Static correlation and electron localization in molecular dimers from the self-consistent RPA and $GW$ approximation

We investigate static correlation and delocalization errors in the self-consistent $GW$ and random-phase approximation (RPA) by studying molecular dissociation of the ${\mathrm{H}}_2$ and LiH molecules. Although both approximations contain topologically identical diagrams, the nonlocality and frequency dependence of the $GW$ self-energy crucially influence the different energy contributions to the total energy as compared to the use of a static local potential in the RPA. The latter leads to significantly larger correlation energies, which allow for a better description of static correlation at intermediate bond distances. The substantial error found in $GW$ is further analyzed by comparing spin-restricted and spin-unrestricted calculations. At large but finite nuclear separation, their difference gives an estimate of the so-called fractional spin error normally determined only in the dissociation limit. Furthermore, a calculation of the dipole moment of the LiH molecule at dissociation reveals a large delocalization error in $GW$ making the fractional charge error comparable to the RPA. The analyses are supplemented by explicit formulas for the $GW$ Green's function and total energy of a simplified two-level model providing additional insights into the dissociation limit.

Figure 1

${\mathrm{H}}_2$ dissociation curves in spin-unrestricted/restricted self-consistent (Usc/sc) $GW$, RPA, and HF. Results are compared to accurate full-CI results from Ref. [41].

Figure 2

(a) $U_{\mathrm{c}}^{GW}$ evaluated with the KS Green's function in PBE and RPA (denoted $G_0{W}_0$@PBE and scRPA) and with $GW$ and HF Green's functions (denoted $G_0{W}_0$@HF and $\mathrm{sc}GW)$. (b) Kinetic energies calculated with different Green's functions as $T\left[G^{GW}\right]$ for $\mathrm{sc}GW,T\left[G^{\mathrm{HF}}\right]$ for $G_0{W}_0$@HF, $T\left[G^{\mathrm{PBE}}\right]$ for $G_0{W}_0$@PBE and $T\left[G^{\mathrm{RPA}}\right]+T_{\mathrm{c}}\left[G^{\mathrm{RPA}}\right]$ for scRPA. We also report the exact kinetic energy from the full-CI calculations of Ref. [42]. (c) Corresponding HOMO-LUMO gaps.

Figure 3

Total energy of ${\mathrm{H}}_2$ evaluated in RPA and $GW$ based on first-order perturbation theory and full-CI results [41].

Figure 4

Natural occupation numbers obtained by diagonalizing the density matrix.

Figure 5

(Left) The four solutions [see Eq. (35)] of quasiparticle excitation energies in the one-shot $GW$ approximation (upper panel) and in the HF and KS-LDA approximations (lower panel) for ${\mathrm{H}}_2$ in a minimal basis. (Right) Quasiparticle amplitudes of the four one-shot $GW$ solutions.

Figure 6

Fractional charge analysis of the LiH molecule. The total energy as a function of fractional charge on the H atom. These curves are obtained by adding the fractional charge curves of the independent atoms in such a way that the total number of electrons is four.

Figure 7

(Left) The dipole moment of the LiH molecule as a function of bond distance in HF, PBE, $\mathrm{sc}GW$, and accurate CCSD calculations. (Right) The dipole moment divided by distance in the same approximations gives an estimate of their fractional charge error.