The four-spin cyclic exchange term J ring of three spin-ladder cuprates (SrCu 2 O 3 , Sr 2 Cu 3 O 5 , and CaCu 2 O 3 ) has been calculated from ab initio quantum chemistry calculations. For the first two compounds, a nonnegligible cyclic exchange is found, aproximately 20% of the magnetic coupling across the rungs, J Ќ , and always larger than the value obtained for two-dimensional La 2 CuO 4 system. In the case of CaCu 2 O 3 , the J ring value is quite small, due to the folding of the Cu-O-Cu rung angle, but the J ring /J Ќ ratio is also 0.2 as in the two other systems. DOI: 10.1103/PhysRevB.67.132409 PACS number͑s͒: 75.30.Et, 71.27.ϩa, 71.70.Gm, 75.50.Ee Spin-ladder cuprates constitute an active research field in the last decade.1,2 They can be viewed as intermediates between the one-dimensional ͑1D͒ antiferromagnets and the still controversial two-dimensional ͑2D͒ square lattices. Ladders composed of Cu and O are specially interesting due to their proximity to high-T c cuprates. Their magnetic properties depend on the number of legs. Even-legged ladders show a spin gap excitation, whereas odd-legged ladders are gapless and behave as a 1D spin chain.1,2 They also present different properties regarding hole doping. It has been suggested that even-legged spin ladders become superconductors upon hole doping, which has been confirmed experimentally 3 in the two-legged ladder Sr 14Ϫx Ca x Cu 24 O 41 under high pressure.The magnetic properties of these compounds are controlled by the effective magnetic coupling constant J, related with the amplitude of the interactions between the spin moments of the Cu ϩ2 ions. Different J constants can be defined, as shown in Fig. 1. The two most important are the coupling along the legs, J ʈ , and across the rungs, J Ќ . The ratio J Ќ /J ʈ is controversial since the interpretation of different experimental data has led to estimates ranging from spatially isotropic, J Ќ /J ʈ ϭ1, to strongly anisotropic couplings, J Ќ /J ʈ ϭ0.5. The strong spatial anisotropy J Ќ /J ʈ ϭ0.5 is in contradiction with geometrical considerations. Since the Cu-O-Cu bonds are quite similar, the exchange pathways are expected to be equivalent, and so, J ʈ ϳJ Ќ . The theoretical calculations of Mizuno, Tohyama, and Maekawa; 4 and de Graaf et al. 5 are in agreement with these considerations.It should be noted that most of the available J Ќ and J ʈ values have been obtained by fitting the experimental data onto a model Heisenberg Hamiltonian, containing just twobody operators. As in the case of the 2D cuprates, 6-10 some authors have recently suggested the necessity of introducing additional interactions in the model Heisenberg Hamiltonian to study the properties of the spin ladders. The most important are the diagonal coupling ͑second-neighbor interactions͒, the interladder exchange and, especially, the four-spin cyclic exchange ͑4SCE͒. In this context, de Graaf et al. 5 have proposed that the omission of the interladder coupling in the analysis of experimental data for SrCu 2 O 3 may be the reason that a...