Ornstein-Zernike function and Coulombic correlation in the homogeneous electron liquid

Adopting the original Ornstein-Zernike (OZ) definition of the direct correlation function $c\left(r\right)$, the present study deals with the deviation $\Delta \left(r\right)$ of $c\left(r\right)$ induced by Coulomb correlation in the homogeneous electron liquid beyond the OZ function $c_{\mathit{FH}}\left(r\right)$ for purely Fermi hole (FH) statistical correlations. It is first stressed that $\Delta \left(r\right)$ at large $r$ is proportional to the Coulomb potential energy $e^2∕r$, suitably scaled with the plasma frequency. Both $\mathbf{r}$ space and $\mathbf{k}$ space formulations are presented. In $\mathbf{k}$ space, direct numerical use is made of inequalities due to Kugler [Phys. Rev. A 1, 1688 (1970)] by employing analytic representations of the pair correlations due to Gori-Giorgi et al. [Phys. Rev. B 61, 7353 (2000)] as a function of the uniform electron density. Then, in $\mathbf{r}$ space, consideration is given to differential equations proposed by Dawson and March [Phys. Chem. Liq. 14, 131 (1984)] and also in the recent study of Nagy and Amovilli [J. Chem. Phys. 121, 6640 (2004)]. In both approaches, one-body potentials enter, into which Coulombic interelectronic repulsions are subsumed. Finally, Gaskell’s [Proc. Phys. Soc. London 77, 1182 (1961)] variational ground-state wave function is shown to be related to the OZ direct correlation function in $\mathbf{k}$ space.

Figure 1

(a) OZ direct correlation function $c_{\mathit{FH}}\left(r\right)$ obtained by a numerical Fourier transform of Eq. (3) when $S\left(k\right)$ has the Fermi hole form [Eq. (5)]. All quantities are in a.u. Note that $c_{\mathit{FH}}\left(r\right)$ is a universal function of $k_{f}r$ in this Fermi hole limit of the homogeneous electron liquid and that $c_{\mathit{FH}}\left(0\right)=-0.86558$. (b) Oscillatory behavior after multiplying $c_{\mathit{FH}}\left(r\right)$ by ${\left(k_{f}r\right)}^2$. It is important to stress how rapidly this plot approaches the limiting behavior $c_{\mathit{FH}}\propto 1∕{\left(k_{f}r\right)}^2$ as $k_{f}r$ become large.

Figure 2

Function $I(k,r_s)$ defined in Eq. (14), which enters inequality (12) for the OZ function $\tilde{c}\left(k\right)$, for (a) $r_s=2$ and (b) $r_s=5$. All quantities are in a.u.

Figure 3

Shape of function $k^{2}B\left(k\right)$ in the formally exact $\mathbf{k}$ space theory set out in Eq. (10) (full line), (a) for $r_s=2$ and (b) $r_s=5$. In the same plots, the exact function is compared with the Fourier transform of ${\nabla }^2\mathcal{B}\left(r\right)$, obtained from the Vashista and Singwi (Ref. 13) approach embodied in Eq. (19) (dashed lines) with $a\pm 1$. The density dependence of $g(r,{\rho }_0)$ involved in that equation is calculated from the fit given by Gori-Giorgi et al. (Ref. 11). All quantities are in a.u.

Figure 4

Plot of function $\varphi$ defined in Eq. (25) against $k_{f}r$ for $r_s=2$ (curve 2) and 5 (curve 5). $\varphi$ is the effective potential, to within an additive constant, which gives the pair function amplitude in the Nagy and Amovilli approach (Ref. 19). All quantities are in a.u.

Figure 5

OZ direct correlation function $c(r,r_s)$ (a) plotted as a function of $k_{f}r$ for $r_s=0.1$, 2, and 5. The Fermi hole form is added for comparison (uppermost curve). In (b), the OZ direct correlation function is multiplied by ${\left(k_{f}r\right)}^2$ for three different densities corresponding to $r_s=0.1$, 2, and 5. Note that $2⩽r_s⩽5$ is the range of the simpler metal densities (e.g., Al and Cs). All quantities are in a.u. One sees the simplicity of the OZ function and especially how quickly it approaches its large $r$ form determined by Coulombic interaction $e^2∕r$.

Figure 6

Kugler’s inequality from Eq. (B1). All quantities are in a.u.