  Photoelectron spectra of early $3d-$transition metal dioxide anions,
ScO$_{2}^-$, TiO$_{2}^-$, VO$_{2}^-$, CrO$_{2}^-$, MnO$_{2}^-$, are calculated
using semilocal and hybrid density functional theory (DFT) and many-body
perturbation theory within the $GW$ approximation using one-shot perturbative
and eigenvalue self-consistent formalisms. Different levels of theory are
compared with each other and with available photoelectron spectra. We show that
one-shot $GW$ with a PBE0 starting point ($G_0W_0$@PBE0) consistently provides
very good agreement for all experimentally measured binding energies (within
0.1-0.2 eV or less), which we attribute to the success of PBE0 in mitigating
self-interaction error and providing good quasiparticle wave functions, which
renders a first-order perturbative $GW$ correction effective. One-shot $GW$
calculations with semilocal exchange in the DFT starting point (e.g.
$G_0W_0$@PBE) do poorly in predicting electron removal energies by underbinding
orbitals with typical errors near 1.5 eV. Higher amounts of exact exchange
(e.g. 50%) in the DFT starting point of one-shot $GW$ do not provide very good
agreement with experiment by overbinding orbitals with typical errors near 0.5
eV. While not as accurate as $G_0W_0$@PBE0, the $G$-only eigenvalue
self-consistent $GW$ scheme with $W$ fixed to the PBE level ($G_nW_0$@PBE)
provides a reasonably predictive level of theory (typical errors near 0.3 eV)
to describe photoelectron spectra of these $3d-$transition metal dioxide
anions. Adding eigenvalue self-consistency also in $W$ ($G_nW_n$@PBE), on the
other hand, worsens the agreement with experiment overall. Our findings on the
performance of various $GW$ methods are discussed in the context of our
previous studies on other transition metal oxide molecular systems.
