Crystal field wavefunction

Due to the so-called /-mixing within the crystal field, multiplets with different / values are coupled. However, similar to the free-ion case, the levels are still designated by the principal 25+1L j component of the crystal-field wavefunction. For the further labeling of levels split by the crystal field, either the irreducible representation /j (Bethe, 1929) to which the particular wavefunction belongs or the crystal quantum number /i defined by Hellwege (1949) are most commonly used. 

For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Figure 4. Calculated HAB values as a function of Fe -Fe separation, based on the structural model given in Figure 1 and the diabatic wavefunctions I/a and f/B. Curves 1 and 2 are based on separate models in which the inner-shell ligands are represented, respectively, by a point charge crystal field model [Fe(H20)62 -Fe(HsO)63 ] and by explicit quantum mechanical inclusion of their valence electrons [Fe(HgO)s2 -Fe(H20)s3+] (as defined by the dashed rectangle in Figure 1). The corresponding values of Kei, the electronic transmission factor, are displayed for various Fe-Fe separations of interest.

Fig. 6. Variation of the crystal-field parameters of LaCl3 Pr3+ under pressure. Solid lines correspond to the conventional one-electron crystal field, utilizing only the 4f2 wavefunctions as the basis set. S denotes the mean deviation as defined in the text. Dashed lines represent the results derived from the inclusion of the 4f15d1 configuration interactions.

Improvement of the crystal-field splitting calculation has been achieved by two different approaches. On one hand the basis set of wavefunctions was extended to include also excited configurations. This approach will be dealt with in sect. 4.4.6. On the other hand, the one-electron approximation has been relaxed to take into account electron correlation effects. The original formulation of the correlation crystal-field parameterization has been proposed by Bishton and Newman (1968). Judd (1977) and Reid (1987) redefined the operators to ensure their mutual orthogonality ... 

Another possibility to address the problem of the correlation crystal fields is an approach based on different wavefunctions for the spin-up and spin-down electrons. This spin-correlated crystal-field model merely doubles the number of crystal-field parameters and thus can be applied in most cases. Shen and Holzapfel (1995c) presented a high pressure study on spin-correlated crystal fields in MFCl Sm2+ (M = Ba, Sr, Ca). In particular, they considered the splitting ratio R of the 5Di and 7Fi multiplets, which should be equal to 0.298 within the conventional one-electron crystal-field theory and independent of the host crystal. In a first step, Shen and Holzapfel (1995c) considered ambient pressure as well as high pressure data of the isoelectronic Eu3+ ion. In this case they found a ratio of R = 0.238, which could be explained by taking into account a spin-correlated crystal-field parameter C2 = —0.007(3). 

The secular determinantal equation is set up in the usual manner, the wavefunctions corrected for the crystal-field interaction are used in the perturbation treatment, energies are generated, and these are used in conjunction with the secular equations to generate new wavefunctions that have now been corrected for spin-orbit coupling. These corrected wavefunctions are used for the calculation of the Zeeman effect. 

Such observations immediately raise the question how reliable are projections of crystal-field effects onto multipoles The analyses of wavefunction-simulated X-ray data of small model compounds have revealed that the interaction density (8p = p(crystal) — p(isolated molecule)) manifests itself in low-order structure factors, and only to an extent that is comparable with the experimental noise [80]. Nevertheless, the multipole refinement was shown to retrieve this low signal (about 1% in F) successfully. A related study on urea, however, demonstrated that this is not the case if random errors of magnitude comparable with the effect of interaction density are added to the theoretical data [81]. The result also implies that indeterminacies associated with the interpretation of non-centrosymmetric structures can severely limit the pseudoatom model in distinguishing between noise and physical effects [82, 83]. 

The most common approach to the interpretation of EPR and Mossbauer spectra of siderophores is the spin Hamiltonian formalism. The wavefunctions are parameterized in terms of a few coupling constants that arise in the spin Hamiltonian description of the electronic states. In this approach, the crystal field potential is generally described by a series of spherical harmonics. The corresponding operators are tabulated. ... 

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