Crystal-field, splittings

Although an octahedron has a much lower symmetry than a sphere it would be reasonable to expect that many-electron wavefunctions would be handled similarly. This is so—symbols such as Eg and A g, like t g, g and aig orbitals, imply, respectively, triple, double and single orbital degeneracy. In each case they are associated with a spin degeneracy which, in each of these three examples, is identical to the spatial degeneracy. However the two vary independently and so symbols such as T g, Eg and A g are perfectly reasonable. 

In crystal field theory a complex is regarded as consisting of a central metal cation surrounded by ionic or dipolar ligands which are electrostatically attracted to the cation. The bonding within the complex arises from the electrostatic attraction between the nucleus of the metal cation and the electrons of the ligands. The interaction between the electrons of the cation and those of the ligands is entirely repulsive. These repulsions will be central 

4 The 6, and d metal orbitals in an octahedral crystal field. Because they are all equivalent (they can be interconverted by rotating around the threefold axis approximately perpendicular to the plane of the paper) they remain triply degenerate. 

5 The d 2-y2 and d 2 orbitals in an octahedral crystal field. Although they look very different, the fact that they can be mixed by an axis relabelling shows that they are a degenerate pair (see the text and Fig. 7.6). 

So far it has been shown that sets of d and f orbitals (and therefore D and F terms) split into subsets in an octahedral crystal field but nothing has been said about the relative energies of these subsets. For the moment, the discussion will be restricted to orbitals because it is easy to give pictures of them. In subsequent sections the discussion will be extended to the corresponding terms. In preparation for this extension, it would be helpful if the reader has some idea of their derivation so that he or she is fully aware, for example, that an F state means seven spatial (as opposed to spin) functions, just like a set of f orbitals. One of the simplest ways of appreciating this is through the Russell-Saunders coupling scheme, that which is adopted to obtain the explicit functions themselves. This scheme is outlined in 

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Orgel diagrams Simple graphs showing the relation between the energies of various electronic slates and the crystal field splitting. 

The effect of the CFSE is expected to be even more marked in the case of the heavier elements because for them the crystal field splittings are much greater. As a result the +3 state is the most important one for both Rh and Ir and [M(H20)6] are the only simple aquo ions formed by these elements. With rr-acceptor ligands the +1 oxidation state is also well known for Rh and Ir. It is noticeable, however, that the similarity of these two heavier elements is less than is the case earlier in the transition series and, although rhodium resembles iridium more than cobalt, nevertheless there are significant differences. One example is provided by the +4 oxidation state which occurs to an appreciable extent in iridium but not in rhodium. (The ease with which Ir, Ir sometimes occurs... 

Data for some typical complexes are given in Table 26.5. The assignments are made, producing values of the inter-electronic repulsion parameter B as well as of the crystal-field splitting, 0Dq. 

Tetrahedral complexes arc also common, being formed more readily with cobali(II) than with the cation of any other truly transitional element (i.e. excluding Zn ). This is consistent with the CFSEs of the two stereochemistries (Table 26.6). Quantitative comparisons between the values given for CFSE(oct) and CFSE(let) are not possible because of course tbc crystal field splittings, Ao and A, differ. Nor is the CFSE by any means the most important factor in determining the stability of a complex. Nevertheless, where other factors are comparable, it can have a decisive effect and it is apparent that no configuration is more favourable than d to the adoption of a tetrahedral as opposed to... 

The difference in energy between the two groups is called the crystal field splitting energy and given the symbol A, (the subscript o stands for octahedral ). 

From the color (absorption spectrum) of a complex ion, it is sometimes possible to deduce the value of AOJ the crystal field splitting energy. The situation is particularly simple in 22Ti3+, which contains only one 3d electron. Consider, for example, the Ti(H20)63+ ion, which has an intense purple color. This ion absorbs at 510 nm, in the green region. The... 

MnFe2- has a crystal field splitting energy, A , of 2.60 X 102 kj/mol. What is the wavelength responsible for this energy ... 

Energy A property of a system which can be altered only by exchanging heat or work with the surroundings activation, 298-300,302 balance, 218-219 crystal field splitting, 418 electrical, 496 exercise and, 219t factor, 452 metabolic, 218 minimum, 165... 

Crystal field splitting parameter, 2, 309 Crystal field theory, 1, 215-221 angular overlap model, 1, 228 calculations, 1, 220 generality, 1,219 low symmetry, 1,220 /-orbital, 1, 231 Crystal hydrates, 2, 305,306 bond distances, 2, 307 Crystals... 

Crystal field effects are of the order of the free ion interaction thus they cannot be treated as a small perturbation as in the lanthanides. Whereas the crystal field splitting in the oxidation state +3 is comparable to that for the lanthanides, it is significantly increased in the progression... 

The poor data on PuF6 are probably best interpreted as a very small TIP of about 150 x 10-6 emu indicating a singlet ground state and a large crystal field splitting of the octahedral compound ( 5). 

Prediction of the energy level structure for Pu2+ (5f ) is of particular interest since no spectra for this valence state of Pu have been reported. On the basis of what is known of the spectra of Am2+ (26), Cf2" (27), and Es2+ (28), there appears to be evidence for a very small crystal-field splitting of the free-ion levels. Such evidence encourages use of a free-ion calculation in this particular case. The parameter values selected are indicated in Table V. Based on the systematics given by Brewer (19), the first f- d transition should occur near 11000 cm-, so the f- -f transitions at higher energies would be expected to be at least partially obscured. A... 

The approach we have adopted for the d configuration began from the so-called strong-field limit. This is to be contrasted to the weak-field scheme that we describe in Section 3.7. In the strong-field approach, we consider the crystal-field splitting of the d orbitals first, and then recognize the effects of interelectron repulsion. The opposite order is adopted in the weak-field scheme. Before studying this alternative approach, however, we must review a little of the theory of free-ion spectroscopy... 

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. 

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