  We present ab initio calculations of the superexchange antiferromagnetic spin
coupling $J$ for two cuprates, Sr$_2$CuO$_3$ and La$_2$CuO$_4$. Good agreement
with experimental estimates is obtained. We find that $J$ increases
substantially as the distance between Cu and apical O is increased. There is an
important synergetic effect of the Coulomb interaction, expanding the Cu $3d$
orbital when an electron hops into this orbital, and the O-Cu hopping, being
increased by this orbital expansion (breathing). This is a new ingredient in
superexchange models. In a model with a fixed basis, breathing effects can be
described as a mixing of $3d$ and $4d$ orbitals or as a single $3d \to 4d$
excitation.

Strongly Correlated Electrons