  The one-band and three-band Hubbard models which describe the electronic
structure of cuprates indicate very different values of effective electronic
parameters, such as the on-site Coulomb energy and the hybridization strength.
In contrast, a comparison of electronic parameters of several cuprates with
corresponding values from spectroscopy and scattering experiments indicates
similar values in the three-band model and cluster model calculations used to
simulate experimental results. The Heisenberg exchange coupling $J$ obtained by
a downfolding method in terms of the three band parameters is used to carry out
an optimization analysis consistent with $J$ from neutron scattering
experiments for a series of cuprates. In addition, the effective one-band
parameters $\tilde{U}$ and $\tilde{t}$ are described using the three band
parameters, thus revealing the hidden equivalence of the one-band and
three-band models. The ground-state singlet weights obtained from an exact
diagonalization elucidates the role of Zhang-Rice singlets in the equivalence.
The results provide a consistent method to connect electronic parameters
obtained from spectroscopy and the three-band model with values of $J$ obtained
from scattering experiments, band dispersion measurements and the effective
one-band Hubbard model.
