Free-ion terms

Figure 3-12. Free ion terms arising from the configuration.

All this is summarized in Fig. 3-12. The energy ordering of the free-ion terms is not determined by consideration of angular momentum properties alone and in general yields only to detailed numerical computation. The ground term - and only the ground term - may be deduced, however, from some simple rules due to Hund. 

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. 

Consider the orbital angular momentum of a free-ion term. Here L = 3 and the orbital degeneracy is 7. Application of Van Vleck s formula (5.8) predicts an effective magnetic moment. 

In tier (1) of the diagram (for the electronic structure of iron(III)), only the total energy of the five metal valence electrons in the potential of the nucleus is considered. Electron-electron repulsion in tier (2) yields the free-ion terms (Russel-Saunders terms) that are usually labeled by term ° symbols (The numbers given in brackets at the energy states indicate the spin- and orbital-multiplicities of these states.)... 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

In order to apply quantum mechanical pertnrbation theory, the free ion term is usually written as... 

Depending upon the size of the crystal field term //cf in comparison to these three free ion terms, different approaches can be considered to the solution of Equation (5.1) by perturbation methods ... 

The splitting of the free-ion term in octahedral symmetry Oh symmetry) reduces the degeneracy of the five d orbitals. Three orbitals have energy lower than the other two. This means that if the orbitals are populated by one electron, three degenerate states are possible, according to the three possible positions for the electron in the low-energy levels (T symmetry) ... 

All lanthanide ions, with the exception of gadolinium(III) and europium(II), are likely to be relaxed by Orbach-type processes at room temperature. In fact, the f" configurations n l) of lanthanides(III) give rise to several free-ion terms that upon strong spin-orbit coupling, provide several closely spaced energy levels. Table III reports the multiplicity of the ground levels, which varies from 6 to 17, and is further split by crystal field effects. 

Free- ion terms Weak crystal field Intermediate crystal field Strong- field terms Strong-field configurations... 

In order to illustrate the splitting of terms of a d" configuration, the states for a d2 ion in several point groups are shown below. The free-ion terms have been given on page 259. 

The energy level diagram we wish to construct will show how the energies of the various states into which the free-ion terms are split depend on the strength of the interaction of the ion with its environment. The separation of the two sets of orbitals into which the group of five d orbitals is split can be taken as our measure of this interaction. Thus our diagram will have the... 

We will now explain the method of constructing an energy level diagram by treating the particular case of a d2 ion in an octahedral environment. For this system the free-ion terms, in order of increasing energy, are... 

To obtain the left (weak interaction) side we look up each of the free-ion terms in Table 9.3 and find that these terms split as follows ... 

An inverse relationship also exists between fields of octahedral and tetrahedral symmetries. We saw earlier in this chapter that crystal fields of these two symmetries produce inverse splitting patterns for one-electron d orbitals. This relationship also holds when electron-electron repulsions are added to the picture any free-ion term will be split into the same new terms (except for g and u designations, which are inappropriate for tetrahedral complexes) by tetrahedral and octahedral fields, but the energy ordering will be opposite for the two symmetries. 

The g subscripts are appropriate only for oaahedra] stereochemistry. The (F) and (P) notations designate the free ion term from which the listed term originates. 

Thc energies of the electronic transitions are labelled according to the parent symmetry of the excited states. P refers to the free ion term which contributes mostly to the indicated state. bThc conditions under which the electronic spectra were obtained S, solution P, diffuse reflectance C, single crystal. 

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