Density-Functional Theory for $f$-Electron Systems: The $\alpha \mathrm{\text{−}}\gamma$ Phase Transition in Cerium

The isostructural $\alpha \mathrm{\text{−}}\gamma$ phase transition in cerium is analyzed using density-functional theory with different exchange-correlation functionals, in particular the PBE0 hybrid functional and the exact-exchange plus correlation in the random-phase approximation [$(\mathrm{EX}+\mathrm{cRPA})@\mathrm{PBE}0$] approach. We show that the Hartree-Fock exchange part of the hybrid functional gives rise to two distinct solutions at zero temperature that can be associated with the $\alpha$ and $\gamma$ phases of cerium. However, despite the relatively good structural and magnetic properties, PBE0 predicts the $\gamma$ phase to be the stable phase at ambient pressure and zero temperature, in contradiction with low temperature experiments. $\mathrm{EX}+\mathrm{cRPA}$ reverses the energetic ordering, which emphasizes the importance of correlation for rare-earth systems.

Figure 1

Cohesive energy [$E_{\mathrm{coh}}=-(E-\Sigma E^{\mathrm{atom}})$] of cerium as a function of the lattice constant ($a_0$). Dashed lines show HSE06 results. The spin moment increases with volume for the LDA and PBE solutions, while in PBE0 and HSE06 it remains approximately constant at zero and one-half for the $\alpha$ and $\gamma$ phases, respectively. Experimental lattice parameters for the two phases at finite temperature [1] are marked by black arrows.

Figure 2

Volume slice through the difference between the bulk Ce electron densities of the $\alpha$ and $\gamma$ phases at the same lattice constant of 4.6 Å, at which both phases have the same energy (see Fig. 1). The $\alpha$ phase has a larger contribution in the interstitial region, whereas the $\gamma$ phase is more localized and exhibits a clear $f$-orbital shape.

Figure 3

PBE0 results for cerium clusters and corresponding bulk values for (a) the cohesive energy ($E_{\mathrm{coh}}$), (b) the lattice constant ($a_0$), and (c) the magnetic moment on the central atom ($m$). All clusters exhibit two solutions that converge to the calculated bulk limit. Experimental results marked on the right axis are taken from Ref. 1.

Figure 4

Calculated $(\mathrm{EX}+\mathrm{cRPA})@\mathrm{PBE}0$ cohesive energy ($E_{\mathrm{coh}}$) for the 19-atom fcc-cerium cluster as a function of $a_0$. The dashed line illustrates the Gibbs construction for the transition pressure in good agreement with the extrapolated experimental $P_t\simeq -0.8\mathrm{GPa}$ ([33]). Arrows on the energy axes: experimental cohesive energy from Ref. 45.