Calculation of pair correlations in a high-density electron gas: Constraints for effective interparticle potentials

The pair-correlation function $g\left(r\right)$ of an interacting, unpolarized electron gas is modelled by using the geminal representation and effective potentials for the two-electron relative motion. We put forward a generalization of the impurity-related Fumi theorem, in order to obtain the scattering-mediated part of the kinetic energy change. Considering the high-density limit, where a perturbatively exact expression for the ground-state energy is available, a rigorous constraint on the effective interactions in the even and odd channels is deduced. The constraint is implemented by using physically motivated one-parametric potentials in these channels.

Figure 1

Pair-correlation function at the origin $g(0,r_s)$ as a function of the electronic density parameter $r_s$ (in atomic units). Results obtained in this work are plotted with a solid line. The dotted line refers to the ladder approximation and the dashed line to the exact result in the high density limit. Solid triangles are the results of a numerical solution of an effective Euler-Lagrange equation.

Figure 2

Difference in the pair-correlation function with respect to the Hartree-Fock case $\Delta g\left(r\right)=g\left(r\right)-g^0\left(r\right)$ in the high-density perturbative limit, and as a function of the distance $r$. Both $\Delta g\left(r\right)$ and $r$ are divided by $r_s$. The solid line is the full $\Delta g\left(r\right)$, and the dashed and dotted lines show the antiparallel $\Delta g_{\uparrow \downarrow }\left(r\right)$ and parallel $\Delta g_{\uparrow \uparrow }\left(r\right)$ contributions to it, respectively. The inset provides a zoom of the $\Delta g\left(r\right)\approx 0$ region, to remark the non-negative values of both the parallel and antiparallel contributions to $\Delta g\left(r\right)$. All quantities in atomic units.

Figure 3

Kinetic-energy change $e_{\mathrm{kin}}(k,n_0)$ as a function of the dimensionless quantity $k∕k_F$ (solid line). The odd channel $e_{\mathrm{kin}}^{-}(k,n_0)$ (dotted line) and even channel $e_{\mathrm{kin}}^{+}(k,n_0)$ (dashed line) contributions to $e_{\mathrm{kin}}(k,n_0)$ are shown in the plot as well. The energies $e_{\mathrm{kin}}(k,n_0)$ are multiplied by $k_F^2∕\left(2\pi \right)$. All quantities in atomic units.