Derivation of spin Hamiltonians from the exact Hamiltonian: Application to systems with two unpaired electrons per magnetic site

The foundations and limits of S=1/2 and S=1 spin Hamiltonians for systems with two unpaired electrons in two well-defined orbitals per site are discussed by merging accurate ab initio calculations in binuclear systems with the effective Hamiltonian theory. It is shown that, beyond the usual JijSi.Sj terms, the effective spin Hamiltonian necessarily introduces four-body spin operators in the S=1/2 case and biquadratic terms in the S=1 formalism. The order of magnitude of these additional terms can be rationalized from a quasidegenerate perturbation theory expansion starting from a Hubbard-type Hamiltonian. This permits to discuss the physical mechanisms governing the reduction from the all electron Hamiltonian to the spin-only Hamiltonians and the conditions under which a further reduction from a spin Hamiltonian to the simplest Heisenberg-Dirac-Van Vleck form is possible. The overall discussion is illustrated by numerical calculations of the magnetic coupling between two Ni2+ cations in the K2NiF4 perovskite and between triply bonded carbon atoms in poly-ynes.