Kernel-corrected random-phase approximation for the uniform electron gas and jellium surface energy

We introduce and test a nonlocal energy-optimized model kernel (NEO) within the adiabatic connection fluctuation-dissipation (ACFD) density-functional theory for the jellium surface and uniform electron gas, as benchmarks for simple metallic systems. Our model kernel is short ranged for the uniform electron gas paradigm system and one-electron self-correlation free. One-electron self-interaction freedom is provided by an iso-orbital indicator. We show how several versions of the NEO kernel perform for the uniform electron gas and jellium surface energies, and in addition we explain the underlying physics of self-interaction-free exchange-only kernels for exponentially decaying surface densities.

Figure 1

The coefficients $a^{NEO-I}$ and $a^{NEO-II}$ defining the kernels NEO-I and NEO-II, for a jellium surface with the electron-density parameter $r_s=6$. The surface is at $z=0$, the bulk is at $z<0$, and the vacuum is at $z>0$.

Figure 2

3D wave-vector analysis of the correlation energy per particle ${\varepsilon }_{c,Q}^{\mathrm{unif}}\left(\overline{n}\right)$ of a uniform electron gas with $r_s=2$ and $r_s=5$, as obtained from Eq. (7) in the RPA and with the use of the NEO-I and NEO-III kernels. The same quantity, as obtained from the Perdew-Wang parametrization of the uniform-gas correlation-hole density [74] (exact) is given for comparison. For the uniform electron gas, the kernels NEO-I and NEO-II coincide.

Figure 3

2D wave-vector analysis of the correlation energy per particle ${\varepsilon }_{c,q}\left[n\right]$ of a jellium slab of width $a=2.23{\lambda }_F$ and $r_s=2.07$, as obtained from Eq. (5) in the RPA and with the use of the NEO-I, NEO-II, and NEO-III kernels. ISTLS calculations (see Ref. [39]) are given for comparison.

Figure 4

2D wave-vector analysis ${\gamma }_{c,q}$ of the correlation surface energy of a jellium slab of width $a=2.23{\lambda }_F$ and $r_s=2.07$. The area under each curve represents the corresponding correlation surface energy ${\sigma }_c$. Units are $\mathrm{erg}/{\mathrm{cm}}^2$ ($1\mathrm{hartee}/{\mathrm{bohr}}^2=1.557\times {10}^6\mathrm{erg}/{\mathrm{cm}}^2$).

Figure 5

Correlation energy per particle (${\varepsilon }_c$ versus $z$, for a jellium slab of width $a=2.23{\lambda }_F$ and $r_s=2.07$). The surface is at $z=0$, the bulk is at $z<0$, and the vacuum is at $z>0$.