<i>Ab initio</i> low-energy effective Hamiltonians for the high-temperature superconducting cuprates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Bi</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Bi</mml:mi><mml:mn>2</mml:mn…

We derive $ab$ $initio$ low-energy effective Hamiltonians (LEH) for high-temperature superconducting (SC) copper oxides Bi$_2$Sr$_2$CuO$_6$ (Bi2201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 10$ K), Bi$_2$Sr$_2$CaCu$_2$O$_8$ (Bi2212, $N_{\ell}=2$, 84$ HgBa$_2$CuO$_4$ (Hg1201, 90$ K) and CaCuO$_2$ (Ca11, $N_{\ell}=\infty$, 110$ with different experimental optimal SC transition temperature exp}$ number $N_{\ell}$ of laminated CuO$_2$ planes between the two neighboring block layers. apply latest methodology multiscale scheme correlated electron systems (MACE), focus on LEH consisting one antibonding (AB) Cu$3d_{x^2-y^2}$/O$2p_{\sigma}$ orbital centered each Cu atom. discuss prominent features this LEH: (1) The ratio $U/|t_1|$ onsite Coulomb repulsion (ECR) $U$ amplitude nearest neighbour hopping $t_1$ increases $T^{\rm exp}_c$ $N_{\ell}$, consistently expected increase in $d$-wave correlation function $P_{dd}$ $U/|t_1|$. One possible cause is replacement apical O atoms by from neighbouring when increases. Furthermore, we show that distance decreases screening (ES) electrons outside (2) For Hg1201 Ca11, hole doping per AB $\delta$ increases, which may partly account disappearance exceeds value experiment. (3) $N_{\ell} \geq 2$, off-site inter-CuO$_2$ plane ECR comparable to intra-CuO$_2$ ECR. contributions both stability state.