Racah Parameters and Quantitative Methods

Although the energies of the spectroscopic states of first row d2 ions were shown in Table 18.4, compilations exist for all gaseous metal ions. The standard reference is a series of volumes published by the National Institute of Standards and Technology (C. E. Moore, Atomic Energy Levels, National Bureau of Standards Circular 467, Vol. I, II, and III, 1949). Table 18.5 shows the energies for the spectroscopic states in dn ions in terms of the Racah B and C parameters. 

It can be seen from Table 18.5 that all excited spectroscopic states having a multiplicity that is different from the ground state have energies that are expressed in terms of both B and C. As we have seen from the previous discussion, spin-allowed transitions occur only between states having the same multiplicity. Therefore, in the analysis of spectra of complexes only B must be determined. It is found for some complexes that C 4B, and this approximation is adequate for many uses. 

The discussion so far has concerned the Racah parameters for gaseous ions. For example, the d2 ion Ti2+ has a 3P state that lies 10,420 cm-1 above the 3F ground state, so 15B = 10,420 cm-1 and B is 695 cm-1 for this ion. Using this value for B results in an approximate value of 2780 cm-1 as the value of C for 

For metal ions having d2, d3, d7, and d8 configurations, the ground state is an F state, but there is an excited P state that has the same multiplicity. For d1 and d7 ions in an octahedral field, the spectroscopic states are the same (except for the multiplicity) as they are for d3 and d8 ions in tetrahedral fields. Therefore, the expected spectral transitions will also be the same for the two types of complexes. The three spectral bands are assigned as follows (Tlg(F) means the Tis state arising from the F spectroscopic state)  

For octahedral complexes of d2 and d7 metal ions (or d3 and d8 ions in tetrahedral complexes), the energies corresponding to vx, v2, and v3 can be shown to be 

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