Attractive forces between ions in quantum plasmas: Failure of linearized quantum hydrodynamics

In a recent Letter [P. K. Shukla and B. Eliasson, Phys. Rev. Lett. 108, 165007 (2012)] the discovery of a new attractive force between protons in a hydrogen plasma was reported that would be responsible for the formation of molecules and of a proton lattice. Here we show, based on ab initio density functional calculations and general considerations, that these predictions are not correct and caused by using linearized quantum hydrodynamics beyond the limits of its applicability.

Figure 1

Coupling parameter $\alpha$ vs Brueckner parameter $r_s$. An attractive proton potential is predicted in the density interval $26.22\ge r_s\ge 0.61$ at zero temperature. The shaded area denotes the range of $r_s$ where the plasmon energy is smaller than the Fermi energy, $\hslash {\omega }_{\mathrm{pe}}<k_{\mathrm{B}}{T}_{\mathrm{F}}$ (weak coupling).

Figure 2

Screened proton potential of SE for three densities. A single negative minimum is observed, shown more clearly in the inset. The location of the minimum and its depth are shown in Figs. 3 and 4, respectively. Thin vertical lines indicate the equilibrium nearest neighbor distance of two protons (cf. Fig. 3). The black arrow marks the range of values of $r$ shown in Fig. 5.

Figure 3

Position of the attractive potential minimum of SE, Ref. [1], in units of the Bohr radius as a function of $r_s$, compared to the mean interparticle distance. Note that the ground state nearest neighbor distance of two protons in the molecular phase ($r_s\gtrsim 2–3$) is in the range of $1.5a_{\mathrm{B}}–2a_{\mathrm{B}}$.

Figure 4

Depth of the attractive potential minimum of SE, Ref. [1]. The deepest value is observed for $r_s\approx 3$ corresponding to $\alpha \approx 0.6$ (cf. Fig. 1).

Figure 5

Electrostatic potential around an H atom immersed in jellium. $r$ is the distance from the proton. DFT data have been calculated in a cubic box (size $15a_0$) using the Perdew-Burke-Ernzerhof (PBE)-generalized gradient approximation (GGA) for the exchange-correlation energy functional, a single $k$ point, and a plane-wave cutoff energy of 200 Ry ($r_s=1.5$) or 300 Ry ($r_s=4$ and $r_s=7$). The DFT data (symbols) are compared to the electrostatic potential from LQHD of Shukla and Eliasson, Ref. [1] (lines). The black arrow marks the voltage range shown in Fig. 2.

Figure 6

Interaction energy of two H atoms immersed in jellium for two densities. While for $r_s=7$ a minimum around the hydrogen molecule bond length in vacuo ($1.4$ bohrs) is observed, for $r_s=1.5$ the molecular bound state is unstable. The DFT data have been calculated in a cubic box (with a size as noted in the inset) using PBE-GGA or LDA for the exchange-correlation energy functional, a single or six $k$ points in the irreducible part of the Brillouin zone, and a plane-wave cutoff energy of 100 Ry.