Octahedral crystal field splitting of spectroscopic terms

4 Octahedral crystal field splitting of spectroscopic terms 

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. 

Octahedral complex ML6 has Oh symmetry. However, for simplicity, we may work with the 0 point group, which has only rotations as its symmetry operations and five irreducible representations, A, A2,. ..,T2. The character table for this group is shown in Table 8.4.1. It is seen that the main difference between the Oh and 0 groups is that the former has inversion center i, while the latter does not. As a result, the Oh group has ten symmetry species Aig, Aiu, A2g, A2u, , T2g, r2u. 

To see how an octahedral crystal field reduces the (2L+l)-fold degeneracy of a spectroscopic term, we first need to determine the character of a rotation operating on an atomic state defined by the orbital angular momentum quantum number L. It can be shown that the character / for a rotation of angle a about the z axis is simply 

With these formulas, it is now straightforward to obtain the x values for all the rotations in the O group and for any L value. Furthermore, we can then reduce the representations with these x values as characters into the irreducible representations of the O group. All these results are summarized in Table 8.4.2. 

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