Ab initio determination of an extended Heisenberg Hamiltonian in CuO2   layers
  Accurate ab initio calculations on embedded Cu_4O_{12} square clusters, fragments of the La_2CuO_4 lattice, confirm a value of the nearest neighbor antiferromagnetic coupling (J=124 meV) previously obtained from ab initio calculations on bicentric clusters and in good agreement with experiment. These calculations predict non negligible antiferromagnetic second-neighbor interaction (J'=6.5 meV) and four-spin cyclic exchange (K=14 meV), which may affect the thermodynamic and spectroscopic properties of these materials. The dependence of the magnetic coupling on local lattice distortions has also been investigated. Among them the best candidate to induce a spin-phonon effect seems to be the movement of the Cu atoms, changing the Cu-Cu distance, for which the variation of the nearest neighbor magnetic coupling with the Cu-O distance is {\Delta J}/{\Delta d_{Cu-O}}\sim 1700 cm^{-1} A^{-1}. 
Carmen J. Calzado and Jean-Paul Malrieu