High-spin case

Field Stabilization Energies, or LFSE s. The variation in LFSE across the transition-metal series is shown graphically in Fig. 8-6. It is no accident, of course, that the plots intercept the abscissa for d, d and ions, for that is how the LFSE is defined. Ions with all other d configurations are more stable than the d, d or d ions, at least so far as this one aspect is concerned. For the high-spin cases, we note a characteristic double-hump trace and note that we expect particular stability conferred upon d and d octahedral ions. For the low-spin series, we observe a particularly stable arrangement for ions. More will be said about these systems in the next chapter. 

Afterward, we have to find approximations for 2-RDM spin components D". Let us now focus on high-spin cases only, such as doublet, triplet, quartet... spins for one, two, three. .. unpaired electrons outside the closed shells. Accordingly, singly occupied orbitals will always have the same spin (Al p = 0 or = 0) so the trace of the one-matrix, Eq. (6), becomes... 

For a d6 ion such as Co3+, a low-spin complex would contain no unpaired electrons, whereas the high-spin case would contain four unpaired electrons. The low-spin complex [Co(NH3)6]C13 has a magnetic moment of 0, whereas that of K3[CoF6] is 4.36 BM in accord with these predictions. 

The complex ion contains Co2+, which has a 3d7 electron configuration. The splitting of the d orbitals will be small since this is a tetrahedral complex, giving the high-spin case with three unpaired electrons. 

For example, a ion could have five unpaired electrons, three in t2g and two in eg orbitals, as a high-spin case or it could have only one unpaired electron, with all five electrons in the t2g levels, as a low-spin case. The possibilities for all cases, d through d , are given in Table 10-5. 

The notation (n RCI)" will be used to designate the occupation restrictions for RCI wavefunctions. For high-spin cases the orbital pairing will be assumed... 

The customary accuracy of MBPT(2) has now been extended to purely analytical second-derivative methods for open shells using UHF or ROHF reference functions. The latter requires some new (noncanonical MBPT/CC) theory recently developed. As discussed previously, unlike ROHF, which is an eigenfunction of spin for high spin cases, UHF reference functions can sometimes suffer from spin contamination. For low order correlated methods like MBPT(2) (unlike CCSD, e.g.), the UHF spin contamination can drastically affect the answers obtained. 

Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

The hexacoordinate complexes with adamantane, norcamphor and camphane exhibit different populations of high- and low-spin state (100%, 46%, 46%) [132]. Structure refinement gives some evidence for a variation of the Fe - O distance the largest value is found for the pure high-spin case (adamantane) [132]. 

However, there is another important difference between the a and (3 electrons in the high-spin case The open a orbitals are populated, the open /3 are not thus the open a orbitals should use orbital energies appropriate for ionization, while the (3 ones should be appropriate for electron affinity. To use distinct orbital energies for active orbitals, depending on whether an electron is added or removed, is a principle that would work for arbitrary spin-coupled cases. It is not possible for a strictly one-electron operator. Several ways to approach this problem have been suggested ... 

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