  It is believed that the density functional theory (DFT) describes most
elements with s, p and d orbitals very well, except some materials that having
strongly localized and correlated valence electrons. In this work, we find that
the widely employed exchange-correlation (xc) functionals, including LDA, GGA
and meta-GGA, underestimate the shear modulus and phase stability of V and Nb
greatly. The advanced hybrid functional that is usually better for correlated
system, on the other hand, completely fails in these two simple metals. This
striking failure is revealed due to the orbital localization error in GGA,
which is further deteriorated by hybrid functionals. This observation is
corroborated by a similar failure of DFT+U and van der Waals functionals when
applied to V and Nb. To remedy this problem, an semi-empirical approach of
DFT+J is proposed which can delocalize electrons by facilitating the on-site
exchange. Furthermore, it is observed that including density derivatives
slightly improves the performance of the semi-local functionals, with meta-GGA
outperforms GGA, and the latter is better than LDA. This discovery indicates
the possibility and necessity to include higher-order density derivatives
beyond the Laplacian level for the purpose to remove the orbital localization
error (mainly from d orbitals) and delocalization error (mainly from s and p
orbitals) completely in V and Nb, so that to achieve a better description of
their electronic structures. The same strategy can be applied to other d
electron system and f electron system.
