Improved method of calculating ab initio high-temperature thermodynamic properties with application to ZrC

Thermodynamic properties of ZrC are calculated up to the melting point ($T^{\mathrm{melt}}\approx 3700\text{K}$), using density functional theory (DFT) to obtain the fully anharmonic vibrational contribution, and including electronic excitations. A significant improvement is found in comparison to results calculated within the quasiharmonic approximation. The calculated thermal expansion is in better agreement with experiment and the heat capacity reproduces rather closely a CALPHAD estimate. The calculations are presented as an application of a development of the upsampled thermodynamic integration using Langevin dynamics (UP-TILD) approach. This development, referred to here as two-stage upsampled thermodynamic integration using Langevin dynamics (TU-TILD), is the inclusion of tailored interatomic potentials to characterize an intermediate reference state of anharmonic vibrations on a two-stage path of thermodynamic integration between the original DFT quasiharmonic free energy and the fully anharmonic DFT free energy. This approach greatly accelerates the convergence of the calculation, giving a factor of improvement in efficiency of $\sim 50$ in the present case compared to the original UP-TILD approach, and it can be applied to a wide range of materials.

Figure 1

Heat capacity at constant pressure (per mole of formula units) within the quasiharmonic approximation for both LDA and GGA results.

Figure 2

Linear thermal expansion within the LDA for different levels of approximation, including quasiharmonic (qh), quasiharmonic + electronic (qh+el), and quasiharmonic + electronic + anharmonic (qh+el+ah), compared against experimental results.

Figure 3

Heat capacity at constant pressure (per mole of formula units) for different levels of approximation, including quasiharmonic (qh), quasiharmonic + electronic (qh+el), and quasiharmonic + electronic + anharmonic (qh+el+ah), compared against experimental results.

Figure 4

Total time taken to compute ${〈E_{\mathrm{low}}^{\mathrm{DFT}}-E^{\mathrm{pot}}〉}_{\lambda }$ for $T=3800\text{K}$ and $V=13.87{\AA }^3/\text{atom}$ using 24 cores plotted against the TI coupling parameter, $\lambda$. Results are shown for both UP-TILD and TU-TILD methods.

Figure 5

${〈E^{\mathrm{pot}}-E^{\mathrm{qh}}〉}_{\lambda }$ for $\lambda =0$, relative to its converged value, plotted against the number of configurations used to evaluate the quantity for both UP-TILD and TU-TILD methods. Inset: Zoomed-in view of the plot at the origin.