Multiplet Effects in X-ray Absorption - Inorganic Chemistry and ...

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42 F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–63Table 8The matrix elements in SO 3 symmetry needed for the calculation of 2pX-ray absorption3d N → 2p 5 3d N +1 in SO 3 symmetryInitial state Transition Final state〈0|0|0〉〈0|1|1〉〈0|0|0〉〈1|0|1〉〈1|1|0〉〈1|0|1〉〈1|1|1〉〈1|1|2〉〈2|0|2〉〈2|1|1〉〈2|0|2〉〈2|1|3〉〈3|0|3〉〈3|1|2〉〈3|0|3〉 ∗〈3|1|3〉〈3|1|4〉〈4|0|4〉 ∗ 〈4|1|3〉 ∗ 〈4|0|4〉 ∗〈4|1|4〉 ∗Boldface and ∗ matrix elements apply to, respectively, a 3d 0 anda3d 8configuration.low-spin transition at 2.25 eV and also the important effectof the 3d spin–orbit coupling. It can be observed that theatomic multiplet spectrum of Co II has a large number ofstates at low energy. All these states are part of the 4 F 9/2configuration that is split by the 3d spin–orbit coupling. Afterapplying a cubic crystal field, most of these multipletstates are shifted to higher energies and only four states remainat low energy. These are the four states of 4 T 1g asindicated in Table 7. These four states all remain within0.1 eV from the E 2 ground state. That this description isactually correct was shown in detail for the 2p X-ray absorptionspectrum of CoO [12], which has a cubic crystalfield of 1.2 eV. At 2.25 eV the high-spin low-spin transitionis evident. A new state is coming from high energyand a G-symmetry state replaces the E 2 symmetrystate at the lowest energy. In fact there is a very interestingcomplication: due to the 3d spin–orbit coupling theG-symmetry states of the 4 T 1g and 2 E g configurations mixand form linear combinations. Around the transition point,this linear combination will have a spin-state that is neitherhigh-spin nor low-spin and in fact a mixed spin-state can befound.1.4.6. The effects on the X-ray absorption calculationsTable 8 gives all matrix element calculations that haveto be carried out for 3d N → 2p 5 3d N+1 transitions in SO 3symmetry for the J-values up to 4.We will use the transitions3d 0 → 2p 5 3d 1 as examples. 3d 0 contains only J = 0symmetry states, indicated in boldface. This limits the calculationfor the ground state spectrum to only one groundstate, one transition and one final state matrix element, givenin boldface. In case of 3d 8 Ni II the ground state has a 3 F 4configuration, indicated as underlined. We are now going toapply the SO 3 → O h branching rule to this table. The J =4 ground state has transitions to J = 3 and 4 final states(Table 8).Table 9The matrix elements in O h symmetry needed for the calculation of 2pX-ray absorption3d N → 2p 5 3d N +1 in O h symmetryInitial state Transition Final state〈A 1 |A 1 |A 1 〉〈A 1 |T 1 |T 1 〉〈A 1 |A 1 |A 1 〉〈T 1 |A 1 |T 1 〉〈T 1 |T 1 |A 1 〉〈T 1 |A 1 |T 1 〉 ∗〈T 1 |T 1 |T 1 〉〈T 1 |T 1 |E〉〈T 1 |T 1 |T 2 〉〈E|A 1 |E〉〈E|T 1 |T 1 〉〈E|A 1 |E〉 ∗〈E|T 1 |T 2 〉〈T 2 |A 1 |T 2 〉 ∗ 〈T 2 |T 1 |T 1 〉 ∗ 〈T 2 |A 1 |T 2 〉 ∗〈T 2 |T 1 |E〉 ∗〈T 2 |T 1 |T 2 〉 ∗〈T 2 |T 1 |A 2 〉 ∗〈A 2 |A 1 |A 2 〉〈A 2 |T 1 |T 2 〉〈A 2 |A 1 |A 2 〉 ∗Boldface and ∗ matrix elements apply to, respectively, a 3d 0 anda3d 8configuration.In octahedral symmetry one has to calculate five matricesfor the initial and final states and thirteen transition matrices.Note that this is a general result for all even numbers of3d electrons, as there are only these five symmetries in O hsymmetry. In the 3d 0 case, the ground state branches to A 1and only three matrices are needed to generate the spectralshape: 〈A 1 |A 1 |A 1 〉 for the 3d 0 ground state, 〈A 1 |T 1 |T 1 〉 forthe dipole transition and 〈T 1 |A 1 |T 1 〉 for the 2p 5 3d 1 finalstate (Table 9). The 3d 0 systems are rather special becausethey are not affected by ground state effects. Table 10 showsthat a 2p 5 3d 1 configuration has twelve representations inSO 3 symmetry that are branched to 25 representations in acubic field. From these 25 representations, only seven are ofinterest for the calculation of the X-ray absorption spectralshape, because only these T 1 symmetry states obtain a finiteintensity.In the 3d 8 case, the ground state branches to T 2g , i.e.3 A 2g = T 1g ⊗A 2g = T 2g . The T 2g ground state yields dipoletransitions to four different final state symmetries, usingT 2g ⊗ T 1u = T 1u + T 2u + E u + A 2u . Consequently the completespectral shape is given by calculating one ground stateTable 10The branching of the J-values in SO 3 symmetry to the representations inO h symmetry, using the degeneracies of the 2p 5 3d 1 final state in X-rayabsorptionJ in SO 3 Degree Branchings Γ in O h Degree0 1 A 1u A 1u[0,4] 21 3 3 × T 1u A 2u[3] 32 4 4 × E u ,4× T 2u T 1u[1,3,4] 73 3 3 × A 2u ,3× T 1u ,3 × T 2u T 2u[2-4] 84 1 A 1u ,E u ,T 1u ,T 2u E u[2,4] 5∑12 25The symmetry in O h is given, including the SO 3 origin of the states insquare brackets.
F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–63 43Fig. 7. The crystal field multiplet calculations for the 3d 0 → 2p 5 3d 1transition in T IIV . The atomic Slater–Condon and spin–orbit couplingparameters were used as given in Table 3. The bottom spectrum is theatomic multiplet spectrum. Each next spectrum has a value of 10Dq thatwas increased by 0.3 eV. The top spectrum has a crystal field of 3.0 eV.matrix (〈T 2g |A 1g |T 2g 〉), four transition matrices and four finalstate matrices and combining all corresponding matricesto yield the intensities and initial and final state energies.Because the 2p 5 3d 9 configuration is equivalent to a 2p 5 3d 1configuration, the degeneracies of the 2p 5 3d 9 final state matricescan also be found in Table 10.Fig. 7 shows the crystal field multiplet calculations for the3d 0 → 2p 5 3d 1 transition in Ti IV . The result of each calculationis a set of seven energies with seven intensities. Theseseven states were broadened by the lifetime broadening andthe experimental resolution. From a detailed comparison toexperiment it turns out that each of the four main lines has tobe broadened differently [13–15]. An additional differencein broadening is found between the t 2g and the e g states.This broadening was ascribed to differences in the vibrationaleffects on the t 2g , respectively, the e g states. Anothercause could be a difference in hybridization effects and infact charge transfer multiplet calculations [16–18] indicatethat this effect is more important than vibrational effects.Fig. 8 compares the crystal field multiplet calculation ofthe 3d 0 → 2p 5 3d 1 transition in Ti IV with the experimental2p X-ray absorption spectrum of FeTiO 3 . The titanium ionsFig. 8. The 2p X-ray absorption spectrum of FeTiO 3 compared with acrystal field multiplet calculation for T IIV with a value of 10Dq of 1.8 eV(reprinted with permission from [14], copyright 1990 American PhysicalSociety).Fig. 9. The crystal field multiplet calculations for the 3d 0 → 2p 5 3d 1transition in Ti IV . The atomic Slater–Condon and spin–orbit couplingparameters were used as given in Table 3. The bottom spectrum is thecrystal field multiplet spectrum with atomic parameters and correspondsto the fifth spectrum in Fig. 8; i.e. 10Dq is 1.5 eV. Each next spectrumhas a value of the Slater integrals further reduced by, respectively, 25,50, 75 and 100%, i.e. the top spectrum is the single particle result.are surrounded by six oxygen atoms in a distorted octahedron.The value of 10Dq was set to 1.8 eV. The calculationis able to reproduce all peaks that are experimentally visible.In particular the two small pre-peaks can be nicely observed.The similar spectrum of SrTiO 3 has an even sharperspectral shape, related to the perfect octahedral surrounding[19,20].Fig. 9 shows the effect of the pd Slater–Condon parameterson the spectral shape of the 3d 0 → 2p 5 3d 1 transitionin T IIV . The bottom calculation is the same as in Fig. 8 andused the 80% reduction of the Hartree–Fock values in orderto obtain a good estimate of the values in the free atom.In most solids the pd Slater–Condon parameters have thesame values as for the free atom, in other words, the solidstate screening of the pd Slater–Condon parameters is almostzero. The five spectra are calculated by using the samevalues for the 3d- and 2p-spin–orbit coupling and the samecrystal field value of 1.8 eV. The Slater–Condon parametersare rescaled to, respectively, 80% (bottom), 60, 40, 20 and0% (top). The top spectrum corresponds to the single particlepicture, where one expects four peaks, respectively, theL 3 –t 2g , the L 3 –e g , the L 2 –t 2g and the L 2 –e g peak, with respectiveintensities given by their degeneracies, i.e. 6:4:3:2.This is exactly what is observed, where it is noted thatthe intensity ratio is a little obscured by the differences inline width. One can conclude that there is a large differencebetween the single particle result (top spectrum) andthe multiplet result (bottom spectrum). The Slater–Condonparameters have the effect to lower the intensity of the t 2gpeaks and shift intensity to the e g peaks. An even largerintensity shift can be observed from the L 3 edge to the L 2edge and a very clear effect is the creation of additional(pre-)peaks, because additional transitions become allowed.
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