5 1 2 3 4 5 6 7 8 r$_{s}$ -0.50 -0.45 -0.40 -0.35 -0.30 -0.25 Relaxation energy (a.u.) This work This work (No extra) Arponen & Pajanne [10] Lantto [11] Boronski & Stachowiak [12] Ortiz [13] LDA Ps energy Exact Ps energy Exact Ps¯ energy FIG. 1: (Color online) Relaxation energy against density parameter. Results obtained with N = 114, 162, and 246 electrons for 1 ≤ r s ≤ 3 . 5, with N = 54, 114, and 162 electrons for 4 ≤ r s ≤ 6 . 5, and with N = 14, 54, and 114 electrons for 7 ≤ r s ≤ 8 were averaged to obtain the final results. Also shown are the relaxation energies obtained by other authors [13–16]. The dashed line following the data of Arponen and Pajanne [13] is the widely used fit of Boro´ nski and Nieminen [6]. The energy of the Ps − ion is taken from Ref. [11]. Fermi wave vector k$_{F}$ we fit ρ ( p ) = w$_{0}$ + w$_{2}$p 2 + w$_{4}$p 4 to our MD data, where w$_{0}$ , w$_{2}$ , and w$_{4}$ are fitting parameters. Above k$_{F}$ the DFT-calculated MD falls off exponentially and so we fit ρ ( p ) = W exp( − sp ) to our data, where W and s are fitting parameters. Electron-positron annihilating-pair MDs at different densities are plotted in Fig. 2. The normalization is chosen such that ∫ ∞ 0 4 πp $^{2}$ρ ( p ) dp = (4 / 3) πk 3 $_{F}$. Our results clearly show the enhancement of the annihilating-pair MD at the Fermi edge predicted by Kahana [18], but our MD data differ quantitatively from the previous results [18, 19] for 1 ≤ r$_{s}$ ≤ 8. In this range, we find the greatest enhancement of the MD at the Fermi edge at r$_{s}$ = 1, whereas the previous works [18, 19] found the enhancement of the MD to increase when the density is lowered. The previous works [18, 19] did not report the weight in the MDs above the Fermi edge. In our calculations the exponential tail of the MD above the Fermi edge carries an increasing amount of weight as the density is lowered, which is responsible