We revisit the problem of the spectra of two holes in a CuO$_{2}$ layer, modeled as a Cu-d$^{8}$ impurity with full multiplet structure coupled to a full O-2p band as an approximation to the local electronic structure of a hole doped cuprate. Unlike previous studies that treated the O band as a featureless bath, we describe it with a realistic tight binding model. While our results are in qualitative agreement with previous work, we find considerable quantitative changes when using the proper O-2p band structure. We also find (i) that only the ligand O-2p orbitals play an essential role, within this impurity model; (ii) that the three-orbital Emery model provides an accurate description for the subspace with $^{1}!A_1$ symmetry, which includes the ground-state in the relevant region of the phase diagram; (iii) that this ground-state has only $sim 50%$ overlap with a Zhang-Rice singlet; (iv) that there are other low-energy states, in subspaces with different symmetries, that are absent from the three-orbital Emery model and its one-band descendants. These states play an important role in describing the elementary excitations of doped cuprates.
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