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

38 F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–63Table 4The 2p X-ray absorption transitions from the atomic ground state to allallowed final state symmetries, after applying the dipole selection rule:J =−1,0or+1Transition Ground Transitions Term symbols3d 0 → 2p 5 3d 1 1 S 0 3 123d 1 → 2p 5 3d 2 2 D 3/2 29 453d 2 → 2p 5 3d 3 3 F 2 68 1103d 3 → 2p 5 3d 4 4 F 3/2 95 1803d 4 → 2p 5 3d 5 5 D 0 32 2053d 5 → 2p 5 3d 6 6 S 5/2 110 1803d 6 → 2p 5 3d 7 5 D 2 68 1103d 7 → 2p 5 3d 8 4 F 9/2 16 453d 8 → 2p 5 3d 9 3 F 4 4 123d 9 → 2p 5 3d 10 2 D 5/2 1 2effects of the neighbors are too large. It turns out that itis necessary to include explicitly both the symmetry effectsand the configuration–interaction effects of the neighbors.Ligand field multiplet theory takes care of all symmetryeffects, while charge transfer multiplet theory allows the useof more than one configuration.1.4. The crystal field multiplet modelCrystal field theory is a well-known model used to explainthe electronic properties of transition metal systems.It was developed in the fifties and sixties, mainly againsta background of explaining optical spectra and EPR data.The starting point of the crystal field model is to approximatethe transition metal as an isolated atom surroundedby a distribution of charges that should mimic the system,molecule or solid, around the transition metal. At first sight,this is a very simplistic model and one might doubt its usefulnessto explain experimental data. However it turned outthat such a simple model was extremely successful to explaina large range of experiments, like optical spectra, EPRspectra, magnetic moments, etc.Maybe the most important reason for the success ofthe crystal field model is that the properties explained arestrongly determined by symmetry considerations. With itssimplicity in concept, the crystal field model could makefull use of the results of group theory. Group theory alsomade possible a close link to atomic multiplet theory.Group theoretically speaking, the only thing crystal fieldtheory does is translate, or branch, the results obtained inatomic symmetry to cubic symmetry and further to anyother lower point groups. The mathematical concepts forthese branchings are well developed [7,8]. In this chapterwe will use these group theoretical results and studytheir effects on the ground states as well as on the spectralshapes.The crystal field multiplet Hamiltonian extends the atomicHamiltonian with an electrostatic field. The electrostaticfield consists of the electronic charge e times a potential thatdescribes the surroundings. This potential φ(r) is written asa series expansion of spherical harmonics Y LM ’s:φ(r) =∞∑L∑L=0M=−Lr L A LM Y LM (ψ, φ)The crystal field is regarded as a perturbation to the atomicresult. This implies that it is necessary to determine the matrixelements of φ(r) with respect to the atomic 3d orbitals〈3d|φ(r)|3d〉. One can separate the matrix elements into aspherical part and a radial part, as was done also for theatomic Hamiltonian. The radial part of the matrix elementsyields the strength of the crystal field interaction. The sphericalpart of the matrix element can be written in Y LM symmetry,which limits the crystal field potential for 3d electrons to:φ(r) = A 00 Y 00 +2∑r 2 A 2M Y 2M +M=−24∑r 4 A 4M Y 4MM=−4The first term A 00 Y 00 is a constant. It will only shift theatomic states and it is not necessary to include this termexplicitly if one calculates the spectral shape.1.4.1. Cubic crystal fieldsA large range of systems consist of a transition metal ionsurrounded by six neighboring atoms, where these neighborsare positioned on the three Cartesian axes, or in otherwords, on the six faces of a cube surrounding the transitionmetal. They form a so-called octahedral field, whichbelongs to the O h point group. The calculation of the X-rayabsorption spectral shape in atomic symmetry involved thecalculation of the matrices of the initial state, the final stateand the transition. The initial state is given by the matrixelement 〈3d N |H ATOM |3d N 〉, which for a particular J-valuein the initial state gives ∑ J 〈J|0|J〉. The same applies forthe final state matrix element and the dipole matrix element.To calculate the X-ray absorption spectrum in a cubic crystalfield, these atomic matrix elements must be branched tocubic symmetry.The symmetry change from spherical symmetry (SO 3 )to octahedral symmetry (O h ) causes the S and P symmetrystates to branch, respectively, to an A 1g andaT 1u symmetrystate. A D symmetry state branches to E g plus T 2g symmetrystates in octahedral symmetry and an F symmetry stateto A 2u + T 1u + T 2u . One can make the following observations:The dipole transition operator has p-symmetry and isbranched to T 1u symmetry, implying that there will be nodipolar angular dependence in O h symmetry. The quadrupoletransition operator has d-symmetry and is split into two operatorsin O h symmetry, in other words, there will be differentquadrupole transitions in different directions.In a similar way, the symmetry can be changed fromoctahedral O h to tetragonal D 4h , with the correspondingdescription with a branching table. An atomic s-orbital isbranched to D 4h symmetry according to the branching seriesS → A 1g → A 1g , i.e. it will always remain the unityelement in all symmetries. An atomic p-orbital is branched