Comparison between exact and semilocal exchange potentials: An all-electron study for solids

The exact-exchange (EXX) potential, which is obtained by solving the optimized-effective potential (OEP) equation, is compared to various approximate semilocal exchange potentials for a set of selected solids (C, Si, BN, MgO, ${\mathrm{Cu}}_2\mathrm{O}$, and NiO). This is done in the framework of the linearized augmented plane-wave method, which allows for a very accurate all-electron solution of electronic structure problems in solids. In order to assess the ability of the semilocal potentials to approximate the EXX-OEP, we considered the EXX total energy, electronic structure, electric-field gradient, and magnetic moment. An attempt to parametrize a semilocal exchange potential is also reported.

Figure 1

The enhancement factors $F_{\text{x}}\left(s\right)$ [see Eq. (7)] of the different exchange functionals considered in this work.

Figure 2

$\mathrm{gBJ}(\gamma ,c,p)$ exchange potentials in C plotted starting at a distance of 0.2 Å from the atom at site $(1/4,1/4,1/4)$ in the [111] direction. The center of the unit cell is at $d=2.32\AA$. The potentials were shifted such that ${\int }_{\text{cell}}{v}_{\text{x}}\left(\mathbf{r}\right)d^{3}r=0$.

Figure 3

(a) BJ and BJUC exchange potentials in C. (b) Ratio $t/D$ for C. The path and potential shift are as in Fig. 1.

Figure 4

DOS of ${\mathrm{Cu}}_2\mathrm{O}$. The Fermi energy is set at zero.

Figure 5

Spin-up DOS of NiO. The Fermi energy is set at zero.

Figure 6

Two-dimensional plots of exchange potentials $v_{\text{x}}$ in a $\left(110\right)$ plane of C. The potentials were shifted such that ${\int }_{\text{cell}}{v}_{\text{x}}\left(\mathbf{r}\right)d^{3}r=0$. The contour lines start at $-2$ Ry (blue color) and end at 1 Ry (red color) with an interval of 0.2 Ry.

Figure 7

Exchange potentials $v_{\text{x}}$ in Si plotted from the vicinity of the atom at site $(1/8,1/8,1/8)$ ($d=0$) to (a) the center of the unit cell ($d=3.53\AA$) or (b) the mid-distance to the atom at site $(5/8,5/8,1/8)$ ($d=2.38\AA$). The potentials were shifted such that ${\int }_{\text{cell}}{v}_{\text{x}}\left(\mathbf{r}\right)d^{3}r=0$.

Figure 8

Exchange potentials $v_{\text{x}}$ in ${\mathrm{Cu}}_2\mathrm{O}$ plotted from the Cu atom at site $(1/2,1/2,0)$ ($d=0$) in the direction of the O atom at site $(3/4,3/4,3/4)$ ($d=3.54\AA$). Logarithmic scales on the $x$ axis were used for panels (a) and (c) which correspond to the vicinity of the Cu and O atoms, respectively. The potentials were shifted such that ${\int }_{\text{cell}}{v}_{\text{x}}\left(\mathbf{r}\right)d^{3}r=0$.

Figure 9

Plots of the radial functions ${\rho }_{20}$ (a) and $v_{\text{x},20}$ (b) versus the distance from the Cu atom in ${\mathrm{Cu}}_2\mathrm{O}$. The functions are multiplied by $r^2$.

Figure 10

Two-dimensional plots of the difference between spin-up and spin-down exchange potentials ($v_{\text{x}}^{\uparrow }-v_{\text{x}}^{\downarrow }$) in a $\left(001\right)$ plane of antiferromagnetic NiO. The contour lines start at $-2$ Ry (blue color) and end at 2 Ry (red color) with an interval of 0.235 Ry. The Ni atom with a full spin-up $3d$ shell is at the left upper corner.

Figure 11

Electron density $\rho$ obtained with different exchange potentials minus ${\rho }^{\text{LDA}}$ plotted in a $\left(110\right)$ plane of Si. The contour lines start at $-0.001$ electrons/${\mathrm{bohr}}^3$ (blue color) and end at 0.001 electrons/${\mathrm{bohr}}^3$ (red color) with an interval of 0.0002 electrons/${\mathrm{bohr}}^3$.

Figure 12

Spin-up electron density ${\rho }_{\uparrow }$ obtained with different exchange potentials minus ${\rho }_{\uparrow }^{\text{LDA}}$ plotted in a $\left(001\right)$ plane of antiferromagnetic NiO. The contour lines start at $-0.005$ electrons/${\mathrm{bohr}}^3$ (blue color) and end at 0.005 electrons/${\mathrm{bohr}}^3$ (red color) with an interval of 0.001 electrons/${\mathrm{bohr}}^3$. The Ni atom with a full spin-up $3d$ shell is at the left upper corner.

Figure 13

The difference between the density ${\rho }_{\text{core}}$ of the core electrons of Cu in ${\mathrm{Cu}}_2\mathrm{O}$ calculated with semilocal functionals and ${\rho }_{\text{core}}$ from EXX-OEP (multiplied by $4\pi r^2$).