  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 90 degrees. 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 90 degrees 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.
