<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>GW</mml:mi><mml:mi mathvariant="normal">Γ</mml:mi><mml:mspace width="0.16em"/><mml:mo>+</mml:mo></mml:mrow></mml:math>Bethe-Salpeter equation approach for photoabsorption spectra: Importance of self-consistent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>GW</mml:mi><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>calculations in small atomic systems

The self-consistent $\mathrm{GW}\mathrm{\ensuremath{\Gamma}}$ method satisfies the Ward-Takahashi identity (i.e., the gauge invariance or the local charge continuity) for arbitrary energy ($\ensuremath{\omega}$) and momentum ($\mathbit{q}$) transfers. Its self-consistent first-principles treatment of the vertex $\mathrm{\ensuremath{\Gamma}}={\mathrm{\ensuremath{\Gamma}}}_{v}$ or ${\mathrm{\ensuremath{\Gamma}}}_{W}$ 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, ${\mathrm{Na}}_{3}, {\mathrm{B}}_{2}$, and ${\mathrm{C}}_{2}{\mathrm{H}}_{2}$ calculated using the standard one-shot ${G}_{0}{W}_{0}+\mathrm{BSE}$ method that results in significantly redshifted PAS by 0.8--3.1 eV, although the PES are well reproduced already in ${G}_{0}{W}_{0}$.