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.

We present a study concerning the effect of the on site $d\text{−}d$ Coulomb interaction energy $U$ on the band-gap states of nonstoichiometric rutile (110) $\mathrm{Ti}{\mathrm{O}}_2$ surface. As well known, the excess electrons resulting from the formation of oxygen vacancies localize on the $\mathrm{Ti}3d$ orbitals forming band-gap states. Local density approximation (LDA) does not give a correct description of these band-gap states, either with or without gradient corrections. The failure of LDA is often attributed to an inadequate treatment of electron correlation in systems with localized orbitals and is commonly corrected with an empirical local Coulomb repulsion term, i.e., the $\mathrm{LDA}+U$ method. This study provides a completely general strategy to estimate the $U$ value in this kind of systems, illustrated here for reduced (110) $\mathrm{Ti}{\mathrm{O}}_2$ surface, well characterized from experiments. From ab initio embedded cluster configuration interaction calculations, combined with the effective Hamiltonian theory, a value of $U$ of $5.5\pm 0.5\mathrm{eV}$ is obtained, in good agreement with those reported for this system from x-ray photoemission spectroscopy experiments $(U=4.5\pm 0.5\mathrm{eV})$. It is observed that when the ab initio estimate of $U$ is injected into the periodic $\mathrm{LDA}+U$ calculations, a correct description of the gap states is obtained from the periodic $\mathrm{LDA}+U$ calculations. Additionally, the results indicate that the position of these states on the band gap strongly depends on the level at which lattice relaxation is taken into account, with significant differences between the density of states curves at the $\mathrm{LDA}+U$ level obtained using the optimal generalized gradient approximation or $\mathrm{LDA}+U$ geometries. These results suggest that this combined strategy could be a useful tool for those systems where electron correlation plays a key role, and no experimental data are available for the on-site Coulomb repulsion.

In magnetic coordination compounds and solids the magnetic orbitals are essentially located on metallic centers but present some delocalization tails on adjacent ligands. Mean field variational calculations optimize this mixing and validate a single band modelization of the intersite magnetic exchange. In this approach, due to the Brillouin’s theorem, the ligand to metal charge transfer (LMCT) excitations play a minor role. On the other hand the extensive configuration interaction calculations show that the determinants obtained by a single excitation on the top of the LMCT configurations bring an important antiferromagnetic contribution to the magnetic coupling. Perturbative and truncated variational calculations show that contrary to the interpretation given in a previous article [C. J. Calzado et al., J. Chem. Phys. 116, 2728 (2002)] the contribution of these determinants to the magnetic coupling constant is not a second-order one. An analytic development enables one to establish that they contribute at higher order as a correlation induced increase in the LMCT components of the wave function, i.e., of the mixing between the ligand and the magnetic orbitals. This larger delocalization of the magnetic orbitals results in an increase in both the ferro- and antiferromagnetic contributions to the coupling constant.

Magnetic interactions in ionic solids are studied using parameter-free methods designed to provide accurate energy differences associated with quantum states defining the Heisenberg constant J. For a series of ionic solids including ${\mathrm{KNiF}}_3,$ ${\mathrm{K}}_2{\mathrm{NiF}}_4,$ ${\mathrm{KCuF}}_3,$ ${\mathrm{K}}_2{\mathrm{CuF}}_4,$ and high- $T_c$ parent compound ${\mathrm{La}}_2{\mathrm{CuO}}_4,$ the J experimental value is quantitatively reproduced. This result has fundamental implications because J values have been calculated from a finite cluster model whereas experiments refer to infinite solids. The present study permits us to firmly establish that in these wide-gap insulators, J is determined from strongly local electronic interactions involving two magnetic centers only thus providing an ab initio support to commonly used model Hamiltonians.