Ab initio low-energy effective Hamiltonians of two typical high-temperature copper-oxide superconductors, whose mother compounds are La2CuO4 and HgBa2CuO4, are derived by utilizing the multiscale 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 first principles and a 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) a one-band Hamiltonian for the antibonding orbital generated from strongly hybridized Cu 3dx2-y2 and O 2pσ orbitals, (2) a two-band Hamiltonian constructed from the antibonding orbital and Cu 3d3z2-r2 orbital hybridized mainly with the apex oxygen pz orbital, and (3) a three-band Hamiltonian consisting mainly of Cu 3dx2-y2 orbitals and two O 2pσ orbitals. Differences between the Hamiltonians for La2CuO4 and HgBa2CuO4, which have relatively low and high critical temperatures Tc, respectively, at optimally doped compounds, are elucidated. The main differences are summarized as follows: (i) the oxygen 2pσ orbitals are farther (∼3.7 eV) below from the Cu dx2-y2 orbital in the case of the La compound than the Hg compound (∼2.4 eV) in the three-band Hamiltonian. This causes a substantial difference in the character of the dx2-y2-2pσ 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. Heavier entanglement of the two bands in the two-band Hamiltonian implies that the two-band rather than the one-band Hamiltonian is more appropriate for the La compound. The relevance of the three-band description is also discussed especially for the Hg compound.
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