from the $TiO_{2}$ dipolar layer - placed at height ($-h_{0}$) from the $FeSe$ plane, vibrate and make dipolar moments $\delta p_{z}=q_{eff}\delta h(x,y,-h_{0})$ perpendicular to the $FeSe$ ($x-y$) plane - see $Fig.1$. It gives rise to a dipolar electric potential $\Phi_{dip}(x,y,-h_{0}-\delta h)$ acting on electrons in the $FeSe$ ($x-y$ plane). Here, $q_{eff}$ is an effective charge per dipole and $\delta h$ is the polar (dominantly oxygen) displacement along the $z$-axis [@Lee-Interfacial-supplement]. Due to some confusion in the literature on the form of $\Phi_{dip}$ [@Lee-Interfacial-supplement] we recalculate it here, in order to know its explicite dependence on the semi-microscopic parameters $\varepsilon_{\parallel}^{eff}$, $\varepsilon_{\perp}^{eff}$, $q_{eff}$, $h_{0}$, $n_{d}$. An elementary electrodynamics approach [@Landau-Electrodynamics] gives for the dipolar potential $\Phi
_{dip}(x,y,-h_{0}-\delta h)$ $$\Phi_{dip}(x,y,-h_{0}-\delta h)=\frac{(\varepsilon_{\parallel}^{eff})^{1/2}}{(\varepsilon_{\perp}^{eff})^{3/2}}(n_{d}q_{eff}h_{0})\times$$$${\displaystyle\iint}
\frac{dx^{\prime}dy^{\prime}\delta h(x^{\prime},y^{\prime},-h_{0})}{\left(
\frac{\varepsilon_{\parallel}^{eff}}{\varepsilon_{\perp}^{eff}}h_{0}^{2}+(x-x^{\prime})^{2}+(y-y^{\prime})^{2}\right)  ^{\frac{3}{2}}},
\label{Phi}$$ where $n_{d}$ is the number of the oscillating $Ti-O$ dipoles per unit $FeSe$ surface. The coefficient in front of the integral is different from that in [@Lee-Interfacial-supplement] - where it is $\varepsilon_{\parallel}^{eff}/(\varepsilon_{\perp}^{eff})^{3/2}(q_{eff}h_{0})$ (probably typos?) and with missed dipole density $n_{d}$. This coefficient does not fulfill the condition $\Phi\sim\varepsilon^{-1}$ in the isotropic case, while $Eq.$(\[Phi\]) does. By introducing $g_{epi}(\mathbf{q})=e\Phi(\mathbf{q})$ the electron-phonon interaction Hamiltonian $H_{epi}=\sum_{\mathbf{q}}e\hat{\Phi}(\mathbf{q})\hat{\rho}(\mathbf{q})$ is rewritten in the form $H_{epi}=\sum_{\mathbf{k},\mathbf{q}}g_{epi}(\mathbf{q})(\hat{b}_{\mathbf{q}}+\hat{b}_{-\mathbf{q}}^{\dagger})\hat{c}_{\mathbf{k}+\mathbf{q}}^{\dagger
}\hat{c}_{\mathbf{k}}$, where $\hat{b}_{-\mathbf{q}}^{\dagger},\hat
{c}_{\mathbf{k}+\mathbf{q}}^{\dagger}$ are boson and fermion 