A general procedure to evaluate many-body spin operator amplitudes from periodic calculations: application to cuprates

A general procedure is presented which permits the form of an extended spin Hamiltonian to be established for a given magnetic solid and the magnitude of its terms to be evaluated from spin polarized, Hartree–Fock or density functional calculations carried out for periodic models. The computational strategy makes use of a general mapping between the energy of pertinent broken-symmetry solutions and the diagonal terms of the spin Hamiltonian in a local representation. From this mapping it is possible to determine not only the amplitude of the well-known two-body magnetic coupling constants between near-neighbor sites, but also the amplitudes of four-body cyclic exchange terms. A scrutiny of the on-site spin densities provides additional information and control of the many broken-symmetry solutions which can be found. The procedure is applied to the La₂CuO₄, Sr₂CuO₂F₂, Sr₂CuO₂Cl₂ and Ca₂CuO₂Cl₂ square lattices and the SrCu₂O₃ ladder compound. It is shown that a proper description of the magnetic structure of these compounds requires that two- and four-body terms are explicitly included in the spin Hamiltonian. The implications for the interpretation of recent experiments are discussed.