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

36 F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–6310 ×9/2 ×8/3. Because there is a symmetry equivalence ofholes and electrons, the pairs 3d 2 –3d 8 ,3d 3 –3d 7 , etc. haveexactly the same term symbols. The general formula to determinethe degeneracy of a 3d N configuration is:( ) 10 10!=N (10 − N)!N!The 2p X-ray absorption edge (2p → 3d transition) isoften studied for the 3d transition metal series, and it providesa wealth of information. Crucial for its understandingare the configurations of the 2p 5 3d N final states. The termsymbols of the 2p 5 3d N states are found by multiplying theconfigurations of 3d N with a 2 P term symbol. The total degeneracyof a 2p 5 3d N state is six times the degeneracy of3d N . For example, a 2p 5 3d 5 configuration has 1512 possiblestates. Analysis shows that these 1512 states are dividedinto 205 term symbols, implying in principle 205 possiblefinal states. Whether all these final states actually have finiteintensity depends on the selection rules.1.3.2. Matrix elementsThe term symbol of a 3d N configuration describes thesymmetry aspects, but it does not say anything about its relativeenergy. The relative energies of the different terms aredetermined by calculating the matrix elements of these stateswith the effective electron–electron interaction H ee ′ and thespin–orbit coupling H ls . The general formulation of the matrixelements of the effective electron–electron interaction isgiven as:〈2S+1 ∣ ∣∣∣ e 2 ∣ 〉 ∣∣∣L 2S+1 J L J = ∑ f k F k + ∑ g k G kr 12kkF i (f i ) and G i (g i ) are the Slater–Condon parameters for, respectively,the radial (angular) part of the direct Coulombrepulsion and the Coulomb exchange interaction. f i and g iare non-zero only for certain values of i, depending on theconfiguration. The direct Coulomb repulsion f 0 is alwayspresent and the maximum value for i equals two times thelowest value of l. The exchange interaction g i is present onlyfor electrons in different shells. For g k , i is even if l 1 + l 2 iseven, and i is odd if l 1 + l 2 is odd. The maximum value ofi equals l 1 + l 2 . For 3d-states, it is important to note that a3d N configuration contains f 0 , f 2 and f 4 Slater–Condon parameters.The final state 2p 5 3d N +1 configuration containsf 0 , f 2 , f 4 , g 1 , and g 3 Slater–Condon parameters.Fora3d 2 configuration, we found the five term symbols1 S, 3 P, 1 D, 3 F and 1 G. f 0 is equal to the number of pairsN(N − 1)/2 ofN electrons, i.e. it is equal to 1 for twoelectrons. The Slater–Condon parameters F 2 and F 4 haveapproximately a constant ratio: F 4 = 0.62, F 2 . In case of the3d transition metal ions, F 2 is approximately equal to 10 eV.This gives for the five term symbols of the 3d 2 configuration,respectively, 3 Fat−1.8 eV, 1 Dat−0.1 eV, 3 Pat+0.2 eV,1 Gat+0.8 eV and 1 Sat+4.6 eV. The 3 F term symbol hasthe lowest energy and it is the ground state of a 3d 2 system.This is a confirmation of the Hunds rules, which will bediscussed below. The three states 1 D, 3 P and 1 G are closein energy some 1.7–2.5 eV above the ground state. The 1 Sstate has a high energy of 6.4 eV above the ground state, thereason being that two electrons in the same orbital stronglyrepel each other.For three and more electrons the situation is considerablymore complex. It is not straightforward to write downan anti-symmetrized three-electron wave function. It can beshown that the three-electron wave function can be built fromtwo-electron wave functions with the use of the so-called coefficientsof fractional parentage. For a partly filled d-band,the term symbol with the lowest energy is given by theso-called Hunds rules. Based on experimental informationHund formulated three rules to determine the ground stateofa3d N configuration. The three Hunds rules are the following.1. Term symbols with maximum spin S are lowest in energy.2. Among these terms, the one with the maximum orbitalmoment L is lowest.3. In the presence of spin–orbit coupling, the lowest termhas J =|L − S| if the shell is less than half full andJ = L + S if the shell is more than half full.A configuration has the lowest energy if the electrons areas far apart as possible. The first Hunds rule ‘maximum spin’can be understood from the Pauli principle: electrons withparallel spins must be in different orbitals, which on overallimplies larger separations, hence lower energies. This is forexample, evident for a 3d 5 configuration, where the 6 S statehas its five electrons divided over the five spin-up orbitals,which minimizes their repulsion. In case of 3d 2 , the secondHunds rule implies that the 3 F term symbol is lower thanthe 3 P-term symbol, because the 3 F wave function tends tominimize electron repulsion.1.3.3. X-ray absorption spectra described with atomicmultipletsWe start with the description of closed shell systems. The2p X-ray absorption process excites a 2p core electron intothe empty 3d shell and the transition can be described as2p 6 3d 0 → 2p 5 3d 1 . The ground state has 1 S 0 symmetry andwe find that the term symbols of the final state are 1 P 1 ,1 D 2 , 1 F 3 , 3 P 012 , 3 D 123 and 3 F 234 . The energies of the finalstates are affected by the 2p3d Slater–Condon parameters,the 2p spin–orbit coupling and the 3d spin–orbit coupling.The X-ray absorption transition matrix elements to be calculatedare:I XAS ∝〈3d 0 [ 1 S 0 ] |⃗r. [ 1 P 1 ] |2p5 3d 1 [ 1,3 PDF] 〉2The 12 final states are built from the 12 term symbols,with the restriction that the states with the same J-valueblock out in the calculation. The symmetry of the dipoletransition is given as 1 P 1 , according to the dipole selectionrules, which state that J =±1 or 0, with the exception of