  The self-consistent GW{\Gamma} method satisfies the Ward-Takahashi identity
(i.e., the gauge invariance or the local charge continuity) for arbitrary
energy ($\omega$) and momentum ($\bf q$) transfers. Its self-consistent
first-principles treatment of the vertex $\Gamma=\Gamma_v$ or $\Gamma_W$ 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, 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 $G_0W_0$.
