  We report computations of the vertical ionization potentials within the $GW$
approximation of the near-complete series of first-row transition metal (V-Cu)
aqua ions in their most common oxidation states, i.e. V$^{3+}$, Cr$^{3+}$,
Cr$^{2+}$, Mn$^{2+}$, Fe$^{3+}$, Fe$^{2+}$, Co$^{2+}$, Ni$^{2+}$, and
Cu$^{2+}$. The $d$-orbital occupancy of these systems spans a broad range from
$d^2$ to $d^9$. All the structures were first optimized at the density
functional theory level using a large cluster of explicit water molecules that
are embedded in a continuum solvation model. Vertical ionization potentials
were computed with the one-shot $G_0W_0$ approach on a range of transition
metal ion clusters (6, 18, 40, and 60 explicit water molecules) wherein the
convergence with respect to the basis set size was evaluated using the systems
with 40 water molecules. We assess the results using three different density
functional approximations as starting points for the vertical ionization
potential calculations, namely $G_0W_0$@PBE, $G_0W_0$@PBE0, and
$G_0W_0$@r$^2$SCAN. While the predicted ground-state structures are similar
with all three exchange-correlation functionals, the vertical ionization
potentials were in closer agreement with the experiment when using the
$G_0W_0$@PBE0 and $G_0W_0$@r$^2$SCAN approaches, with the r2SCAN based
calculations being significantly less expensive. Computed bond distances and
vertical ionization potentials for all structures were compared with available
experimental data and are in good agreement.
