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We formulate a method of deriving effective low-energy Hamiltonian fornonperiodic systems such as interfaces for strongly correlated electron systemsby extending conventional multi-scale $\textit{ab initio}$ scheme forcorrelated electrons (MACE). We apply the formalism to copper-oxide high$T_{\rm c}$ superconductors in an example of the interface between overdopedLa$_{2-x}$Sr$_{x}$CuO$_{4}$ and Mott insulating La$_{2}$CuO$_{4}$ recentlyrealized experimentally. We show that the parameters of the $E_g$ Hamiltonianderived for the La$_{2}$CuO$_{4}$/La$_{1.55}$Sr$_{0.45}$CuO$_{4}$ superlatticediffer considerably from those for the bulk La$_{2}$CuO$_{4}$, particularlysignificant in the partially-screened Coulomb parameters and the level offsetbetween the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbitals, $\Delta E$. Inaddition, we investigate the effect of the lattice relaxation on the $E_g$Hamiltonian by carefully comparing the parameters derived before and after thestructure optimization. We find that the CuO$_{6}$ octahedra distort after therelaxation as a consequence of the Madelung potential difference between theinsulator and metal sides, by which the layer dependence of the hopping andCoulomb parameters becomes more gradual than the unrelaxed case. Furthermore,the structure relaxation dramatically changes the $\Delta E$ value and theoccupation number at the interface. This study not only evidences theimportance of the ionic relaxation around interfaces but also provides a set oflayer-dependent parameters of the $E_g$ Hamiltonian, which is expected toprovide further insight into the interfacial superconductivity when solved withlow-energy solvers.

$\textit{Ab initio}$ derivation of effective Hamiltonian for La$_{2}$CuO$_{4}$/La$_{1.55}$Sr$_{0.45}$CuO$_{4}$ heterostructure