First-principles study of the structural, elastic, and electronic   properties of the cubic perovskite BaHfO$_3$
  First principles study of structural, elastic, and electronic properties of the cubic perovskitetype BaHfO$_3$ has been performed using the plane wave ultrasoft pseudo-potential method based on density functional theory with revised Perdew-Burke-Ernzerhof exchange-correlation functional of the generalized gradient approximation (GGA-RPBE). The calculated equilibrium lattice constant of this compound is in good agreement with the available experimental and theoretical data reported in the literatures. The independent elastic constants (\emph{C}$_{11}$, \emph{C}$_{12}$, and \emph{C}$_{44}$), bulk modules \emph{B} and its pressure derivatives $B^{\prime}$, compressibility $\beta$, shear modulus \emph{G}, Young's modulus \emph{Y}, Poisson's ratio $\nu$, and Lam\'{e} constants ($\mu, \lambda$) are obtained and analyzed in comparison with the available theoretical and experimental data for both the singlecrystalline and polycrystalline BaHfO$_3$. The band structure calculations show that BaHfO$_3$ is a indirect bandgap material (R-$\Gamma$ = 3.11 eV) derived basically from the occupied O 2\emph{p} and unoccupied Hf 5\emph{d} states, and it still awaits experimental confirmation. The density of states (total, site-projected, and \emph{l}-decomposed) and the bonding charge density calculations make it clear that the covalent bonds exist between the Hf and O atoms and the ionic bonds exist between the Ba atoms and HfO$_3$ ionic groups in BaHfO$_3$. From our calculations, it is shown that BaHfO$_3$ should be promising as a candidate for synthesis and design of superhard materials due to the covalent bonding between the transition metal Hf 5\emph{d} and O 2\emph{p} states. 
Hongsheng Zhao, Aimin Chang and Yunlan Wang