Unified description of ground and excited states of finite systems: The self-consistent $GW$ approach

$GW$ calculations with a fully self-consistent Green's function $G$ and screened interaction $W$—based on the iterative solution of the Dyson equation—provide a consistent framework for the description of ground- and excited-state properties of interacting many-body systems. We show that for closed-shell systems self-consistent $GW$ reaches the same final Green's function regardless of the initial reference state. Self-consistency systematically improves ionization energies and total energies of closed-shell systems compared to $G_0{W}_0$ based on Hartree-Fock and (semi)local density-functional theory. These improvements also translate to the electron density, as exemplified by an improved description of dipole moments, and permit us to assess the quality of ground-state properties such as bond lengths and vibrational frequencies.

Figure 1

Total energy (a) and ionization energy (b) of N${}_2$ at each iteration of the sc-$GW$ loop for a HF and PBE input Green's function, for the augmented correlation-consistent polarized valence quadrupole zeta (aug-cc-pVQZ) (Ref. 34) basis set. The (absolute values of the) differences arising from HF and PBE initializations vanish exponentially for both total energy (c) and ionization energy (d).

Figure 2

Difference between Galitskii-Migdal total energies ($E_{\mathrm{GM}}$) and full configuration interaction values ($E_{\mathrm{CI}}$) (Refs. 35, 36, 37), with $E_{\mathrm{GM}}$ evaluated from sc-$GW$, $G_0{W}_0$@HF, and $G_0{W}_0$@PBE. PBE total energies are included for comparison. The calculations were performed using the augmented correlation-consistent polarized valence five zeta (aug-cc-pV5Z) basis set. (Ref. 34)

Figure 3

Left: The spectral function of benzene calculated with a Tier 2 basis set. Vertical dashed lines are located at experimental vertical ionization energies from Ref. 46. Right: Comparison of experimental (Ref. 46) and theoretical vertical ionization energies (VIEs) extracted from the spectral function of benzene for sc-$GW$, $G_0{W}_0$@HF, and $G_0{W}_0$@PBE.

Figure 4

First vertical ionization energy (VIE) for 30 closed-shell molecules composed of two to eight atoms. Experimental values are taken from Ref. 39. Results from PBE total energy differences ($\Delta$SCF-PBE) are included for comparison.