Molecular bonding with the RPAx: From weak dispersion forces to strong correlation

In a recent paper [Phys. Rev. B 90, 125150 (2014)], we showed that the random phase approximation with exchange (RPAx) gives accurate total energies for a diverse set of systems including the high and low density regime of the homogeneous electron gas, the ${\mathrm{N}}_2$ molecule, and the ${\mathrm{H}}_2$ molecule at dissociation. In this paper, we present results for the van der Waals bonded ${\mathrm{Ar}}_2$ and ${\mathrm{Kr}}_2$ dimers and demonstrate that the RPAx gives superior dispersion forces as compared to the RPA. We then show that this improved description is crucial for the bond formation of the ${\mathrm{Mg}}_2$ molecule. In addition, the RPAx performs better for the ${\mathrm{Be}}_2$ dissociation curve at large nuclear separation but, similar to the RPA, fails around equilibrium due to the build up of a large repulsion hump. For the strongly correlated LiH molecule at dissociation we have also calculated the RPAx potential and find that the correlation peak at the bond midpoint is overestimated as compared to the RPA and the exact result. The step feature is missing and hence the delocalization error is comparable to the RPA. This is further illustrated by a smooth energy versus fractional charge curve and a poor description of the LiH dipole moment at dissociation.

Figure 1

Convergence of the RPA (top panel) and RPAx (bottom panel) correlation energy $E_c$ of the ${\mathrm{Be}}_2$ molecule at an interatomic distance $d=2.45$ Å, with respect to the size of the supercell. The numbers close to the black circles indicate the linear size in bohr of the cubic supercell. Different schemes for treating the $\mathbf{q}\to \mathbf{0}$ limit are compared. See text for details.

Figure 2

Convergence of exact-exchange energy $E_x$ of the ${\mathrm{Be}}_2$ molecule at an interatomic distance $d=2.45$ Å, with respect to the size of the supercell. The numbers close to the black circles indicate the linear size in bohr of the cubic supercell. Different schemes for treating the integrable divergence in the exact-exchange expression are compared. See text for details.

Figure 3

Dissociation curve of ${\mathrm{Ar}}_2$ molecule. The plot compares results from PBE, RPA, and RPAx calculations.

Figure 4

Dissociation curve of ${\mathrm{Kr}}_2$ molecule. The plot compares results from PBE, RPA, and RPAx calculations.

Figure 5

Relative importance of the scf for Ar and Kr dimers.

Figure 6

Dissociation curve of ${\mathrm{Be}}_2$ molecule. The plot compares results from PBE, RPA, and RPAx calculations.

Figure 7

Dissociation curve of ${\mathrm{Mg}}_2$ molecule. The plot compares results from PBE, RPA, and RPAx calculations.

Figure 8

Left and middle panels: Fractional charge analysis of the H and Li atoms. Right panel: Total energy of the LiH molecule as a function of fractional charge on the H atom.

Figure 9

Binding energy curve of the LiH molecule. The plot compares results from PBE, RPA, RPAx, and tRPAx calculations.

Figure 10

(Left) The KS potentials of a model LiH at $R=3.5$ Å. $x=0$ corresponds to the bond midpoint. Inset shows the corresponding potential energy curves. (Right) The dipole moment as a function of nuclear separation of the exact density and the self-consistent densities within EXX, RPA, and RPAx.