4 (4 / 3) πr 3 s = V/ ( N − 1), where V is the volume of the cell and N is the number of electrons. Our calculations were performed in simple cubic cells and we used the Ewald interaction to describe the Coulomb interactions. We have also restricted our attention to paramagnetic HEGs, although it is straightforward to apply our method to spin-polarized HEGs. Our relaxation energies were extrapolated to basis-set completeness using the empirically derived expression ΔΩ$_{E}$ cut = ΔΩ ∞ + κE − 4 / 3 cut , where κ is a fitting parameter and E$_{cut}$ is the plane-wave cutoff energy. We estimate the residual finite-size errors in our DFT relaxation energies to occur in the third significant figure. The fit to our data shown in Fig. 1 is where A − 1 = − 0 . 28877, A$_{0}$ = − 0 . 22339, A$_{1}$ = 0 . 011536, B$_{1}$ = 0 . 012331, and B$_{2}$ = 0 . 020016. This fitting form tends to the energy of the Ps − ion at low density [11] and could be used as the LDA electron-positron correlation functional in a two-component DFT calculation for a positron in a real system. Equation (1) does not yield the exact high-density behavior calculated within the random phase approximation, although this is only relevant for r$_{s}$ < 0 . 1 [12]. Our relaxation-energy results are in reasonable agreement with the many-body-theory results of Refs. [13] and [15], but are in clear disagreement with the quantum Monte Carlo (QMC) data of Ref. [16]. The orbitals in the trial wave functions used in the QMC calculations reported in Refs. [16] and [17] were single plane waves for the electrons and the positron, which do not allow for the strong pairing that occurs between the electrons and the positron at low density. In common with the most accurate many-body-theory results, our relaxation energy is higher than the energy of a Ps atom at the lowest density considered of r$_{s}$ = 8. Neglecting the extra interaction ˆ V$_{e}$ as suggested by Leung et al. [9] leads to relaxation energies that increase monotonically towards the DFT-LDA energy of a Ps atom. The annihilating-pair MD depends sensitively on the accuracy of the correlated electron-positron pairing φ$_{i}$ ( x ) and we expect that our fully self-consistent treatment of the pairing orbitals will be more accurate than previous approaches [18, 19]. The MD at each momentum p was extrapolated to basis-set completeness using the empirically determined expression ρ$_{E}$ cut ( p ) = ρ ∞ ( p ) + α ( p ) /E$_{cut}$ + β ( p ) /E 3 / 2 $_{cut}$, where α ( p ) and β ( p ) are fitting parameters. As the system size is increased, the momenta at which the MD is defined become more finely spaced and the finite-size errors at each point are reduced. We have therefore fitted a model curve to our MD data obtained at the largest system size available at each density. We have verified that our results are well-converged with respect to system size. Below the ΔΩ( r$_{s}$ ) = A − $_{1}$r − 1 s + A$_{0}$ + A$_{1}$r$_{s}$ − 0 . 262005 B$_{2}$r 2 s 1 + B$_{1}$r$_{s}$ + B$_{2}$r 2 s , (3)