Exact-exchange Kohn-Sham potential, surface energy, and work function of jellium slabs

Exact-exchange self-consistent calculations of the Kohn-Sham potential, surface energy, and work function of jellium slabs are reported in the framework of the optimized effective potential (OEP) scheme of density functional theory. In the vacuum side of the jellium surface and at a distance $z$ that is larger than the slab thickness, the exchange-only Kohn-Sham potential is found to be imagelike $\left(\sim -e^2/z\right)$ but with a coefficient that differs from that of the classical image potential $V_{\text{im}}\left(z\right)=-e^2/4z$. The three OEP contributions to the surface energy (kinetic, electrostatic, and exchange) are found to oscillate as a function of the slab thickness, as occurs in the case of the corresponding calculations based on the use of single-particle orbitals and energies obtained in the local-density approximation (LDA). The OEP work function presents large quantum size effects that are absent in the LDA and which reflect the intrinsic derivative discontinuity of the exact Kohn-Sham potential.

Figure 1

Main features of the jellium-slab model of metal surfaces. Top panel: normalized jellium density $\left[n_{+}\left(z\right)\right]$ and the self-consistent OEP electron density $n\left(z\right)$ for two different values of the electron-density parameter $r_s$. Lower panel: OEP Hartree, exchange, and Kohn-Sham potentials for $r_s=2.07$. Dotted lines denote KS eigenvalues, ${\epsilon }_F$ is the Fermi energy, and $W$ is the work function. $d=4{\lambda }_F$.

Figure 2

LDA and OEP self-consistent electron densities and its difference for $d=8{\lambda }_F$ and $r_s=2.07$. Note that $n^{\text{LDA}}\left(z\right)$ is slightly more diffuse than $n^{\text{OEP}}\left(z\right)$, as $n^{\text{LDA}}\left(z\right)-n^{\text{OEP}}\left(z\right)>0$ for $z$ outside the jellium edge (in the vacuum).

Figure 3

Kinetic surface energy, as a function of slab width $d$, for $r_s=2.07$, from Eq. (43), with $l=K$. Full line, OEP results; dotted line, LDA results. The two arrows on the right denote the extrapolated asymptotic values ${\sigma }_K\left(\text{OEP}\right)\to -4720\text{erg}/{\text{cm}}^2$, ${\sigma }_K\left(\text{LDA}\right)\to -4832\text{erg}/{\text{cm}}^2$.

Figure 4

Electrostatic surface energy, as a function of slab width $d$, for $r_s=2.07$, from Eq. (43), with $l=\text{el}$. Full line, OEP results; dotted line, LDA results. The two arrows on the right denote the extrapolated asymptotic values ${\sigma }_{\text{el}}\left(\text{OEP}\right)\to 1103\text{erg}/{\text{cm}}^2$, ${\sigma }_{\text{el}}\left(\text{LDA}\right)\to 1172\text{erg}/{\text{cm}}^2$.

Figure 5

Exchange surface energy, as a function of slab width $d$, for $r_s=2.07$, from Eq. (43), with $l=x$. Full line, OEP results; dotted line, LDA results; dash-dotted line, standard LDA-exchange results. The three arrows on the right denote the extrapolated asymptotic values ${\sigma }_x\left(\text{OEP}\right)\to 2316\text{erg}/{\text{cm}}^2$, ${\sigma }_x\left(\text{LDA}\right)\to 2390\text{erg}/{\text{cm}}^2$, and ${\sigma }_x^{\text{LDA}}\to 2767\text{erg}/{\text{cm}}^2$.

Figure 6

Slab work function versus slab width $d$ for $r_s=2.07$. Full line, OEP result; dotted line, LDA result. Occupation events corresponding to transitions from a slab with $m$ occupied SDL toward $m+1$ occupied SDL are denoted as $m\to m+1$.

Figure 7

Self-consistent OEP exchange potential, around the $6\to 7$ SDL transition, for $r_s=2.07$. The origin of coordinate $z$ for each slab has been taken at the slab center. The position of the right slab edge has been indicated by a vertical dashed line for each case.

Figure 8

Left: electronic structure of the slab for ${\alpha }_6=1^{-}$. Right: electronic structure of the slab for ${\alpha }_7=0^{+}$. The work-function $W$ jumps discontinuously from its left large value toward the smaller right value. Slab edge is at $z=0$.

Figure 9

Exchange-driven discontinuity $C_x\left(m\right)$ for increasing number of occupied slab levels, as follows from Eq. (51).