Bond Breaking and Bond Formation: How Electron Correlation is Captured in Many-Body Perturbation Theory and Density-Functional Theory

For the paradigmatic case of ${\mathrm{H}}_2$ dissociation, we compare state-of-the-art many-body perturbation theory in the $GW$ approximation and density-functional theory in the exact-exchange plus random-phase approximation (RPA) for the correlation energy. For an unbiased comparison and to prevent spurious starting point effects, both approaches are iterated to full self-consistency (i.e., sc-RPA and sc-$GW$). The exchange-correlation diagrams in both approaches are topologically identical, but in sc-RPA they are evaluated with noninteracting and in sc-$GW$ with interacting Green functions. This has a profound consequence for the dissociation region, where sc-RPA is superior to sc-$GW$. We argue that for a given diagrammatic expansion, sc-RPA outperforms sc-$GW$ when it comes to bond breaking. We attribute this to the difference in the correlation energy rather than the treatment of the kinetic energy.

Figure 1

$\Phi$ functional for RPA and $GW$ correlation energies [Eq. (8)]. The arrowed lines correspond to the interacting Green function $G$ in $GW$, and the KS Green function $G_s$ in RPA. Dashed lines denote the bare Coulomb interaction, and the minus sign of the prefactor comes from the rules for evaluating Feynman diagrams [18, 19].

Figure 2

Total energy (eV) of the ${\mathrm{H}}_2$ molecule as a function of bond length (Å). Different flavors of $GW$ and RPA are compared to PBE, rPT2, and accurate full configuration interaction calculations taken from Ref. 32. Hartree-Fock (HF) and exact-exchange OEP (OEPx) are identical for ${\mathrm{H}}_2$ and are included for comparison. All calculations were performed using a Gaussian cc-pVQZ [39] basis set.

Figure 3

Left panel: Difference between the sc-$GW$ and sc-RPA total energy ($\Delta E_{\mathrm{tot}}$), the correlation energy ($\Delta E_c=E_c^{GW}-U_c^{\mathrm{RPA}}$), and the remaining terms [$\Delta (E_{\mathrm{tot}}-E_c)$]. Right panel: Breakdown of the remaining term into the difference of the Hartree ($\Delta E_H$), the external ($\Delta E_{\mathrm{ext}}$), the exchange ($\Delta E_x$), and the kinetic energy ($\Delta T$).

Figure 4

Left panel: HOMO-LUMO gap extracted from the sc-$GW$ spectral function and from HF, PBE, and sc-RPA eigenvalues. The HOMO-LUMO gap evaluated from PBE and HF total energy differences ($\Delta$SCF) is included for comparison. Full-CI calculations were done with an aug-cc-pVDZ basis set. Right panel: Inverse of the RPA Coulomb correlation energy $U_c^{\mathrm{RPA}}$ for a two-level model as a function of the HOMO-LUMO gap.