Periodic density functional theory (DFT)+U calculations using sufficiently large supercells to explicitly account for dopant disorder are reported for undoped Ca${}_{2}$CuO${}_{2}$Cl${}_{2}$ and for the doped Ca${}_{2\ensuremath{-}x}$Na${}_{x}$CuO${}_{2}$Cl${}_{2}$ ($x$ $=$ 0.125 and $x$ $=$ 0.250). For the undoped material, the charge-transfer antiferromagnetic character is properly reproduced. For the doped systems, a metallic character is predicted with the conduction band dominated by the O(2p) states, with doping having a small effect on the antiferromagnetic order of the Cu${}^{2+}$ cations within the CuO${}_{2}$ planes while maintaining the diamagnetic character in the rest of centers. The analysis of the spin density plots for the doped material evidences the appearance of a new feature reminiscent of the so-called Zhang-Rice singlet, thus providing unbiased independent support to the corresponding model Hamiltonian. However, it is also worth pointing out that the present DFT picture provides only an average static representation of what is expected to be a dynamic electronic structure.