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