  We report an all-electron, atomic orbital (AO) based, two-component (2C)
implementation of the $GW$ approximation (GWA) for closed-shell molecules. Our
algorithm is based on the space-time formulation of the GWA and uses analytical
continuation of the self-energy, and pair-atomic density fitting (PADF) to
switch between AO and auxiliary basis. By calculating the dynamical
contribution to the $GW$ self-energy at a quasi-one-component level, our 2C
$GW$ algorithm is only about a factor of two to three slower than in the scalar
relativistic case. Additionally, we present a 2C implementation of the simplest
vertex correction to the self-energy, the statically screened $G3W2$
correction. Comparison of first ionization potentials of a set of 67 molecules
with heavy elements (a subset of the SOC81 set) calculated with our
implementation against results from the WEST code reveals mean absolute
deviations of around 70 meV for $G_0W_0$@PBE and $G_0W_0$@PBE0. These are most
likely due to technical differences in both implementations, most notably the
use of different basis sets, pseudopotential approximations, different
treatment of the frequency dependency of the self-energy and the choice of the
2C-Hamiltonian. Finally, we assess the performance of some (partially
self-consistent) variants of the GWA for the calculation of first IPs by
comparison to vertical experimental reference values. $G_0W_0$PBE0 (25 \% exact
exchange) and $G_0W_0$BHLYP (50 \% exact exchange) perform best with mean
absolute deviations (MAD) of about 200 meV. Eigenvalue-only self-consistent
$GW$ (ev$GW$) and quasi-particle self-consistent $GW$ (qs$GW$) significantly
overestimate the IPs. Perturbative $G3W2$ corrections improve the agreement
with experiment in cases where $G_0W_0$ alone underestimates the IPs. With a
MAD of only 140 meV, 2C-$G_0W_0$PBE0 + $G3W2$ is in best agreement with the
experimental reference values.
