Anisotropic superexchange for nearest and next-nearest coppers in chain, ladder, and lamellar cuprates

We present a detailed calculation of the magnetic couplings between nearest-neighbor and next-nearest-neighbor coppers in the edge-sharing geometry, ubiquitous in many cuprates. In this geometry, the interaction between nearest-neighbor coppers is mediated via two oxygens, and the Cu-O-Cu angle is close to (Formula presented) The derivation is based on a perturbation expansion of a general Hubbard Hamiltonian, and produces numerical estimates for the various magnetic energies. In particular we find the dependence of the anisotropy energies on the angular deviation away from the (Formula presented) geometry of the Cu-O-Cu bonds. Our results are required for the correct analysis of the magnetic structure of various chain, ladder, and lamellar cuprates.