  At optimal doping, different cuprate compounds can exhibit vastly different
critical temperatures for superconductivity ($T_c$), ranging from about 20 K to
about 135 K. The precise properties of the lattice that determine the magnitude
of the $T_c$ are currently unknown. In this paper, we investigate the
dependence of the optimal doping $T_c$ on the parameters of the Emery ($d-p$)
model for the CuO$_2$ planes in the cuprates. We show that the best scaling is
obtained not with the parameters of the model written in the real
($d/p$-orbital) space, but rather written in the space of effective Wannier
orbitals. In this basis, one obtains a model of three sublattices coupled
through all possible 4-point interactions. We identify multiple predictor
variables that fit the experimental $T_c$ to about $\pm4-5$ K and that
remarkably depend on the leading attractive coupling constants in the
transformed Hamiltonian.
