Exchange-correlation hole of a generalized gradient approximation for solids and surfaces

We propose a generalized gradient approximation (GGA) for the angle- and system-averaged exchange-correlation hole of a many-electron system. This hole, which satisfies known exact constraints, recovers the Perdew-Burke-Ernzerhof for solids (PBEsol) exchange-correlation energy functional, a GGA that accurately describes the equilibrium properties of densely packed solids and their surfaces. We find that our PBEsol exchange-correlation hole describes the wave-vector analysis of the jellium exchange-correlation surface energy in agreement with a sophisticated time-dependent density-functional calculation (whose three-dimensional wave-vector analysis we report here).

Figure 1

Gradient correction to the correlation hole, $B_c\left(r_s,\zeta ,v\right)$, versus $v$ for $r_s=2$. PBE and PBEsol are compared here. The solid line represents the Langreth-Mehl (LM) RPA contribution, which should be recovered at large $v$.

Figure 2

The exponent $\mathcal{H}\left(s\right)$ of Eq. (10) versus the reduced gradient $s$. The solid line represents the numerical solution of the implicit equation for $\mathcal{H}\left(s\right)$ [Eq. (A4) of Ref. 6]. The dashed line represents the fit of Eq. (14).

Figure 3

Dimensionless exchange-hole shape $J^{\text{PBEsol}}\left(s,y\right)$ [see Eq. (10)] versus $y=k_{F}u$ for $s$ between 0 and 3 in steps of 0.5. For comparison with $J^{\text{PBE}}\left(s,y\right)$, see Fig. 2 of Ref. 6. When $s=0$, $J^{\text{PBEsol}}\left(s,y\right)$ yields $J^{\text{LSDA}}\left(y\right)$.

Figure 4

The PBEsol enhancement factor $F_{\text{xc}}$ for the spin-unpolarized case $\left(\zeta =0\right)$ as a function of the reduced gradient $s$ for several values of $r_s$. The lines represent the enhancement factor obtained from the PBEsol xc energy functional of Ref. 12. The dots represent the enhancement factor obtained from our PBEsol angle-averaged xc-hole density through Eq. (2).

Figure 5

The PBEsol enhancement factor $F_{\text{xc}}$ for the fully spin-polarized case $\left(\zeta =1\right)$ as a function of the reduced gradient $s$ for several values of $r_s$. The lines represent the enhancement factor obtained from the PBEsol xc energy functional of Ref. 12. The dots represent the enhancement factor obtained from our PBEsol angle-averaged xc-hole density through Eq. (2).

Figure 6

PBE, PBEsol, and exact wave-vector-resolved exchange surface energies ${\gamma }_x\left(k\right)$ versus $k/2k_F$ for a jellium slab of thickness $a=2.23{\lambda }_F$ and $r_s=2.07$. The semilocal PBE and PBEsol calculations have been performed from nonoscillatory parametrizations of the dimensionless exchange-hole shapes $J^{\text{PBE}}$ and $J^{\text{PBEsol}}$, respectively. The area under each curve represents the corresponding exchange surface energy: ${\sigma }_x^{\text{PBE}}=2164\text{erg}/{\text{cm}}^2$, ${\sigma }_x^{\text{PBEsol}}=2424\text{erg}/{\text{cm}}^2$, and ${\sigma }_x^{\text{exact}}=2348\text{erg}/{\text{cm}}^2$.

Figure 7

Exact-RPA and benchmark TDDFT wave-vector-resolved xc surface energies ${\gamma }_{\text{xc}}\left(k\right)$ versus $k/2k_F$ for a jellium slab of thickness $a=2.23{\lambda }_F$ and $r_s=2.07$. The area under each curve represents the corresponding xc surface energy: ${\sigma }_{\text{xc}}^{\text{RPA}}=3091\text{erg}/{\text{cm}}^2$ and ${\sigma }_{\text{xc}}^{\text{TDDFT}}=3090\text{erg}/{\text{cm}}^2$. The straight dotted line represents the universal low wave-vector limit of Eq. (19).

Figure 8

PBE, PBEsol, and benchmark TDDFT wave-vector-resolved xc surface energies ${\gamma }_{\text{xc}}\left(k\right)$ versus $k/2k_F$ for a jellium slab of thickness $a=2.23{\lambda }_F$ and $r_s=2.07$. The semilocal PBE and PBEsol calculations have been performed from nonoscillatory parametrizations of the dimensionless exchange-hole shapes $J^{\text{PBE}}$ and $J^{\text{PBEsol}}$, respectively. The area under each curve represents the corresponding xc surface energy: ${\sigma }_{\text{xc}}^{\text{PBE}}=2885\text{erg}/{\text{cm}}^2$, ${\sigma }_{\text{xc}}^{\text{PBEsol}}=3027\text{erg}/{\text{cm}}^2$, and ${\sigma }_{\text{xc}}^{\text{TDDFT}}=3090\text{erg}/{\text{cm}}^2$. The straight dotted line represents the universal low wave-vector limit of Eq. (19).

Figure 9

PBE, PBEsol, exact-RPA, and benchmark TDDFT wave-vector-resolved correlation surface energies ${\gamma }_c\left(k\right)$ versus $k/2k_F$ for a jellium slab of thickness $a=2.23{\lambda }_F$ and $r_s=2.07$. The area under each curve represents the corresponding correlation surface energy: ${\sigma }_c^{\text{PBE}}=720\text{erg}/{\text{cm}}^2$, ${\sigma }_c^{\text{PBEsol}}=604\text{erg}/{\text{cm}}^2$, ${\sigma }_c^{\text{RPA}}=743\text{erg}/{\text{cm}}^2$, and ${\sigma }_c^{\text{TDDFT}}=742\text{erg}/{\text{cm}}^2$.