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At optimal doping, different cuprate compounds can exhibit vastly differentcritical temperatures for superconductivity ($T_c$), ranging from about 20 K toabout 135 K. The precise properties of the lattice that determine the magnitudeof the $T_c$ are currently unknown. In this paper, we investigate thedependence 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 isobtained not with the parameters of the model written in the real($d/p$-orbital) space, but rather written in the space of effective Wannierorbitals. In this basis, one obtains a model of three sublattices coupledthrough all possible 4-point interactions. We identify multiple predictorvariables that fit the experimental $T_c$ to about $\pm4-5$ K and thatremarkably depend on the leading attractive coupling constants in thetransformed Hamiltonian.

Simple predictors of $T_c$ in superconducting cuprates reveal role of interactions between effective Wannier orbitals in the $d-p$ 3-band model