The $t$--$J$ model for Cu$_2$O$_5$ coupled ladders in Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$*

Starting from the proper charge transfer model for Cu$_2$O$_5$ coupled ladders in Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$ in the strongly correlated regime, we derive the low energy Hamiltonian for this system with superexchange $J$ and the effective $d-d$ hopping $t$. It occurs that the widely used ladder $t$--$J$ model is not sufficient and has to be supplemented by the Coulomb repulsion between doped holes in the neighboring ladders. Next, we show how a simple mean-field solution of this extended $t$--$J$ model may explain the onset of the experimentally observed charge density wave with the odd (three or five) period in Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$. This suggests that it is not the bare ladder $t$--$J$ model but the extended $t$--$J$ model which should be used as the starting point for the explanation of the superconductivity observed under pressure in the Cu$_2$O$_5$ coupled ladders in Sr$_{0.4}$Ca$_{13.6}$Cu$_{24}$O$_{41}$.