  Three-types (three-band, two-band and one-band) of effective Hamiltonians for
the HgBa$_2$CuO$_4$ and three-band effective Hamiltonian for La$_2$CuO$_4$ are
derived beyond the level of the constrained-GW approximation combined with the
self-interaction correction (cGW-SIC) derived in Hirayama et al. Phys. Rev. B
98, 134501 (2018) by improving the treatment of the interband Hartree energy.
The charge gap and antiferromagnetic ordered moment show good agreement with
the experimental results when the present effective Hamiltonian is solved,
indicating the importance of the present refinement. The obtained Hamiltonians
will serve to clarify the electronic structures of these copper oxide
superconductors and to elucidate the superconducting mechanism.
