Benchmark of $GW$ methods for azabenzenes

Many-body perturbation theory in the $GW$ approximation is a useful method for describing electronic properties associated with charged excitations. A hierarchy of $GW$ methods exists, starting from non-self-consistent $G_0$$W_0$, through partial self-consistency in the eigenvalues and in the Green's function (sc$G{W}_0$), to fully self-consistent $GW$ (sc$GW$). Here, we assess the performance of these methods for benzene, pyridine, and the diazines. The quasiparticle spectra are compared to photoemission spectroscopy (PES) experiments with respect to all measured particle removal energies and the ordering of the frontier orbitals. We find that the accuracy of the calculated spectra does not match the expectations based on their level of self-consistency. In particular, for certain starting points $G_0$$W_0$ and sc$G{W}_0$ provide spectra in better agreement with the PES than sc$GW$.

Figure 1

Schematic illustration of the molecules studied here: (a) benzene, (b) pyridine, (c) pyridazine, (d) pyrimidine, and (e) pyrazine.

Figure 2

Relative OSIE with respect to the HOMO for benzene, pyridine, pyridazine, pyrimidine, and pyrazine. Visualizations of the HOMO to HOMO-3 orbitals at the PBE ordering are also shown.

Figure 3

PBEh and HF eigenvalues, as estimated based on a PBE calculation using Eqs. (2) and (3), for benzene, pyridine, pyridazine, pyrimidine, and pyrazine.

Figure 4

Spectra of benzene, calculated with DFT and $G_0$$W_0$, broadened by a 0.4 eV Gaussian, compared to gas phase PES (Ref. 96). Illustrations of the frontier orbitals are also shown.

Figure 5

Spectra of pyridine, calculated with DFT and $G_0$$W_0$, broadened by a 0.3 eV Gaussian, compared to gas phase PES. (Ref. 96). Illustrations of the frontier orbitals are also shown.

Figure 6

Spectra of pyridazine, calculated with DFT and $G_0$$W_0$, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 7

Spectra of pyrimidine, calculated with DFT and $G_0$$W_0$, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 8

Spectra of pyrazine, calculated with DFT and $G_0$$W_0$, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 9

Spectra of benzene, calculated with $GW$ at different levels of self-consistency, broadened by a 0.4 eV Gaussian, compared to gas phase PES (Ref. 96). Illustrations of the frontier orbitals are also shown.

Figure 10

Spectra of pyridine, calculated with $GW$ at different levels of self-consistency, broadened by a 0.4 eV Gaussian, compared to gas phase PES (Ref. 96). Illustrations of the frontier orbitals are also shown.

Figure 11

Spectra of pyridazine, calculated with $GW$ at different levels of self-consistency, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 12

Spectra of pyrimidine, calculated with $GW$ at different levels of self-consistency, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 13

Spectra of pyrazine, calculated with $GW$ at different levels of self-consistency, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 14

Feynman diagram for the 2OX. Arrows represent the Green's function, and dashed lines represent the (bare) Coulomb interaction.

Figure 15

Spectra of benzene, calculated using $G_0$$W_0$ with second-order exchange, based on different DFT starting points, broadened by a 0.4 eV Gaussian, compared to gas phase PES (Ref. 96). Illustrations of the frontier orbitals are also shown.

Figure 16

Spectra of pyridine, calculated using $G_0$$W_0$ with second-order exchange, based on different DFT starting points, broadened by a 0.4 eV Gaussian, compared to gas phase PES (Ref. 96). Illustrations of the frontier orbitals are also shown.

Figure 17

Spectra of pyridazine, calculated using $G_0$$W_0$ with second-order exchange, based on different DFT starting points, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 18

Spectra of pyrimidine, calculated using $G_0$$W_0$ with second-order exchange, based on different DFT starting points, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 19

Spectra of pyrazine, calculated using $G_0$$W_0$ with second-order exchange, based on different DFT starting points, broadened by a 0.3 eV Gaussian, compared to gas phase PES (Ref. 100). Illustrations of the frontier orbitals are also shown.

Figure 20

The IP of pyrimidine obtained with $G_0$$W_0$ based on PBE, PBEh, and HF, with sc$G{W}_0$ based on PBE and HF, and with sc$GW$ as a function of the basis set size. The computed IPs are also compared to experiment (Ref. 100).

Figure 21

Spectra of pyrimidine obtained with $G_0$$W_0$ based on PBE, PBEh, and HF at the tier 2 and tier 4 levels, compared to PES (Ref. 100). Illustrations of the frontier orbitals are also shown.