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Electron transfer in mixed-valence tetranuclear iron clusters. Orbital effects and magnetic properties
ChemicalPhysics North-Holland
177 (1993) 15-22
Electron transfer in mixed-valence tetranuclear iron clusters. Orbital effects and magnetic properties J.J. Borr&Almenar
‘, E. Coronado a, R. Georges b and C.J. Gomez-Garcia a
’ DepartamentoQuimicaInorgctnica,.Universidadde Valencia,DoctorMoliner SO,46100 Burjasot,Spain b LaboratoireChimie du Solide, Universitkde Bordeaux I, 351. Coursde la Liberation,33405 Talence,France Received 12 January 1993; in final form 14 June 1993
The energy levels and magnetic properties of mixed-valence iron clusters with cubane structure are studied from a model that takes into account the different electron transfer paths between the es-type metal orbitals via the p orbitals of the bridging atoms. The interplay between double exchange and superexchange on the magnetic properties are discussed, including distortion effects of the cubane structure from Td to Dsd symmetries.
1. Introduction The study of the electronic and magnetic properties of mixed-valence (MV) clusters presenting simultaneously magnetic exchange interactions and electron transfers is a subject of current interest in both magnetochemistry and biochemistry [ 1,2 1. To date most of the works in this area have been focused on the simplest MV system: the dinuclear entity. In this case the double exchange theory predicts that the nature of the ground state results from the interplay between the electron transfer, that stabilizes the higher spin states, and both antiferromagnetic exchange and vibronic interactions that favor the lower spin states [ 3,4]. Therefore, in presence of strong electron transfer a ferromagnetic ground state is to be expected for the MV dimer. This result cannot be generalized to more extended MV systems. For example, in the trinuclear MV systems it has been shown that the nature of the ground spin state depends on the topology of the interacting ions, and on the sign of the transfer integrals [ 51. Thus, for a linear trimer the electron transfer stabilizes a ferromagnetic ground state, irrespective of its sign. Conversely, in a triangular entity the nature of the ground state depends on the sign of the transfer integral. For negative values, the ferromagnetic state is stabilized, while for positive values the ground state is formed by a mixture of spin states comprised be-
tween Sminand S,,. This last situation favors that minor effects, as for example superexchange interactions or distortions, become quite significant in determining the spin-state ordering of the MV triangle. Very recently this problem has been extended to the tetranuclear MV clusters with metal configurations d1d1d*d2 [ 61 and dsd5dsd6 [ 71 and geometries comprised between D4,,and Td limits. These kinds of systems are of current interest as models of four ironsulfur cubes of biological relevance [ 8 1. In particular, in the last case the metal configuration corresponds to that of oxidized high-potential iron proteins (the cubane cluster is in the 3Fe(III)-lFe(I1) state). In these tetranuclear clusters the effect of electron delocalization predicts similar results to those found in the trinuclear MV systems. Indeed, only for a square planar geometry the electron transfer stabilizes a ferromagnetic ground state, irrespective of the sign of the transfer integral (fig. 1a). For Td or intermediate DZdgeometries the nature of the ground state depends on the sign of the transfer integrals; this is ferromagnetic for negative values, while it exhibits different spin values for positive transfer integrals, being formed by a mixture of spin states in the Td limit (fig. lb). Notice that in these models a single transfer integral has been assumed to account for the electron transfer within each pair of metal ions. Indeed, the orbital degeneracy of the metal ions as well as the role of the bridging atoms on the electron
0301-0104/93/S 06.00 0 1993 Elsevier Science Publishers B.V. AU rights reserved.
J.J. Borr&.+Almenar et al. /Chemical Physics177 (1993) IS-22
16
2. Model
D
4h
E/t -
t/t=0
We focus on a tetranuclear unit formed by three Fe(II1) and one Fe(I1) in the high-spin configuration. This cluster has a tetrahedral Fe4X4 core, where X generally refers to a dianion of the type S*- or O*which bridges the iron atoms. These metal ions are occupying tetrahedral coordination sites, and therefore only the two es-type metal orbitals are expected to be involved in the delocalization process, which can be described by the transfer Hamiltonian:
2.0l.Oo.o-l.O-2.o-
A=
-3.o0
4
8 12 2S+l
16
20
Td -
t’/t= 1
2.01.0-
I
t r,m~,n= QSi,m>,n3
o.o-
-1.0
- - - _
-_--____I-
-2.0 1 -3.0’ 0
t,,??$J,nc;tm c,,,,
4
8 12 2s+1
16
(1)
where i,j ( = 1,2,3,4) refer to the four sites and m, n to the e, orbitals. C&, creates an electron on orbital I m) of site i, while C,, destroys an electron on orbital 1n) of sitej. t,,,J,n are the corresponding metalmetal transfer integrals, also called double exchangeintegrals, which depend on the geometry of the cluster since they are nearly proportional to the related metal-metal overlap integrals,
(a)
Elt
c r,m.J.n
20
(b) Fig. 1. Energy spectrum of the tetrameric iron cluster for D,,, (a) and Td (b) symmetries as derived from the non degenerate orbital model; t and t’ referto the two types of metal-metal transfer parameters.
transfer process have not been considered. These effects are examined in the present work in the partio ular case of a T,, geometry for the metal atoms. The interplay between double exchange and superexchange on the magnetic properties are discussed, including distortion effects of the cubane structure from Td to Dza symmetries.
(2)
where Q is the proportionality constant and has energy dimensions. In principle, there are twenty four transfer integrals belonging to three types, according with the nature of the involved eg orbitals of the metal: x2-y*+x*-y*, z*+z*, and x2-y*+z*. When the crossing terms are neglected and only one type of transfer integral is considered (for example, the x2-y*+x*-y*), the model reduces to the nondegenerate orbital one previously reported [ 7 1. The model developed here takes into account all the available transfer integrals between the e, orbitals, including the crossing terms. We have solved the problem within a basis involving only the high-spin configurations on each metal ion, excluding so the high-energy low-spin ones. Since the electron transfer process likely occurs via the bridging anions, the metal-metal transfer integrals should be directly directed with the intermediate metal-anion transfer integrals. In order to parametrize the different transfer integrals, we start from an idealized cubane-type structure in which each metal is located in a tetrahedral site FeX, sharing two
J.J. Bornis-Almenaret al. /Chemical PhysicsI77 (1993) 15-22
bridging anions with each one of its neighbors (fig. 2). From this geometry it is straightforward to evaluate the relative values of the metal-metal transfer integrals in terms of the intermediate metal-anion transfer integrals. Thus, if we consider as anion orbitals the p ones, three different transfer integrals pe, can be distinguished for each metal site, namely b, b’ and b”. These are schematized in fig. 3 and table 1; b accounts for the p-dx+,2 transfers, while b’ and b” are the px (or p,,)-dr2 and pZ-d,z transfers, respectively. Geometrical considerations allow to show that when a Td symmetry is assumed for the metal cluster b’ and b” are related with b in a simple way,
17
Due to the covalent contribution of the Fe-X bond, the e, orbitals actually contain some contribution from the anion orbitals and therefore they can be expressed as (4) where the mixture coefficients C,,, may be assumed to be proportional to the ratio between the involved p-e, transfer integrals and the energy U, where Ucorresponds to the energy difference between p and eB orbitals. From table 1 and eq. (3) we can calculate these coefficients in terms of b/U, and from these, the metal-metal overlap integrals defined as Si,mj,n= ( &i 15). These are summarized in table 2 in units of Qb2/27Uz. In the following we will refer to this quantity as effective transfer parameter, P. Finally, expressing the transfer Hamiltonian (eq. ( 1) ) in terms of the S.hmJ,nby the virtue of its relationship with the tr,mXn(eq. (2) ) and diagonalizing it, we obtain the energy levels of the tetrahedral cluster in P units. Notice that from the S’j,mj,nwe can learn the relative magnitude of the transfer integrals in the cluster, but this gives no information on their signs. In order words we dont know if the tr,mJ,nand P parameters are of the same or of reverse sign.
3. Energy levels and magnetic properties of the regular cubane unit Fe,X.,
Fig. 2. Idealized structure of the cubane cluster [Fe&] 3+.
The energy spectrum of the cubane cluster in presence of double exchange is displayed in fig. 4. As we can see, the spin states are dispersed in the diagram
s4
s4 W,z+
-, PJ
We
+ PJ
Fig. 3. Types of transfers between the sulfur’s p orbitals and the Q-type metal’s orbitals in the cubane structure.
J.J. Borr&s-Almenar et al. /Chemical Physics177 (1993) 15-22
18
Table 1 Metal-sulfur transfer integrals. The numbering of the sulfur atoms is defined in fig. 3 e,\p
d,a_,a 41
S1
s2
S3
S4
x
Y
z
x
Y
2
X
Y
z
X
Y
z
b -6’
-b -b’
0 _ b”
-b b’
b b’
0 -6”
-b b’
-b -b’
0 6”
b -b’
b b’
0 b”
Table 2 Metal-metal overlap integrals in P units. The numbering of the metal atoms is defined in fg 2 (dx+.),
(dx+&
(dx+A
(dxz-_ya)4
0 42 -36 42 0 -1oJT 16J5 -42fi
42 0 42 -36 42fi
-36 42 0 42 16fi -42fi
42 -36 42 $3
E/P
-
-200 0
I
I
4
8 12 2S+l
I
I
I
16
20
Fig. 4. Influence of the electron delocalization: energy spectrum of the cubane cluster as a function of the spin multiplicity. For simplicity the degeneracy of each energy level is not represented!
forming unresolved energy bands; the ground spin state has S= 17/2 for both positive and negative P. This result contrast with the very simple spin state structure derived from the non-degenerate orbital model (see fig. 1). Despite these differences, some resemblances between the two energy spectra can be noticed: (i) both are unsymmetric relative to the
-16J5 423 0
(dzz), 4;Jj 16fi lOJ3 0 10 -68 10
(drl)z
(dzzlp
-1oJ5 -4;Jj
16fi 1ofi 455
-16fi 10 0 10 -68
-68 10 0 10
(dzz14 -42fi -16J5 -1oJj 0 10 -68 10 0
change of sign of the transfer parameter; (ii) depending on the sign of the transfer parameter, both models predict or a tendency of the lowest lying spin states to be degenerated, or a tendency to stabilize the higher spin states. Thus, in the present model these quasidegenerate states have S values in the range 9/2 to 17/2, while in the non degenerate model the ground state is formed by a wide mixture of spin states with S comprised between l/2 and 17/2. When the sign of the transfer parameter is reversed the present model stabilizes a state with S= 17/2, while the nondegenerate model stabilizes the ferromagnetic spin state S= 19/2. Let us now examine the influence of antiferromagnetic exchange interactions on these energy levels. For sake of simplicity we assume six equal J values for describing the nearest-neighbor exchange interactions in the cluster (the Heisenberg exchange Hamiltonian is defined as -2JS,S,). Then, the exchange splitting of the spin states is simply given by Ek= -J&(&+ I), where Sk refers to the spin value of the state k. In fig. 5 are plotted the energies of the low lying spin states of the cubane for increasing values of the ratio 1JI /P. We observe that for both positive and negative P the spin of the ground state strongly decreases as the antiferromagnetic exchange
J. J. Borr&s-Almenar et al. / ChemicalPhysicsI 77 (I 993) 15-22
E/P -80
-160/
0.0
0.4
0.8
1.2
1.6 IJVP
1.2
1.6
(a)
0.0
0.4
0.8
19
As a result, the range in which the S= 17/2 state is the ground state is reduced to the range of 0 to w 0.1, while the S=9/2 extends now in the range of ~0.1 to % 1.0. Finally, for larger exchange values the S= l/2 becomes the ground state. These variable spin-state structures are reflected in the magnetic properties, which are strongly dependent on the ratio lJ1 /P. These are given as plots of the normalized product xNT versus the reduced temperature kT/ IJI in fig. 6. xN is defined to give the uniform value j&T= 1 in the high temperature limit. Thus, for positive and strong double exchange, XNT sharply increases when the temperature is lowered showing a well-defined plateau (xNT~2.5) at low temperature, when only the S= 17/2 state is populated (fig. 6a). In the other limit (strong antiferromagnetic exchange, IJI /P> 2), the antiferromagnetic state S= l/2 is the ground state, and then XNT decreases with T down to the constant value l/43 at low-temperature. In between the curves are featureless reaching intermediate values in the low temperature limit, in agreement with the stabilization of the
,J,,,p,
Fig. 5. Influence of the magnetic exchange: correlation diagram of the low-lying spin states of the cubane cluster as a function of
0.5
IJllf’.
0.0 0
is increased. Thus, for positive P the S= 1712 state is the ground state when IJI /P is smaller than OS. For IJI /P values of x0.5-0.9 the ground state is the S= 912. The S= 512 state becomes the ground state when lJ1 /P is in the range of k: 0.9 to R 1.6, and finally when IJI /Pexceeds cz 1.8, the S= l/2 becomes the ground state. It is to be noticed that only when the ground state is the S= 17/2, this is well separated in energy from the excited states; in the other cases the excited states are very close, and for some particular values of 1JI /P accidental degeneracies of the levels may occur. For negative P a similar evolution of the energy levels is observed. The only significant difference arises from the fact that the separation between the S= 17/2 and the excited states is smaller.
10
20
30
4a kTNl
50
(a)
(W Fig. 6. Magnetic properties of the cubane cluster in electron delocalization and Heisenberg exchange.
presenceof
20
J.J. Bonds-Almenar et al. /Chemical Physics 177 (1993) X5-22
intermediate spin states. For negative P the evolution of the curves is similar to that observed in the preceding case (fig. 6b). The most significant differences are observed in the strong double exchange regime, and in the range in which the S=9/2 state is the ground state. In the former case the characteristic plateau is shifted to lower temperatures, as a result of the approach of the higher spin excited states (S> 9/2) to the ground state S= 17/2. In the later case the magnetic properties strongly depend on the value of lJ1 /P exhibiting either maxima of ,&T (in the range 0.1 < 1JI /PC 0.4) or minima (in the range 0.5 < IJI /P-c 1). This is a consequence of the crossing of excited spin states in this region giving rise to irregular spin state structures.
4. Influence of the distortions In the above we have used the metal-metal overlap integrals as deduced from the ideal Td geometry defined in fig. 2 (table 2) to obtain the electronic and magnetic properties of the cubane cluster. Bearing in mind that these overlap integrals depend on the geometry of the cluster, it is to be expected that an elongation of the tetrahedron along one of the C, axis should reduce the overlap integrals involved in four of the six sides of the tetrahedron, increasing the remaining two ones (see fig. 7). Obviously, the compresion along the C, axis should have the oppo(a) Compression
(b) Elongation c,
Fig. 7. Distorted structures of the Fe, tetrahedron showing both the compression and elongation pathways.
site effect. Therefore, a rough estimate of the influence of the symmetry reduction (from Td to D2,, geometries) on the energy levels and magnetic properties of the cluster may be obtained by modifying in a convenient manner these overlap integrals. In our case the elongation effect has been introduced reducing the twelve overlap integrals associated to the sides 1-3, 1-4, 2-3 and 2-4 by a factor 0.9, and increasing those of the sides l-2 and 3-4 by a factor 1.1. Conversely, for the compresion case, the former set of integrals has been multiplied by 1.1, and the second one by 0.9. Despite the strong changes introduced in the overlap integrals (of x IO%), the relative positions of the lowest lying energy levels remain almost unchanged for both positive and negative P (fig. 8). The only noteworthy difference is the stabilization (destabilization) of the ferromagnetic SE=19/2 state with respect to the other low-lying energy levels when an elongation (compression) of the tetrahedron is considered (in the particular case of strong and negative P this ferromagnetic state even becomes the ground state). According with the above observation, the effect of these distortions on the magnetic properties should be very small for both positive and negative P (fig. 9).
5. Concluding remarks We have shown in this work how the different pathways involved in the electron transfer process affect the electronic structure and magnetic properties of mixed-valence tetranuclear iron clusters with cubane-type structure. We notice that the simple picture of the transfer process given by a model involving one orbital (of spherical symmetry) per center, is considerably complicated when the two e&ype orbitals of the metals and the p orbitals of the bridging atoms are taken into account. The most significant features are the following: (i) A complex spin state structure is observed in which the electron transfer tends to stabilize the higher spin states either for positive as well as for negative values of the metal-metal transfer integrals. Thus, in both cases the ground state has a spin S= 17/2. This situation is basically different from that predicted assuming one orbital per center.
21
J. J. Borrtfs-Almenaret al. / ChemicalPhysics177 (1993) 15-22
o.oI 0
10
20
30
40
kT/IJI
4u
kT/IJI
(4 0.0
0.5
1.0
1.5
,J,/p
(4 EAPI -90 -100 -110 0.01..,,,....,..,.,....,....
-120
U
10
20
30 (b)
Fig. 9. Influence of the distortions on the magnetic behavior of the cluster in the strong double-exchange limit.
@I Fig. 8. Energies of the low lying spin states of the elongated clus ter as a function of IJI /P.
(ii) As in the non-degenerate orbital model, this spin state structure is unsymmetric relative to the sign reverse of the transfer parameter. Thus, while for P> 0 the ground S= 17/2 state is well separated in energy from the excited ones, for PC 0 the upper levels (between S= 912 and 1912) become very close in energy to the ground state one. In this last situation minor effects, as for example superexchange interactions or distortions of the cluster, should determine the spin state ordering, and therefore a large spin state variability for the ground state is to be expected. (iii) As far as the magnetic properties are concerned, the above features allow to predict that in these clusters the magnetic behaviors will be strongly dependent on the relative magnitudes (and signs) of
transfer and exchange parameters. In presence of antiferromagnetic exchange intermediate or antiferromagnetic ground states should be observed. (iv) With respect to the distortions of the cluster from the ideal Td geometry it has been found that this effect has a very limited influence on the spin state structures and magnetic properties of the cluster.
5. Acknowledgements
This work was supported by the Direction General de Investigacidn Cientfica y T&cnica (PB9 l-0652) and by a Program of Integrated Actions with France. CJGG and JJBA are grateful for a fellowship from the Commision of the European Communities. We also thank the Calcul Center of the University of Valencia for the free use of his computational resources.
22
J. J. Borrds-Almenaret al. /Chemical PhysicsI77 (1993) 15-22
References
[ 1 ] G. Blondin and J.J. Girerd, Chem. Rev. 90 ( 1990) 1359, and references therein; G.Blondin, S.A. Borshch and J.J. Girerd, Commun. Inorg. Chem. 12 (1992) 315. [2] B. Bersuker and S. A. Borshch, in: Advances in chemical physics, Vol. 8 1, eds. I. Prigogine and S.A. Rice (Wiley, New York, 1992). [ 31 C. Zener, Phys. Rev. 82 ( 195 1) 403; P.W. Anderson andH. Hasegawa, Phys. Rev. 100 (1955) 675; P.G. de Gennes, Phys. Rev. 118 ( 1960) 141. [4] M. Drillon, G. Pourroy and J. Darriet, Chem. Phys. 88 (1984) 27; S.A. Borshch, I.N. Kotov and LB. Bersuker, Chem. Phys. Letters 111 (1984) 264.
[ 5 ] G. Pourroy, E. Coronado, M. Drillon and R. Georges, Chem. Phys. 104 (1986) 73; M.I. Belinskii, Mol. Phys. 60 (1987) 793; C.J. Gdmez-Garcia, E. Coronado and G. Pourroy, J. Appl. Phys. 67 (1990) 5992; C.J. G6mez-Garcfa, E. Coronado, R. Georges and G. Pourroy, PhysicaB 182 (1992) 18. [ 61 A.V. Palii, S. M. Cktrovsky and B.S. Tsukerblat, New J. Chem, 16 (1992) 943; S.A. Borshch, E.L. Bominaar and J.J. Girerd, New J. Chem. 17 (1993) 39. [7] J.J. Borras-Almenar, E. Coronado, R. Georges and C.J. Gbmez-Garcia, Chem. Phys. 166 ( 1992) 139; J. Magn. Magn. Mater. 104-107 (1992) 955. [8] E. Mtlnck, V. Papaefthymiou, K.K. Surerus and J.J. Girerd, in: Metal clusters in proteins, ed. L. Que, ACS Symp. Ser. No. 372 (1988).
177 (1993) 15-22
Electron transfer in mixed-valence tetranuclear iron clusters. Orbital effects and magnetic properties J.J. Borr&Almenar
‘, E. Coronado a, R. Georges b and C.J. Gomez-Garcia a
’ DepartamentoQuimicaInorgctnica,.Universidadde Valencia,DoctorMoliner SO,46100 Burjasot,Spain b LaboratoireChimie du Solide, Universitkde Bordeaux I, 351. Coursde la Liberation,33405 Talence,France Received 12 January 1993; in final form 14 June 1993
The energy levels and magnetic properties of mixed-valence iron clusters with cubane structure are studied from a model that takes into account the different electron transfer paths between the es-type metal orbitals via the p orbitals of the bridging atoms. The interplay between double exchange and superexchange on the magnetic properties are discussed, including distortion effects of the cubane structure from Td to Dsd symmetries.
1. Introduction The study of the electronic and magnetic properties of mixed-valence (MV) clusters presenting simultaneously magnetic exchange interactions and electron transfers is a subject of current interest in both magnetochemistry and biochemistry [ 1,2 1. To date most of the works in this area have been focused on the simplest MV system: the dinuclear entity. In this case the double exchange theory predicts that the nature of the ground state results from the interplay between the electron transfer, that stabilizes the higher spin states, and both antiferromagnetic exchange and vibronic interactions that favor the lower spin states [ 3,4]. Therefore, in presence of strong electron transfer a ferromagnetic ground state is to be expected for the MV dimer. This result cannot be generalized to more extended MV systems. For example, in the trinuclear MV systems it has been shown that the nature of the ground spin state depends on the topology of the interacting ions, and on the sign of the transfer integrals [ 51. Thus, for a linear trimer the electron transfer stabilizes a ferromagnetic ground state, irrespective of its sign. Conversely, in a triangular entity the nature of the ground state depends on the sign of the transfer integral. For negative values, the ferromagnetic state is stabilized, while for positive values the ground state is formed by a mixture of spin states comprised be-
tween Sminand S,,. This last situation favors that minor effects, as for example superexchange interactions or distortions, become quite significant in determining the spin-state ordering of the MV triangle. Very recently this problem has been extended to the tetranuclear MV clusters with metal configurations d1d1d*d2 [ 61 and dsd5dsd6 [ 71 and geometries comprised between D4,,and Td limits. These kinds of systems are of current interest as models of four ironsulfur cubes of biological relevance [ 8 1. In particular, in the last case the metal configuration corresponds to that of oxidized high-potential iron proteins (the cubane cluster is in the 3Fe(III)-lFe(I1) state). In these tetranuclear clusters the effect of electron delocalization predicts similar results to those found in the trinuclear MV systems. Indeed, only for a square planar geometry the electron transfer stabilizes a ferromagnetic ground state, irrespective of the sign of the transfer integral (fig. 1a). For Td or intermediate DZdgeometries the nature of the ground state depends on the sign of the transfer integrals; this is ferromagnetic for negative values, while it exhibits different spin values for positive transfer integrals, being formed by a mixture of spin states in the Td limit (fig. lb). Notice that in these models a single transfer integral has been assumed to account for the electron transfer within each pair of metal ions. Indeed, the orbital degeneracy of the metal ions as well as the role of the bridging atoms on the electron
0301-0104/93/S 06.00 0 1993 Elsevier Science Publishers B.V. AU rights reserved.
J.J. Borr&.+Almenar et al. /Chemical Physics177 (1993) IS-22
16
2. Model
D
4h
E/t -
t/t=0
We focus on a tetranuclear unit formed by three Fe(II1) and one Fe(I1) in the high-spin configuration. This cluster has a tetrahedral Fe4X4 core, where X generally refers to a dianion of the type S*- or O*which bridges the iron atoms. These metal ions are occupying tetrahedral coordination sites, and therefore only the two es-type metal orbitals are expected to be involved in the delocalization process, which can be described by the transfer Hamiltonian:
2.0l.Oo.o-l.O-2.o-
A=
-3.o0
4
8 12 2S+l
16
20
Td -
t’/t= 1
2.01.0-
I
t r,m~,n= QSi,m>,n3
o.o-
-1.0
- - - _
-_--____I-
-2.0 1 -3.0’ 0
t,,??$J,nc;tm c,,,,
4
8 12 2s+1
16
(1)
where i,j ( = 1,2,3,4) refer to the four sites and m, n to the e, orbitals. C&, creates an electron on orbital I m) of site i, while C,, destroys an electron on orbital 1n) of sitej. t,,,J,n are the corresponding metalmetal transfer integrals, also called double exchangeintegrals, which depend on the geometry of the cluster since they are nearly proportional to the related metal-metal overlap integrals,
(a)
Elt
c r,m.J.n
20
(b) Fig. 1. Energy spectrum of the tetrameric iron cluster for D,,, (a) and Td (b) symmetries as derived from the non degenerate orbital model; t and t’ referto the two types of metal-metal transfer parameters.
transfer process have not been considered. These effects are examined in the present work in the partio ular case of a T,, geometry for the metal atoms. The interplay between double exchange and superexchange on the magnetic properties are discussed, including distortion effects of the cubane structure from Td to Dza symmetries.
(2)
where Q is the proportionality constant and has energy dimensions. In principle, there are twenty four transfer integrals belonging to three types, according with the nature of the involved eg orbitals of the metal: x2-y*+x*-y*, z*+z*, and x2-y*+z*. When the crossing terms are neglected and only one type of transfer integral is considered (for example, the x2-y*+x*-y*), the model reduces to the nondegenerate orbital one previously reported [ 7 1. The model developed here takes into account all the available transfer integrals between the e, orbitals, including the crossing terms. We have solved the problem within a basis involving only the high-spin configurations on each metal ion, excluding so the high-energy low-spin ones. Since the electron transfer process likely occurs via the bridging anions, the metal-metal transfer integrals should be directly directed with the intermediate metal-anion transfer integrals. In order to parametrize the different transfer integrals, we start from an idealized cubane-type structure in which each metal is located in a tetrahedral site FeX, sharing two
J.J. Bornis-Almenaret al. /Chemical PhysicsI77 (1993) 15-22
bridging anions with each one of its neighbors (fig. 2). From this geometry it is straightforward to evaluate the relative values of the metal-metal transfer integrals in terms of the intermediate metal-anion transfer integrals. Thus, if we consider as anion orbitals the p ones, three different transfer integrals pe, can be distinguished for each metal site, namely b, b’ and b”. These are schematized in fig. 3 and table 1; b accounts for the p-dx+,2 transfers, while b’ and b” are the px (or p,,)-dr2 and pZ-d,z transfers, respectively. Geometrical considerations allow to show that when a Td symmetry is assumed for the metal cluster b’ and b” are related with b in a simple way,
17
Due to the covalent contribution of the Fe-X bond, the e, orbitals actually contain some contribution from the anion orbitals and therefore they can be expressed as (4) where the mixture coefficients C,,, may be assumed to be proportional to the ratio between the involved p-e, transfer integrals and the energy U, where Ucorresponds to the energy difference between p and eB orbitals. From table 1 and eq. (3) we can calculate these coefficients in terms of b/U, and from these, the metal-metal overlap integrals defined as Si,mj,n= ( &i 15). These are summarized in table 2 in units of Qb2/27Uz. In the following we will refer to this quantity as effective transfer parameter, P. Finally, expressing the transfer Hamiltonian (eq. ( 1) ) in terms of the S.hmJ,nby the virtue of its relationship with the tr,mXn(eq. (2) ) and diagonalizing it, we obtain the energy levels of the tetrahedral cluster in P units. Notice that from the S’j,mj,nwe can learn the relative magnitude of the transfer integrals in the cluster, but this gives no information on their signs. In order words we dont know if the tr,mJ,nand P parameters are of the same or of reverse sign.
3. Energy levels and magnetic properties of the regular cubane unit Fe,X.,
Fig. 2. Idealized structure of the cubane cluster [Fe&] 3+.
The energy spectrum of the cubane cluster in presence of double exchange is displayed in fig. 4. As we can see, the spin states are dispersed in the diagram
s4
s4 W,z+
-, PJ
We
+ PJ
Fig. 3. Types of transfers between the sulfur’s p orbitals and the Q-type metal’s orbitals in the cubane structure.
J.J. Borr&s-Almenar et al. /Chemical Physics177 (1993) 15-22
18
Table 1 Metal-sulfur transfer integrals. The numbering of the sulfur atoms is defined in fig. 3 e,\p
d,a_,a 41
S1
s2
S3
S4
x
Y
z
x
Y
2
X
Y
z
X
Y
z
b -6’
-b -b’
0 _ b”
-b b’
b b’
0 -6”
-b b’
-b -b’
0 6”
b -b’
b b’
0 b”
Table 2 Metal-metal overlap integrals in P units. The numbering of the metal atoms is defined in fg 2 (dx+.),
(dx+&
(dx+A
(dxz-_ya)4
0 42 -36 42 0 -1oJT 16J5 -42fi
42 0 42 -36 42fi
-36 42 0 42 16fi -42fi
42 -36 42 $3
E/P
-
-200 0
I
I
4
8 12 2S+l
I
I
I
16
20
Fig. 4. Influence of the electron delocalization: energy spectrum of the cubane cluster as a function of the spin multiplicity. For simplicity the degeneracy of each energy level is not represented!
forming unresolved energy bands; the ground spin state has S= 17/2 for both positive and negative P. This result contrast with the very simple spin state structure derived from the non-degenerate orbital model (see fig. 1). Despite these differences, some resemblances between the two energy spectra can be noticed: (i) both are unsymmetric relative to the
-16J5 423 0
(dzz), 4;Jj 16fi lOJ3 0 10 -68 10
(drl)z
(dzzlp
-1oJ5 -4;Jj
16fi 1ofi 455
-16fi 10 0 10 -68
-68 10 0 10
(dzz14 -42fi -16J5 -1oJj 0 10 -68 10 0
change of sign of the transfer parameter; (ii) depending on the sign of the transfer parameter, both models predict or a tendency of the lowest lying spin states to be degenerated, or a tendency to stabilize the higher spin states. Thus, in the present model these quasidegenerate states have S values in the range 9/2 to 17/2, while in the non degenerate model the ground state is formed by a wide mixture of spin states with S comprised between l/2 and 17/2. When the sign of the transfer parameter is reversed the present model stabilizes a state with S= 17/2, while the nondegenerate model stabilizes the ferromagnetic spin state S= 19/2. Let us now examine the influence of antiferromagnetic exchange interactions on these energy levels. For sake of simplicity we assume six equal J values for describing the nearest-neighbor exchange interactions in the cluster (the Heisenberg exchange Hamiltonian is defined as -2JS,S,). Then, the exchange splitting of the spin states is simply given by Ek= -J&(&+ I), where Sk refers to the spin value of the state k. In fig. 5 are plotted the energies of the low lying spin states of the cubane for increasing values of the ratio 1JI /P. We observe that for both positive and negative P the spin of the ground state strongly decreases as the antiferromagnetic exchange
J. J. Borr&s-Almenar et al. / ChemicalPhysicsI 77 (I 993) 15-22
E/P -80
-160/
0.0
0.4
0.8
1.2
1.6 IJVP
1.2
1.6
(a)
0.0
0.4
0.8
19
As a result, the range in which the S= 17/2 state is the ground state is reduced to the range of 0 to w 0.1, while the S=9/2 extends now in the range of ~0.1 to % 1.0. Finally, for larger exchange values the S= l/2 becomes the ground state. These variable spin-state structures are reflected in the magnetic properties, which are strongly dependent on the ratio lJ1 /P. These are given as plots of the normalized product xNT versus the reduced temperature kT/ IJI in fig. 6. xN is defined to give the uniform value j&T= 1 in the high temperature limit. Thus, for positive and strong double exchange, XNT sharply increases when the temperature is lowered showing a well-defined plateau (xNT~2.5) at low temperature, when only the S= 17/2 state is populated (fig. 6a). In the other limit (strong antiferromagnetic exchange, IJI /P> 2), the antiferromagnetic state S= l/2 is the ground state, and then XNT decreases with T down to the constant value l/43 at low-temperature. In between the curves are featureless reaching intermediate values in the low temperature limit, in agreement with the stabilization of the
,J,,,p,
Fig. 5. Influence of the magnetic exchange: correlation diagram of the low-lying spin states of the cubane cluster as a function of
0.5
IJllf’.
0.0 0
is increased. Thus, for positive P the S= 1712 state is the ground state when IJI /P is smaller than OS. For IJI /P values of x0.5-0.9 the ground state is the S= 912. The S= 512 state becomes the ground state when lJ1 /P is in the range of k: 0.9 to R 1.6, and finally when IJI /Pexceeds cz 1.8, the S= l/2 becomes the ground state. It is to be noticed that only when the ground state is the S= 17/2, this is well separated in energy from the excited states; in the other cases the excited states are very close, and for some particular values of 1JI /P accidental degeneracies of the levels may occur. For negative P a similar evolution of the energy levels is observed. The only significant difference arises from the fact that the separation between the S= 17/2 and the excited states is smaller.
10
20
30
4a kTNl
50
(a)
(W Fig. 6. Magnetic properties of the cubane cluster in electron delocalization and Heisenberg exchange.
presenceof
20
J.J. Bonds-Almenar et al. /Chemical Physics 177 (1993) X5-22
intermediate spin states. For negative P the evolution of the curves is similar to that observed in the preceding case (fig. 6b). The most significant differences are observed in the strong double exchange regime, and in the range in which the S=9/2 state is the ground state. In the former case the characteristic plateau is shifted to lower temperatures, as a result of the approach of the higher spin excited states (S> 9/2) to the ground state S= 17/2. In the later case the magnetic properties strongly depend on the value of lJ1 /P exhibiting either maxima of ,&T (in the range 0.1 < 1JI /PC 0.4) or minima (in the range 0.5 < IJI /P-c 1). This is a consequence of the crossing of excited spin states in this region giving rise to irregular spin state structures.
4. Influence of the distortions In the above we have used the metal-metal overlap integrals as deduced from the ideal Td geometry defined in fig. 2 (table 2) to obtain the electronic and magnetic properties of the cubane cluster. Bearing in mind that these overlap integrals depend on the geometry of the cluster, it is to be expected that an elongation of the tetrahedron along one of the C, axis should reduce the overlap integrals involved in four of the six sides of the tetrahedron, increasing the remaining two ones (see fig. 7). Obviously, the compresion along the C, axis should have the oppo(a) Compression
(b) Elongation c,
Fig. 7. Distorted structures of the Fe, tetrahedron showing both the compression and elongation pathways.
site effect. Therefore, a rough estimate of the influence of the symmetry reduction (from Td to D2,, geometries) on the energy levels and magnetic properties of the cluster may be obtained by modifying in a convenient manner these overlap integrals. In our case the elongation effect has been introduced reducing the twelve overlap integrals associated to the sides 1-3, 1-4, 2-3 and 2-4 by a factor 0.9, and increasing those of the sides l-2 and 3-4 by a factor 1.1. Conversely, for the compresion case, the former set of integrals has been multiplied by 1.1, and the second one by 0.9. Despite the strong changes introduced in the overlap integrals (of x IO%), the relative positions of the lowest lying energy levels remain almost unchanged for both positive and negative P (fig. 8). The only noteworthy difference is the stabilization (destabilization) of the ferromagnetic SE=19/2 state with respect to the other low-lying energy levels when an elongation (compression) of the tetrahedron is considered (in the particular case of strong and negative P this ferromagnetic state even becomes the ground state). According with the above observation, the effect of these distortions on the magnetic properties should be very small for both positive and negative P (fig. 9).
5. Concluding remarks We have shown in this work how the different pathways involved in the electron transfer process affect the electronic structure and magnetic properties of mixed-valence tetranuclear iron clusters with cubane-type structure. We notice that the simple picture of the transfer process given by a model involving one orbital (of spherical symmetry) per center, is considerably complicated when the two e&ype orbitals of the metals and the p orbitals of the bridging atoms are taken into account. The most significant features are the following: (i) A complex spin state structure is observed in which the electron transfer tends to stabilize the higher spin states either for positive as well as for negative values of the metal-metal transfer integrals. Thus, in both cases the ground state has a spin S= 17/2. This situation is basically different from that predicted assuming one orbital per center.
21
J. J. Borrtfs-Almenaret al. / ChemicalPhysics177 (1993) 15-22
o.oI 0
10
20
30
40
kT/IJI
4u
kT/IJI
(4 0.0
0.5
1.0
1.5
,J,/p
(4 EAPI -90 -100 -110 0.01..,,,....,..,.,....,....
-120
U
10
20
30 (b)
Fig. 9. Influence of the distortions on the magnetic behavior of the cluster in the strong double-exchange limit.
@I Fig. 8. Energies of the low lying spin states of the elongated clus ter as a function of IJI /P.
(ii) As in the non-degenerate orbital model, this spin state structure is unsymmetric relative to the sign reverse of the transfer parameter. Thus, while for P> 0 the ground S= 17/2 state is well separated in energy from the excited ones, for PC 0 the upper levels (between S= 912 and 1912) become very close in energy to the ground state one. In this last situation minor effects, as for example superexchange interactions or distortions of the cluster, should determine the spin state ordering, and therefore a large spin state variability for the ground state is to be expected. (iii) As far as the magnetic properties are concerned, the above features allow to predict that in these clusters the magnetic behaviors will be strongly dependent on the relative magnitudes (and signs) of
transfer and exchange parameters. In presence of antiferromagnetic exchange intermediate or antiferromagnetic ground states should be observed. (iv) With respect to the distortions of the cluster from the ideal Td geometry it has been found that this effect has a very limited influence on the spin state structures and magnetic properties of the cluster.
5. Acknowledgements
This work was supported by the Direction General de Investigacidn Cientfica y T&cnica (PB9 l-0652) and by a Program of Integrated Actions with France. CJGG and JJBA are grateful for a fellowship from the Commision of the European Communities. We also thank the Calcul Center of the University of Valencia for the free use of his computational resources.
22
J. J. Borrds-Almenaret al. /Chemical PhysicsI77 (1993) 15-22
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