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 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.