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

48 F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–63Hamiltonian. Using the two configuration description ofFig. 14, one finds for Co II two final states 2p 5 3d 8 and2p 5 3d 9 L. These states mix in a manner similar to the twoconfigurations in the ground state and as such give rise to afinal state Tanabe–Sugano diagram. All final state energiesare calculated from the mixing of the two configurations.This calculation is only possible if all final state parametersare known. The following rules are used.(a) The 2p3d Slater–Condon parameters are taken from anatomic calculation. For trivalent ions and higher valences,these atomic values are sometimes reduced.(b) The 2p and 3d spin–orbit coupling are taken from anatomic calculation.(c) The crystal field values are assumed to be the same asin the ground state.(d) The energies of the configurations, i.e. the charge transferenergy, are given by the values of U dd and U pd . Effectively∆¯F = ∆¯I + U dd − U pd . Because in generalU pd is approximately 1–2 eV larger than U dd , one oftenassumes ∆¯F = ∆¯I − 1or−2eV.(e) The hopping parameter t is assumed to be equal in theinitial and final states.Detailed analysis of X-ray absorption and resonant X-rayemission spectra has shown that the crystal field values aresmaller by 10–20% in the final state [34]. The same observationhas been made for the hopping parameters [35]. Onecan understand these trends from the (slight) compressionof the 3d wave function in the final state. From the presenceof the 2p core hole one would expect a significant compressionof the 3d wave function, but the effect of the 2p corehole is counteracted by the effect of the extra 3d-electronin the final state. Because we have seen that U dd is a bitsmaller than U pd this counteracting action is not completeand there will be a small compression of the 3d wave function.In conclusion it can be said that ∆¯, t and 10Dq will allbe slightly smaller in the final state. Because the reductionof these parameters has counteracting effects on the spectralshape, in most simulations one varies only ∆¯ and keeps tand 10Dq constant.1.5.3. The X-ray absorption spectrum with charge transfereffectsThe essence of the charge transfer model is the use oftwo or more configurations. Ligand field multiplet calculationsuse one configuration for which it solves the effectiveatomic Hamiltonian plus the ligand field Hamiltonian, so thefollowing matrices:I XAS,1 ∝〈3d N |p|2p 5 3d N+1 〉 2∣ ∣∣∣H INIT,1 =〈3d N e 2∣ 〉 ∣∣∣+ ς d l d · s d + H LFM 3d Nr 12∣ ∣∣∣H FINAL,1 =〈2p 5 3d N+1 e 2∣ ∣∣∣+ ς p l p · s p + ς d l d · s d + H LFMr 12〉× 2p 5 3d N+1The charge transfer model adds a configuration 3d N+1to the 3dL¯N ground state. In case of a transition metal oxide,ina3d N+1 configuration an electron has been moved fromthe oxygenL¯2p-valence band to the metal 3d-band. One cancontinue with this procedure and add 3d N+2L¯2 configuration,etc. In many cases two configurations will be enoughto explain the spectral shapes, but in particular for high valencestates it can be important to include more configurations[36,37]. As far as X-ray absorption and X-ray emissionis concerned, the consequences for the calculations arethe replacement of 3d N with 3d N + 3d N+1 plus the correspondingchanges in the final state. ThisL¯adds a secondinitial state, final state and dipole transition:I XAS,2 ∝〈3d N+1 L¯|p|2p 5 3d N+2 L¯〉 2〈H INIT,2 = 3d N+1 e 2∣ 〉 ∣∣∣L¯ ∣ + ς d l d s d + H LFM 3d N+1r 12L¯〈H FINAL,2 = 2p 5 3d N+2 e 2∣ ∣∣∣L¯ ∣ + ς p l p s p + ς d l d s d + H LFMr 12〉× 2p 5 3d N+2 L¯The two initial states and two final states are coupledby monopole transitions, i.e. configuration interaction. Themixing parameter t couples both configurations and ∆ is theenergy difference. The Hamiltonian is abbreviated with t/∆to describe the monopole interaction:〈 ∣H MIXI1,I2 = 3d N t〉∣ ∣ 3d N+1∆L¯〈 ∣H MIXF1,F2 = 2p 5 3d N+1 t〉∣ ∣ 2p 5 3d N+2∆L¯The X-ray absorption spectrum is calculated by solvingthe equations given above. If a 3d N+2 LL ′ configuration isincluded its energy is 2∆¯ + U dd , where U dd is the correlationenergy between two 3d-electrons [28]. The formaldefinition of U dd is the energy difference one obtains whenan electron is transferred from one metal site to another, i.e.a transition 3d N + 3d N → 3d N+1 + 3d N−1 . The numberof interactions of two 3d N configurations is one more thanthe number of interactions of 3d N +1 plus 3d N −1 , implyingthat this energy difference is equal to the correlation energybetween two 3d-electrons.By analyzing the effects of charge transfer it is found that,for systems with a positive value of ∆, the main effects onthe X-ray absorption spectral shape are:(1) the formation of small satellites; and(2) the contraction of the multiplet structures.