If the metal centers contain several unpaired electrons, in principle each unpaired electron on one subunit can couple with each unpaired electron on the other unit if a bridging ligand orbital exists that overlaps with both metal orbitals involved. As described inSection 2.42.2the corresponding superexchange pathway is mediated by CT, MMCT, and DCT states, and spectroscopic informa- tion from the CT spectrum can be used to determine the relative contribution of this specific pathwayto the cumulative ground-state coupling constant 2JGS.
This has been employed in the study of linear and bent-oxo bridged FeIIIdimers which are models for the met form of hemerythrin, an oxygen transport protein found in marine inverte- brates.7Core atoms and coordinate frames for the bent, tribridged structure (1) and the mono- bridged, almost linear structure (2) are given inScheme 2. The 300 and 10 K absorption spectra of (1) and (2) are shown inFigure 5A and 5B, respectively. These spectra can be divided into three different regions. Region I exclusively contains ligand-field (LF) bands whereas region II shows both LF- and low-energy CT bands. In region III only CT bands are present. The spectra in part exhibit spectacular temperature effects. InFigure 5A for (1), most bands decrease in intensity or disappear upon cooling except for the most intense peak at 21,000 cm1which almost doubles in intensity at 10 K. For structure (2) (Figure 5B), the intensity at 21,000 cm1decreases and that of the band at 25,000 cm1increases at low temperature. In region III, variable temperature dependen- cies are found for (2).
These findings can be understood in terms of dimer exchange interactions. Both (1) and (2) contain high-spin FeIII centers which are antiferromagnetically coupled, i.e., the ground state (GS) splits into six total spin states due to antiferromagnetic coupling between two s=5/2 monomers to give the total spin S=0, 1, 2, 3, 4, 5 states with S=0 being lowest in energy. A phenomenological description of the GS spin ladder shown in the bottom ofFigure 6is provided by the Heisenberg Hamiltonian H=2JGS sA.sB with sA=sB=5/2; experimentally determined JGS values are 120 cm1 for (1) and 95 cm1 for (2). Also shown in Figure 6 (middle) are the dimer states resulting from a LF transition of one center to a quartet LF state. In this case coupling of thes=5/2 GS on one half of the dimer and ans=3/2 excited LF state on the other half generatesS=1, 2, 3, 4 dimer states with a splitting described by a LF excited stateJLF. Du e toþ/combinations of local excitations at A and B, each individualSstate again splits into two states represented byþand, giving a total of 8 dimer states for each quartet LF transition in
O
Fe Fe
N O N
C O
O C
O N
N′ N
N 120
z′
x′ y′
(1)
O
Fe Fe
N N
N
N O
O N
O O
O 165
(2)
x
y z
Scheme 2
the monomer. Finally, the top portion of Figure 6gives the excited states resulting from a spin- allowed CT of the monomer. These dimer CT states are obtained by coupling a 6 CT state generated by excitation of one electron from a ligand orbital into one of the singly occupied metal orbitals of FeAor FeBwith the 6A1ground state on the other half of the dimer. In this case,þ/
combinations lead to 12 CT states in the dimer. At low temperature, when only the S=0 component of the GS is populated, there can be spin-allowed (S=0) transitions for CT but not for LF states whereas both spin-allowed LF and CT transitions are possible from higher S states of the GS at elevated temperatures. This allows assignment of a spin multiplicity to each spin-allowed LF or CT transition in Figure 5 based on the temperature dependence of its intensity.
In order to further elucidate the origin of CT bands in 5, VBCI calculations have been performed on (1) and (2). For simplicity, a linear dimer with C2v molecular symmetry (20) taken as a model for (2) will be considered first. The analysis will then be extended to a singly bridged, bent Fe–O–Fe dimer (10) which is taken as a model for (1). Figure 7 shows the metal- bridging ligand orbital interactions in the linear (20; left) and the bent structure (10; right). In the linear dimer (20), oxo!Fe CT transitions are possible frompx!dxz(‘‘out-of-plane (o.o.p.)’’;
the Fe–O–Fe plane for (20) isyz),py!dyz(‘‘in-plane (i.p.)’’) andpz! dz2
(‘‘’’). Based on the scheme given in Figure 6, each of these three orbitally allowed CT transitions gives rise to
2Sþ1A1CTand 2Sþ1B2CTstates (S=0,. . .,5). However, not all of these states are important. The
1A1 GS can have CT transitions to the states of 1B2 (parallel polarization) and 1A1 symmetry (perpendicular polarization with respect to Fe–Fe). VBCI matrices of all 1A1and 1B2states are necessary to describe this CT spectrum, and the 1A1matrix is needed to account for the1A1GS
Region I Region II Region III
Region I Region II Region III 10K
295K
250K 10K x100
x50 x4
x 20 (a)
(b)
20,000
15,000
10,000
8,000 7,000 6,000 5,000 5,000
4,000 3,000
2,000 1,000 0
5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000
5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 Energy (cm–1)
Energy (cm–1) ε(M–1cm–1)ε(M–1cm–1)
0
Figure 5 Absorption spectra for (1) (a) and (2) (b). Arrows indicate the change in intensity with decreasing temperature. (reproduced by permission of the American Chemical Society fromInorg. Chem.1995,34, 688).
550 Valence Bond Configuration Interaction Model
stabilization (note that due to the linear symmetry the individual and CT states do not mix with each other but all contribute to the ground state AF coupling). The high-spin matrices11A1
and 11B2 are relevant since from DFT transition-state calculations of the high-spin states, 11A1 and11B2, VBCI parameters can be extracted and the total splitting of the GS 11B21A1=30 JGSallows the determination ofJGS(cf.Section 2.42.2). The results of this protocol are shown in Figure 8. Based on the two high-spin o.o.p. transitions px!dxz (11B2!11A1, 11B2), the two high-spin i.p. transitions py!dyz (11B2!11A1, 11B2) and the two high-spin transitions pz!dz2
(11B2!11A1, 11B2) the six VBCI parameters hdxzpx, hdyzpy, hdz2pz, x, y, and z were obtained by fitting the appropriate VBCI matrices to the calculated transition energies (right half ofFigure 8). Values forUandEDCTwere obtained from photoelectron spectroscopy (7 eV7) and by doubling the corresponding CT energies, respectively. These parameters were then used in the
1A1 and 1B2 VBCI matrices to obtain the singlet energy level scheme given on the left half of Figure 8.
The important results from this treatment of the linear dimer (20) are:
The1A1GS level is stabilized by10,530 cm1and the11B2level by8,460 cm1, giving aJGS value of69 cm1(GSAF; expt. value for (2):95 cm1).
The i.p. CT levels 1A1and 1B2 are greatly lowered in energy with respect to their S=5 high-spin counterparts. Defining in analogy to the ground state 30JCT=E(11B2) E(1A1) gives for these states a CT excited coupling constant (ESAF) ofJCT=600 cm1, about nine times larger than GSAF.
Due to the large transition energy for thetransition, the correspondingpz!dz2CT states lie at energies above the MMCT and DCT states of the transitions.
The normalized, squared coefficients of the CT states in the1A1GS wavefunction have been taken as a measure of the relative importance of the various superexchange pathways.
S=5
S=5 S=4 S=3 S=2 S=1 S=0 S=4 S=3 S=2
S=0
S=4 S=3 S=2
S=1
2J GS
1A1 6A1
6A1– 6A1
9A1
5A1 3B2 11B2
7B2 4JLF
2JCT
11Γ
11Γ
9Γ
9Γ
7Γ
7Γ
5Γ
5Γ
3Γ
3Γ
– 4Γ
6A1– 6Γ
1Γ
1Γ
9Γ
9Γ
7Γ
7Γ
5Γ
5Γ
3Γ
3Γ
30JCT
+ –
–
– – – – + + +
+ +
+ –
–
–
– +
+
+
Figure 6 Energy level diagrams giving the GS, ligand field (LF) and charge transfer (CT) excited state manifolds of binuclear FeIIIh.s. dimers (taken from ref.7).
This gives 40% and 45% CT character in the ground state deriving from the o.o.p and i.p. transitions, respectively, and 13.8% from the transition, indicating the dominant role of superexchange pathways to the AF coupling in linear-oxo dimers.
This analysis has then been extended to the bent Fe–O–Fe dimer10(Figure 7, right). While the o.o.p. interactions are identical to the linear dimer 20, the i.p. (dyz) and (dz2) orbital interactions become mixed. For the i.p. oxo py orbital, there are now two possible CT transitions: py!dyz and py!dz2. Thus a new CT transition, py!dz2needs to be added to the VBCI matrices described above. In addition, this opens new possibilities for MMCT (dz2$dyz) and DCT (dz2 py!dz2) states. Likewise, there is a new CT transition frompz!dyz
which has to be taken into account. This interaction also generates new MMCT and DCT states.
The new interactions mediate the ‘‘mixed/’’ superexchange pathways.
dxz px dxz
dyz dyz
dz2 dz2
py
pz
Linear Bent
o.o.p top view
i.p side view
Figure 7 Diagrams for the major superexchange pathways in the FeIII–O–FeIIIunit. The linear pathways are given on the left and the pathway in the bent dimer on the right (taken from ref.7).
552 Valence Bond Configuration Interaction Model
In order to also apply the VBCI model to this case, the high-spin problem again has been treated first. In particular, the ten 11A1, 11B2CT excited states resulting from the five possible oxo!Fe transitions px!dxz (o.o.p. ), py!dyz (i.p. ), py!dz2 (mixed /), pz!dyz (mixed /) andpz!dz2() were calculated by DFT, and by fitting the appropriate11A1and11B1VBCI matrices to the resulting energies, the corresponding five VBCI transfer integrals and five-values have been determined. The corresponding energy level scheme is shown on the right-hand side of Figure 9.UandEDCThave been parameterized as in the linear case.
Using these VBCI parameters in the treatment of the singlet, 1A1 and 1B2, states, the energy levels shown on the left-hand side ofFigure 9are obtained. The important points to observe from this diagram are:
The1A1GS is lowered by10,465 cm1in energy and the11B2state by8,345 cm1, giving a GSJ value of 71 cm1(GSAF; expt. value:120 cm1).
The singlet i.p. and o.o.p transitions from oxopx,yto dxz,yzare greatly lowered in energy with respect to their undecet counterparts (ESAF).
The1A1and1B2CT states deriving from the in-planepy!dyztransition are at lower energy than their o.o.p.px!dxzcounterparts. In agreement with this VBCI prediction, the lowest-energy CT transitions in the absorption spectrum of (1) at 18,400 cm1(shoulder in mull spectrum; not
1A1, 11A1,
1A1
1B2
1A1,1B2
11B2
11A1,11B2
1B2
1A1
1A1
1A1
1A1
1B2
11B2
11A1
11B2
11A1
∆=110,000cm-1 pz→ dz2
π DCT STATES
MMCT STATES
∆=34,000, 34,500 cm-1 px→ dxz, py→ dyz
NO CI VBCI VBCI NO CI
Linear dimer JGS = - 69 cm-1
CT
11B2 G.S.
11B2 - 30JGS
Figure 8 Energy level scheme giving the result of the VBCI analysis on the linear structure 20. The S=0 states are given on the left and theS=5 states on the right (taken from ref.7).
visible inFigure 5A) and 20,500 cm1are assigned to1A1!1A1and1A1!B2components of the py!dyz CT transition, respectively. In contrast, the lowest component of the o.o.p. px!dxz
transition is only observed at 25,960 cm1, more than 5,000 cm1 above its i.p. counterpart.
This ‘‘inverted’’ ordering of the i.p. and o.o.p. transitions is due to additional i.p. mixed/ interactions (vide supra) which significantly decrease the energy of the i.p. states.
Based on the normalized CT contributions in the GS wavefunctions, the relative importance of the i.p. and o.o.p. superexchange pathways has decreased in the bent with respect to the linear dimer. This loss ofinteraction is compensated by an increase of the importance of the pathway, which is due to the energetic lowering of the pz!dz2 CT state, and in particular by the new, mixed / pathways py!dz2 andpz!dyzwhich in the bent geometry account for 4 and 12% CT character in the GS wavefunction, respectively. The combined effects of increase of interaction and additional, mixed / interactions lead to the observed increase of AF coupling upon lowering the Fe–O–Fe angle in -oxo FeIIIdimers.
An analogous treatment has been applied to Mn oxo dimers.32
2.42.5 EVALUATION OF U,, AND hdpFROM PHOTOELECTRON SPECTROSCOPY As shown in the previous sections, diagonalization of the VBCI matrices involves the CT energies , transfer integralshd(orhdp) and Mott–Hubbard gapsU. While experimental information on
G.S.
∆∆=70,500 cm-1
∆=60,000 cm-1
∆=41,000 cm-1
∆=33,000 cm-1
NO CI VBCI VBCI NO CI
11B2 G.S.
-30J GS 11B2
JGS=71cm-1 1A1
1A1, 1B2
1A1, 1B2
1A1, 1B2
1A1, 1B2 1A1, 1B2
1A1
1A1
1A1
1A1
1A1 1A1,
1B2
1B2
1B2
1B2 1B2
∆=110,000cm-1
∆=110,000 cm-1 pz→ dz2
pz→ dz2
py→ dz2
py→ dyz
px→ dxz pz→ dyz
px→ dxz py→ dyz pz→ dz2
pz→ dyz
py→ dz2
11A1, 11B2 11A1, 11B2 11A1, 11B2 11A1, 11B2 11A1, 11B2 11A1
11B2
11A1 11B2
11A1 11B2
11A1 11B2
11A1 11B2
Figure 9 Energy level scheme giving the result of the VBCI analysis on the bent structure 10. TheS=0 states are given on the left and theS=5 states on the right (taken from ref.7).
554 Valence Bond Configuration Interaction Model
andhdpcan be obtained from CT spectra,Uin general is not accessible by this technique as the corresponding transitions are high energy and probably very diffuse. For these reasons,Uvalues derived from photoelectron spectroscopy (PES) have been used in Sections 2.42.3 and 2.42.4.
Whereas we have calculated and hdp from DFT on the high-spin states, these can also be derived from PES data using the so-called impurity or cluster CI models of PES.23,33–35 These calculations involve diagonalization of the Anderson Hamiltonian with part or all of the transla- tional symmetry of the lattice being neglected in favor of an explicit treatment of on-site electron–
electron interactions. While in the impurity approach the bandwidth of thed-states is neglected but the finite bandwidth of the O 2p states is taken into account, the cluster CI (or local cluster) model totally neglects the bandwidth of bothdandpstates and only includes an oxygen–oxygen nearest-neighbor hybridizationTpp via differences in energy for the different symmetry levels. In addition, the oxygen levels are shifted upon metal–oxygen mixing. These calculations have proven very useful for the interpretation of photoelectron and inverse photoelectron spectroscopic data of ionic solids and in addition have allowed to understand their magnetic properties. In this section the relationship of these treatments to the VBCI model of isolated dimers and the evaluation of VBCI/cluster–CI parameters from the analysis of PES data will briefly be discussed.
An example is provided by a planar array of alternating Cuand O centers derived from, e.g., a Y–Ba–Cuoxide high-Tc superconductor (Figure 10).23 The basic assumption of the cluster CI model is that a (CuO4)6square–planar cluster can be considered as a separable unit of this CuO plane (Figure 10, solid lines). If this condition is not met (i.e., if the bandwidth ofp states is large with respect to the O 2p!Cu3dCT energy), the model breaks down. The electronic structure of the local cluster is described with the model Hamiltonian (Equation (14))
Hˆ ẳX
m
"dðmịdmỵdm ỵX
m
"pðmịpmỵpm ỵ X
m
Tpdðmịðdmỵpm ỵpmỵdmị ỵ X
m;m0;n;n0
Uðm;m0;n;n0ịdmỵdm0dnỵdn0
ð14ị Here the operatordþm creates a Cu3d hole with an energy "d(m) and the operatorpþm a ligand O2p hole with an energy "p(m). The oxygen–oxygen–hybridization is already included in the different "p(m). The third term describes the mixing between the Cu3d states and the ligand orbitals;Tdp(m) is the transfer integral for the Cu3d–O 2pligand hybridization (mixing). The last term describes the two-particle Cu3dCoulomb- and exchange interactionsU; the indicesm,m0,n, and n0 denote orbital and spin quantum numbers. The energies "d(m), "p(m) and the integrals Tdp(m) do not depend on spin. For each symmetry/spin label m, diagonalization of the 22 interaction matrix
3d9
3d10L
E Tdp Tdp E
ð15ị
px dx2-y2
Figure 10 Section from a square Cu–O lattice with CuO4cluster (solid lines) and relevant orbital. Neigh- boring cluster dotted lines.
leads to a one-hole ground state (GS) and a one-hole excited state (CT). Here, |3d9i refers to a state with one hole in thed-shell, and |3d10Lito a state with a hole in thepshell.is given by the energy difference between metal and ligand orbital, ="(d)"(p). Since the transfer integrals Tdp=Tiare of different magnitude for different combinations ofdandporbitals, this leads to the splitting of the (one-hole) ligand-field states. AsTx2y2,p=T(b1)=T1is the largest transfer matrix element, the GS of lowest energy has the hole in the b1g=dx2y2orbital.
Photoemission leads to removal of an electron leading to a second hole in the CuO4 cluster.
Ionization of a b1g electron, e.g., produces final states with a second hole in a b1g level. This generates three possible configurations |d8>, |d9L> and |d10L2> at energies U, 1 and 21, respectively. The d8andd10L2configurations each produce one singlet state while the d9Lstate produces both a singlet state and a triplet state. The singlets all have the same spin and spatial symmetry and can undergo CI which involves the parametersU,1and T1.
Diagonalization of the respective matrix gives the energy level scheme for the final states. The same can be done with the states resulting from removal of an electron from a non-b1gorbital; in this case the final state CI matrices are only slightly larger. The theoretical PE spectrum can then be evaluated by forming the electric dipole operator matrix elements between the ground state and the various final states. It has been pointed out that nonlocal effects are important for a detailed understanding of the local cluster energy level scheme, cf. ref. 35. A fit to the measured spectrum will afford values for the various CT energiesi, the transfer matrix elementsTiandU.30 The parameters derived from this treatment of the local CuOncluster can then be employed in cluster–CI calculations to evaluate the coupling that arises between two Cu centers due to virtual charge fluctuations (superexchange). In the simplest case, the cluster–CI calculation is based on a three-center linear unit with a central ligand atom and two cations (Figure 10), i.e. the mixing of the copper centers of two neighboring CuO4clusters (solid and dotted lines inFigure 10) with the terminal oxygen atoms is neglected and only their interaction via one-bridging oxygenporbital is considered. The determination of the two-hole wavefunctions, the interaction matrices and the energy level scheme is entirely analogous to the VBCI procedure for isolated Cu dimers outlined in Section 2.42.2. Importantly, the transfer integral t used in the cluster CI superexchange calculation is related to the transfer integral T1=T(b1g) of the single CuO4 cluster by t=1/2 T1. In the case of a 3D cubic lattice (rock salt structure), the superexchange transfer integral t is related to the transfer integralT1=T(eg) of the CuO6cluster byt=T1/ ffiffiffi
p3
. Using these values and Uandparameters corrected for crystal-field and electrostatic effects, the Ne´el temperatures of transition-metal oxides have been successfully reproduced.23