Effects beyond the random-phase approximation in calculating the interaction between metal films

The performance of the adiabatic-connection fluctuation-dissipation theorem is discussed through the implementation of a non-local energy optimized exchange-correlation kernel to account for short-range correlation effects. We evaluate the jellium surface energy, through a painstaking extrapolation of single slab calculations, as well as the binding and interaction energies between two and three jellium slabs. Whereas total electron correlation energies are rather sensitive to the details of the kernel, any physically well-motivated approximation within our framework describes binding energies (including surface energies) within the same level of accuracy.

Figure 1

Several parametrizations of the static XC kernel $f_{\mathrm{XC}}^{\mathrm{hom}}\left(q\right)$ of the HEG with $r_s=4$. Thick solid line: energy optimized non-local Hubbard-like kernel; thin solid line: energy optimized Dobson-Wang local kernel; dash-dotted line: ALDA; dashed line: parametrization by Corradini et al.

Figure 2

The absolute error $\Delta {\epsilon }_{\mathrm{C}}={\epsilon }_{\mathrm{C}}^{\mathrm{ACFDT}}-{\epsilon }_{\mathrm{C}}^{\mathrm{DQMC}}$ in the correlation energy per electron of the homogeneous electron gas using different XC kernels. ${\epsilon }_{\mathrm{C}}^{\mathrm{DQMC}}$ is taken from the accurate parametrization by Perdew and Wang (Ref. 20 of the DQMC data by Ceperley and Alder (Ref. 41). Note that the errors of the ALDA and RPA have been divided by two. The superior performance of the OH kernel (thick solid line) is evident for the whole range of electron densities. Note that the Dobson-Wang (DW) energy optimized prescription[8] is constructed so as to give zero absolute error $\Delta {ϵ}_{\mathrm{C}}$ at all $r_s$ values, if carried out exactly. However, the simple analytic kernel given in Ref. 8 was a numerical fit to a limited range of data in the metallic range $2<r_s<5$. A better fit to the exact DW kernel could presumably be constructed for the range $r_s<2$.

Figure 3

The absolute convergence of the correlation energy per electron of the HEG $(r_s=4)$ versus the reciprocal of frequency cutoff $u_{\max }$ in Eq. (1) for several ACFDT prescriptions (${\epsilon }_{\mathrm{F}}$ is the Fermi energy of the HEG). $\Delta {\epsilon }_{\mathrm{C}}$ denotes the error in ${\epsilon }_{\mathrm{C}}$ relative to its converged value at $u_{\max }^{-1}=0$. The excellent convergence of the RPA and OH approximations contrasts significantly with the numerical problems arising from the use of a static HEG kernel with a non-zero short wavelength term.

Figure 4

Interaction energy per electron for two interacting jellium slabs of thickness $L=3a_0$ and background density $r_s=1.25$ as a function of the distance $a$. The inset shows in more detail the binding curve around the mechanical equilibrium distance. Thick solid line: OH1; thin solid line: RPA; dashed line: KS-LDA. We can see how the KS-LDA underestimates the binding energy, whereas the RPA predicts a smaller bond-length than the OH1 and KS-LDA.

Figure 5

As in Fig. 4, but representing the interaction force per electron. Whereas the three models behave similarly if $a<a_{\mathrm{eq}}$, the KS-LDA deviates from the ACFDT results if $a>a_{\mathrm{eq}}$.

Figure 6

Local-field corrections to the $q_{\Vert }$-dependent distribution ${\gamma }_{\mathrm{C}}\left(q_{\Vert }\right)-{\gamma }_{\mathrm{C}}^{\mathrm{RPA}}\left(q_{\Vert }\right)$ for a jellium slab of thickness $L=12.52r_s$ $(r_s=2)$. Solid line: OH1, dashed line: Cor.

Figure 7

Local-field corrections to the frequency-dependent distribution ${\gamma }_C\left(\mathrm{i}u\right)-{\gamma }_{\mathrm{C}}^{\mathrm{RPA}}\left(\mathrm{i}u\right)$ for a jellium slab of thickness $L=12.52r_s$ $(r_s=2)$. Solid line: OH1, dashed line: Cor.

Figure 8

The finite-size XC surface energy for $r_s=2$ using the KS-LDA, RPA, and OH1 approximations. The open circles represent the averages over each oscillation, whereas the closed circles are the estimations of the surface energy following the procedure by Pitarke and Eguiluz (Ref. 5). In each panel, the hollow square marks the single slab geometry $(L=12.52r_s)$ used to estimate the local field corrections to the RPA.

Figure 9

Correlation contribution to the interaction energy for two jellium slabs of thickness $L=12.8\mathrm{a.u.}$ and background density $r_s=4$, as a function of the distance $a$. Thick solid line: OH1; thin solid line: RPA; dashed line: KS-LDA. The latter, obtained by subtracting the exact exchange energy from the KS-LDA exchange-correlation one, goes very quickly to zero. The ACFDT results exhibit a clear power-law decay (the fitted power is $p=5∕2$) which reflects the existence of long-distance vdW forces. We also plot the numerical fit (dashed-dotted line) described in the text of the OH1 results obtained from values between $a=20$ and $a=25$. The inset shows the difference between the OH1 and RPA energies, which decays as $a^{-7∕2}$.

Figure 10

The non-additive difference $\Delta {\epsilon }_{L,\mathrm{C}}^{\mathrm{int}3}\left(a\right)={\epsilon }_{L,\mathrm{C}}^{\mathrm{int}3}\left(a\right)-{\epsilon }_{L,\mathrm{C}}^{\mathrm{int}3\mathrm{w}2}\left(a\right)$ between the correlation contributions to the interaction energy per particle for three equally spaced slabs ($L=12.8\mathrm{a.u.}$ and $r_s=4$) as calculated from a full three-body calculation and from two-body interactions (3w2). Solid line: ACFDT-OH1, dashed line: KS-LDA. For distances less than $5\mathrm{a.u.}$ the differences are mainly due to the distinct overlapping in the two- and three-slab systems. However, if $a\gtrsim 7$ such differences are only due to the the presence of non-additive terms in the actual dispersion forces for the three-slab system. The inset shows the total correlation contributions ${\epsilon }_{L,\mathrm{C}}^{\mathrm{int}3}\left(a\right)$ (solid line) and the additive part ${\epsilon }_{L,\mathrm{C}}^{\mathrm{int}3\mathrm{w}2}\left(a\right)$ (dots) using the ACFDT-OH1.