting, electronic topological transition, and the band Jahn-Teller effect.