Screened Coulomb interaction of localized electrons in solids from first principles

We report the implementation of a first-principles approach for calculating the screened Coulomb and exchange energies for localized electrons in solids. Our method is based on the pseudopotential plane-wave formalism. The localized orbitals are represented by maximally localized Wannier functions, and the screening effects are calculated within the constrained random phase approximation. As first applications of this development, we investigate the onsite Coulomb $U$ and exchange $J$ for the $3d$ electrons in ZnO, NiO, and CuGaS${}_2$. Both the bare (unscreened) and the screened $U$ and $J$ matrices are presented. We find that it is very important for these parameters to be calculated self-consistently. Intrachannel (i.e., $d$-$d$) and energy-dependent screening effects are also discussed.

Figure 1

The structure and computational components of the newly developed package for calculating screened Coulomb $U$ and exchange $J$ using a combined MLWF and cRPA approach.

Figure 2

Convergence behavior of the calculated $U$ and $J$ parameters for the semicore $d$ electrons in $zb$-ZnO. The self-consistency is reached when $U_{\text{out}}=U_{\text{in}}$ and $J_{\text{out}}=J_{\text{in}}$.

Figure 3

The LDA (left panels) and LDA $+U$ (right panels) band structures of $zb$-ZnO. Projections of the band wave functions onto O $2p$ (top panels) and Zn $3d$ (bottom panels) orbitals are shown as vertical bars superimposed on the band structures.

Figure 4

Two representative Zn $d$-like MLWF optimized using the wannier90 package. (a) and (b): $d$-like MLWF with the $e$ symmetry; (c) and (d): $d$-like MLWF with the $t_2$ symmetry. The MLWF on the left are constructed with the LDA solution, whereas those on the right are constructed with the $\mathrm{LDA}+U$ solution.

Figure 5

Comparison between the auxiliary functions $F_{ii}(\mathbf{G},\mathbf{q}=0)$ (dotted line) and $F_{ij}(\mathbf{G},\mathbf{q}=0)$ (dashed line) involved in the calculation of $U$ and $J$ parameters. The diagonal part of the polarization potential (solid line) is also plotted to illustrate the effectiveness of the screening effects on the Coulomb $U$ and exchange $J$.

Figure 6

Band structure of antiferromagnetic NiO calculated using the $\mathrm{LDA}+U$ method. The black dots are the band structure reconstructed using the MLWF.

Figure 7

Frequency dependence of the screened Coulomb $U$ for $d$ electrons in NiO calculated with the cRPA method. The horizontal dashed line indicates the bare Coulomb energy.

Figure 8

The $\mathrm{GGA}+U$ band structure of CuGaS${}_2$. The inner and outer energy windows used for generating MLWF are indicated by the dashed and solid double arrows.

Figure 9

Two representative Cu $3d$-like MLWF in CuGaS${}_2$: (a) with the $e$ symmetry and (b) with the $t_2$ symmetry.