  We determine the ground-state phase diagram of the three-band Hubbard model
across a range of model parameters using density matrix embedding theory. We
study the atomic-scale nature of the antiferromagnetic (AFM) and
superconducting (SC) orders, explicitly including the oxygen degrees of
freedom. All parametrizations of the model display AFM and SC phases, but the
decay of AFM order with doping is too slow compared to the experimental phase
diagram, and further, coexistence of AFM and SC orders occurs in all parameter
sets. The local magnetic moment localizes entirely at the copper sites. The
magnetic phase diagram is particularly sensitive to $\Delta_{pd}$ and $t_{pp}$,
and existing estimates of the charge transfer gap $\Delta_{pd}$ appear too
large in so-called minimal model parametrizations. The electron-doped side of
the phase diagram is qualitatively distinct from hole-doped side and we find an
unusual two-peak structure in the SC in the full model parametrization.
Examining the SC order at the atomic scale, within the larger scale $d_{x^2 -
y^2}$-wave SC pairing order between Cu-Cu and O-O, we also observe a local
$p_{x (y)}$ [or $d_{xz (yz)}$]-symmetry modulation of the pair density on the
Cu-O bonds. Our work highlights some of the features that arise in a three-band
versus one-band picture, the role of the oxygen degrees of freedom in new kinds
of atomic-scale SC orders, and the necessity of re-evaluating current
parametrizations of the three-band Hubbard model.
