Tendency towards Local Spin Compensation of Holes in the High-$T_c$ Copper Compounds

Using an Anderson impurity description of the high-$T_c$ superconductors, we study the character of the states closest to the Fermi level. In a kind of phase diagram involving the Cu $d-d$ Coulomb interaction ($U$), the charge-transfer energy ($\Delta$), and the hybridization ($T$), we show under which conditions the singlet, triplet, or delocalized ligand hole states are closest to the Fermi level. For reasonable values of $T$ the lowest-energy hole states are local singles (${}_{}{}^1{}_{}{}^{}A_{1g}^{}{}_{}{}^{}$) of primarily $d^9\mathit{L}$ character for $U>\mathrm{}\Delta$ and local triples (${}_{}{}^3{}_{}{}^{}B_{1g}^{}{}_{}{}^{}$) of primarily $d^8$ character for $U<\mathrm{}\Delta$.