All-electron $GW$+Bethe-Salpeter calculations on small molecules

Accuracy of the first-principles $GW$+Bethe-Salpeter equation (BSE) method is examined for low-energy excited states of small molecules. The standard formalism, which is based on the one-shot $GW$ approximation and the Tamm-Dancoff approximation (TDA), is found to underestimate the optical gap of ${\mathrm{N}}_2$, CO, ${\mathrm{H}}_2\mathrm{O},{\mathrm{C}}_2{\mathrm{H}}_4$, and ${\mathrm{CH}}_2\mathrm{O}$ by about 1 eV. Possible origins are investigated separately for the effect of TDA and for the approximate schemes of the self-energy operator, which are known to cause overbinding of the electron-hole pair and overscreening of the interaction. By applying the known correction formula, we find the amount of the correction is too small to overcome the underestimated excitation energy. This result indicates a need for fundamental revision of the $GW$+BSE method rather than adjustment of the standard one. We expect that this study makes the problems in the current $GW$+BSE formalism clearer and provides useful information for further intrinsic development beyond the current framework.

Figure 1

The HOMO and LUMO energies of DFT-LDA vs PW cutoff energy for a ${\text{N}}_2$ molecule.

Figure 2

The HOMO and LUMO energies of $G_0{W}_0$@LDA vs PW cutoff energy for a ${\text{N}}_2$ molecule.

Figure 3

The HOMO and LUMO energies of $G_0{W}_0$@LDA vs cutoff energy for the Fock term. The calculation is done for a ${\text{N}}_2$ molecule.

Figure 4

The HOMO and LUMO energies of $G_0{W}_0$@LDA vs cutoff energy for the screening term. The calculation is done for a ${\text{N}}_2$ molecule.

Figure 5

The HOMO and LUMO energies of $G_0{W}_0$@LDA vs number of levels for a ${\text{N}}_2$ molecule.

Figure 6

Comparison of the experimental and theoretical ionization potential. $G_0{W}_0$@LDA, $G_0{W}_0$@${\mathrm{HF}}_{\mathrm{diag}}$, and EOM-CCSD calculations are shown with squares, asterisks, and rhombuses respectively, and HOMO levels of LDA (triangles) are shown for comparison.