Vibronic Model for Cooperative Spin-Crossover in Pentanuclear

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Vibronic Model for Cooperative Spin-Crossover in Pentanuclear {[MIII(CN)6]2[M0 II(tmphen)2]3} (M/M0 = Co/Fe, Fe/Fe) Compounds Sophia Klokishner,*,† Serghei Ostrovsky,† Andrei Palii,† Michael Shatruk,‡ Kristen Funck,§ Kim R. Dunbar,§ and Boris Tsukerblat|| †

Institute of Applied Physics, Academy of Sciences of Moldova, Kishinev, Moldova Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida, United States § Department of Chemistry, Texas A&M University, College Station, Texas, United States Chemistry Department, Ben-Gurion University of the Negev, Beer-Sheva, Israel

)

‡

ABSTRACT:

In this paper we present a microscopic model for the cooperative spin-crossover (SCO) phenomenon in crystals containing cyanobridged pentanuclear clusters {[MIII(CN)6]2[M0 II(tmphen)2]3} (M/M0 = Co/Fe, 1; Fe/Fe, 2) with a trigonal bipyramidal (TBP) structure. The low-spin to high-spin (lshs) transition is considered as a cooperative phenomenon that is driven by the interaction of electronic shells of FeII ions with totally symmetric deformation of the local environment that is extended over the crystal lattice via the acoustic phonon field. Due to proximity of FeII ions in the cluster, the short-range intracluster interaction between these ions via the optical phonon field is included as well. The model takes into consideration also the spinorbit coupling operating within the cubic 5T2(t42e2) term of the hs-FeII ions. For cluster 2 the Heisenberg-type exchange interaction between the hs-FeII and ls-FeIII ions is taken into account. The competition between short- and long-range interactions is shown to determine the type and temperature of the SCO transition in cluster systems. The proposed model explains, in a unique way, the temperature dependence of the static magnetic susceptibility and the M€ossbauer spectra of the title compounds. The approach described in this paper represents a theoretical tool for the study of spin-crossover systems based on metal clusters.

1. INTRODUCTION Cyanide-bridged homo- and heteronuclear compounds have been shown to exhibit a remarkable diversity of structural, magnetic, electrochemical, and magneto-optical properties. In recent decades, the interest in studying cyanide-based compounds has been renewed due to the discovery of high TC cyanide-based magnets.13 This recognition has inspired researchers to explore polynuclear cyanide clusters with the goal of constructing finite systems that demonstrate such fascinating phenomena as single-molecule magnet (SMM) behavior411 and temperature-induced,12 light-induced, and charge transferinduced13,14 spin transitions. The pursuit of cyanide complexes with unique magnetic, thermal, and photophysical properties is a compelling topic because these complexes have potential applications in quantum computing and high-density data storage and as sensors and switches in molecular electronic devices. At the same time, polynuclear cyanide clusters offer an opportunity for r 2011 American Chemical Society

studying the factors that control slow paramagnetic relaxation of magnetization and spin conversion between close energy levels. Finally, these clusters represent an important bridge between mononuclear complexes and extended solids and, as such, are helpful for understanding the origin of cooperativity in spin transitions in solids. Efforts in this vein have led to the investigation of a series of discrete cyano-bridged pentanuclear transition metal clusters of general formula {[MIII(CN)6]2[M0 II(tmphen)2]3}1214(tmphen = 3,4,7,8-tetramethyl-1,10-phenanthroline) that consist of a trigonal bipyramidal (TBP) core and exhibit diverse and interesting properties. The main attractive feature of this extensive family of cyanide clusters is the possibility to Received: July 26, 2011 Revised: September 23, 2011 Published: September 26, 2011 21666

dx.doi.org/10.1021/jp207111r | J. Phys. Chem. C 2011, 115, 21666–21677

The Journal of Physical Chemistry C systematically alter physical and magnetic characteristics while maintaining identical geometry and ligand composition by varying only the metal ions. Thus the {[MnIII(CN)6]2[MnII(tmphen)2]3} cluster,9 the first representative of this family, demonstrates SMM properties. More recently the TBP compound {[Co(tmphen)2]3[Fe(CN)6]2} has been found to exhibit a temperature-driven charge-transfer-induced spin transition13,14 (CTIST) [(ls-Fe II )2 (ls-Co III )2 (hs-Co II )] T [(ls-Fe III )2 (hs-CoII)3] (where ls = low-spin and hs = high-spin) accompanied by a significant increase of the effective magnetic moment of the cluster. Another avenue in the domain of cyanide-bridged TBP clusters is their spin-crossover (SCO) behavior. As described in ref 12, the FeII ions were incorporated into the equatorial metal positions of TBP clusters with the expectation that the octahedral nitrogen environment of the FeII ions would facilitate SCO behavior. Indeed, it was found that a thermally induced spin transition from the high-spin to the low-spin state of FeII in {[MIII(CN)6]2[M0 II(tmphen)2]3} (M/M0 = Co/Fe, 1; Fe/ Fe, 2) compounds occurs, as evidenced by a combination of M€ossbauer spectroscopy, magnetic data, and single-crystal X-ray studies. In this paper we will focus on these two compounds in order to advance the understanding of underlying mechanisms of the SCO phenomenon in such clusters. The SCO phenomenon was discovered more than 70 years ago15 and refers to the transition between high- and low-spin states for certain metal ions with d4d7 electronic configurations in an octahedral ligand environment. SCO-related phenomena can occur if the cubic crystal field parameter is near the critical value at which the energies of low- and high-spin configurations cross in the TanabeSugano diagrams.16 In many cases, the properties of SCO compounds cannot be understood solely on the basis of static crystal field theory, that is, by assuming a fixed gap (a definite value of the cubic crystal field) between the lowand high-spin levels. The type of SCO transition in solids depends on the interplay between the strength of inter-ion interactions and the energy gap between the low- and high-spin states. SCO compounds have been the subject of many experimental and theoretical studies.1722 The most pronounced effects in spectroscopic and thermodynamic properties have been identified for FeII complexes, in which the ls-FeII(t62) to hs-FeII(t42e2) transformation is accompanied by a spin change from S = 0 to S = 2 (Figure 1). The underlying physics of spin transitions mainly relates to the large, totally symmetric relaxation of ligands accompanying promotion of the two electrons from nonbonding t2 to antibonding e orbitals in each FeII ion. The local distortions in a crystal are linked by the lattice deformations, which produce effective interactions between electronic shells of the metal ions via the phonon field, thus resulting in a cooperative spin transition. For the description of spin-crossover iron(II) compounds, the semiempirical thermodynamic approach,2325 the Wajnflasz and

Figure 1. Occupation of t2g and eg orbitals in the ls-1A1g and hs-5T2g states of FeII ions.

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Pick (WP) model and its extensions, called “Ising-like models”, are very often applied.2633 Kambara34 and then Sasaki and Kambara35,36 suggested a cooperative microscopic model in which the spin conversion was shown to be induced by intermolecular spin coupling mediated by the lattice vibrational mode and lattice strain. Temperature dependence of the high-spin fraction based on this model reproduced the essential features of the observed thermally induced one- and two-step spin conversion phenomena. An important feature of the molecular SCO compounds is the presence of two types of vibrations that play physically different roles, namely, the high-frequency molecular vibrations and a band of intermolecular vibrations. The former vibrations are directly coupled to the electronic shells, while the low-frequency vibrations with significant dispersion lead to interactions between the electronic shells, thus giving rise to cooperativity. This peculiarity of SCO systems was taken into account in the framework of the microscopic model developed in refs 3739, in which the medium between molecules was assumed to be “softer” than that within the molecules. This model goes beyond the semiempirical thermodynamic2325 and WP approach26 because it provides the possibility to account for all relevant interactions (internal and external) and write them down in the genuine basis of electronic states. In all the aforementioned theoretical papers, however, the subjects of study were crystals in which there is one iron ion exhibiting SCO, coupled with other ions in the crystal through long-range, electron-deformational interaction. There have been far fewer experimental reports on the SCO phenomenon in cluster compounds than in mononuclear complexes.4044 The case of SCO in cluster systems considered in refs 4044 and in this paper is more complicated because, in these systems, the role of intra- and intercluster interactions in the spin transformation should be elucidated. Apart from the exchange interaction between the cluster ions in states with nonvanishing spins, short range interactions of other origin (direct interaction of electric quadrupole moments, interaction via the field of optical phonons) may affect the spin transition along with long-range interactions. In the [FeII(tmphen)2]3[CoIII(CN)6]2 compound (FeII3CoIII2),12 the CoIII ions in the carbon environment are diamagnetic, and there is no exchange interaction between ls-CoIII and hs-FeII ions or between hs-FeII ions in equatorial positions. For this reason the FeII3CoIII2 crystal represents an ideal system for studying intra- and intercluster interactions governing SCO. Thus, the solution to the problem of spin transitions in the FeII3CoIII2 compound involves, first, understanding the role of short- and long-range interactions in this phenomenon. The cluster [FeII(tmphen)2]3[FeIII(CN)6]212 shows a different picture. In this cluster, hs-FeII and ls-FeIII ions are coupled by an exchange interaction. In spite of the fact that the exchange interaction of hs-FeII and ls-FeIII ions through the cyanide bridge is sufficiently weak compared to those observed in oxide clusters, it is interesting to probe whether this interaction will affect the spin transformation. Another peculiarity of compound 2 is the presence of orbitally degenerate ls-FeIII ions in the strong crystal field of C-bound cyanides and FeII ions in the nearly axial nitrogen environment, which imparts an unquenched orbital angular momentum of these ions in the high-spin state. The effects of orbital degeneracy on the spin-crossover transformation in the [FeII(tmphen)2]3[FeIII(CN)6]2 crystal will be examined in this paper as well. In this context our main aim is to develop a microscopic approach to the problem of SCO in crystals containing clusters as a structural element and to reveal 21667

dx.doi.org/10.1021/jp207111r |J. Phys. Chem. C 2011, 115, 21666–21677

The Journal of Physical Chemistry C the mechanisms underlying the SCO phenomenon in crystals composed of cyanide clusters {[MIII(CN)6]2[M0 II(tmphen)2]3} (M/M0 = Co/Fe, 1; Fe/Fe, 2). Within the developed theoretical approach, we will explain experimental data on the magnetic susceptibility and M€ossbauer spectra of compounds 1 and 2.

2. MODEL In cyano-bridged TBP clusters 1 and 2,12 three FeII ions occupy equatorial sites of the TBP structure with two [M0 (CN)6]3 anions located in the apical positions (Figure 2a). Three cyanide ligands of each hexacyanometalate unit act as bridges, whereas the other three point away from the cluster. The three equatorial FeII metal ions are in a pseudo-octahedral coordination environment with two tmphen molecules acting as bidentate ligands and two CN ligands bridging the Fe and axial M 0 ions. The tmphen ligands are involved in intra- and intermolecular ππ interactions, the consequence of which is

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formation of dimers of TBP clusters in the solid state (Figure 2b), which then also form ππ contacts between the dimers (Figure 2c). Not surprisingly, the octahedral nitrogen environment of the FeII ion resulted in the observation of SCO behavior for compounds 1 and 2. The occurrence of lshs transitions was established by a combination of M€ossbauer spectroscopy, magnetic measurements, and single-crystal X-ray diffraction.12 For both clusters, the χT product increases by ∼9 emu 3 K/mol between 150 and 375 K, thus indicating lshs transitions at FeII sites. As was mentioned earlier, there is no exchange interaction in cluster 1 for the pairs ls-CoIIIls-FeII and ls-CoIIIhs-FeII or between hs-FeII ions in the equatorial positions. On the contrary, in compound 2 the hs-FeII and ls-FeIII ions are coupled by exchange interactions mediated by the cyanide bridges. At the same time, ls-FeIII ions in the strong crystal field of C-bound cyanides and hs-FeII ions in the nearly axial nitrogen environment possess unquenched orbital angular momenta.

Figure 2. (a) Molecular structure of pentanuclear clusters {[MIII(CN)6]2[M0 II(tmphen)2]3} (M/M0 = Co/Fe, 1; Fe/Fe, 2).12). The trigonalbipyramidal cyano-bridged core of the cluster is emphasized with a polyhedron. Color scheme: Fe = red, Co = green, N = blue, C = gray (hydrogen atoms are omitted for clarity). (b, c) Packing of dimers of TBP clusters in crystal. ππ interactions is indicated with arrows. 21668


The Journal of Physical Chemistry C

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The influence of orbital degeneracy on SCO will be examined in the following section. In a strong crystal field of C-bound cyanide ligands, the ground terms of the CoIII and FeIII ions are the low-spin orbital singlet 1A1(t62) (S = 0) and the orbital triplet 2T2(t52), respectively. Depending on the temperature and strength of cooperative interactions, the ground state of an FeII ion in a crystal field of intermediate strength induced by the nitrogen atoms can be a low-spin term, 1A1(t62), or a high-spin term, 5 T2(t42e2). Both magnetic measurements and M€ossbauer spectroscopy for hydrate crystals of 1 and 212 demonstrate the presence of some amount of FeII ions in the high-spin configuration even at very low temperatures. Hereafter we will denote the fraction of FeII ions in the high-spin state at all temperatures by x and the concentration of ions that undergo the lshs transition as (1 x). The fraction pi of TBP clusters in which i of three FeII ions are in the high-spin configuration in the whole temperature range is estimated as pi ¼ C3i xi ð1 xÞ3 i

ð1Þ

where = r!/i!(r i)! and i = 0, 1, 2, 3. To construct the Hamiltonian of the system, we consider single-ion terms separately from the terms arising from inter-ion interactions. 2.1. Single-Ion Interactions. The Hamiltonian describing single-ion terms can be written as Cri

H0 ¼ λk

∑kα

s αk Bl αk þ Δ B

∑kα ððl kαZ Þ2 2=3Þ

B þ μB H

∑kα ðg0 Bs kα klBkα Þ λ1 k1 ∑k ðSBk1 Bl k1 þ BS k2 Bl k2 Þ

þ μB H B

∑k ½g0 ðSBk1 þ BS k2 Þ k1 ð Bl k1 þ

Bl k2 Þ

∑k Hk ð2Þ

where α denotes the hs-FeII ions in the kth TBP cluster, the first term is the spinorbit coupling operating within the cubic 5 T2(t42e2) term of the hs-FeII ion, and the second term describes the axial crystal field splitting of the 5T2(l = 1) term into an orbital singlet (ml = 0) and an orbital doublet (ml = (1). The third term refers to the Zeeman interaction for the hs-FeII ion and contains both spin and orbital contributions; μB is the Bohr magneton, and g0 is the spin Lande factor. The fourth and fifth terms represent spinorbital and Zeeman interactions for two ls-FeIII ions in cluster 2. It should be noted that the spinorbital interaction plays a double role in spin-crossover. On the one hand, this interaction splits the 5T2(t42e2) term, and the lowest component of this term approaches to the ground 1A1(t62) level, thus facilitating spin crossover. On the other hand, the degeneracy of the lowest hs-level is smaller than 15, thus retarding the lshs transformation. On the basis of the structural data, which show that the carbon environment of FeIII ions is only slightly distorted, we assume that these surroundings can be described approximately by the cubic symmetry. Finally, k and k1 are the orbital reduction factors for hs-FeII and ls-FeIII ions, respectively. The ground Kramers doublet and the excited quadruplet arising from the splitting of the 2T2 term by the spinorbit interaction are separated by a quite large gap of 3|λ1|/2 e 730 cm1 (λ1 = 486 cm1 for a free ls-FeIII ion).45 Nevertheless, the contribution of the excited quadruplet to the magnetic properties of a

single ls-FeIII ion will be shown to have a tangible influence on the magnetic behavior and M€ossbauer spectra. 2.2. Inter-ion Interactions. In compounds 1 and 2, throughspace interactions between the equatorial hs-FeII ions are expected to be negligible due to large metalmetal distances (>6 Å). Since in complex 1 the apical positions are occupied by diamagnetic ls-CoIII ions, magnetic exchange in ls-CoIII hs-FeII pairs is not operative. On the contrary, in complex 2 the superexchange interaction through the cyanide bridges couples the hs-FeII ions in equatorial and ls-FeIII ions in apical positions. The characteristic values of this interaction are on the order of several cm1.46,47 As long as the ground states of interacting ls-FeIII and hs-FeII ions are orbitally degenerate, the exchange interaction should contain, in general, both orbital and spin contributions (see review articles).48,49 Nevertheless, to simplify the present consideration and to avoid overparameterization, we will use the Lines model proposed in his study of the magnetic exchange between hs-CoII ions50 (a more detailed background of the Lines model is discussed in refs 48 and 49). Furthermore, we will neglect the essentially anisotropic orbitally dependent terms48,49 and retain only the isotropic part of the exchange interaction between hs-FeII and ls-FeIII ions in cluster 2. The Hamiltonian of exchange interaction for the kth cluster is as follows: k ¼ 2Jex Hex

∑α

s kα ðS Bk1 þ B S k2 Þ B

ð3Þ

where sα = 2 is the spin of the hs-FeII ion and S1 = S2 = 1/2 are the spins of the ls-FeIII ions. The summation in eq 3 is extended over the hs-FeII ions appearing in the kth cluster due to the spin transition and those which are in the high-spin state over the entire temperature range. As far as the orbitally degenerate ions are involved in the SCO transformation along with the totally symmetric vibrations, the JahnTeller modes5153 should be taken into account as well.3436 In structural phase transformations54 where the ordering of orbitals of the same type (t2 or e) occurs, the JahnTeller coupling plays a key role. On the contrary, SCO transitions are accompanied by electronic t2 T e jumps in which the metal ligand distance is essentially changed. Under these circumstances, the interaction with totally symmetric modes dominates while the JahnTeller effect (even being operative) plays a secondary role. Although the experimental X-ray data do not give absolutely clear evidence in favor of this theoretical concept, one can see that the FeN bonds increase with temperature,12 and at low and room temperatures their lengths are typical for ls- and hs-FeII ions, respectively, and at the same time the changes of crystal symmetry in the course of SCO transformation were not indicated.12 Therefore, following refs 3739, we assume that the mechanism responsible for the lshs transition is the interaction of FeII ions with the spontaneous totally symmetric lattice strain. The lattice acoustic and optical phonon modes and the macroscopic strain that corresponds to the limit of long-wavelength crystal vibrations are considered separately in the theory of the structural phase transitions.54 The strain has even parity and corresponds to the change in shape of the sample. The treatment of strain separately from the phonons is helpful because of the problem of applying proper boundary conditions when the crystal is strained. The problem was solved by Kanamori,55 who treated the strain first and then treated all other phonons using periodic boundary conditions on the strained crystal. Since spin transformation in SCO systems is accompanied by 21669



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the appearance of lattice strain, we use Kanamori’s approach in the subsequent consideration, and in fact, we replace the interaction with long-wavelength acoustic phonons by the interaction with strain. In a molecular crystal, lattice vibrations can be divided into molecular (quasi-localized) and intermolecular types. The molecular vibrations (which have high frequencies) are directly coupled to the electronic shells, whereas the intermolecular vibrations (low frequencies) transmit the local strains throughout the system. This can be approximately taken into account by assuming that the medium between the clusters is more soft than that within the clusters. Therefore, as in refs 3739, a distinction is made between the intra- and intercluster space. We introduce the internal molecular fully symmetric strain: pffiffiffi ε1 ¼ ðε1xx þ ε1yy þ ε1zz Þ= 3 Similarly, one can write down the external (intermolecular volume) strain: pffiffiffi ε2 ¼ ðε2xx þ ε2yy þ ε2zz Þ= 3 The bulk moduli corresponding to these strains will be denoted as c1 and c2, respectively. For each cluster, the interaction of SCO FeII ions with ε1 strain is considered. The contribution of uniform strains ε1 and ε2 to the potential energy of the crystal can be set by the following expression: Hst ¼

1 1 nmc1 Ω0 ε21 þ nmc2 ðΩ Ω0 Þε22 2 2 þ ε1 υhs

∑ I1αk þ ε1 υls k,∑α I2αk

k, α

ð4Þ

where the first two terms describe the elastic energy of the deformed crystal and the third and fourth terms correspond to the coupling of d-electrons with ε1 deformation. n is the number of FeII ions that undergo the lshs transition in a complex (n = 1, 2, 3; α = 1, ..., n); m is the number of TBP complexes whose FeII ions are involved in the spin transition (k = 1, ..., m); Ω0 is the volume occupied by an Fe ion and its nearest environment; Ω is the unit cell volume per FeII ion; υhs and υls are the constants of interaction of the FeII ion with totally symmetric strain ε1 in the high- and low-spin states, respectively; Iαk 1 is a diagonal 16 16 matrix with 15 eigenvalues equal to 1 and one vanishing eigenvalue corresponding to states 5T2 and 1A1, respectively; and the matrix Iαk 2 is defined on the same basis and has only one diagonal nonvanishing matrix element equal to 1 and corresponding to the 1A1 state. It is pointed out that eq 4 does not account for clusters in which none of the FeII ions undergoes the lshs transition. Introducing two new effective coupling parameters, υ1 = (υhs υls)/2 and υ2 = (υhs + υls)/2, we rewrite eq 4 in the following form: Hst ¼

1 1 nmc1 Ω0 ε21 þ nmc2 ðΩ Ω0 Þε22 2 2 þ υ 1 ε1

∑ ταk þ υ2 ε1nm

k, α

ð5Þ

Here, in the basis of states 5T2 and 1A1, the 16 16 matrix ταk is diagonal and has 15 eigenvalues equal to 1 and one eigenvalue equal to 1. The transition from 1A1(t62) state to 5T2(t42e2) state is accompanied by elongation of the ironnitrogen bonds due to occupancy of the excited e-orbitals. Correspondingly, this elongation facilitates the deformation of the soft intercluster space.

If there is no elongation of the FeN bonds in the cluster, the intercluster space will not be deformed. For a uniform crystal compression (or extension), the relationship of ε2 to ε1 is roughly taken to be ε2 ≈ ε1c1/c2. In fact, the eigenvalues of the adiabatic Hamiltonian, eq 5, represent adiabatic potential sheets corresponding to the high- and low-spin states of the crystal. In order to find the equilibrium positions of the nuclei in these states, the minimization of the right part of eq 5 over the strain is performed. This procedure leads to the following result: ε1 ¼ A¼

A υ1 nm

∑ ταk Aυ2

k, α

c2 c1 ½c2 Ω0 þ c1 ðΩ Ω0 Þ

ð6Þ

Finally, after substitution of ε1 from eq 6 into eq 5, one obtains Hst ¼ B

ταk ταk ταk ∑ ∑ ∑ 2nm k, α k, α k , α J

0 0

0

ð7Þ

0

where B = Aυ1υ2 and J = Aυ12. The first term in eq 7 redefines the effective energy gap Δ0 between the high- and low-spin states of the FeII ion in the cubic crystal field. The second term in eq 7 represents an infinite range interaction between the FeII ions that undergo the spin conversion. This interaction arises from the coupling of FeII ions to lattice strain. The model of the elastic continuum introduced above satisfactorily describes only the long-wave acoustic vibrations of the lattice.56 Therefore, the obtained intermolecular interactions correspond to interaction via the field of long-wave acoustic phonons. The parameter of inter-ion interaction J is independent of the distance between the ions k,α and k0 ,α0 since in long-wave acoustic vibrations the ions in the unit crystal cell vibrate in the same direction with the same amplitude. In addition, FeII ions inside the cluster are coupled by shortrange (intracluster) interactions. Possible mechanisms of shortrange interactions are related to direct interaction of electric quadrupole moments and to interaction via the field of optical phonons. The effect of interactions through the field of optical phonons is found to be larger than that of electric quadrupole quadrupole interactions. 57 The Hamiltonian describing shortrange interactions between FeII ions within the TBP can be written as Hsr ¼ J0

∑k α∑< α ταk ταk

0

ð8Þ

0

The Hamiltonian in eq 8 takes into account interactions between FeII ions participating in SCO and interaction of these ions with FeII ions in the high-spin state that do not undergo SCO, as well as interactions between the latter. It should be mentioned that eq 7, in comparison to eq 8, accounts only for FeII ions participating in spin transition. The Hamiltonian for the whole crystal can be written as H ¼ H0 þ Hsr þ Hst þ Hex þ

Δ0 2

∑k ∑α ταk

ð9Þ

where Δ0 is the effective energy gap between high- and lowspin states of the FeII ion in the cubic crystal field, Hex = ∑kHkex. 21670



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3. MAGNETIC SUSCEPTIBILITY 3.1. General Expressions in the Mean Field Approximation. We applied the molecular field approximation for deforma-0

tional long-range interactions and replaced the quantity ταk τk0 α in eq 7 by 0

0

ταk ταk0 ¼ Æτæταk0 þ Æτæταk Æτæ2

ð10Þ

~ B TÞταk =Tr½expð H=k ~ B TÞ Æτæ τ ¼ Tr½expð H=k ð11Þ ~ is the Hamiltonian of where kB is the Boltzmann constant and H the crystal in the molecular field approximation, that is, just the Hamiltonian (eq 9) in which the interaction Hst is substituted by ~ st ¼ ðJτ þ BÞ H

∑kα ταk

In the framework of the molecular field approximation, the ~ of the crystal can be written as a sum of oneHamiltonian H cluster Hamiltonians: 0 ~ k ¼ Jτ þ B Δ0 H ταk J0 ταk ταk 2 α α < α0 α !α αk kα 2 λk s k l k I1 þ Δ ððlZ Þ 2=3ÞI1αk B

∑

∑α

k þ Hex þ μB H B

∑

∑α

!

∑α ðg0 Bs kα k l kα ÞI1αk

Figure 3. Possible electronic configurations of clusters containing (A) three SCO FeII ions, (B) two SCO FeII ions, and (C) one SCO ion. For the sake of clarity, MIII (M = Co, Fe) ions are not shown.

! ! þ μB H B½g0 ðS Bk1 þ B S k2 Þ k1 ð l k1 þ l k2 Þ ! ! Bk1 l k1 þ B S k2 l k2 Þ λ1 k1 ðS ð12Þ

In these equations, the value ZðHβ Þ ¼ p2

The last two terms in eq 12 refer to the Zeeman and spinorbital interactions of ls-FeIII ions in compound 2, while this term is not ~ k describe operative for compound 1. In fact, the Hamiltonians H clusters with different numbers of SCO FeII ions, and k denotes the clusters in the crystal. With the aid of eq 12, the selfconsistent eq 11 for the order parameter τ̅ can be rewritten in the following form: ~ 1 =kB TÞτα1 Tr½expð H τ ¼ p3 þ p2 ~ 1 =kB T Tr½expð H þ p1

~ 2 =kB TÞτα2 Tr½expð H ~ 2 =kB TÞ Tr½expð H

þ p0

~ 3 =kB TÞτα3 Tr½expð H ~ 3 =kB TÞ Tr½expð H

ð13Þ

~ 1, H ~ 2, and H ~ 3 are the mean field Hamiltonians of the Here H individual clusters containing one, two, and three SCO FeII ions, respectively (Figure 3). These Hamiltonians can be easily derived from eq 12. For calculation of temperature dependence of the order parameter τ̅ , the self-consistent procedure is applied. Calculations of magnetic properties are based on the Hamiltonian given in eq 12. Molar magnetization Mβ and molar susceptibility χββ were calculated from the following expressions: Mβ ¼ NA kB T

∂ ln ZðHβ Þ , ∂Hβ

χββ ¼

Mβ Hβ

ð14Þ

∑i exp½ E1i ðHβ Þ=kB T

þ p1

∑i exp½ E2i ðHβ Þ=kBT

þ p0

∑i exp½ E3i ðHβ Þ=kBT þ Zhs p3

designates the partition function that accounts for contributions from clusters with different numbers of SCO ions, as well as the contribution Zhs from FeII ions in the high-spin state over the entire temperature range; Eji(Hβ) (β = X, Y, Z) are the energies of the cluster containing j SCO ions in the molecular field approximation in the presence of an external magnetic field Hβ, and NA is Avogadro’s number. 3.2. Discussion of Experimental Data. In order to interpret the temperature-dependent magnetic behavior of 1 and 2, as reflected in the observation of spin transitions at FeII centers, we begin with a discussion of characteristic parameters J and B. The operator for interaction of an iron ion with internal molecular strain can be presented as Hstr ¼

∂Wðr, RÞ ∂ε1

ε1 ¼ 0

ε1

ð15Þ

where W(r,R) is the potential energy (crystal field) of the electronic shell of the FeII ion. The matrix elements υhs and υls of the operator for interaction with strain in the high- and 21671



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low-spin states are as follows: ∂Wðr, RÞ dR υls ¼ Ælsj jlsæ ∂R dε1 R ¼ Rls R ¼ Rls υhs

∂Wðr, RÞ dR ¼ Æhsj jhsæ ∂R dε1 R ¼ Rhs R ¼ Rhs

ð16Þ

Here Rhs and Rls are the metalligand distances in the low- and high-spin states, and υhs and υls are proportional to the mean values of the derivatives of crystal field energies in these states. For totally symmetric internal strain one can obtain dR Rls dR Rhs ¼ pffiffiffi , ¼ pffiffiffi ð17Þ dε1 R ¼ Rls dε1 R ¼ Rhs 3 3 and the parameters υhs and υls that contribute to υ1 and υ2 (eq 5) can be expressed as 1 ∂Wðr, RÞ υls ¼ pffiffiffi Ælsj jlsæRls , ∂R 3 R ¼ Rls 1 ∂Wðr, RÞ jhsæRhs ð18Þ υhs ¼ pffiffiffi Æhsj ∂R 3 R ¼ Rhs Since, for an octahedral complex FeX6, the derivative W(r,R) of the potential energy can be written as [∂W(r,R)]/∂R = [5W(r, R)]/R, the values υhs and υls corresponding to the electronic configurations t42e2 and t62 are proportional to the cubic crystal field parameters Dqhs and Dqls: pffiffiffi pffiffiffi υls ¼ 120Dqls = 3 υhs ¼ 20Dqhs = 3 Although these expressions are obtained within the point-charge crystal field approximation, they provide a reasonable estimation for the parameters. Finally, the parameters υ1 and υ2 are represented as pffiffiffi υ1 ¼ 10ðDqhs 6Dqls Þ= 3 pffiffiffi υ2 ¼ 10ðDqhs þ 6Dqls Þ= 3 ð19Þ For crystal field parameters Dqhs = 1176 cm1 and Dqls = 2055 cm1 20 (that are within the range of typical parameters for such systems), one obtains υ1 = 6.4 104 cm1 and υ2 = 7.8 104 cm1. For SCO compounds, the main change in volume occurs in the intermolecular space, and therefore c1 . c2. In compound 1, the unit cell volume per Fe ion is Ω = 1026 Å3, and Ω0 is approximately 64 Å3. For the specified values of υ1 and υ2, c2 ≈ (0.050.1)c1, and the typical value of c2 ≈ 1011 dyn/cm2 for SCO compounds, the parameters J and B fall into the range of 2080 cm1 and 95 to 24 cm1, respectively. Because of the structural similarity of 1 and 2, the volumes Ω and Ω0 in these compounds and the elastic moduli c1 and c2 are assumed to be approximately equal. Therefore, the performed estimation of parameters J and B is valid for compound 2 as well. Later, the energy gap between the high- and low-spin states will be redetermined with proper account for the term 2B (see eq 12), which depends on the parameters υ1 and υ2. In all subsequent calculations we use the gap Δhsls = Δ0 2B. In the best-fit procedure we assumed that J < J0, which is in accord with the results reported in ref 57, where the optical phonons were shown to promote larger interaction constants, because the atoms are moving against each other and generate large electric fields. Best-fit calculations have also been performed for different

Figure 4. Temperature dependence of the χT product for compound 1 (water-containing FeII3CoIII2 crystals). (O) Experimental data;12 (—) theoretical fit with λ = 103 cm1, x = 10%, Δhsls = 640 cm1, Δ = 180 cm1, J = 35 cm1, J0 = 45 cm1, and k = 1.

starting values of the fitting parameters. It is worth noting that different sets of initial parameters led to the same best-fit values. The uniqueness of the fitting parameter set is also supported by the fact that each of the parameters governs the magnetic behavior in a definite temperature range. So the gap Δhsls determines the temperature at which χT starts increasing, while the parameters J and J0 are responsible for χT curve steepness. At T < 100 K, the χT values depend on the high-spin fraction x present at any temperatures. Finally, the low-temperature decrease of χT is caused by splitting (Δ) of the high-spin term in the axial crystal field. Figure 4 shows experimental data for compound 1 (watercontaining FeII3CoIII2 crystals12) together with calculated χT versus T curves. One can see that good agreement with the experimental data is obtained. Relative error, calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ∑ððχTÞtheor =ðχTÞexp 1Þ2 =N , is about 5.8 102. Below 100 K, the χT values show that the FeII ions are mainly in the low-spin state, with a small admixture of high-spin ions. In the temperature range 150300 K, the χT product gradually increases, thus indicating the low- to high-spin transition at the FeII centers. The parameter J of the long-range cooperative electronlattice interaction obtained from the best-fit procedure falls within the limits estimated above. Relatively small values of the parameters J and J0, as compared to the gaps Δ0 2B and Δ, are also in agreement with the observed gradual temperature dependence of χT and noticeable increase of χT at temperatures higher than 150 K. Finally, the percentage of FeII ions (x = 10%) in the high-spin state at any given temperature, as estimated from the best-fit procedure, is very close to that obtained from the M€ossbauer spectra.12 For comparison, the result of fitting the χT curve without inclusion of the long- and short-range interactions is shown in Figure 5 for the case of Δ < 0. It is seen that in this approximation the calculated curve differs significantly from the experimental one at both low and high temperatures, and the obtained value of k = 0.6 is too small for hs-FeII ions. At arbitrary temperature, the investigated compound represents a paramagnetic mixture of TBPs with different numbers of FeII ions in the high-spin configuration. We denote the fraction of TBPs with i FeII ions in the high-spin state at temperature T by ni(T) (i = 0, 1, 2, 3). Actually the model accounts for four types of clusters that differ in the number of FeII ions undergoing SCO. The contribution to the fraction n0(T) comes only from clusters containing three spin-crossover ions. The fraction n1(T) of TBPs

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Figure 5. Temperature dependence of the χT product for compound 1 (water-containing FeII3CoIII2 crystals). (O) Experimental data;12 (—) theoretical fit with λ = 103 cm1, x = 10%, Δhsls = 620 cm1, Δ = 136 cm1, and k = 0.6 with neglect of cooperative long- and shortrange interactions.

Figure 6. Thermal variation of Fe3Co2 TBPs with different numbers of hs-FeII ions, calculated with the same set of parameters as in Figure 4.

with one hs-FeII ion originates from clusters containing three and two SCO ions. At the same time, clusters with three, two, and one SCO ions can form the fraction n2(T). Finally, all types of clusters give rise to the fraction n3(T). Figure 6 represents the thermal variation of fractions ni(T) (i = 0, 1, 2, 3) for compound 1. In calculations of ni(T) values (Figure 6), the eigenvalues of the ~ 2, and H ~ 3 obtained with the set of the best-fit ~ 1, H Hamiltonians H parameters for compound 1 and the probabilities pi determined by eq 1 have been used. It is seen from Figure 6 that at low temperatures, T < 100 K, the majority of TBPs contain FeII ions in the low-spin configuration [fraction n0(T)]. At temperatures less than T = 75 K, the fraction n1(T) that refers to TBPs with one FeII in the high-spin state remains constant and approximately equal to 0.25. At low temperatures, the probability of finding TBPs with two or three hs-FeII ions is negligible. Beginning from T = 75 K, the number of clusters with two and three FeII ions in the high-spin configuration begins to increase as the temperature is raised. At T = 225 K, the number of clusters with two and three ions in the high-spin state exceeds that of clusters with one ion in this state. At the same time, the fraction n0(T) decreases abruptly for T > 150 K. At 300 K the percentage of clusters containing one, two, and

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Figure 7. Temperature dependence of the χT product for compound 2 (water-containing FeII3FeIII2 crystals). (O) Experimental data;12 solid lines 1 and 2 represent a theoretical fit with λ = 103 cm1, λ1 = 486 cm1, Δhsls = 700 cm1, J = 38 cm1, J0 = 45 cm1, Δ = 100 cm1, and x = 5%. In calculations of curve 1, both the ground Kramers doublet and the excited quadruplet for ls-FeIII ions were taken into account; curve 2 corresponds to the consideration of these ions as pseudospins 1/2.

three FeII ions in the high-spin state is ∼87%. In fact, at high temperatures the course of χT is mainly determined by these clusters. The temperature dependence of traction ni(T) also supports the fact that, at low temperatures, the clusters containing three FeII ions in the low-spin state are the lowest in energy. At room temperature, the ground state of the system corresponds to the configuration in which all three FeII ions are in the high-spin state. The obtained results (Figures 46) clearly demonstrate that the magnetic behavior of the system under examination cannot be explained by a simple Boltzmann repopulation of the cluster energy levels. Gradual SCO takes place since the sum of the fractions ni(T) (i = 13) does not increase abruptly with temperature. The magnetic behavior of complex 2 (water-containing FeII3FeIII2 crystals)12 was analyzed by neglecting intracluster Heisenberg exchange interactions between FeII and FeIII ions. The result of the best-fit procedure is presented as curve 1 in Figure 7. The relative error R is found to be 1.5 102. The obtained energy gap Δhsls between the high- and low-spin configurations for complex 2 is slightly larger than the corresponding gap for compound 1, whereas the parameters of short- and long-range interactions are approximately the same. The difference in the gap Δhsls leads to lower χT values for compound 2 as compared to compound 1 at temperatures higher than 150 K. It should be mentioned that the obtained values for Δhsls are in agreement with those reported in refs 30 and 58 for extended SCO FeII compounds. A comparison of curves 1 and 2 in Figure 7 shows that, with neglect of the contribution of excited quadruplets of ls-FeIII ions to the magnetic properties, the calculated χT values are lower. This difference increases with temperature and amounts to 0.74 cm3 3 K 3 mol1 at room temperature due to appreciable mixing of the ground and excited states of a single ls-FeIII ion by the external magnetic field. In section 4 it will be demonstrated that taking account of this effect allows us to correctly determine the partial contributions of ls-FeIII, ls-FeII, and hs-FeII ions to the M€ossbauer spectra. Since typical values of the exchange parameters in cyanide-bridged complexes are on the order of several cm1, we also calculated 21673



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Table 1. M€ ossbauer Parameters for FeII Ions in Coma pound 1 T, K

δ,12 mm/s

4.2

0.45

4.2

0.46

Γ, mm/s 0.145

II

ls-Fe

II

2.85

0.165

hs-Fe

220

0.41

0.44

0.18

ls-FeII

220

1.04

2.6

0.22

hs-FeII

0.38

300

0.37

0.99

2.3

0.23 0.24

T, K

Fe ion

1.12

300 a

ΔEQ,12 mm/s

Table 2. M€ ossbauer Parameters for FeII and FeIII Ions in Compound 2a

II

ls-Fe

hs-Fe

δ,12 mm/s

ΔEQ,12 mm/s

Γ, mm/s

Fe type

4.2

0.44

0.44

0.225

ls-FeII

4.2 4.2

1.12 0.02

2.85 0.9

0.225

hs-FeII ls-FeIII

220

0.38

0.4

0.195

ls-FeII

220

0.99

2.56

0.21

hs-FeII

220

0.11

0.82

0.3

ls-FeIII

300

0.36

0.27

0.265

ls-FeII

300 300

0.98 0.14

2.06 0.7

0.3 0.285

hs-FeII ls-FeIII

II

Water-containing FeII3CoIII2 crystals.

the χT product with the set of the best-fit parameters and Jex = 3 and +3 cm1. We found that, at temperatures higher than 50 K, the small exchange interaction has no effect on the magnetic properties of complex 2. In general, magnetic exchange leads to a splitting of the energy levels arising from the high-spin configuration and can affect the spin transition. Nevertheless, for cyanide-bridged clusters this splitting is small and cannot lead to noticeable changes in magnetic properties in the temperature range over which the spin transition occurs. In this temperature range, the χT products calculated with and without inclusion of the exchange interaction nearly coincide. On the contrary, at low temperatures the magnetic behavior crucially depends on the value and sign of the exchange parameter.

a

Water-containing FeII3FeIII2 crystals.

€SSBAUER SPECTRA 4. Mo M€ossbauer spectra provide a direct measure of the population of the high- and low-spin states and serve as a reliable test for the theoretical background of the SCO phenomenon. We assume that each type of iron ion in the sample is represented in the M€ossbauer spectrum by a quadrupole doublet, an assumption valid for 1 at all temperatures and for 2 above 50 K. The M€ossbauer spectrum observed is obtained by summing up the spectra yielded by different cluster electronic states in molecular field, taking into account their equilibrium populations for a given value of the molecular field at a certain temperature. The shape function of the M€ossbauer doublet produced by each cluster nucleus is determined by superposition of the Lorenz curves: Fi ðΩÞ ¼

1 Z

∑( ∑ν exp

Eν Γ ν 2 kT Γ þ ½Ω δi ( ðΔEνQ i =2Þ2

ð20Þ Here ν denotes the states of the cluster, and ΔEνQi and δνi are the quadrupole splitting and isomer shift on ion i of a cluster in the νth electronic state of the cluster. The slight temperature dependence of these parameters in the water-containing FeII3CoIII2 and FeII3FeIII2 crystals12 is due to mixing of the ls-1A1 and hs-5T2 states by spinorbital coupling, population of the excited orbital states of the hs-FeII ion, and second-order Doppler shift. These effects are not considered explicitly; instead, in calculations of the M€ossbauer spectra we use the values of parameters δ and ΔEQ at each temperature determined in ref 12 (Tables 1 and 2). We also use the experimental half-widths Γof the M€ossbauer lines, which are slightly temperature-dependent (Tables 1 and 2). At the same time, the point is underscored that

Figure 8. M€ossbauer spectra of compound 1 (water-containing FeII3CoIII2 crystals) at T = 4.2, 220, and 300 K. (b) Experimental data;12 (thick solid lines) theoretical fit with the best-fit parameters Δhsls = 640 cm1, Δ = 180 cm1, J = 35 cm1, J0 = 45 cm1, and k = 1. Contributions from ls- and hs-FeII ions are shown as dashed and dotted lines, respectively.

the model takes into account the main effect that leads to temperature-dependent M€ossbauer spectra: namely, the temperature dependence of cluster energies Eν in the molecular field. The experimental and simulated M€ossbauer spectra of complex 1 at 4.2, 220, and 300 K are shown in Figure 8. The 4.2 K spectrum represents a superposition of two doublets characteristic 21674



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Table 3. Percentages of ls- and hs-FeII Ions Determined from Observed and Calculated M€ ossbauer Spectra of Compound 1a at Different Temperatures experimental12 T, K 4.2

a

II

theoretical II

hs-Fe , %

ls-Fe , %

II

hs-Fe , %

ls-FeII, %

10

90

8.4

91.6

220

31

69

33.8

66.2

300

66

34

63.9

36.1

Water-containing FeII3CoIII2 crystals.

of ls-FeII and hs-FeII ions. The intensity of the doublet arising from ls-FeII ions significantly exceeds that of the doublet arising from hs-FeII ions. With the temperature increase, the amount of hs-FeII ions increases. This leads to noticeable changes in the M€ossbauer spectra. At 220 K, 1/3 of the FeII ions are in the highspin state, while at 300 K, approximately 2/3 of the FeII ions become high-spin, and the doublet corresponding to hs-FeII ions dominates the spectrum. Nevertheless, even at room temperature the contribution of the low-spin fraction to the whole spectrum is noticeable. The percentages of hs- and ls-FeII ions determined from our model are in good agreement with the experimental ones (Table 3). It should be emphasized that this result can be interpreted solely as a consequence of the SCO that represents an essentially cooperative phenomenon. Indeed, the simple Boltzmann population of the levels that originate from the high- and low-spin states of the FeII ion, given a fixed (temperature-independent) crystal field gap, cannot reproduce the experimentally measured change in the relative populations of the low- and high-spin states as functions of temperature. In simulation of the M€ ossbauer spectra of compound 2 (water-containing FeII3FeIII2 crystals) we also included the contribution of the spectra of ls-FeIII ions besides those of hs- and ls-FeII ions. The M€ossbauer spectra of compound 2 can be obtained with the aid of the following expression: FC ðΩÞ ¼ nðFeIIhs Þ

∑(

þ nðFeIIls Þ

þ nðFeIII ls Þ

Γ2hs

∑( ∑(

Γhs 2 II ð1Þ þ Ω δFe hs ( ΔEhs =2

Γ2ls þ

Γ2FeIII þ

Figure 9. M€ossbauer spectra of compound 2 at T = 4.2, 220, and 300 K. (b) Experimental data; (thick solid lines) theoretical curves with the best-fit parameters Δhsls = 700 cm1, J = 38 cm1, J0 = 45 cm1, Δ = 100 cm1, and x = 5%. Contributions from hs-FeII, ls-FeIII, and ls-FeII ions are shown as dotted, dashed, and thin solid lines, respectively.

Table 4. Percentages of ls- and hs-FeII Ions Determined from Observed and Calculated M€ ossbauer Spectra of Compound 2a at Different Temperatures

Γls

2 II ð1Þ Ω δFe ( ΔE =2 ls ls

ΓFeIII III

ð2Þ

Ω δFe ( ΔEls =2 ls

experimental

2

ð21Þ II II Here n(FeIII ls ) = 0.4, n(Fehs), and n(Fels ) are the fractions of lsIII II II Fe , hs-Fe , and ls-Fe in the crystal at a definite temperature. Temperature dependence of the n(FeIIhs) and n(FeIIls ) fractions can be easily calculated with the aid of eq 13. At any temperature the condition n(FeIIhs) + n(FeIIls ) = 0.6 holds. Experimental and calculated M€ossbauer spectra for compound 2 are presented in Figure 9. Partial contributions from ls-FeII and hs-FeII ions are given in Table 4. The theoretical model provides a reasonable estimation of the partial contributions of ls- and hs-FeII ions to M€ossbauer spectra of compound 2 at all temperatures and, hence, it correctly describes the low-spin T high-spin transformation in this compound. Below 50 K, the spin

a

theoretical

T, K

hs-FeII, %

ls-FeII, %

hs-FeII, %

ls-FeII, %

4.2 220

Vibronic Model for Cooperative Spin-Crossover in Pentanuclear

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