interaction is purely coulombic, $$g_{\alpha }({\bf m-n})=\frac{\kappa _{\alpha }}{|{\bf m-n}|^{2}},$$ where $\alpha =x,y,z$. To account for the fact that $c$ axis-polarized phonons couple to the holes stronger than others due to a poor screening [@ale5; @alekor; @alekor2], we choose $%
\kappa _{x}=\kappa _{y}=\kappa _{z}/\sqrt{2}$. The direct hole-hole repulsion is $$V_{c}({\bf n-n^{\prime }})=\frac{V_{c}}{\sqrt{2}|{\bf n-n^{\prime
}}|}$$ so that the repulsion between two holes in the nearest neighbor ($NN$) configuration is $V_{c}$. We also include the bare $NN$ hopping $T_{NN}$, the next nearest neighbor ($NNN$) hopping across copper $T_{NNN}$ and the $NNN$ hopping between the pyramids $T_{NNN}^{\prime }$.

The polaron shift is given by the lattice sum Eq.(3), which after summation over polarizations yields $$E_{p}=2\kappa _{x}^{2}\omega _{0}\sum_{{\bf m}}\left( \frac{1}{|{\bf m-n}%
|^{4}}+\frac{h^{2}}{|{\bf m-n}|^{6}}\right) =31.15\kappa
_{x}^{2}\omega _{0}, \label{televen}$$ where the factor $2$ accounts for two layers of apical sites. For reference, the Cartesian coordinates are ${\bf n}=(n_{x}+1/2,n_{y}+1/2,0)$, ${\bf m}%
=(m_{x},m_{y},h)$, and $n_{x},n_{y},m_{x},m_{y}$ are integers. The polaron-polaron attraction is $$V_{ph}({\bf n-n^{\prime }})=4\omega \kappa _{x}^{2}\sum_{{\bf m}}\frac{%
h^{2}+({\bf m-n^{\prime }})\cdot ({\bf m-n})}{|{\bf m-n^{\prime }}|^{3}|{\bf %
m-n}|^{3}}.  \label{ttwelve}$$ Performing the lattice summations for the $NN$, $NNN$, and $NNN^{\prime}$ configurations one finds $V_{ph}=1.23\,E_{p},$ $0.80\,E_{p}$, and $0.82\,E_{p}$, respectively. As a result, we obtain a net inter-polaron interaction as $%
v_{NN}=V_{c}-1.23\,E_{p}$, $v_{NNN}=\frac{V_{c}}{\sqrt{2}}-0.80\,E_{p}$, 