  The magnetic interactions in one-dimensional, two-dimensional (2D) and ladder
cuprates are evaluated systematically by using small Cu-O clusters. We find
that the superexchange interaction J between nearest neighbor Cu spins strongly
depends on Cu-O structure through the Madelung potential, and in 2D and ladder
cuprates there is a four-spin interaction Jcyc, with magnitude of 10% of J. We
show that Jcyc has a strong influence on the magnetic excitation in the
high-energy region of 2D cuprates.
