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GWΓ + Bethe-Salpeter equation approach for photoabsorption spectra: Importance of self-consistent GWΓ calculations in small atomic systems

The self-consistent GWΓ method satisfies the Ward-Takahashi identity (i.e., the gauge invariance or the local charge continuity) for arbitrary energy ($ω$) and momentum ($\bf q$) transfers. Its self-consistent first-principles treatment of the vertex $Γ=Γ_v$ or $Γ_W$ is possible to first order in the bare ($v$) or dynamically-screened ($W$) Coulomb interaction. It is developed within a linearized scheme and combined with the Bethe-Salpeter equation (BSE) to accurately calculate photoabsorption spectra (PAS) and photoemission (or inverse photoemission) spectra (PES) simultaneously. The method greatly improves the PAS of Na, Na$_3$, B$_2$, and C$_2$H$_2$ calculated using the standard one-shot $G_0W_0$ + BSE method that results in significantly redshifted PAS by 0.8-3.1 eV, although the PES are well reproduced already in $G_0W_0$.