Charge-Transfer-Induced Spin Transitions in ... - ACS Publications

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Charge-Transfer-Induced Spin Transitions in Crystals Containing Cyanide-Bridged Co−Fe Clusters: Role of Intra- and Intercluster Interactions Marianna A. Roman, Oleg S. Reu, and Sophia I. Klokishner* Institute of Applied Physics, Academy of Sciences of Moldova, Kishinev, Moldova ABSTRACT: A theoretical model has been developed to explain at the electronic level the charge-transfer-induced spin transition (CTIST) in crystals based on cyano-bridged binuclear Fe−Co clusters. The CTIST is considered as a cooperative phenomenon (phase transformation) driven by the long-range electrondeformational interaction via the acoustic phonons field that is taken into account within the mean field approach. The model for CTIST includes also the metal−metal electron transfer and intracluster magnetic exchange. The conditions that favor the CTIST are discussed. The qualitative explanation of the experimental data is given.

1. INTRODUCTION Apart from recent success in preparing new metal-cyanide solids, low-dimensionality cyanide arrays and clusters have also received a great deal of attention.1−6 The primary motivation of these efforts is to obtain finite systems that demonstrate at the molecular level such fascinating phenomena as single molecule magnet behavior,7−14 temperature-induced,15 and light-induced spin crossover and charge transfer16 induced spin-transitions (CTIST). In this context coordination complexes exhibiting two close-lying electronic states differing in charge distribution represent an appealing field of investigation because they offer an opportunity for studies of the factors that control spin conversion between these states. Despite the myriad of known cyanide-bridged discrete clusters, there are limited examples that involve CN− ligands spanning Co and Fe centers; exceptions include a series of dinuclear compounds17−25 and a molecular square.26 The primary reason for the lack of interest to these materials as magnetic ones is the fact that Co and Fe ions in the presence of CN− ligands typically undergo redox processes to produce diamagnetic CoIII and FeII centers. The iron−cobalt compounds have nevertheless attracted attention because of the characteristic metal-to-metal charge-transfer band in the visible region. This characteristic property offers much promise for designing optically switchable materials,27−30 as demonstrated by Bernhardt et al., who have reported pronounced color changes in dinuclear Co/Fe complexes as a function of the metal oxidation states.31 The first photomagnetic effect in Prussian blue analogues was reported by Hashimoto and coworkers in 1996 in the analogue of chemical formula K0.4Co1.3[Fe(CN)6]·5H2O32,33 that represents a coordination polymer. Then a vast family of extended solids that display thermal phase transitions associated with a metal-to-metal electron transfer have been discovered.34−39 Correspondingly, © XXXX American Chemical Society

the theoretical modeling was aimed at the study of the energy spectra and observed characteristics of extended Prussian blue compounds. In the framework of the Blume−Capel model40,41 the magnetic properties of Prussian blue analogues have been examined in theoretical papers42,43 focusing on the condition of the reversible switching between nonmagnetic LS state and magnetic metastable-ordered HS state, which was observed in these compounds. A unified theoretical description of the thermodynamic properties of spin-crossover solids and Prussian blue analogues is given in paper.44 Calculations of the band structure and density of states by DFT and extended Hückel methods for Prussian blue analogs have been reported in papers.45−47 In ref 48 a periodic DFT method using the B3LYP hybrid exchange−correlation potential was applied to the RbMn[Fe(CN)6] compound to evaluate the suitability for studying, and predicting, the photomagnetic behavior of Prussian blue analogues and related materials. Meanwhile, the examined compounds represent extended solids in which Fe and Co ions sequentially alternate. Recently, Dunbar with co-workers has obtained a new family of compounds containing as a structural element pentanuclear clusters {[MIII(CN)6]2[M′II(tmphen)2]3} (M′, M are transition metal ions) with a trigonal bipyramidal structure, which are quite attractive from the point of view of their ability to cardinally change the physical characteristics within the same geometrical cluster structure and ligand composition by varying only the metal ions. Thus, the MnIII2MnII3 cluster12,49,50 being the first representative of this family demonstrates the singlemolecule magnet properties. For the cluster compounds [FeII(tmphen)2]3[MIII(CN)6]2 (M = Fe, Co) spin-crossover Received: May 10, 2012 Revised: July 25, 2012

A

dx.doi.org/10.1021/jp304560u | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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behavior has been observed.51,52 In refs 16 and 53, the compound {[Co(tmphen)2]3[Fe(CN)6]2} has been reported. Characterization by structural, optical, magnetic, and Mössbauer techniques led to conclude that this compound undergoes a driven by temperature CTIST [(ls-FeII)2(ls-CoIII)2(hs-CoII)] ↔ [(ls-FeIII)2(hs-CoII)3] (where the abbreviations ls and hs refer to low-spin and high-spin states of Fe and Co ions, respectively), which is accompanied by a significant change of the effective cluster magnetic moment. As compared with Prussian blue analogs, the CTIST in cluster compounds occurs inside a cluster. Because the clusters are well separated, there is no electron transfer between them. At the same time, the cooperativity of the transition is provided by intercluster interaction. The steepness of the magnetic susceptibility curve points to appreciable intermolecular interactions. In the recent paper54 the mentioned above members of the family of trigonal bipyramidal complexes [FeII(tmphen)2]3[MIII(CN)6]2 (M = Fe, Co) or Fe3Fe2, Fe3Co2 and {[Co(tmphen)2]3[Fe(CN)6]2} (Co3Fe2) have been investigated for optical and photomagnetic properties. It was demonstrated that the spin crossover of the FeII centers in the Fe3Fe2, Fe3Co2 analogues and the chargetransfer events in the Co3Fe2 complex may occur under irradiation as well. In ref 55 an Fe2Co trinuclear cluster {[FeTp(CN)3]2Co(Meim)4}·6H2O was demonstrated to exhibit a thermally induced, reversible electron transfer with a thermal hysteresis and photoinduced electron transfer by excitation of the charge-transfer band. In ref 56 it is underlined that the evergreen domain of spin transitions induced by charge transfer is continuing to provide fascinating results, and the investigation of these phenomena in molecules is very important because these molecules could become switchable components in molecular electronics and, thanks to the significant change in the spin state induced by the charge transfer, in spintronics as well. In spite of this fact, neither theoretical models nor the interpretation of the physical properties of molecular systems exhibiting CTIST exist. Therefore, in the present paper we develop a microscopic theoretical approach to the examination of the CTIST under action of temperature in cluster Fe−Co compounds. To elucidate with utmost clarity the main electronic mechanisms governing this transition for the first step, we consider a model crystal consisting of binuclear Fe−Co clusters. Along with intracluster electron transfer and exchange interaction in the Fe−Co pair the model takes into account the cooperative electron-deformational interaction arising from the transition [(ls-FeII) (ls-CoIII)] ↔ [(ls-FeIII) (hs-CoII)]

Figure 1. Schemes of orbitals for the Co−Fe pair: (a) ls-CoIII ion, lsFeII ion; (b) ls-CoII, ls-FeIII ion; (c) hs-CoII, ls-FeIII ion.

the occupation schemes lowest in energy look as follows: two of the low-lying orbitals φaα (α = 3−5) are doubly occupied and the other three orbitals are singly occupied (Figure 1c). As in the ls-CoIII ion, the three low-lying orbitals φbβ (β = 3−5) of the ls-FeII ion in the carbon surrounding are doubly occupied. Finally, the ls-FeIII ion can be considered as having a hole on one of the φbβ (β = 3−5) orbitals. The electronic Hamiltonian of a Fe−Co pair determining the wave functions and the eigenvalues of the pair in the fixed nuclear configuration includes two parts: He = h 0 + g

(1)

where the one electronic operator h0 involves kinetic and potential energies of all electrons, g is the two-particle (interelectronic repulsion). As a basis set, we take the states of a molecule with localized electrons arising from its three configurations: I.

ls‐CoIII(a), ls‐Fe II(b)

II.

ls‐CoII(a), ls‐Fe III(b)

III. hs‐CoII(a), ls‐Fe III(b)

(2)

The operator of the full spin of the cluster is defined as

2. THE MODEL 2.1. Hamiltonian of an Isolated Fe−Co Pair. We start with the model of a single Fe−Co pair and assume that as in the {[Co(tmphen)2]3[Fe(CN)6]2} compound16,53 the Co ion is surrounded by six nitrogens, whereas the Fe ion is in the strong crystal field of six carbons. Because the symmetry of the nearest surrounding of the Co ion is lower than the cubic one,16 we also suppose that the cubic orbitals t2and e are split by the low symmetry crystal field and give rise to two group of orbitals φa3, φa4, φa5 and φa1,φa2 (Figure 1). The energy gap between the group of low-lying orbitals φa3, φa4, φa5 and the higher in energy orbitals φa1,φa2 will be considered well above the separation of states in each group. The six electrons of the lsCoIII ion are paired up in the orbitals φaα (α = 3−5) (Figure 1a). The unpaired electron in the ls-CoII ion may occupy either of the two orbitals φa1or φa2 (Figure 1b). For the hs-CoII ion

S ⃗ = Sa⃗ + Sb⃗

(3)

where Sa and Sb are the individual spins of the Co and Fe ions. In the state originating from configuration I the full spin S of the molecule is vanishing. In the states of configuration II the full spin acquires the values 0, 1. In the states of configuration III the full spin values are 1, 2. For the wave functions of the localized molecular states we accept the notation j(α ,β)SM ⟩

(4)

where j refers to the configuration from which the state originates, j = 1−3, and M is the projection of the full spin S. The indices α and β take on the value 0 for the unique state arising from configuration I, but for the states of configuration II α = 1, 2, β = 3−5. These α and β values correspond to the label of the d-orbital with the unpaired electron in the ls-CoII B



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Table 1. Quasidiagonal Square Block of the Electronic Hamiltonian He of the Co−Fe Pair (α = 1, 2, β = 3−5) |1(0,0)00⟩ 0

|2(1,3)00⟩ t13

t13

Δ1 + 2 j1

3

t14

0

|2(1,4)00⟩ t14

|2(1,5)00⟩ t15

|2(2,3)00⟩ t23

|2(2,4)00⟩ t24

|2(2,5)00⟩ t25

0

0

0

0

0

0

0

0

Δ1 +

3 j 21

0 3 j 21

t15

0

0

Δ1 +

0

0

t23

0

0

0

Δ1 + 2 j1

0

0

t24

0

0

0

0

Δ1 + 2 j1

0

t25

0

0

0

0

0

Δ1 + 2 j1

3

∑

⟨1(00)00|He|2(α ,β)00⟩ =

CSSM |Φaj(α)(Sama) Φbj(β)(Sbmb)| amaSbmb

ma , mb

(5)

+

here ma and mb are the projections of spins Sa and Sb, respectively, CSM SamaSbmb are the Clebsch−Gordan coefficients, Φj(α) (S m ) and Φj(β) a a a b (Sbmb) are wave functions of the Co and Fe complying with the configuration j. In the case of ls-CoIII and ls-FeII ions (j = 1) each of the wave functions Φj(α) a (Sama) and Φj(β) b (Sbmb) contains only a single sixth-order Slater determinant. For the ls- and hs-CoII ions the wave functions represent seventh order determinants, a fifth-order determinant corresponds to the wave function of the ls-Fe(III) ion. In the Appendix the procedure of obtaining the expressions for the wave functions Φ3(α) a (Sama) (α = 3−5) of the hs-Co(II), Φj(β) (1/2m ) (j = 2, 3, β = 3−5) of the ls-Fe(III) ion and b bβ cluster wave functions |j(α,β)SM⟩ is described. The energies of the states |2(α,β)SM⟩ (α = 1, 2, β = 3−5) and |3(α,β)SM⟩ (α, β = 3−5) will be read off from the energy of the state |1(00)00⟩. In the basis of localized states |j(α,β)SM⟩ the diagonal part H(1) e of the electronic Hamiltonian He matrix can be written in the form: H(1) e

⎛0 0 0 ⎞ ⎛0 0 0 ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ 0 Δ1I1 0 ⎟ − 2Sa⃗ Sb⃗ ⎜ 0 j1 I1 0 ⎟ ⎜ ⎟ ⎜0 0 j I ⎟ ⎝ 0 0 Δ2 I2 ⎠ ⎝ 2 2⎠

3

∑

1 (2⟨φbβ ||φaα⟩ 2

(4⟨φbβ φai||φaαφai⟩ − 2⟨φbβ φai||φaiφaα⟩)

i=3−5

+ 2⟨φbβ φbβ ||φaαφbβ ⟩ +

∑

(4⟨φbβ φbj||φaαφbj⟩ − 2⟨φbβ φbj||φbjφaα⟩)) = tαβ

j = 3,4

(8)

here α = 1, 2, β = 3−5, ⟨φbβ ||φ aα⟩ = ⟨φ bβ|h 0 |φ aα⟩, ⟨φbβφai||φaαφai⟩ = ⟨φbβφai|g|φaαφai⟩. The matrix elements tαβ depend on the character of the orbitals φaα and φbβ and the symmetry of the nearest surrounding of the Co and Fe ions. Thus, the electron transfer mixes 7 states |1(00)00⟩ and |2(α,β)00⟩ (α = 1, 2, β = 3−5). So the matrix of the Hamiltonian He has a single quasidiagonal square block of the seventh order containing nondiagonal matrix elements. This block corresponds to the molecular spin S = 0, and it is presented in Table 1. The remaining localized functions |2(α,β)1M⟩ and |3(α,β)SM⟩ give only diagonal matrix elements of the Hamiltonian He, and hence the energies of these states are determined with the aid of eqs 6 and 7. 2.2. Intermolecular Electron-Deformational Interaction. Although temperature does not have a notable effect on the Fe−C distances, it influences significantly the geometry of the Co ions in {[Co(tmphen)2]3[Fe(CN)6]2.16,53 The Co centers in this compound are bound to the nitrogen end of the bridging CN− ligand, which can stabilize ls-CoIII, ls-CoII, hsCoII. The three possible electronic configurations for CoII/III ions can generally be distinguished by structural data; e.g., Co− N bond distances follow the trend: ls-CoIII (∼1.9 Å) < ls-CoII (∼2 Å) < hs-CoII (∼2.1 Å). Thus, the charge-transfer-induced spin transition in the {[Co(tmphen)2]3[Fe(CN)6]2 compound is accompanied by the deformation of the nearest cobalt surrounding and, consequently, the deformation of the intercluster volume occurs as well. To demonstrate with utmost clarity the influence of these effects on the magnetic properties, we consider a model crystal consisting of N binuclear Fe−Co clusters. Each crystal cell contains one molecule. Because under charge-transfer-induced spin transitions a change in crystal symmetry is not observed, the spontaneous all-round full symmetric lattice strain is only assumed to arise from this transformation. In molecular crystals

(6)

where “0” on the matrix diagonals stands for the state |1(0,0)00⟩, I1 and I2 are the six-by-six and nine-by-nine unit matrices acting in the |2(α,β)SM⟩ (α = 1, 2, β = 3−5) and |3(α,β)SM⟩ (α = 3−5, β = 3−5) spaces, respectively. In the accepted model of electronic orbitals Δ1 and Δ2 are the energies of the states |2(α,β)SM⟩ and |3(α,β)SM⟩ with allowance only for crystal field splitting and intra- and intercenter Coulomb interactions. Finally, the second term in eq 6 gauges the exchange splitting of the localized states Ei(S) = −ji (Si(Si + 1) − Sa(Sa + 1) − Sb(S b+1)) (i = 1, 2)

3

|2(α,β)00⟩ (α = 1, 2, β = 3−5) are connected by electron transfer. The calculation of the nondiagonal elements of the Hamiltonian He is carried out using the method based on the angular momentum technics and proposed in refs 57−59. For these matrix elements we find

and ls-FeIII ions, respectively. Finally, for j = 3 α acquires the values 3−5 that point out to the singly occupied d-orbital of the lower group of orbitals in the hs-CoII ion, whereas β = 3−5. In the accepted spin-addition scheme the general form of the molecular wave function |j(α,β)SM⟩ can be presented as follows |j(α ,β)SM ⟩ =

0

(7)

where j1 and j2 are the multielectronic exchange parameters for states |2(α,β)SM⟩ and |3(α,β)SM⟩, respectively. Meanwhile, not all of the localized states represent the eigenstates of the electronic Hamiltonian He . The states |1(0,0)00⟩ and C



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⎛0 0 0 ⎞ ⎜ ⎟ 0 τ1n = ⎜ 0 I1 0 ⎟ ⎜⎜ ⎟ 0⎟ ⎝ 0 0 I2 ⎠

the intermolecular volume is far softer than that of the molecule itself. With this in mind we introduce as in refs 59−61 the internal molecular ε1 and external (intermolecular volume) ε2 strains and corresponding to these strains bulk moduli c1 and c2. Under deformations ε1 and ε2 each element of the molecular and intermolecular volumes remains similar to itself. In each cluster we consider the interaction of the Co ion with the arising ε1 strain. The operator of interaction of the Co ion with the full symmetric strain is written in the form: Hes = V ( r ⃗)ε1

⎛1 0 0 ⎞ ⎜ ⎟ 0 In = ⎜ 0 I1 0 ⎟ ⎜⎜ ⎟ 0⎟ ⎝ 0 0 I2 ⎠

(9)

Nc1Ω 0ε12 Nc 2(Ω − Ω 0)ε2 2 + 2 2

Hst =

+ ε1υ1 ∑ τ1n + ε1υ2 ∑ τ2n + Nw1ε1 n

⟨1(0,0)00|V ( r ⃗)|1(0,0)00⟩ = w1,

(α = 1, 2, β = 3−5) ⟨3(α′,β′)S′M′|V ( r ⃗)|3(α ,β)SM ⟩ = w3δαα ′δββ ′δSS ′δMM ′

ε1 = −

(10)

Nc1Ω 0ε12 Nc 2(Ω − Ω 0)ε2 2 + + ε1 ∑ σ1n 2 2 n

Hst = −B1 ∑ τ1n − B2 ∑ τ2n − n

−

J2 2N

n

∑ τ1nτ1m − n,m

J3 2N

J1 N

∑ τ1nτ2m n,m

∑ τ2nτ2m n,m

(18)

Here the first two terms arising from the interaction of the cobalt ions with the spontaneous deformation act as additional fields applied to each cobalt ion and redefine in fact the energy gaps Δ1 and Δ2 (eq 6) corresponding with the states |2(α,β) SM⟩ and |3(α,β)SM⟩, B1 = A w1υ1, B2 = Aw1υ2, J1 = Aυ1υ2, J2 = Aυ12, J3 = Aυ22. The last three terms in (18) arise from the coupling of the cobalt ions to the strain and represent an infinite-range interaction between all cobalt ions in the crystal. It is worth noting that the introduced model of the elastic continuum satisfactorily describes only the interaction of the cobalt ions with the long-wave acoustic vibrations of the lattice. Therefore, the obtained interion interaction in fact corresponds to the exchange via the field of long-wave acoustic phonons. In the cases c1 = c2 and c1 ≫ c2 the Hamiltonians accepted in the cooperative Jahn−Teller problem63 and in the model of elastic interactions64 can be, respectively, obtained with the aid of eq 18. The Hamiltonian for the whole crystal can be presented as

(11)

(12)

where the unit matrices I01 and I02 have the dimensions 24 and 72, respectively. Then we present the matrix σn1 in the following form: σ1n = υ1τ1n + υ2τ2n + w1I n

(17)

Then if the value of ε1 is substituted back into eq 15, we obtain

here Ω0 and Ω are the molecular and unit cell volumes. The first two terms in eq 11 describe the elastic energy of deformed crystal, n numbers the binuclear Co−Fe clusters in the crystal. Finally, the third term in eq 11 corresponds to the coupling of the d-electrons of the Co ion with the uniform deformation ε1. In the basis of localized molecular states |j(α,β)SM⟩ the matrix σn1 is diagonal ⎛ w1 0 0 ⎞ ⎟ ⎜ 0 σ1n = ⎜ 0 w2I1 0 ⎟ ⎟ ⎜ ⎟ ⎜ 0 w3I20 ⎠ ⎝0

A(w1N + υ1 ∑n τ1n + υ2 ∑n τ2n)

N c2 A= c1[c 2 Ω 0 + c1(Ω − Ω 0)]

where μ = ⟨φaα|V(r)⃗ |φaα⟩ (α = 1, 2) and λ = ⟨φaα|V(r)⃗ |φaα⟩ (α = 3−5) are the one-electron matrix elements of the operator V(r)⃗ in the basis of the d-orbitals of the Co ion. The interaction constants with the full-symmetric mode are approximately taken the same for the cobalt orbitals φaα with α = 1, 2 and α = 3−5, respectively, because the energy gap between the both groups of orbitals is supposed to be much greater than the splittings in each group. Within the framework of Kanamori’s approach62 the contribution of uniform strains ε1 and ε2 to the crystal Hamiltonian can be set out by the operator Hst =

(15)

Substituting eq 16 into eq 15 and bearing in mind that the strain is a macroscopic variable, we find it by minimizing Hst:

(α = 3−5, β = 3−5)

w3 = 5λ + 2μ

n

For an uniform crystal compression (or extension) the relation ε2 to ε1 is roughly supposed to be c ε2 = ε1 1 c2 (16)

⟨2(α′,β′)S′M′|V ( r ⃗)|2(α ,β)SM ⟩ = w2δαα ′δββ ′δSS ′δMM ′

w2 = 6λ + μ

(14)

Then the operator (11) takes on the form

In the basis of localized states |j(α,β)SM⟩ the matrix elements of the operator V(r)⃗ can be easily expressed in terms of oneelectron contributions:

w1 = 6λ

⎛0 0 0 ⎞ ⎜ ⎟ 0 τ2n = ⎜ 0 I1 0 ⎟ ⎜⎜ ⎟ 0⎟ ⎝ 0 0 − I2 ⎠

H=

∑ Hne + Hst n

(19)

Hne

where is the Hamiltonian for a single Fe−Co pair (eq 1). 2.3. Crystal Hamiltonian in the Molecular Field Approximation. The various states of the Fe−Co pair in the crystal are determined by Hamiltonian (19). To find these states, the mean field approximation is applied. Passing to this

(13)

where υ1 = (w2 + w3 − 2w1)/2, υ2 = (w2 − w3)/2, and the matrices τn1, τn2, and In are determined as follows: D



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approximation we substitute the quantities τn1 τm2 , τn1 τm1 and τn2 τm2 with τinτjm = τi̅τjm + τinτj̅ − τi̅τj̅

The distortions equations:

τn1

and

τm2

i , j = 1, 2

3 ε1 = −(J1 + J2 )τ1̅ − (J1 + J3)τ2̅ + Δ̃1 + j1 2

⎞ 1 ⎛⎜ 3 −(J + J2 )τ1̅ − (J1 + J3)τ2̅ + Δ̃1 + j1 ∓ U ⎟ ⎠ 2⎝ 1 2 (g2,3 = 1)

(20)

ε2,3 =

satisfy the system of self-consistent

( ( )) Tr(exp(− ))

1 ε4 = −(J1 + J2 )τ1̅ − (J1 + J3)τ2̅ + Δ̃1 − j1 (g4 = 18) 2 5 ε5 = −(J2 − J1)τ1̅ + (J3 − J1)τ2̅ + Δ̃2 + j2 (g5 = 27) 2 3 ε6 = − (J2 − J1)τ1̅ + (J3 − J1)τ2̅ + Δ̃2 − j2 (g6 = 45) 2

H̃

τi̅ =

Tr exp − kT τin

i = 1, 2

H̃ kT

(21)

H̃ is the Hamiltonian of the system in the mean field approximation; it decomposes into the sum of single cluster Hamiltonians: H̃ =

(g1 = 5)

(23)

∑ H̃ n

where

n

H̃ n = hn − J1(τ1̅ τ2n + τ2̅ τ1n) − J2 τ1̅ τ1n − J3τ2̅ τ2n

U=

(22)

(24)

where the Hamiltonian hn describes the states of an isolated Fe−Co pair and accounts for the redefined energies Δ̃1 = Δ1 − B1 − B2 and Δ̃2 = Δ2 − B1 + B2 of the localized states |2(α,β) SM⟩ and |3(α,β)SM⟩ as well as for the exchange interaction in the pairs ls-CoII−ls-FeIII, hs-CoII−ls-FeIII and the electrontransfer coupling the states |1(00)00⟩ and |2(α, β)00⟩ (α = 1, 2, β = 3 − 5) (see eqs 6 and 7 and Table 1). The eigenvalues of the Hamiltonian (22) look as follows: J2 τ1̅ + J1τ2̅ ε ε (J1 + J2 ) ∑i = 1,4 gi exp − kTi + (J2 − J1) ∑i = 5,6 gi exp − kTi + −

( )

( )

⎛ 3 ⎞⎟2 ⎜ − (J + J )τ − (J + J )τ + Δ̃ + j + 24t 2 1 2 1 ̅ ̅ 1 2 1 3 ⎝ 2 1⎠

the numbers gi given in the parentheses indicate both the spin and the orbital degeneracy of the states of the Co−Fe pair in the molecular field. While deriving the expressions (23) in a first step, we put the parameters tαβ (α = 1, 2, β = 3−5) (see eq 8 and Table 1) equal to tαβ = t. With the aid of eqs 21−23 the following system of self-consistent equations for the order parameters τ1̅ and τ2̅ is obtained

g2 2

((J + J ) − 1

2

(J1 + J2 )ε1 U

g3

) exp(− ) + ((J + J ) + ε2 kT

2

1

(J1 + J2 )ε1

2

U

) exp(− ) ε3

kT

Z

=0 J1τ1̅ + J3τ2̅ −

( ε)

( ε)

(J1 + J3) ∑i = 1,4 gi exp − kTi − (J3 − J1) ∑i = 5,6 gi exp − kTi +

g2 2

((J + J ) − 1

3

(J1 + J3)ε1 U

) exp(−

ε2 kT

)+

g3 2

((J + J ) + 1

(J1 + J3)ε1

3

U

) exp(−

ε3 kT

)

Z

=0

(25)

connected with the equilibrium positions of the nuclei in the lsCoII and hs-CoII states by the relations

where 6

Z=

v2 − v1 =

∑ gi exp(−εi /kT )

6f ℏω ΔR1

v3 − v1 =

6f ℏω ΔR 2 (26)

i=1

here ν1, ν2, and ν3 are the constants characterizing the coupling of the ls-CoIII, ls-CoII, and hs-CoII ion with the full symmetric vibration of the ligand surrounding, respectively, f is the mean force constant for this vibration. The corresponding constants w2−w1 and w3−w1 of interaction of the Co ion with the full symmetric strain ε1 (eqs 10) can be expressed through the parameters v2−v1 and v3−v1 as follows:59

is the partition function. The temperature behavior of the order parameters τ1̅ and τ2̅ is determined by the competition of intraand intercluster interactions. To elucidate the conditions that favor the charge-transfer-induced spin transition, first we evaluate and discuss the values of the above-mentioned parameters. 2.4. Values of the Characteristic Parameters. We start with the estimation of the parameters J 1 , J 2 , and J 3 characterizing the electron-deformational interaction. Under the charge-transfer-induced spin transition the Co ion changes its state from ls−CoIII to ls−CoII or to hs−CoII. These transitions are accompanied by the elongations ΔR1 = 0.1 Å and ΔR2 = 0.2 Å of the average Co−N bond lengths,16 respectively. For a pseudooctahedral surrounding of the Co ion (CoN6) the bond changes ΔR1 and ΔR2 are approximately

w2 − w1 = (ν2 − ν1)

2f R 0 = 2 3 fR 0ΔR1 ℏω

w3 − w1 = (ν3 − ν1)

2f R 0 = 2 3 fR 0ΔR 2 ℏω

(27)

where R0 is the mean value of the ligand radia for the CoN6 complex. Correspondingly, the parameters υ1 = (w2 + w3 − 2w1)/2 and υ2 =(w2 − w3)/2 are E


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μ=

3 fR 0(ΔR1 + ΔR 2) 3 fR 0(ΔR1 − ΔR 2)

υ2 =

Figure 2. Thermal variation of the magnetic moment for different sets of parameters: (1) t = 300 cm−1, Δ1 = 300 cm−1, Δ2 = 600 cm−1; (2) t = 50 cm−1, Δ1 = 300 cm−1, Δ2 = 600 cm−1; (3) t = 300 cm−1, Δ1 = 600 cm−1, Δ2 = 300 cm−1; (4) t = 50 cm−1, Δ1 = 600 cm−1, Δ2 = 300 cm−1.

3. RESULTS AND DISCUSSION 3.1. Temperature Dependence of the Magnetic Moment. From the crystallographic data presented in ref 16 it follows that the symmetries of the ligand surroundings of the Co and Fe ions in the {[Co(tmphen) 2]3 [Fe(CN)6 ]2 } compound can be described by the point groups C2v and Cs, respectively. The low symmetry crystal fields acting on the FeIII and CoII ions split the ground cubic terms 2T2 and 4T1 of these ions into three well-separated orbital singlets. Thus, the orbital moments of the FeIII and CoII ions are quenched by the crystal field. Therefore, the ions FeIII and CoII can be considered as pure spin ones. At the same time the g-factors for the ground states of FeIII and CoII differ from the spin only value due to the mixing of these states and the orbitally nondegenerate excited states by spin−orbital interaction. This leads in its turn to the difference between the spin-only χT value of 6.375 emu K mol−1 and the observed room temperature value of 8.3 emu K mol−1 for the pentanuclear {[Co(tmphen)2]3[Fe(CN)6]2} cluster. Bearing in mind the above said, we examine the magnetic properties of the crystal consisting of Fe−Co pairs in the spin model. For this crystal the energy levels in the mean field approximation are used for calculation of the magnetic moment with the aid of the Van Vleck formula: 2

μ = gS μ B

2

( ε) ε ∑i gi exp(− kT )

(30)

To understand the scenario of the CTIST, we start with the inspection of the influence of the characteristic parameters of the system on its magnetic properties. With this aim we consider four qualitatively different situations: (i) Δ1 < Δ2, t ∼ Δ1; (ii) Δ1 < Δ2, t ≪Δ1; (iii) Δ2 < Δ1, t ∼ Δ2; (iv) Δ2 < Δ1, t ≪ Δ2. Such situations can be realized, for instance, when 4 of 6 nitrogen atoms surrounding the cobalt ion belong to ligands other than the tmphen ligand. Because the observed CTIST16,53 takes place in the temperature range 4.2−293 K, we assume that the energy gaps Δ1 and Δ2 fall inside the range 0−600 cm−1. In the case when the localized states arising from configuration II are lower in energy than those arising from configuration III and the transfer parameter is of the order of the gap Δ1 the spin conversion goes very slow with temperature rise. Actually up to the temperature 125 K the effective magnetic moment is negligible (Figure 2, curve 1). The further

(28)

Then, we evaluate the molecular field parameters J1 = Aυ1υ2, J2 = Aυ12, and J3 = Aυ22 characterizing the electron-deformational interaction. In estimations we take into account that the main change in the volume in spin-crossover compounds falls on the intermolecular space, i.e., c1 ≫ c2. For typical values c2 = 1011 dyn/cm2, c2 ≈ 0.1c1, R0 = 2 Å, f = 105 dyn/cm, Ω ≈ 1130 Å3 (volume per a Co−Fe pair16), Ω0 = 8 Å3, ΔR1 = 0.1 Å, and ΔR2 = 0.2 Å16 the parameters J1, J2, and J3 take on the values: J1 = −17.4 cm−1, J2 = 50 cm−1, J3 = 6.1 cm−1. The parameter J2 significantly exceeds in magnitude the parameters J1 and J3. Therefore, further on we will examine ferro-type ordering of the distortions. In the ls-CoII−ls-FeIII and hs-CoII−ls-FeIII pairs the main contribution to the Heisenberg exchange coupling comes from the interaction of a half-filled orbital on one of the ions with an empty orbital on another ion or/and from the interaction of a half-filled orbital with a full orbital. Therefore, according to Goodenough−Kanamori rules65 the parameters j1 and j2 are assumed to be positive (ferromagnetic exchange). Taking into account that the typical values for the parameters of superexchange mediated by the cyanide bridges are of the order of several wavenumbers66 in the subsequent calculations we shall take for simplicity j1 = j2 = 5 cm−1. As for spincrossover CoII complexes, the high-spin state lies within a range 300−2000 cm−1 above the low-spin state,67 the gap Δ1 will be varied so that to provide the energy difference Δ2 − Δ1 falling in this range. The parameter t values will be also changed to clarify the effect of overlap of the orbitals of the cobalt and iron ions.

2

8χT

increase in temperature leads to the rise of the magnetic moment. However, the room temperature value of the magnetic moment is about 2 μB thus testifying incomplete spin conversion. For the same order of localized levels and transfer parameter value small as compared with the gap Δ1 (Figure 2, curve 2) the magnetic moment values noticeably exceed those presented by curve 1 starting from temperature 37 K. When the localized states arising from configuration III (eq 2) are lower in energy than those from configuration II, while the transfer parameter t ∼ Δ2, the magnetic moment is vanishing up to temperature 75 K, then it starts increasing. The corresponding curve in the temperature range 75− 250 K lies in between curves 1 and 2. At temperatures higher than 250 K the magnetic moment values corresponding with curve 3 exceed those presented in curve 2. For the fourth set of parameters the magnetic moment abruptly increases beginning from temperature 25 K, at T = 150 K it attains its maximum value 4.2 μB. The obtained results can be explained analyzing the energy level schemes complying with the magnetic moment curves (Figure 3). When the transfer parameter is compared with the gap Δ1 and exceeds significantly the parameters of cooperative electron-deformational interaction, the energies of the cluster levels in the mean field approximation do not depend on temperature (Figure 3a). Level 2 (the ground one) and level 4 (the first excited one) belong to spin values S = 0 and S = 1, respectively. The energy gap between levels 2 and 4 is mainly

∑i giSi(Si + 1) exp − kTi i

(29)

The temperature dependence of the magnetic susceptibility can be easily obtained from the relation F



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Figure 3. Energy levels of the Co−Fe cluster in the mean field approximation for the same sets of parameters as in Figure 2.

determined by the transfer parameter value t = 300 cm−1, whereas the energies of levels 1 and 4 are approximately equal due to the small values of the exchange parameter j1 in cyanide bridged systems. Because the degeneracy of level 4, g4 = 18, the contribution of this level with spin S = 1 to the magnetic moment becomes noticeable in spite of the large gap between levels 2 and 4. At the same time the maximum magnetic moment value achieved at 300 K is 2 μB. For Δ1 < Δ2 the decrease in the transfer parameter leads to a slight temperature dependence of the level energies, the energy of level 4 (Figure 3b) lowers that leads to higher magnetic moment values as compared with those given by curve 1. When the localized states arising from configuration III are lower in energy than those of configuration II and t ∼ Δ2 level 6 is the first excited one (Figure 3c), the energy gap between this level and the ground one is about 800 cm−1 and up to T = 250 K curve 3 in Figure 2 lies between curves 1 and 2. An abrupt increase of the magnetic moment takes place for the case when the transfer parameter is small compared with the gap Δ2; the energies ε6 and ε5 of the first and second excited levels corresponding to S = 2 and 1 fall with temperature increase. Thus, for Δ2 < Δ1 and t ≪ Δ2, the highest values of the magnetic moment can be achieved. 3.2. Temperature Behavior of Mössbauer Spectra. Now we address the question of the possible types of Mössbauer spectra. For the iron ion in the dimeric cluster the operator of quadrupole interaction between the electronic shell and nucleus can be written as68

e 2Q n

V (r) =

4I(2I − 1)

2 Q e[3Iẑ − I(I + 1)]

(31)

where Qn is the nuclear quadrupole moment, I is the nuclear spin, Iẑ is the operator of the z-component of the nuclear spin, Qe =

∑ (3zi 2 − ri 2)ri−5 i

is the operator of the gradient of the electric field acting on the nucleus. We introduce also the isomer shift operator δ̂ =

2 πzne 2 ∑ δ( ri ⃗)(R exc 2 − R gr 2) 5 i

(32)

here δ(r1⃗ ) is the δ function, Rexc and Rgr are the nucleus radii for the excited and ground states, respectively. The mean values of operators (31) and (32) provide the quadrupole splitting and the isomer shift value of the Mössbauer spectra of the iron ion in the ith state of the Fe−Co cluster. For eigenstates 1 and 4−6 of the cluster Hamiltonian (22) these mean values coincide with those for the FeIII ion: δ i = δ(Fe III), ΔEiQ =

1 2 e Q nQ (Fe III) 2

(i = 1, 4−6) (33)

For the tunnel states (i = 2, 3) the values δ and are calculated with the aid of corresponding wave functions obtained by diagonalization of the matrix given in Table 1 and δ(FeII) = 0.02 mm/s, ΔEQ(FeII) = 0.2 mm/s, δ(FeIII) = −0.02 mm/s, and ΔEQ(FeIII) = 0.98 mm/s.16 The total i

G

ΔEQi



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Mössbauer spectrum was obtained by summing the spectra yielded by different cluster states in the molecular field taking into account their equilibrium populations. The form-function of the Mössbauer spectrum is determined by a superposition of the Lorentz curves: F(Ω) =

1 Z

⎛

∑ exp⎜⎝− i

Ei ⎞ Γ ⎟ kT ⎠ Ω − δ ± 1 ΔE Q i i 2

(

2

)

+ Γ2

gi (34)

where gi is the degeneracy of the ith level and Z is the partition function. In calculations for the half-width of the line the value Γ = 0.14 mm/s was taken. In Figures 4−7 below we show

Figure 5. Temperature dependence of the Mössbauer spectrum in the case of t = 50 cm−1, Δ1 = 300 cm−1, Δ2 = 600 cm−1. Relative contributions from the levels in the mean field approximation: (a) T = 50 K, 2: 99.9%; 1,4: 0.1%; (b) T = 77 K, 1,4: 3.87%; 5,6: 0.05%; 2: 96.02%; 3: 0.06%. (c) T = 150 K, 1,4: 44.5%; 5,6: 9%; 2: 45.3%; 3: 1.2%; (d) T = 300 K, 1,4: 48.1%; 5,6: 40.8%; 2: 9.5%; 3: 1.6%.

Figure 4. Temperature dependence of the Mössbauer spectrum in the case of t = 300 cm−1, Δ1 = 300 cm−1, Δ2 = 600 cm−1. Relative contributions from the levels in the mean field approximation: (a) T = 77 K, 2: 100%; (b) T = 200 K, 1,4: 3.6%; 5,6: 1.4%; 2: 95%; (c) T = 250 K, 1,4: 11.3%; 5,6: 6.7%; 2: 82%; (d) T = 300 K, 1,4: 20.6%; 5,6: 16.6%; 2: 62.8%.

various examples of the temperature dependence of the Mössbauer spectra. Along with the key parameters in the captions for Figures 4−7 we also give the percentage contribution to the total intensity (area) of the overall spectrum from the states 1,4; 5,6; 2 and 3. The temperature spectrum transformations in the case of Δ1 = 300 cm−1, Δ2 = 600 cm−1, and t = 300 cm−1 are shown in Figure 4. The Mössbauer spectra parameters for states 2 and 3 remain practically unchanged with temperature, their numerical values δ2 = 0.004 mm/s, δ3 = −0.004 mm/s, ΔEQ2 = 0.51 mm/s, and ΔEQ3 = 0.67 mm/s differ from those for states 1 and 4−6 because states 2 and 3 represent a superposition of states of FeIII and FeII. At 77 K the spectrum consists of one doublet with the parameters equal in fact to δ2 and ΔEQ2 . Only at 250 K does the contribution from levels 1 and 4−6 become noticeable and make up about 18%. The total spectrum broadens insignificantly. However, at 300 K the intensity of the spectra arising from levels 1 and 4−6 increases, and in the total spectra 2 doublets can be distinguished. At 300 K the contribution of

Figure 6. Temperature dependence of the Mössbauer spectrum in the case of t = 300 cm−1, Δ1 = 600 cm−1, Δ2 = 300 cm−1. Relative contributions from the levels in the mean field approximation: (a) T = 77 K, 2: 100%; (b) T = 200 K, 1,4: 0.7%; 5,6: 22.4%; 2: 76.9%; (c) T = 250 K, 1,4: 2.4%; 5,6: 47.8%; 2: 49.8%; (d) T = 300 K, 1,4: 4.2%; 5,6: 64.6%; 2: 31.2%.

states 1 and 4−6 amounts to 37%. For the same order of localized levels (Δ1 = 300 cm−1, Δ2 = 600 cm−1) and transfer parameter t = 50 cm−1 ΔEQ2 and δ2 approximately keep the H



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ΔEQ(FeII). With a temperature increase the population of states with higher energies occurs, and at T = 150 K the contribution of FeIII ions to the total spectrum amounts to 86.9%. Finally, at room temperature the spectrum represents a doublet arising from FeIII with the area of 95.7%. Thus, the charge-transferinduced spin transition in this case may be considered as an almost complete one. The calculations also apparently show that, when Δ1 > Δ2 and t ≪ Δ2, the increase of Δ1 leads to the full conversion of ls-FeII ions into ls-FeIII ions. The simulation of the Mössbauer spectra clearly demonstrates that a complete charge-transfer-induced spin transition may take place under following conditions: (i) small electrontransfer parameter as compared with the energy gaps Δ1 and Δ2 between the localized states arising from configurations I, II and I, III, respectively; (ii) the states of configuration II (ls-CoII, lsFeIII) are higher in energy than those of configuration III (hsCoII, ls-FeIII). It is worth noting that the experimental data on Mössbauer spectra of the red crystalline phase16 of the {[Co(tmphen)2]3[Fe(CN)6]2 cluster compound speak in favor of this scenario: actually all ls-FeII ions present in the crystal at 4.2 K transform into ls-FeIII ions at high temperatures. The experimental Mössbauer spectra do not evidence the presence of cluster states representing a linear combination of states of the FeIII and FeII ions. It is worth noting that for the same set of parameters the calculated magnetic susceptibility (Figure 2, curve 4) shows an abrupt increase. Thus, the suggested model captures the essential features of the observed phenomenon.

Figure 7. Temperature dependence of the Mössbauer spectrum in the case of t = 50 cm−1, Δ1 = 600 cm−1, Δ2 = 300 cm−1. Relative contributions from the levels in the mean field approximation: (a) T = 50 K, 5,6: 0.7%; 2: 99.3%; (b) T = 77 K, 5,6: 19.3%; 2: 80.7%; (c) T = 150 K, 1,4: 1.1%; 5,6: 85.8%; 2: 13.1%; (d) T = 300 K, 1,4: 5.7%; 5,6: 90.1%; 2: 4%; 3: 0.2%.

■

CONCLUDING REMARKS In summary, we have analyzed the conditions favorable for charge-transfer-induced spin transitions in model crystals containing as a structural element cyanide bridged Co−Fe dimeric clusters. The following relevant electronic processes and interactions were taken into account: (i) intracluster metal−metal electron transfer; (ii) intramolecular magnetic exchange interactions; (iii) intermolecular electron-deformational interactions. It was shown that depending on the ratio between the transfer parameter and energy gaps between the ground configuration presented by diamagnetic ls-CoIII and lsFeII ions and excited configurations containing paramagnetic ions ls-FeIII and ls-CoII or hs-CoII the charge-transfer-induced spin transition can be a gradual or an abrupt one. The cooperative electron-deformational interaction noticeably affects the spin transformation in the case when the parameters of this interaction are of the same order of magnitude with the parameter t characterizing the electron transfer from the iron to the cobalt ion. The intracluster exchange interaction in the cyanide-bridged ls-FeIII−ls-CoII and ls-FeIII−hs-CoII pairs does not influence the spin transition because it is of the order of several wavenumbers. The energy diagram that gives the possibility to explain qualitatively the observed Mössbauer spectra transformations16 corresponds to the case of the transfer parameter small as compared with the energy gaps between the localized states of configurations I, II, III under the condition that the states of configuration III are lower in energy than those of configuration II. At the same time to discuss the real system consisting of pentamers Fe2Co316 in more detail all electronic states arising from configurations (CoIII)2CoII(FeII)2, (CoII)2CoIII(FeII)(FeIII), and (CoII)3(FeIII)2 referred in ref 16 as configurations 1A, 1B, and 1C should be included in the model. Because the transfer parameter t is proportional to the overlap integral S

values 0.3 mm/s and 0.015 mm/s, respectively, in the range 50−300 K (Figure 5). These values differ from ΔEQ(FeII) and δ(FeII), thus testifying that the ground cluster state represents a linear combination of states arising from configurations I and II with major contribution coming from configuration I. At the same time the spectrum at temperature 50 K represents a single doublet. Already at 150 K the contribution of states 1 and 4−6 considerably increases, which is seen from Figure 5. There obviously appears a spectrum consisting of two doublets with the intensity ratio 1.2:1. Finally, at room temperature the spectrum mainly arises from excited states 1 and 4−6, the partial contribution from these states to the total spectrum intensity making up 88.9%. Consequently, the contribution of state 2 significantly diminishes. Qualitatively, another picture (Figure 6) arises when the order of localized states originating from configurations II and III is reversed (Δ1 > Δ2). For the transfer parameter comparable with the gap Δ2 the low temperature spectrum contains one Mössbauer doublet with averaged parameters: δ2 = 0.01 mm/s, ΔEQ2 = 0.44 mm/s. Already at 200 K the spectrum contains two doublets arising from states 2 and 5, 6, respectively. The contribution of states 5, 6 becomes noticeable due to the their high degeneracy. At 200 K the intensity of the spectrum component complying with these states makes up 22.4%, and along with the central lines associated with state 2 the spectrum contains two weakly distinct shoulders. With temperature increase the partial spectrum arising from states 5 and 6 increases in intensity up to 64.6%. Nevertheless, four lines in the spectrum are well seen. Finally, Figure 7 shows the spectra transformations for the same order of localized states and transfer parameter small as compared with the gaps Δ1 and Δ2. In this case the low temperature spectrum contains one doublet with quadrupole splitting and isomer shift differing negligibly from δ(FeII) and I



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required. We are going to address these problems in the near future. Meanwhile, it should be stressed that in spite of several restricting assumptions made in the consideration above carried out, the suggested model for a crystal containing dimeric Fe− Co clusters reflects the main qualitative features of the observed phenomenon.

between the orbitals of the cobalt and iron ion (see eq 8), the probabilities of electron transfer between the states of configurations 1A and 1B, 1B and 1C, respectively, will be approximately of the order of S2. At the same time, the probability of electron transfer between the states of configurations 1A and 1C is of the order of S4 and small as compared with those for the transfers 1A → 1B, 1B → 1C. Thus, the general model for the crystal consisting of pentanuclear clusters should take into account the most probable electron transfers 1A → 1B, 1B → 1C as well as all spin states of the pentanuclear cluster. DFT calculations of the transfer and exchange parameters as well as of the energy gaps between the states of configurations 1A, 1B and 1C are also ⎛3 ⎞ Φa3(3)⎜ ma⎟ = ⎝2 ⎠

∑ ma1,ma2, ma3,ma4

■

APPENDIX

The hs-CoII ion possesses three unpaired electrons and, therefore, the wave functions Φ3(α) a (Sama) (α = 3−5) are built from Slater determinants in two steps using the successive coupling of electronic spins to give the total spin Sα = 3/2:

⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ (3/2)ma 1m ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C(1/2) ma3(1/2)ma2C1m(1/2)ma1 φa5⎝ ⎠ φa5⎝ − ⎠ φa 4 ⎝ ⎠ φa 4 ⎝ − ⎠ φa3(ma3) φa2(ma2 ) φa1(ma1) 2 2 2 2

(A1)

Φ3(α) a (3/2ma) can be obtained from eq A1 by means of the substitutions a5 ↔ a3 and a4 ↔ a3, respectively. The wave III function Φj(3) b (Sbmb) (j = 2, 3) of the ls-Fe ion can be written as

where |...| stands for the Slater determinant, φaα(maα) = φα(r ⃗ − R⃗ a)χ(1/2,maα), R⃗ a is the position vector of the Co ion, χ(1/ 2,maα) is the monoelectronic spin wave function, maα is the electron spin projection. For α = 4, 5 the wave functions

⎛1 ⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ Φbj(3)⎜ mb3⎟ = φb5⎜ ⎟ φb5⎜ − ⎟ φb4⎜ ⎟ φb4⎜ − ⎟ φb3(mb3) ⎝2 ⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠

where φbβ(mbβ) = φb(r ⃗ − R⃗ b) χ(1/2, mbβ), Rb is the position vector of the iron ion. From eq A2 the wave functions Φj(β) b (1/ 2mbβ) with β = 4, 5 can be easily derived by means of the

(A2)

substitutions b5 ↔ b3 and b4 ↔ b3. With the aid of eq 5 one can obtain the wave functions of the Fe−Co cluster with localized electrons in the following form:

⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ |1(0,0)00⟩ = φa5⎜ ⎟ φa5⎜ − ⎟ φa4⎜ ⎟ φa4⎜ − ⎟ φa3⎜ ⎟ φa3⎜− ⎟ φb5⎜ ⎟ φb5⎜− ⎟ φb4⎜ ⎟ φb4⎜ − ⎟ φb3⎜ ⎟ φb3⎜ − ⎟ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ (A3)

|2(1,3)SM ⟩ =

SM ∑ C1/2 m

a11/2mb3

allm

⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ φa5⎜ ⎟ φa5⎜ − ⎟ φa4⎜ ⎟ φa4⎜ − ⎟ φa3⎜ ⎟ φa3⎜ − ⎟ φa1(ma1) φb5⎜ ⎟ φb5⎜ − ⎟ φb4⎜ ⎟ φb4⎜ − ⎟ φb3(mb3) ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ (A4)

|3(3,3)SM ⟩ =

SM ∑ C1/2 m

a11/2mb3

allm

⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ ⎛1⎞ ⎛ 1⎞ φa5⎜ ⎟ φa5⎜ − ⎟ φa4⎜ ⎟ φa4⎜ − ⎟ φa3(ma3) φa2(ma2) φa1(ma1) φb5⎜ ⎟ φb5⎜ − ⎟ φb4⎜ ⎟ φb4⎜ − ⎟ φb3(mb3) ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ ⎝2⎠ ⎝ 2⎠ (A5)

3−5), |3(3,β)SM⟩ (β = 4, 5), and |3(α,3)SM⟩ (α = 4, 5),

■

|3(α,β)SM⟩ (α = 4, 5, β = 4, 5) can be obtained from (A4) and

Notes

The wave functions |2(1,β)SM⟩ (β = 4, 5), |2(2,β)SM⟩ (β =

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. The authors declare no competing financial interest.

(A5), respectively, by means of the following substitutions: (i)

■

b3 → b4 or b3 → b5; (ii) a1 → a2, b3 → b4, or b3 → b5; (iii) b3 → b4 or b3 → b5; (iv) a3 → a4 or a3 → a5; (v) a3 → a4, b3 → b4, or

REFERENCES

(1) Ohba, M.; Okawa, H.; Fukita, N.; Hashimoto, Y. J. Am. Chem. Soc. 1997, 119, 1011−1019.

b3 → b5; a3 → a5, b3 → b4, or b3 → b5. J



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