Phase diagram of the $CuO_{3}$ chains in $YBa_{2}Cu_{3}O_{6+X}$ and $PrBa_{2}Cu_{3}O_{6+X}$

We use a mapping of the multiband Hubbard model for $CuO_{3}$ chains in $RBa_{2}Cu_{3}0_{6+x}$ (R=Y or a rare earth) onto a $t-J$ model and the description of the charge dynamics of the latter in terms pf s spinless model, to study the electronic structure of the chains. We briefly review results for the optical conductivity and we calculate the quantum phase diagram of quarter filled chains including Coulomb repulsion up to that between next-nearest-neighbor $Cu$ atoms $V_{2}$, using the resulting effective Hamiltonian, mapped onto an XXZ chain, and the method of crossing of excitation spectra. The method gives accurate results for the boundaries of the metallic phase in this case. The inclusion of $V_{2}$ greatly enhances the region of metallic behavior of the chains.