  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.

Superconductivity;Strongly Correlated Electrons