Exact diagonalization study of the hole distribution inCuO3chains within the four-band dp model

We consider the hole distribution in ${\mathrm{CuO}}_3$ chains as a function of the total hole number 0⩽ n ⩽2 per chain unit within the standard dp model containing one orbital per site. The nonequivalency of the apical and the chain oxygen sites is taken into account explicitly. Using slightly modified standard ${\mathrm{CuO}}_2$ plane parameter sets, we compare the results of exact diagonalization studies of periodic and open ${\mathrm{CuO}}_3$ chains with experimental data for ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ (YBCO) and Ca(Sr${)}_2$ ${\mathrm{CuO}}_3$, and with theoretical results obtained within the following often used approximations: the Hartree-Fock and the local ansatz approximations, as well as with the exclusion of double occupancies at Cu sites and also that at oxygen ones (spinless fermion picture). We have found out that the ratio of the hole densities on the apical and the chain oxygen sites is sensitive to the magnitude of the nearest-neighbor Coulomb interaction ${\mathrm{V}}_{\mathrm{pd}}$ and the difference of their site energies ${\mathrm{\Delta }}_{\mathrm{pp}}$. Adopting values ${\mathrm{V}}_{\mathrm{pd}}$∼ 1 eV and ${\mathrm{V}}_{\mathrm{pp}}$∼0.${5\mathrm{V}}_{\mathrm{pd}}$, we estimate ${\mathrm{\Delta }}_{\mathrm{pp}}$=1.5 to 2.5 eV for YBCO from the comparison with O 1s x-ray absorption spectroscopy and ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ nuclear magnetic resonance data. Using experimental values of the corresponding binding energies and the components of the electric field gradient tensor, the site energy of the apical oxygen ${2\mathrm{p}}_{\mathrm{z}}$ states relative to the planar copper Cu(2) ${3\mathrm{d}}_{{\mathrm{x}}^2\mathrm{-}{\mathrm{y}}^2}$ states is estimated as ∼5 to 6 eV. With increasing hole doping and/or the strength of ${\mathrm{V}}_{\mathrm{pd}}$, the holes are increasingly localized at the apical oxygen sites. Various chain aspects of proposed scenarios for the ${\mathrm{PrBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ problem are briefly discussed.