  Starting from a single band Hubbard model in the Wannier function basis, we
revisit the problem of the ligand contribution to exchange and derive explicit
formulae for the exchange integrals in metal oxide compounds in terms of atomic
parameters that can be calculated with constrained LDA and LDA+U. The analysis
is applied to the investigation of the isotropic exchange interactions of
LiCu$_{2}$O$_{2}$, a compound where the Cu-O-Cu angle of the dominant exchange
path is close to 90$^{\circ}$. Our results show that the magnetic moments are
localized in Wannier orbitals which have strong contribution from oxygen atomic
orbitals, leading to exchange integrals that considerably differ from the
estimates based on kinetic exchange only. Using LSDA+U approach, we also
perform a direct {\it ab-initio} determination of the exchange integrals
LiCu$_{2}$O$_{2}$. The results agree well with those obtained from the Wannier
function approach, a clear indication that this modelization captures the
essential physics of exchange. A comparison with experimental results is also
included, with the conclusion that a very precise determination of the Wannier
function is crucial to reach quantitative estimates.
