Accurate effective harmonic potential treatment of the high-temperature cubic phase of Hafnia

HfO$_2$ is an important high-$\kappa$ dielectric and ferroelectric, exhibiting a complex potential energy landscape with several phases close in energy. It is, however, a strongly anharmonic solid, and thus describing its temperature-dependent behavior is methodologically challenging. We propose an approach based on self-consistent, effective harmonic potentials and higher-order corrections to study the potential energy surface of anharmonic materials. The introduction of a reweighting procedure enables the usage of unregularized regression methods and efficiently harnesses the information contained in every data point obtained from density functional theory. This renders the approach highly efficient and a promising candidate for large-scale studies of materials and phase transitions. We detail the approach and test it on the example of the high-temperature cubic phase of HfO$_2$. Our results for the thermal expansion coefficient, $\alpha_V \approx 3.3\times 10^{-5}$ K$^{-1}$, are in agreement with existing experimental ($\alpha_V \approx 4\pm1\times 10^{-5}$ K$^{-1}$) and theoretical ($\alpha_V \approx 5\pm1\times 10^{-5}$ K$^{-1}$) work. Likewise, the bulk modulus agrees well with experiment. We show the detailed temperature dependence of these quantities.