Accurate estimates of the magnetic coupling in binuclear complexes can be obtained from ab initio configuration interaction (CI) calculations using the difference dedicated CI technique. The present paper shows that the same technique also provides a way to analyze the various physical contributions to the coupling and performs numerical analysis of their respective roles on four binuclear complexes of Cu ${\mathrm{(d}}^9)$ ions. The bare valence-only description (including direct and kinetic exchange) does not result in meaningful values. The spin-polarization phenomenon cannot be neglected, its sign and amplitude depend on the system. The two leading dynamical correlation effects have an antiferromagnetic character. The first one goes through the dynamical polarization of the environment in the ionic valence bond forms (i.e., the ${\mathrm{M}}^{+}\cdots {\mathrm{M}}^{-}$ structures). The second one is due to the double excitations involving simultaneously single excitations between the bridging ligand and the magnetic orbitals and single excitations of the environment. This dispersive effect results in an increase of the effective hopping integral between the magnetic orbitals. Moreover, it is demonstrated to be responsible for the previously observed larger metal-ligand delocalization occurring in natural orbitals with respect to the Hartree–Fock ones.

Most interpretations of the magnetic coupling J between two unpaired electrons rest upon simple valence models that involve essentially the ferromagnetic direct exchange contribution, $K_{\mathrm{ab}},$ and the antiferromagnetic effect of the delocalization resulting from the interaction between neutral and ionic determinants, $t_{\mathrm{ab}},$ whose energy difference is U. Ab initio valence-only calculations give very poor estimates of J, whatever the definition of the magnetic orbitals, and large CI expansions are required to evaluate it properly. It is, however, possible to define valence effective Hamiltonians from the knowledge of the eigenenergies and the eigenvectors of these accurate CI calculations. When applied to four different complexes, this strategy shows that spin polarization may change the sign of the direct exchange interaction, $K_{\mathrm{ab}},$ and that dynamical correlation results in a dramatic reduction of the effective repulsion U. The present article also shows how $K_{\mathrm{ab}},$ $t_{\mathrm{ab}},$ and U effective parameters can be extracted from density functional theory (DFT) calculations and that the typical overestimation of J in DFT can be attributed to an excessive lowering of the effective on-site repulsion.