Orbital localization error of density functional theory in shear properties of vanadium and niobium

It is believed that the density functional theory (DFT) describes most elements with s, p, and d orbitals very well, except some materials that have strongly localized and correlated valence electrons. In this work, we find that the widely employed exchange-correlation (XC) functionals, including local-density approximation (LDA), generalized gradient approximation (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 systems, 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, a semiempirical 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 semilocal 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 of removing the orbital localization error (mainly from d orbitals) and delocalization error (mainly from s and p orbitals) completely in V and Nb so that a better description of their electronic structures is achieved. The same strategy can be applied to the other d electron system and f electron system.