Effects of self-consistency and plasmon-pole models on $GW$ calculations for closed-shell molecules

We present theoretical calculations of quasiparticle energies in closed-shell molecules using the $GW$ method. We compare three different approaches: a full-frequency $G_0{W}_0$ ($\mathrm{FF}\text{−}{G}_0{W}_0$) method with density functional theory (DFT-PBE) used as a starting mean field; a full-frequency $G{W}_0$ ($\mathrm{FF}\text{−}G{W}_0$) method where the interacting Green's function is approximated by replacing the DFT energies with self-consistent quasiparticle energies or Hartree-Fock energies; and a $G_0{W}_0$ method with a Hybertsen-Louie generalized plasmon-pole model (HL $\mathrm{GPP}\text{−}{G}_0{W}_0$). While the latter two methods lead to good agreement with experimental ionization potentials and electron affinities for methane, ozone, and beryllium oxide molecules, $\mathrm{FF}\text{−}{G}_0{W}_0$ results can differ by more than one electron volt from experiment. We trace this failure of the $\mathrm{FF}\text{−}{G}_0{W}_0$ method to the occurrence of incorrect self-energy poles describing shake-up processes in the vicinity of the quasiparticle energies.

Figure 1

Graphical solution of the quasiparticle equation for the HOMO (a) and the LUMO (b) of methane. The quasiparticle energies $E_n$ are given by the values of $\omega$ at the intersections of $\omega -{\epsilon }_n$ and $\text{Re}{\Sigma }_n\left(\omega \right)-V_n^{\mathrm{xc}}$. Shown are self-energies from full-frequency $G_0{W}_0$ theory, full-frequency $G{W}_0$ theory, and $G_0{W}_0$ theory with the generalized Hybertsen-Louie plasmon-pole approximation. All calculations employed a DFT-PBE starting point.

Figure 2

Graphical solution of the quasiparticle equation for the HOMO (a) and the LUMO (b) of the beryllium oxide molecule. The quasiparticle energies $E_n$ are given by the values of $\omega$ at the intersections of $\omega -{\epsilon }_n$ and $\text{Re}{\Sigma }_n\left(\omega \right)-V_n^{\mathrm{xc}}$. Shown are self-energies from full-frequency $G_0{W}_0$ theory, full-frequency $G{W}_0$ theory, and $G_0{W}_0$ theory with the generalized Hybertsen-Louie plasmon-pole approximation. All calculations employed a DFT-PBE starting point.

Figure 3

(a) Graphical solution of the quasiparticle equation for the HOMO of ozone. The quasiparticle energies $E_n$ are given by the values of $\omega$ at the intersections of $\omega -{\epsilon }_n$ and $\text{Re}{\Sigma }_n\left(\omega \right)-V_n^{\mathrm{xc}}$. Shown are self-energies from full-frequency $G_0{W}_0$ theory, full-frequency $G{W}_0$ theory, and $G_0{W}_0$ theory with the generalized Hybertsen-Louie plasmon-pole approximation. All calculations employed a DFT-PBE starting point. (b) Resulting spectral functions for the HOMO of ozone. Arrows denote the position of shake-up features. Note that some solutions of the quasiparticle equation do not give rise to peaks in the spectral function because they are suppressed by strong peaks in the imaginary part of the self-energy. The solutions which give rise to peaks in the spectral functions are marked by black dots.

Figure 4

Imaginary part of the inverse dielectric matrix for the BeO molecule in a supercell calculation. Shown are the full-frequency result (FF) and the Hybertsen-Louie generalized plasmon-pole (HL GPP) model for $\mathit{G}={\mathit{G}}^{\prime }=\left[001\right]2\pi /a_0$ and $\mathit{G}={\mathit{G}}^{\prime }=\left[400\right]2\pi /a_0$ (multiplied by a factor of 10) with $a_0$ denoting the linear dimension of the supercell. Arrows denote the positions of the effective excitations in the generalized plasmon-pole model. The molecular axis is along the $z$ direction.