Wave functions, electronic localization, and bonding properties for correlated materials beyond the Kohn-Sham formalism

Many-body theories such as dynamical mean field theory (DMFT) have enabled the description of the electron-electron correlation effects that are missing in current density functional theory (DFT) calculations. However, there has been relatively little focus on the wave functions from these theories. We present the methodology of the newly developed elk-triqs interface and how to calculate the DFT with DMFT ($\mathrm{DFT}+\mathrm{DMFT}$) wave functions, which can be used to calculate $\mathrm{DFT}+\mathrm{DMFT}$ wave-function-dependent quantities. We illustrate this by calculating the electron localization function (ELF) in monolayer ${\mathrm{SrVO}}_3$ and ${\mathrm{CaFe}}_2{\mathrm{As}}_2$, which provides a means of visualizing their chemical bonds. Monolayer ${\mathrm{SrVO}}_3$ ELFs are sensitive to the charge redistribution between the DFT, one-shot $\mathrm{DFT}+\mathrm{DMFT}$, and fully charge self-consistent $\mathrm{DFT}+\mathrm{DMFT}$ calculations. In both tetragonal and collapsed tetragonal ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ phases, the ELF changes weakly with correlation-induced charge redistribution of the hybridized As $p$ and Fe $d$ states. Nonetheless, the interlayer As-As bond in the collapsed tetragonal structure is robust to the changes at and around the Fermi level.

Figure 1

(a) The unit cell of the monolayer ${\mathrm{SrVO}}_3$, where each monolayer is separated by $20\AA$ of vacuum. (b) The comparison of the Wannier V $t_{2\mathrm{g}}$ spectral functions calculated in the one-shot and fully charge self-consistent (FCSC) $\mathrm{DFT}+\mathrm{DMFT}$ methods by the wien2k-triqs and elk-triqs code combinations.

Figure 2

The monolayer ${\mathrm{SrVO}}_3$ (a) $xy$- and (d) $xz$-plane ELFs, slicing through the center of the V and O atoms. The fourfold symmetry of the planes has been exploited to show the results from the GGA (PBE), one-shot (OS), and fully charge self-consistent (FCSC) $\mathrm{DFT}+\mathrm{DMFT}$ calculations. (b) and (e) are the differences [for example, $\text{FCSC-GGA}={\eta }^{\mathrm{FCSC}}\left(\mathbf{r}\right)-{\eta }^{\mathrm{GGA}}\left(\mathbf{r}\right)]$ in the ELFs from the different theoretical techniques in the $xy$ and $xz$ planes, respectively. (c) and (f) show the charge density differences [for example, $\text{FCSC-GGA}={\rho }^{\mathrm{FCSC}}\left(\mathbf{r}\right)-{\rho }^{\mathrm{GGA}}\left(\mathbf{r}\right)]$ between the different theoretical techniques in the $xy$ and $xz$ planes, respectively. The gray solid and magenta dot-dashed contours in (c) and (f) show the positive and negative charge density difference isovalues of $6\times {10}^{-3}$. The charge density differences are in units of electrons per cubic bohr.

Figure 3

(a) and (e) show the structures of the tetragonal (TET) and collapsed tetragonal (CT) phases, respectively. The parallelepiped Wigner-Seitz unit cell is shown along with the dashed line indicating the $xz$ plane on which the ELFs were calculated, and in (e) the interlayer As-As bond is also indicated by the double arrow. (b), (c), and (d) are total and partial density of states for the GGA (PBE), SCAN, and fully charge self-consistent (FCSC) $\mathrm{DFT}+\mathrm{DMFT}$ calculations in the TET structure. (f), (g), and (h) show the same quantities as (b), (c), and (d), respectively, in the CT structure.

Figure 4

The band structures according to the GGA (PBE) and SCAN functionals and the $A(\mathbf{k},\omega )$ of the fully charge self-consistent $\mathrm{DFT}+\mathrm{DMFT}$ calculations for the (a) tetragonal (TET) and (b) collapsed tetragonal (CT) structures. The high-symmetry points correspond to a simple tetragonal unit cell. The natural log color scale and range were used for clarity of the bands.

Figure 5

The 2D ELFs in the $xz$ plane [as indicated in Figs. 3 and 3] for the (a) tetragonal (TET) and (b) collapsed tetragonal (CT) structures. The ELFs for the GGA (PBE), SCAN, and fully charge self-consistent (FCSC) $\mathrm{DFT}+\mathrm{DMFT}$ calculations are shown in quadrants, as this plane is fourfold symmetric. The dot-dashed contours are of $\eta (\mathbf{r})=0.5$.