  The pressure-induced transition of vanadium from BCC to rhombohedral
structures is unique and intriguing among transition metals. In this work, the
stability of these phases is revisited by using density functional theory. At
finite temperatures, a novel transition of rhombohedral phases back to BCC
phase induced by thermal electrons is discovered. This reentrant transition is
found not driven by phonons, instead it is the electronic entropy that
stabilizes the latter phase, which is totally out of expectation. Parallel to
this transition, we find a peculiar and strong increase of the shear modulus
C44 with increasing temperature. It is counter-intuitive in the sense that it
suggests an unusual harding mechanism of vanadium by temperature. With these
stability analyses, the high-pressure and finite-temperature phase diagram of
vanadium is proposed. Furthermore, the dependence of the stability of RH phases
on the Fermi energy and chemical environment is investigated. The results
demonstrate that the position of the Fermi level has a significant impact on
the phase stability, and follows the band-filling argument. Besides the Fermi
surface nesting, we find that the localization/delocalization of the d orbitals
also contributes to the instability of rhombohedral distortions in vanadium.
