  We developed a semiclassical approximation method in combination with an
adaptive moment estimation optimizer (SCA + ADAM) approach based on the PyTorch
plus CUDA library on a the graphics processing unit (GPU). This method was
employed to evaluate one-particle properties of the Hubbard model with
long-range spatial correlations within an appropriate computing duration. The
method was applied to the ionic Hubbard model on a two-dimensional square
lattice with long-range spatial correlations. The computation time was
evaluated as a function of the lattice size on the central processing unit and
GPU. Herein, we also discuss the density of states and antiferromagnetic (AF)
order parameter in the Hubbard model without the ionic potential and compare
the results with those of the Hartree-Fock approximation. Finally, we present
the one-particle properties and order parameter in charge density wave, AF
metal and AF insulator of the ionic Hubbard model.
