  Ab initio low-energy effective Hamiltonians of two typical high-temperature
copper-oxide superconductors, whose mother compounds are La$_2$CuO$_4$ and
HgBa$_2$CuO$_4$, are derived by utilizing the multi-scale ab initio scheme for
correlated electrons (MACE). The effective Hamiltonians obtained in the present
study serve as platforms of future studies to accurately solve the low-energy
effective Hamiltonians beyond the density functional theory. It allows further
study on the superconducting mechanism from the first principles and
quantitative basis without adjustable parameters not only for the available
cuprates but also for future design of higher Tc in general. More concretely,
we derive effective Hamiltonians for three variations, 1)one-band Hamiltonian
for the antibonding orbital generated from strongly hybridized Cu
$3d_{x^2-y^2}$ and O $2p_\sigma$ orbitals 2)two-band Hamiltonian constructed
from the antibonding orbital and Cu $3d_{3z^2-r^2}$ orbital hybridized mainly
with the apex oxygen $p_z$ orbital 3)three-band Hamiltonian consisting mainly
of Cu $3d_{x^2-y^2}$ orbitals and two O $2p_\sigma$ orbitals. Differences
between the Hamiltonians for La$_2$CuO$_4$ and HgBa$_2$CuO$_4$, which have
relatively low and high critical temperatures, respectively, at optimally doped
compounds, are elucidated. The main differences are summarized as i) the oxygen
$2p_\sigma$ orbitals are farther(~3.7eV) below from the Cu $d_{x^2-y^2}$
orbital for the La compound than the Hg compound(~2.4eV) in the three-band
Hamiltonian. This causes a substantial difference in the character of the
$d_{x^2-y^2}-2p_\sigma$ antibonding band at the Fermi level and makes the
effective onsite Coulomb interaction U larger for the La compound than the Hg
compound for the two- and one-band Hamiltonians. ii)The ratio of the
second-neighbor to the nearest transfer t'/t is also substantially different
(~0.26) for the Hg and ~0.15 for the La compound in the one-band Hamiltonian.
