Many-body renormalization of forces in $f$-electron materials

We present the implementation of dynamical mean-field theory (DMFT) in the CASTEP ab initio code. We explain in detail the theoretical framework for DFT+DMFT and we demonstrate our implementation for three strongly-correlated systems with $f$-shell electrons: $\gamma$-cerium, cerium sesquioxide ${\mathrm{Ce}}_2{\mathrm{O}}_3$, and samarium telluride SmTe by using a Hubbard I solver. We find very good agreement with previous benchmark DFT+DMFT calculations of cerium compounds, while for SmTe we show the improved agreement with the experimental structural parameters as compared with LDA. Our implementation works equally well for both norm-conserving and ultrasoft pseudopotentials, and we apply it to the calculation of total energy, bulk modulus, equilibrium volumes, and internal forces in the two cerium compounds. In ${\mathrm{Ce}}_2{\mathrm{O}}_3$ we report a dramatic reduction of the internal forces acting on coordinates not constrained by unit cell symmetries. This reduction is induced by the many-body effects, which can only be captured at the DMFT level. In addition, we derive an alternative form for treating the high-frequency tails of the Green function in Matsubara frequency summations. Our treatment allows a reduction in the bias when calculating the correlation energies and occupation matrices to high precision.

Figure 1

DFT+DMFT execution flowchart, containing both the inner self-consistency loop (DMFT at fixed charge density) as well as the outer one (Kohn-Sham equations at DFT+DMFT charge density).

Figure 2

Density of states of $\gamma$-Cerium, calculated by CASTEP's DFT+DMFT implementation and using the Hubbard I impurity solver. $G_{\mathrm{imp}}$ labels the impurity Green function derived DOS of Ce $f$ states, and $G_{\mathrm{latt}}$ labels the $G^B(k,\omega )$ derived DOS.

Figure 3

$\gamma$-Cerium's total energy $E_{\mathrm{tot}}$ as a function of lattice constant $a$, calculated both with DFT and with DFT+DMFT. Arrows show the experimental, DFT, and DFT+DMFT values of the equilibrium lattice constant. Curves show Birch-Murnaghan fits to the calculated points.

Figure 4

Density of states of ${\mathrm{Ce}}_2{\mathrm{O}}_3$ calculated by CASTEP's DFT+DMFT implementation and using the Hubbard I solver. $G_{\mathrm{imp}}$ labels the impurity Green function derived DOS of Ce $f$ states, and $G_{\mathrm{latt}}$ labels the $G^B(\mathrm{k},\omega )$ derived DOS.

Figure 5

${\mathrm{Ce}}_2{\mathrm{O}}_3$'s total energy $E_{\mathrm{tot}}$ as a function of lattice constant $a$, calculated both with DFT and with DFT+DMFT. Arrows show the experimental, DFT, and DFT+DMFT values of the equilibrium lattice constant. Curves show Birch-Murnaghan fits to the calculated data points.

Figure 6

Density of states of SmTe calculated by CASTEP's DFT+DMFT implementation and using the Hubbard I solver. $G_{\mathrm{imp}}$ labels the impurity Green function derived DOS of Sm $f$ states, and $G_{\mathrm{latt}}$ labels the $G^B(\mathrm{k},\omega )$ derived DOS.

Figure 7

SmTe's total energy $E_{\mathrm{tot}}$ as a function of lattice constant $a$, calculated both with DFT and with DFT+DMFT. Arrows show the experimental, DFT, and DFT+DMFT values of the equilibrium lattice constant. Curves show Birch-Murnaghan fits to the calculated data points.

Figure 8

The total energy as a function of the $z$-position increments $\mathrm{\Delta }z$ of Ce (upper panel) and oxygen (lower panel) for three different increments: $\mathrm{\Delta }z=4\%,2\%,1\%$ in units of $c$-axis lattice spacing. $a$ was kept equal to $3.81\text{Å}$. The energies are shifted in order to fit the graph.

Figure 9

A graphical comparison of forces calculated at the lattice constant $a=3.81\text{Å}$ within DFT (left panel) and DFT+DMFT (right panel). Forces acting on ${\mathrm{Ce}}_{1\left(2\right)}$ and ${\mathrm{O}}_{2\left(3\right)}$ atoms are shown. The lengths of arrow are proportional to the forces. Notice much smaller forces in the case of DMFT.