D orbitals in an octahedral crystal field

Figure 9.8 Splitting of d orbitals in an octahedral crystal field.

Figure 1-4. The splitting of the d orbitals in an octahedral crystal field. The total splitting is given by the quantity Aoct.

Rg. 7.4 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. 

Hg. 7.5 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). 

Fig. 7.12 (a) Ground and (b) an excited state with the same spin multiplicity, derived from the d configuration in an octahedral crystal field. In the ground state there is a hole in the e orbitals in the excited state the hole is in the tzi orbitals. 

Fig. 7.15 (a) Ground and (b, c) two excited state configurations of a d ion in an octahedral crystal field. The triple orbital degeneracy of the ground state may be associated with the three possible orbital sites for the hole in (a). 

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. 

The ground term of the cP configuration is F. That of is also F. Those of and d are " F. We shall discuss these patterns in Section 3.10. For the moment, we only note the common occurrence of F terms and ask how they split in an octahedral crystal field. As for the case of the D term above, which splits like the d orbitals because the angular parts of their electron distributions are related, an F term splits up like a set of / orbital electron densities. A set of real / orbitals is shown in Fig. 3-13. Note how they comprise three subsets. One set of three orbitals has major lobes directed along the cartesian x or y or z axes. Another set comprises three orbitals, each formed by a pair of clover-leaf shapes, concentrated about two of the three cartesian planes. The third set comprises just one member, with lobes directed equally to all eight corners of an inscribing cube. In the free ion, of course, all seven / orbitals are degenerate. In an octahedral crystal field, however, the... 

A low-spin to high-spin transition relates to the crystal field splitting of the d-orbitals in an octahedral or tetrahedral crystal field. However, even in cases where the energy difference between two spin states is much larger, electronic transitions are observed. An atom with total spin quantum number S has (22 + 1) orientations. In a magnetic field the atom will have a number of discrete energy levels with... 

As shown in previous sections, under an octahedral crystal field, the five d orbitals are split into two sets of orbitals, /2g and eg. In terms of spectroscopic states, we can readily see that the 2D term arising from configuration d1 will split into 2r2g and 2E% states, arising from configurations t g and e, respectively. But what about the spectroscopic terms such as F and G arising from other d" configurations In this section, we will see how these terms split in an octahedral crystal field. 

This representation can be easily reduced to the irreducible representations A2 + T1 + T2. Note that if this F term arises from a configuration with electrons occupying d orbitals, which are even functions, the electronic states split from this term in an octahedral crystal field are then A2g, 7 ig, and 7 2g. (On the other hand, if the F term comes from a configuration with f electron(s) such as f1, the electronic states in an octahedral crystal field will then be A2u, 7 iu, and 7 2U. This is because f orbitals are odd functions.) All the results given in Table 8.4.2 can be obtained in a similar fashion. 

Figure 2.3 Orientations of ligands and d orbitals of a transition metal ion in octahedral coordination, (a) Orientation of ligands with respect to the cartesian axes (b) the x-y plane of a transition metal ion in an octahedral crystal field. The orbital is cross-hatched the dx2 y2 orbital is open ligands are black circles.

Figure 1.6 Crystal field splitting of metal ion d orbitals in an octahedral field of ligands.

A Copper(II) is a Jahn-Teller ion and normally sits in a distorted lattice site. The configuration of copper is which leads to an unpaired electron in the e level in an octahedral crystal field. The Cg level consists of the degenerate and d, orbitals. According to the Jahn-Teller theorem, the system distorts to remove the degeneracy and lower the overall energy of the system. Hence one set of bonds (four in plane, or two axial) become longer than the others and the copper ion has a distorted geometry. 

FIGURE 22.17 Crystal field splitting between d orbitals in an octahedral complex. 

Fig. 20.2 The changes in the energies of the electrons occup5hng the d orbitals of an M ion when the latter is in an octahedral crystal field. The energy changes are shown in terms of the orbital energies.

Similar energy level diagrams may be drawn for dn systems in tetrahedral crystal fields. There is an interesting relationship between these and the ones for certain systems in octahedral fields. We have already seen that the splitting pattern for the d orbitals in a tetrahedral field is just the inverse of that for the d orbitals in an octahedral field. A similar inverse relationship exists between the energy level diagrams of dn systems in tetrahedral and octahedral fields. The components into which each Russell-Saunders state is split are reversed in their energy order in the tetrahedral compared to the octahedral... 

In an octahedral crystal field, the ground state is and spin-orbit coupling splits this into Fg, F, and 2rg, with Fg lowest. Since a Kramer s doublet is lowest, no Jahn-Teller distortions are possible. Axial fields split the ground state into A 2 and E. Since there are Kramer s doublets in either level, an ESR spectrum is obtainable, but spin-orbit coupling may mix these states and cause short spin-lattice relaxation times. In practice, the ground state is 2 and the system behaves in a similar fashion to the d Of,) case. Distortions are unlikely and spectra readily seen. The g values vary in a complex manner (14). 

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