High-Pressure Phases of Calcium: Density-Functional Theory and Diffusion Quantum Monte Carlo Approach

The phase diagram of Ca is examined using a combination of density-functional theory (DFT) and diffusion quantum Monte Carlo (DMC) calculations. Gibbs free energies of several competing structures are computed at pressures near 50 GPa. Existing disagreements for the stability of Ca both at low and room temperature are resolved with input from DMC. Furthermore, DMC calculations are performed on 0 K crystalline structures up to 150 GPa and it is demonstrated that the widely used generalized gradient approximation of DFT is insufficient to accurately account for the relative stability of the high-pressure phases of Ca. The results indicate that the theoretical phase diagram of Ca needs a revision.

Figure 1

Phonon dispersion curves at 45 GPa and 0 K of (a) sc [21] and (b) $Cmmm$ from linear response theory. Solid circles are renormalized phonon frequencies from MD at 300 K near 45 GPa.

Figure 2

(a) Double-well potentials associated with atomic displacements along the transverse soft modes at the $M$ and $Y$ points in sc (solid line) and $Cmmm$ (dashed line), respectively. The ground state ionic wave functions for these modes are shown as well. (b) Gibbs free energies from DFT at 300 K of sc and $Cmmm$ relative to $I{4}_1/amd$, with entropy obtained from VDOS (circles) and thermodynamic integration (crosses).

Figure 3

(a) DMC energies for sc, $I{4}_1/amd$, and $Cmmm$ configurations from MD at 300 K. Here $999\mathrm{eV}/\mathrm{\text{atom}}$ is an arbitrary reference energy. The inset shows the DFT electronic density of states of sc and $I{4}_1/amd$ structures at 50 GPa. (b) DMC-corrected Gibbs free energies at 300 K of sc and $Cmmm$ relative to $I{4}_1/amd$ showing that the sc structure is preferred. DFT results are shown for comparison.