Simple dynamic exchange-correlation kernel of a uniform electron gas

We propose a simple dynamic exchange-correlation kernel of the uniform electron gas. We model the reduction of the electron-electron interaction due to the short-range exchange-correlation effects by introducing a frequency-dependent error-function effective interaction. By imposing the fulfillment of the compresibility and the third-frequency-moment sum rules, as well as the correct asymptotic behavior at large wave vectors, we find an accurate and simple dynamic exchange-correlation kernel that accurately reproduces the wave-vector analysis and the imaginary-frequency analysis of the correlation energy of the uniform electron gas.

Figure 1

Static xc kernel $f_{\mathit{xc}}(n;q,0)$ versus $q$, for an electron density $n$ equal to that of valence electrons in aluminum $(r_s=2.07)$ (Ref. 24), as obtained from Eqs. (4) (thin solid line) and (12) (thick solid line) with $\omega =0$. Also shown are the parametrization of Ref. 20 (dashed line) and the static $(\omega =0)$ form of the xc kernel of Ref. 23 (dashed-dotted line).

Figure 2

Long-wavelength xc kernel $f_{\mathit{xc}}(n;0,\omega )$ versus $r_s$, as obtained from Eq. (12) with $q\to 0$ (solid lines) for various values of the imaginary frequency $iu$. Here, ${\omega }_p=\sqrt{4\pi n}$ represents the plasma frequency. Crosses and stars represent the results obtained from Eqs. (6, 8), respectively. Equation (4) yields a long-wavelength xc kernel $f_{\mathit{xc}}(n;0,\omega )$ that is qualitatively similar to the more accurate form represented in this figure.

Figure 3

Wave-vector analysis of the correlation energy per electron of the uniform electron gas at $r_s=2.07$, as obtained from Eq. (12). The “exact” wave-vector analysis is obtained from the Perdew-Wang parametrization of the xc hole (Ref. 31). Also shown are the results we have obtained from our static xc kernel $f_{\mathit{xc}}(n;q,0)$ of Eq. (12) with $\omega =0$.

Figure 4

As in Fig. 3, but for $r_s=4$.

Figure 5

As in Fig. 3, but for $r_s=6$.

Figure 6

Imaginary-frequency analysis (low-$u$ regime) of the correlation energy per electron of the uniform electron gas at $r_s=4$. The RA imaginary-frequency analysis is obtained from the Richardson-Ashcroft parametrization (Ref. 20), as in Ref. 30. Also shown is the result we have obtained from our static xc kernel $f_{\mathit{xc}}(n;q,0)$.

Figure 7

As in Fig. 6, but for the high-$u$ regime

Figure 8

Deviation of our ACFDT correlation energy per particle ${\epsilon }_c\left(n\right)$ with respect to the Perdew-Wang parametrization ${\epsilon }_c{\left(n\right)}^{\mathit{\text{exact}}}$ (Ref. 15) versus $r_s$. Also shown are the ACFDT results we have obtained from the Corradini and Richardson-Ashcroft parametrizations of Refs. 20, 23, respectively.