Splitting of d Orbitals in Octahedral Symmetry

We are concerned with what happens to the (spectral) d electrons of a transition-metal ion surrounded by a group of ligands which, in the crystal-field model, may be represented by point negative charges. The results depend upon the number and spatial arrangements of these charges. For the moment, and because of the very common occurrence of octahedral coordination, we focus exclusively upon an octahedral array of point charges. 

The set of five d orbitals share a common radial part like that sketched in Fig. 

Their angular parts are shown in Fig. 3-1. Let us consider the six point charges in an octahedral array to be disposed along the positive and negative x, y and axes to which these d orbitals are referred. This is conveniently drawn, as shown in Fig. 

by placing the charges at the centres of each face of a cube, itself centred on the metal atom. By comparing the orbitals in Fig. 3-1 with the crystal field of point charges in Fig. 3-2, we observe that some orbitals are more directed towards the point charges than others. The dyi and d 2 y2 orbitals are directed exactly towards the six charges while the d y, d, and dy have lobes which lie between the jc, y and z 

Each lobe of the d 2 yi orbital interacts predominantly with one point charge. The repulsive effects relate to the electron density within any given orbital so we might describe the interaction in units of lobe repulsion and say that, for the dp. yi orbital, this amounts to 4 = 16 repulsion units (4 squared because electron density oc jF). 

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Figure 3-4. Barycentre splitting of the d orbitals in octahedral symmetry.

As the ligand-field splitting of the d orbitals in lower symmetry complexes lead to lower degeneracies and so a need for more parameters to describe them (see Table 6.6), so the d-d transitions of complexes of other than octahedral and tetrahedral geometries can only be discussed by introducing these parameters. For square planar complexes, for instance, two additional parameters must be introduced. 

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 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) ... 

Figure 1.18 shows energy levels for d orbitals in crystal fields of differing symmetry. The splitting operated by the octahedral field is much higher than that of the tetrahedral field (A = lower than the effect imposed by the square... 

We may use the set of five d wave functions as a basis for a representation of the point group of a particular environment and thus determine the manner in which the set of d orbitals is split by this environment. Let us choose an octahedral environment for our first illustration. In order to determine the representation for which the set of d wave functions forms a basis, we must first find the elements of the matrices which express the effect upon the set of wave functions of each of the symmetry operations in the group the characters of these matrices will then be the characters of the representation we are seeking. 

The wave functions for s, P, D. F. etc. terms have the same symmetry as the wave functions for the corresponding sets of s, p. d,f, etc. orbitals. This means that a D term is split by an octahedral field in exactly the same manner as a set of d orbitals... 

Fig. 1. Site geometries and d orbital splittings for copper(II) in octahedral and tetragonal environments (symmetry properties of the orbitals are indicated in parentheses)...

The particular splitting pattern of the d orbitals in Figure 4.5 is characteristic of cubic (octahedral type) crystal fields. In the tetrahedral type, the gg and ordering would be reversed. In other symmetries, the d splitting is as shown in Figure 4.6. 

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