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

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40 F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–63Table 6The high-spin and low-spin distribution of the 3d electrons for the configurations 3d 4 to 3d 7Configurations High-spin Low-spin 10Dq (D) Exchange (J) J/D3d 4 t 3 2g+ e1 g+ t 3 2g+ t1 2g− 1D 3J te 33d 5 t 3 2g+ e2 g+ t 3 2g+ t2 2g− 2D 6J te + J ee − J tt ∼33d 6 t 3 2g+ e2 g+ t1 2g− t 3 2g+ t3 2g− 2D 6J te + J ee − 3J tt ∼23d 7 t 3 2g+ e2 g+ t2 2g− t 3 2g+ t3 2g− e1 g+ 1D 3J te + J ee − 2J tt ∼2The fourth column gives the difference in crystal field energy, the fifth column the difference in exchange energy. For the last column, we have assumedthat J te ∼ J ee ∼ J tt = J.into 3 A 2g + 3 T 1g + 3 T 2g , following the branching rules asdescribed above. At higher crystal field strengths states startto change their order and they cross. Whether states actuallycross each other or show non-crossing behavior depends onwhether their symmetries allow them to form a linear combinationof states. This also depends on the inclusion of the3d spin–orbit coupling. The right part of the figure shows theeffect of the reduction of the Slater–Condon parameters. Fora crystal field of 1.5 eV the Slater–Condon parameters werereduced from their atomic value, indicated with 80% of theirHartree–Fock value to 0%. The spectrum for 0% has all itsSlater–Condon parameters reduced to zero, In other words,the 3d3d coupling has been turned of and one observes theenergies of two non-interacting 3d-holes. This single particlelimit has three configurations, respectively, the two holes ine g e g ,e g t 2g and t 2g t 2g states. The energy difference betweene g e g and e g t 2g is exactly the crystal field value of 1.5 eV.This figure shows nicely the transition from the single particlepicture to the multiplet picture for the 3d 8 ground state.The ground state of a 3d 8 configuration in O h symmetryalways remains 3 A 2g . The reason is clear if one comparesthese configurations to the single particle description of a3d 8 configuration. In a single particle description a 3d 8 configurationis split by the cubic crystal field into the t 2g andthe e g configuration. The t 2g configuration has the lowestenergy and can contain six 3d electrons. The remaining twoelectrons are placed in the e g configuration, where both havea parallel alignment according to Hunds rule. The result isthat the overall configuration is t 2g 6 e g+ 2 . This configurationidentifies with the 3 A 2g configuration.Both configurations e g and t 2g can split by the Stoner exchangesplitting J. This Stoner exchange splitting J is givenas a linear combination of the Slater–Condon parameters asJ = (F 2 + F 4 )/14 and it is an approximation to the effectsof the Slater–Condon parameters and in fact, a secondparameter C, the orbital polarization, can be used in combinationwith J. The orbital polarization C is given as C =( 9 F 2 − 5 F 4 )/98. We assume for the moment that the effectof the orbital polarization will not modify the ground states.In that case, the (high-spin) ground states of 3d N configurationsare simply given by filling, respectively, the t 2g+ ,e g+ ,t 2g− and e g− states. For example, the 4 A 2g ground state of3d 3 is simplified as t 3 2g+ and the 3 A 2g ground state of 3d 8as t 3 2g+ e g+ 2 t 3 2g− , etc.For the configurations 3d 4 ,3d 5 ,3d 6 and 3d 7 there aretwo possible ground state configurations in O h symmetry.A high-spin ground state that originates from the Hundsrule ground state and a low-spin ground state for which firstall t 2g levels are filled. The transition point from high-spinto low-spin ground states is determined by the cubic crystalfield 10Dq and the exchange splitting J. The exchangesplitting is present for every two parallel electrons. Table 6gives the high-spin and low-spin occupations of the t 2g ande g spin-up and spin-down orbitals t 2g+ ,e g+ ,t 2g− and e g− .The 3d 4 and 3d 7 configuration differ by one t 2g versus e gelectron hence exactly the crystal field splitting D. The 3d 5and 3d 6 configurations differ by 2D. The exchange interactionJ is slightly different for e g e g ,e g t 2g and t 2g t 2g interactionsand the fifth column contains the overall exchangeinteractions. The last column can be used to estimate thetransition point. For this estimate the exchange splittingswere assumed to be equal, yielding the simple rules thatfor 3d 4 and 3d 5 configurations high-spin states are foundif the crystal field splitting is less than 3J. In case of 3d 6and 3d 7 configurations the crystal field value should be lessthan 2J for a high-spin configuration. Because J can beestimated as 0.8 eV, the transition points are approximately2.4 eV for 3d 4 and 3d 5 , respectively, 1.6 eV for 3d 6 and3d 7 . In other words, 3d 6 and 3d 7 materials have a tendencyto be low-spin compounds. This is particularly true for 3d 6compounds because of the additional stabilizing nature ofthe 3d 61 A 1g low spin ground state.1.4.4. Symmetry effects in D 4h symmetryIn D 4h symmetry the t 2g and e g symmetry states split furtherinto e g and b 2g , respectively, a 1g and b 1g . Depending onthe nature of the tetragonal distortion either the e g or the b 2gstate have the lowest energy. All configurations from 3d 2 to3d 8 have a low-spin possibility in D 4h symmetry. Only the3d 2 configuration with the e g state as ground state does notpossess a low-spin configuration. The 3d 1 and 3d 9 configurationscontain only one unpaired spin thus they have nopossibility to form a low-spin ground state. It is important tonote that a 3d 8 configuration as, for example, found in Ni IIand Cu III can yield a low-spin configuration. Actually thislow-spin configuration is found in the trivalent parent compoundsof the high T C superconducting oxides [9,10]. TheD 4h symmetry ground states are particularly important for
F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–63 41Table 7The branching of the spin-symmetry states and its consequence on the states that are found after the inclusion of spin–orbit couplingConfigurations Ground state in SO 3 HS ground state in O h Spin in O h Degree Overall symmetry in O h3d 0 1 S 0 1 A 1g A 1g 1 A 1g3d 1 2 D 3/2 2 T 2g E 2g 2 E 1g + G g3d 2 3 F 2 3 T 1g T 1g 4 E g + T 1g + T 2g + A 1g3d 3 4 F 3/2 4 A 2g G g 1 G g3d 4 5 D 0 5 E g E g + T 2g 5 A 1g + A 2g + Eg + T 1g + T 2g3 T 1g T 1g 4 Eg + T 1g + T 2g + A 1g3d 5 6 S 5/2 6 A 1g G g + E 1g 2 G g + E 1g2 T 2g E 2g 2 G g + E 1g3d 6 5 D 2 5 T 2g E g + T 2g 6 A 1g + E g + T 1g + T 1g + T 2g + T 2g1 A 1g A 1g 1 A 1g3d 7 4 F 9/2 4 T 1g G g 4 E 1g + E 2g + G g + G g2 E g E 2g 1 G g3d 8 3 F 4 3 A 2g T 1g 1 T 2g3d 9 2 D 5/2 2 E g E 2g 1 G gThe fourth column gives the spin-projection and the fifth column its degeneracy. The last column lists all the symmetry states after inclusion of spin–orbitcoupling.those cases where O h symmetry yields a half-filled e g state.This is the case for 3d 4 and 3d 9 plus low-spin 3d 7 . Theseground states are unstable in octahedral symmetry and willrelax to, for example, a D 4h ground state, the well-knownJahn-Teller distortion. This yields the Cu II ions with all statesfilled except the 1 A 1g -hole.1.4.5. The effect of the 3d spin–orbit couplingAs discussed above the inclusion of 3d spin–orbit couplingwill bring one to the multiplication of the spin andorbital moments to a total moment. In this process one losesthe familiar nomenclature for the ground states of the 3d Nconfigurations. In total symmetry also the spin moments arebranched to the same symmetry group as the orbital moments,yielding for NiO a 3 A 2g ground state with an overallground state of T 1g ⊗ A 2g = T 2g . It turns out that in manysolids it is better to omit the 3d spin–orbit coupling becauseit is effectively ‘quenched’. This was found to be the casefor CrO 2 . A different situation is found for CoO, where theexplicit inclusion of the 3d spin–orbit coupling is essentialfor a good description of the 2p X-ray absorption spectralshape. In other words, 2p X-ray absorption is able to determinethe different role of the 3d spin–orbit coupling in,respectively, CrO 2 (quenched) and CoO (not quenched).Table 7 gives the spin-projection to O h symmetry. Theground states with an odd number of 3d electrons have aground state spin moment that is half-integer [7,11]. Table 7shows that the degeneracy of the overall symmetry states isoften not exactly equal to the spin number as given in thethird column. For example, the 3 T 1g ground state is split intofour configurations, not three as one would expect. If the 3dspin–orbit coupling is small (and if no other state is closein energy), two of these four states are quasi-degenerate andone finds three states. This is in general the case for allsituations. Note that the 6 A 1g ground state of 3d 5 is splitinto two configurations. These configurations are degenerateas far as the 3d spin–orbit coupling is concerned. Howeverbecause of differences in the mixing of excited term symbolsa small energy difference can be found. This is the originof the small but non-zero zero field splitting in the EPRanalysis of 3d 5 compounds.Fig. 6 shows the Tanabe–Sugano diagram for a 3d 7 configurationin O h symmetry. Only the excitation energiesfrom 0.0 to 0.4 eV are shown to highlight the high-spin toFig. 6. The Tanabe–Sugano diagram for a 3d 7 configuration in O h symmetry.The atomic states of a 3d 7 configuration are split by electrostaticinteractions and within the first 0.4 eV above the ground state only the 4 Fstates are found, split by the atomic 3d spin–orbit splitting. The horizontalaxis gives the crystal field in eV and the 4 F ground state is split intothe 4 T 1g ground sate and the 4 T 2g and 4 A 2g excited states that quicklymove up in energy with the crystal field. The 4 T 1g state is split intofour sub-states as indicated with the grey block. These sub-states are, respectively,the (double group symmetry [11,12]) E 2g ground state, a G gstate, another G g state and a E 1g state. At a crystal field value of 2.25 eVthe symmetry changes to low-spin and the G g states mix with the 2 E glow-spin G g state.
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