Beyond the Valence Space

It is obvious that this cannot be the whole story. The calculated magnetic coupling of the Cu + complex is just 10 % of the experimental and DDCl values. Hence, it is unavoidable to go beyond this valence-only description and incorporate more physical mechanisms in the description. 

The Ih, Ip, Ih-lp excitations In the first step towards the full DDCI result, we analyze the role of the Ih, 1 p and h- p determinants as illustrated in Fig. 5.6. The determinants on the left are pure single excitations and those on the right are single excitations combined with an excitation within the CAS. Because of the Brillouin theorem the contributions of the pure single excitations are strictly zero for the spin state for which the orbitals have been optimized and tiny contributions are observed for the other spin states given that the optimal orbitals for the different spin states are in principle very similar. 

The situation is quite different for the single excitations that are combined with electron replacements in the CAS. The 1/t-lp excitation in the determinants marked as spin polarization not only excites one of the electrons from orbital h to orbital p but also changes the spin of the excited electron. These so-called triplet excitations have to be compensated by a simultaneous spin change in the active space to maintain the spin of the electronic state under consideration. This gives rise to a triplet coupled 

6 The h-p and the a-b electron pairs are triplet coupled (S = 1) in the determinants that cause spin polarization in the ligands. Which values can be assigned to the total spin by coupling the two S = 1 electron pairs Are aU spin states relevant to the binuclear Cu + system under study  

The second type of important Ih-lp determinants combines a spin-conserving h to p excitation with an electron replacement from a to b (or vice versa) in the active space. The resulting determinants can be considered as single excitations with respect to the ionic determinants, but Brillouin s theorem does not apply because 

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The observed changes suffered by the parameters upon dressing them with the effects that go beyond the valence space can at least partially be rationalized by looking at the interaction of the model space determinants with those in the external space. The interaction of the spin-conserving Ih-lp excitations with the neutral determinants is (nearly) zero due to Brillouin s theorem. On the contrary, the interaction with the ionic determinants is strong (see the right part of Fig. 5.10). Hence, this class of external determinants largely decreases the on-site repulsion U as previously seen in Exercise 6.7 and confirmed here in the example. 

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