Inspired by the work of de Loth and coworkers[41] and triggered by the accurate magnetic couplings provided by DDCI wave functions,[58] Malrieu, Caballol, and coworkers analyzed the magnetic coupling in great detail in a series of articles published between 2002 and 2011.[52, 59-63] These studies have been reviewed recently,[53] and here, we will only illustrate the main conclusions with an example to show the power of the applied methodology for analysis. First, the main mechanisms that contribute to the coupling between two spins will be exposed and next it will be shown how a simpler description can be recovered through an effective Hamiltonian.

#### Mechanisms

Experimentally, it has been established that the singlet-triplet splitting in [N,N′,N″-trimethyl-1,4,7-triazacyclononane_{2}Cu_{2}(μ-1,3-N_{3})_{2}]^{2+} (Cu-azido) is around 800 cm^{−1} with the open-shell singlet as lowest state. To a good approximation, the replacement of the bulky external ligands with NH_{3} groups does not alter the coupling of the spin moments of the Cu^{2+} ions and this model will be used to analyze the origin of the 800 cm^{−1} energy gap. The orthonormal orbitals *a* and *b*, given in Figure 2, define a subspace of the full valence space that can be used to obtain the basic wave functions of the singlet (S) and triplet (T) states:

- (5)

The second term is known as the kinetic exchange and favors the antiparallel alignment of the spin moments, contrary to the first term, the direct exchange, which stabilizes the triplet state.

The CI calculation in the space spanned by the neutral and ionic determinants gives direct access to the numerical estimates of these two antagonist interactions. The direct exchange contributes +12 cm^{−1} to the energy difference, whereas the kinetic exchange is much larger and of opposite sign, −94 cm^{−1}. The sum of the two terms is approximately −80 cm^{−1},that is, just 10% of the total splitting and hence there must be other important contributions involving determinants with holes or electrons outside the small valence subspace considered up to now.

The first class of external determinants that play an important role are those included in the CAS+S calculation and comprise spin and charge polarization effects. 2 The charge polarization can be defined as the response of the electrons in the closed shells to the electron transfer from one magnetic site to the other. The minimal space description includes the charge transfer process through the presence of the ionic determinants in the wave function, but the energy of these determinants is largely overestimated and the process is not as effective as it should be. The determinants obtained from a single excitation from inactive to virtual orbitals do not interact with the neutral determinants, but do lower the effective energy of the ionic determinants, which gain weight in the wave functions. As the ionic determinants can only contribute to the singlet wave function [cf. Eq. (7)], the charge polarization favors the antiparallel coupling of the spin moments. For the example molecule the magnetic coupling is enhanced by −221 cm^{−1}.

The spin polarization introduces extra spin density around the magnetic center through determinants with one hole in the inactive and one electron in the virtual orbitals (see Fig. 3). In these determinants, the unpaired electrons external to the reference space are coupled to a triplet, whereas the two electrons in the reference space are also parallel aligned. These two triplet coupled electron pairs can be coupled to singlet or to triplet. 3 This means that the effect of the spin polarization is not easily anticipated, it can both contribute in a ferro- and an antiferromagnetic manner. For the Cu-azido complex considered here, the contribution of the spin polarization is antiferromagnetic, −59 cm^{−1}.

Adding the two contributions included in CAS+S calculation to the estimate obtained with the reference wave function, the magnetic coupling becomes −362 cm^{−1}. This is already half way to the experimental estimate, but some effects are still missing. In the first place, one may add the double ligand-to-metal charge transfer (LMCT) and the double metal-to-ligand CT (MLCT) determinants to the wave function. 4 The contribution of these determinants is in general antiferromagnetic, but rather modest in size given the high energy of the determinants. In Cu-azido the contribution is just −13 cm^{−1}.

Much more important are the determinants that combine a single excitation from inactive to virtual orbitals with either an inactive to magnetic or a magnetic to virtual electron replacement. In many applications of the DDCI method to magnetic coupling problems, only the inclusion of this class of determinants provides (nearly) quantitative agreement with experiment. As can be seen in Figure 3, these classes of determinants combine LMCT and MLCT excitations with single excitations from inactive to virtual orbitals. The latter are connected to the orbital optimization process. The coupling of the single inactive-to-virtual excitations to the LMCT and MLCT excitations provide relaxed CT excitations. The direct interaction of the determinants with those of the reference space is small, but coupling between external determinants in the CI makes that the MLCT and LMCT gain important weight in the wave function.[52] The nature of the contribution is variable although the relaxation of the LMCT tends to increase the antiferromagnetic character of the coupling and the relaxed MLCT usually acts in the opposite direction. The addition of these two classes of determinants to the CI expansion strengthens the magnetic coupling by −427 cm^{−1} and brings the theoretical estimate (−802 cm^{−1}) in good agreement with experimental data that indicate that *J* is at least −800 cm^{−1}. The disentanglement of the two types of determinants is rather complicated because the full contribution only appears when the determinants are coupled to all other external determinants, that is, the contributions are severely underestimated when attempting to separately include the relaxed MLCT or relaxed LMCT determinants.

The results of the earlier discussion of the mechanisms are summarized in Figure 4. The graph shows the importance of including the different mechanism in the wave function. Because of the nonadditivity of the different contributions, it is not possible to rigorously separate them and assign an absolute importance to each mechanism individually. The graph clearly shows how the first two contributions due to the reference space give at most a qualitative description of the coupling and that the quantitative correct picture only emerges, when all determinants that span the DDCI space are considered.

#### Valence-only model

The earlier described procedure provides us with a detailed decomposition of the magnetic coupling, but this may not be the most appealing representation of the mechanism. To make the communication more fluid with the areas of chemistry outside the circle of quantum chemists, it may be preferable to avoid the extensive use of determinants and other jargon and try to come back to the intuitive picture provided by the minimal valence space. This consists of two rather simple antagonist contributions, namely the direct exchange integral and the kinetic or super exchange. It was shown that the bare description in the reference space gives a reasonable first approximation but does not correctly reproduce the magnitude of the coupling.

The comparison of the numerical matrix elements of the effective Hamiltonian with the matrix given in Eq. (6) allows us to identify the effective values of *K*_{ab}, *t*_{ab}_{,} and *U*. The diagonalization of the effective Hamiltonian gives the same energy splitting between singlet and triplet as in the full DDCI calculation, but now we can directly relate the magnetic coupling of the correct strength with concepts as simple as direct and kinetic exchange using Eq. (9) with the parameters derived from the effective Hamiltonian procedure. These parameters are called effective parameters because they are no longer the bare parameters and they no longer have the clear-cut physical meaning they had in the bare reference space calculation, but are screened with the other effects due to determinants outside this minimal valence space.

This screening may lead to unexpected, or maybe better to say, counterintuitive values of the parameters. The most striking one is that the effective direct exchange integral may become negative favoring antiferromagnetic alignment of spins. The general tendency is that is smaller than the bare *U* of the minimal space calculation and that the effective hopping parameter normally increases: .