Chapter 1 8 Interpretation of Spectra In the previous chapter the topic of spectral studies on coordination compounds was introduced only briefl y in connection with ligand fi eld theory, and some of the attendant problems that are associ- ated with interpreting the spectra were described. In this chapter, a more complete description will be presented of the process of interpreting spectra of complexes. It is from the analysis of spectra that we obtain information about energies of spectroscopic states in metal ions and the effects produced by different ligands on the d orbitals. However, it is fi rst necessary to know what spectroscopic states are appropriate for various metal ions. The analysis then progresses to how the spectroscopic states for the metal ions are affected by the presence of the ligands and how ligand fi eld parameters are determined for spectral data. 18.1 SPLITTING OF SPECTROSCOPIC STATES As we have seen, an understanding of spin-orbit coupling is necessary to determine the spectroscopic states that exist for various electron confi gurations, d n (see Section 2.6). Because they will be needed frequently in this chapter, the spectroscopic states that result from spin-orbit coupling in d n ions that have degenerate d orbitals are summarized in Table 18.1 . The spectroscopic states shown in Table 18.1 are those that arise for the so-called free or gaseous ion. When a metal ion is surrounded by ligands in a coordination compound, those ligands generate an electrostatic fi eld that removes the degeneracy of the d orbitals. The result is that eg and t 2g subsets of orbitals are produced. Because the d orbitals are no longer degenerate, spin-orbit coupling is altered so that the states given in Table 18.1 no longer apply to a metal ion in a complex . However, just as the d orbitals are split in terms of their energies, the spectroscopic states are split in the ligand fi eld. The spec- troscopic states are split into components that have the same multiplicity as the free ion states from which they arise. A single electron in a d orbital gives rise to a 2 D term for the gaseous ion, but in an 1 octahedral fi eld the electron will reside in a t2 g orbital, and the spectroscopic state for the t 2g confi gura- 2 2 tion is T2 g . If the electron were excited to an eg orbital, the spectroscopic state would be Eg . Thus, tran- 2 2 sitions between T2 g and Eg states would not be spin forbidden because both states are doublets. Note that lowercase letters are used to describe orbitals , whereas capital letters describe spectroscopic states . 645 646 CHAPTER 18 Interpretation of Spectra Table 18.1 Spectroscopic States for Gaseous Ions Having d n Electron Confi gurations a . Ion Spectroscopic states d 1 , d 9 2D d 2 , d 8 3F , 3 P , 1G , 1D , 1S d 3 , d 7 4F , 4P , 2H , 2G , 2 F , 2 2D , 2 P d 4 , d 6 5D , 3 H , 3G , 2 3F , 3D , 2 3P , 1I , 2 1G , 1F , 2 1D , 2 1S d 5 6S , 4G , 4F , 4D , 4P , 2I , 2H , 2 2G , 2 2F , 3 2G , 3 2D , 2 P , 2S a 2 3 F means two distinct 3 F terms arise, etc. Table 18.2 Splitting of Spectroscopic States in a Ligand Field a . Gaseous Ion spectroscopic Components in an state octahedral fi eld Total degeneracy S A 1 g 1 P T 1 g 3 D E g ϩ T 2 g 5 F A 2 g ϩ T 1 g ϩ T 2 g 7 G A 1 g ϩ E g ϩ T 1 g ϩ T 2 g 9 H E g ϩ 2 T 1 g ϩ T 2 g 11 I A 1 g ϩ A 2 g ϩ E g ϩ T 1 g ϩ 2 T 2 g 13 a Ligand fi eld states have the same multiplicity as the spectroscopic state from which they arise. 2 3 A gaseous ion having a d confi guration gives rise to a F ground state as a result of spin-orbit coupling. 2 Although they will not be derived, in an octahedral ligand fi eld the t2 g confi guration gives three dif- 3 3 3 ferent spectroscopic states that are designated as A2 g , T1 g , and T2 g . These states are often referred to as ligand fi eld states . The energies of the three states depend on the strength of the ligand fi eld, but the relationship is not a simple one. The larger the ligand fi eld splitting, the greater the difference between the ligand fi eld states of the metal. For the time being, we will assume that the function is linear, but it will be necessary to refi ne this view later. Table 18.2 shows a summary of the states that result from splitting the gaseous state terms of metal ions in an octahedral fi eld produced by six ligands. Figure 18.1 shows the approximate energies of the ligand fi eld spectroscopic states as a function of the n Δ fi eld strength for all d ions. In the drawings, the states are assumed to be linear functions of o , but 1 2 2 this is not correct over a wide range of fi eld strength. For the d ion, the D ground state is split into T2g 2 Δ and Eg states in the ligand fi eld. As the fi eld strength, o , increases, a center of energy is maintained 18.1 Splitting of Spectroscopic States 647 3 A2g 1 2 2 3 4T d Eg d d 1g 3 T2g 2 3 4 4 D F F T2g 2 3T T2g 1g 4 A2g Δ Δ Δ o o o d 4 5 d 5 T2g 6A 5D 6S 2g 5 Eg Δ Δ o o 4 A2g 3 6 5 7 8 T1g d Eg d d 4 T2g 3F 5 4 3 D F T2g 4 5 T1g T2g 3 A2g Δ Δ Δ o o o 9 5 10 d T2g d 2 1 1 D S A1g 5 Eg Δ Δ o o ■ FIGURE 18.1 The splitting patterns for ground-state D and F terms in an octahedral fi eld. 2 2 for the energies of the T2g and Eg states in exactly the same way that a center of energy is maintained by the t2g and e g orbital subsets. Therefore, in order to give no net change in energy, the slope of the 2 ϩ Δ 2 Ϫ Δ line for the Eg state is (3/5) o , whereas that of the T2g state is (2/5) o . Note that the ground-state terms for d n ions (except for d 5 ) are all either D or F terms and that the state splitting occurs so that the center of energy is maintained. For the ligand fi eld states that are produced by splitting the 3 F term (which results from a d 2 confi guration), the center of energy is also preserved even though there are three states in the ligand fi eld. By looking at Table 18.2, it can be seen that all of the states that arise from splitting the D and F ground states for the gaseous ions have T , E , or A designations. When describing the splitting of the d orbitals in an octahedral fi eld (see Chapter 17), the “ t ” orbitals were seen to be triply degenerate, whereas the “ e ” orbitals were doubly degenerate. We can consider the spectroscopic states in the ligand fi eld to have the same degeneracies as the orbitals , 648 CHAPTER 18 Interpretation of Spectra 2 which makes it possible to preserve the center of energy. For example, the lines representing the T2g 2 Ϫ Δ ϩ Δ and Eg states have slopes of (2/5) o and (3/5) o , respectively. Taking into account the degenera- cies of the states, the energy of sum of the two subsets is 325[(Ϫϩ/)ΔΔ ] 235 [ϭ (/) ] 0 οο This result shows that even though the two states have energies that depend on the magnitude of the ligand fi eld splitting, the overall energy change is 0. For purposes of determining the center of energy, an “ A ” state can be considered as being singly degenerate. When the 3 F state corresponding to the d 2 3 3 3 confi guration is split into A2 g , T1 g , and T2 g components, the slopes of the lines must be such that the overall energy change is 0. Thus, the slopes of the lines and the multiplicities are related by 335[3Ϫϩϩϩϩϭ(/)ΔΔοο ] [(/) 15 ] 165[( / ) Δ ο ] 0 ()TT () () A which shows that the three ligand fi eld states yield the energy of the 3 F gaseous ion state as the center of energy. From the diagrams shown in Figure 18.1 , it can be seen that the splitting pattern for a d 4 ion is like that for d 1 ion except for being inverted and the states having the appropriate multiplicities. Likewise, the splitting pattern for a d 3 ion is like that for d 2 except for being inverted and the multiplicity being different. The reason for this similarity is the “ electron-hole ” behavior that is seen when the spectro- scopic states for confi gurations such as p 1 and p 5 are considered. Both give rise to a 2 P spectroscopic state, and only the J values are different.