Diversity of Spin Crossover Transitions in Binuclear Compounds

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Diversity of Spin Crossover Transitions in Binuclear Compounds: Simulation by Microscopic Vibronic Approach Andrew Palii, Serghei M. Ostrovsky, Oleg S. Reu, Boris Tsukerblat, Silvio Decurtins, Shi-Xia Liu, and Sophia I. Klokishner J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b04834 • Publication Date (Web): 15 Jun 2016 Downloaded from http://pubs.acs.org on June 20, 2016

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The Journal of Physical Chemistry

Diversity of Spin Crossover Transitions in Binuclear Compounds: Simulation by Microscopic Vibronic Approach. Andrew Palii,1,2# Serghei Ostrovsky,1 Oleg Reu,1 Boris Tsukerblat,3 Silvio Decurtins,4 Shi-Xia Liu,4 and Sophia Klokishner 1#

1

Institute of Applied Physics, Academy of Sciences of Moldova, Academy Str. 5, MD-2028 Kishinev, Moldova 2

Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region, Russian Federation

3

Department of Chemistry, Ben-Gurion University of the Negev, PO Box 653 Beer-Sheva, 84105 Israel

4

Departement für Chemie und Biochemie, Universität Bern, Freiestrasse 3, CH-3012 Bern, Switzerland

Corresponding authors’ e-mail: [email protected]; [email protected] Phone: +(373)-22-738604

ABSTRACT A new microscopic approach to the problem of cooperative spin crossover in molecular crystals containing binuclear complexes as structural units has been developed. The cooperative interaction arising from the coupling of the molecular mode to the acoustic phonons represents the cornerstone of the approach. Another important ingredient of the model is the Coulomb interaction between the spin crossover ions in the binuclear cluster that is different in the ls-ls, hs-ls and hs-hs states (ls – low spin, hs –high spin). The suggested approach makes it possible to reproduce all types of transitions observed in binuclear compounds including two step transitions. The applicability of the developed approach is illustrated by the interpretation of the experimental data on spin transitions for the binuclear [{Fe(bpym)(NCSe)2}2bpym] and [{Fe(bt)(NCS)2}2bpym] cluster compounds.

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1. Introduction More than 80 years ago the phenomenon of spin crossover (SCO) was discovered.1-4 Since that time hundreds of SCO compounds have been synthesized and characterized with the aim to obtain compounds demonstrating spin transitions in the vicinity of room temperature5-10 that gives prospects for their practical applications. In this view, the second important requirement for the spin transition is the presence of wide hysteresis loops in the mentioned temperature range. With this in mind, researchers turned their interest to bi- and polynuclear SCO compounds. In the 90’s and later on, a number of binuclear compounds manifesting SCO has been reported. The characteristic feature of these compounds is the presence of binuclear complexes in which both ions may manifest SCO and are linked through a bridging ligand. Dinuclear iron(II) complexes of the type [{Fe(L)(NCX)2}2(bpym)] (L = 2,2’-bipyrimidine (bpym), 2,2’-bithiazoline (bt); X = S, Se) were first investigated by Real et al. 11-13 The first twostep SCO presented by the binuclear complex [{Fe(bt)(NCS)2}2bpym] was reported in

12

. The

experimental data on the Mössbauer spectra and variable-temperature magnetic susceptibility (χ) for this compound confirm the SCO behavior, and apparently show that the transition takes place in two steps centered around 163 and 197 K, respectively. As to the shape of the transitions, the first one is more abrupt when compared with the second one.12 This publication

12

was the first

one to explain the appearance of the second step in the temperature dependence of the magnetic susceptibility within the thermodynamic approach, taking into account the inter-center Coulomb interaction. Later on in Ref.

14

it was demonstrated that for [{Fe(bpym)(NCSe)2}2bpym], at a

temperature of around 120 K a dramatic decrease of the values occurs due to a thermally induced spin transition; the values reach a plateau in the range of 30-120 K, and then with lowering the temperature decrease again. To answer the question whether the plateau originates from only hs-ls pairs or from a 1:1 mixture of hs-hs and ls-ls pairs, Mössbauer spectra of [{Fe(bpym)(NCSe)2}2bpym] have been measured in an applied magnetic field.15,16 It was revealed that the observed plateau in the temperature dependence of is predominantly due to the presence of hs-ls pairs, while the sharp decrease of below 30 K is due to the zero-field splitting of the remaining hs-Fe(II) ions. The nature of the plateau of the dinuclear SCO complex [{Fe(phdia)(NCS)2}2 (phdia)] (phdia = 4,7-phenanthroline-5,6-diamine) exhibiting a two-step spin transition at Tc1=108K and Tc2=80K, and displaying 2K and 7K wide thermal hysteresis loops in the upper and the lower steps, respectively, was investigated in Ref.

17

. Dinuclear complexes with formula

{[Fe(bztpen)]2[-N(CN)2]}(PF6)3∙nH2O ( = 1, 0) have been synthesized by Real et al.18 The 2 ACS Paragon Plus Environment

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anhydrous form of this complex shows a gradual two-step spin transition with no observed hysteresis in the solid state. Both steps are approximately 100K wide, centered at ≈ 200K and ≈350K with a plateau of χT, approximately 80K wide while separating the transitions. It was demonstrated by Garcia et al

19

that the triple helicate dinuclear iron (II) complex [Fe2(L)3]-

(ClO4)4⋅2H2O exhibits a thermochromic two-step spin conversion at Tc1=240 K and Tc2=120 K, but does not switch its spin state further below 20 K as proven by Mössbauer spectroscopy. The families of doubly pyrazolate (µ-bpypz) and triazolate (µ-bpytz) bridged binuclear Fe(II) complexes have been synthesized and characterized in Ref.

20-23

and

24

, respectively. To

study the influence of the electronic and structural properties of the bridging ligand on the magnetic

behavior

and

to

elucidate

the

effect

of

cooperative

interactions,

the

[{Fe(dpia)(NCS)2}2(4,4’-bpy)] (dpia=di(2-picolyl)amine, 4,4’-bpy=4,4’-bipyridine) complex was examined.25 A comprehensive overview of the experimental work reported in the field of molecular SCO phenomena in dinuclear compounds is given in two review papers 26,27. Summarizing briefly the cited publications, one can conclude that different types of spin transitions have been observed in binuclear SCO compounds. Among them the most frequently detected ones are: (i) the one step ls-ls to hs-hs transition in which both ions of the cluster pass from the ls state to the hs state (these transitions may be accompanied by a hysteresis loop as well); (ii) two-step transitions with a plateau: particularly noteworthy are the two-step transitions, displaying wide thermal hysteresis loops in the upper and the lower steps (two-step transitions with a hysteresis loop in one of the steps are also observed); and, finally, (iii) partial transitions in which only one of the ions in the cluster changes its spin state, while another remains in the hs-state in the whole temperature range. In certain instances, the transition may be incomplete at one or both extremes of the spin transition curve. During the last years the number of reported binuclear iron(II) complexes manifesting SCO has noticeably increased. In spite of the progress in the synthesis and characterization of binuclear SCO compounds, only a few theoretical papers dealing with this problem have been published. First of all one should mention the work by Real et al 12 that produced a strong impact in the field. In this study, the two-step SCO in binuclear systems was suggested to be the result of the inequivalence of the intradimer Coulomb repulsion energies for the (ls)a – (ls)b, (ls)a – (hs)b and (hs)a – (hs)b pairs of ions a and b in the dimer. In a related study, the crystal was divided into two sublattices and different types of intra- and intersublattice interactions were introduced.28 The employed Hamiltonian includes the ferromagnetic type intrasublattice interaction, the intersublattice ferromagnetic type interaction for the same type of metallic centers belonging to different clusters as well as the antiferromagnetic type intercluster interaction. To reproduce the observed temperature dependence of the hs-fraction in the 3 ACS Paragon Plus Environment


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[{Fe(bt)(NCS)2}2bpym] crystal manifesting a two-step SCO it was also necessary to increase the ratio of the degeneracies of the hs- and ls-states by a factor of / , where and are the frequencies of molecular vibrations in the ℎ and states, respectively. Later on, the electronic structures of four binuclear iron(II) complexes displaying SCO were studied by DFT calculations.29 The results of the calculations confirmed the conclusion of the phenomenological model, that the enthalpy of the − ℎ state must be lower than the average enthalpy of the − and ℎ − ℎ states to facilitate a two-step spin transition. In Ref. 30, the model suggested in Ref.

28

was generalized by taking into account the effect of intramolecular vibrations. For a

binuclear cluster with ions belonging to the states − , − ℎ or ℎ − ℎ , different vibrational frequencies , , , and , have been introduced. The model reproduces a gradual and an incomplete spin transition and does not describe the double stepped transition. The authors of Ref. 30 explain this fact by the narrow range of parameter values that allow for the equi-energy scheme. The approach developed in Ref.

31

represents an extension of the vastly

employed Ising-like model, initially suggested for mononuclear SCO systems. The phenomenological cooperativity was introduced as a bilinear intermolecular interaction between molecular Ising spins and the problem of interacting spins was treated in the mean field approximation. The effect of the intramolecular isotropic magnetic exchange interaction on the type of the spin transition was elucidated as well. Within the framework of the model suggested in Ref.

31

, some well-known experimental results including the two-step transitions have been

interpreted. As it has been mentioned, the described models of cooperative SCO in binuclear compounds 12,28,30,31 operate with a crystal divided into two sublattices with an antiferromagnetic interaction between them and intra- and intersublattice ferromagnetic interactions. The origin of these interactions has not been discussed. Varying the ratio of the characteristic parameters of intra- and intersublattice interactions, the abrupt, partial and two-step transition in binuclear SCO compounds can be reproduced. In a recent paper 32, the meta-hybrid GGA functional TPSSh 33,34 has been applied for the analysis of the spin transformation in the binuclear [{Fe(bt)(NCX)2}2( -bpym)] and [{Fe(pypzH)(NCX)}2(µ-pypz)2] (X=S,Se,BH3) compounds in the frame of DFT. For the [{Fe(bt)(NCS)2}2( -bpym)] and [{Fe(bt)(NCSe)2}2( -bpym)] systems displaying a two-step SCO between the temperatures T1/21 and T1/22 corresponding to the transitions from ls-ls to hs-ls and hs-ls to hs-hs the following inequality T1/21 < T1/22 was obtained. This inequality is in qualitative agreement with the observed one, however, the difference between the observed and calculated temperatures is sufficiently large. Therefore, the conclusion of the authors that the presence or absence of a two-step transition seems mainly to be controlled by the local electronic structures, needs additional strong justification. Thus, in all above cited theoretical papers 4 ACS Paragon Plus Environment

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concerning binuclear clusters, the origin of the intercluster interactions has been scarcely discussed

12,28,30,31

or a conclusion has been made that these interactions are not operable.32

Moreover, the models suggested in

12,28,30,31

do not reproduce the two-step transitions with

thermal hysteresis loops in the upper and the lower steps. Recently we have suggested a new approach to the problem of SCO

35

in mononuclear iron(II) compounds and proved that the

cooperative interaction arising from the coupling of the molecular vibrational mode of each SCO center to phonons is the main factor responsible for this phenomenon. For a single impurity ion in a crystal, the latter coupling was for the first time described by V. Hyzneakov et al in their publications.36,37 In fact, as we have demonstrated

35

this single site interaction also fits the

situation in SCO compounds, in which each SCO ion interacts both with molecular vibrations and phonons. In Ref.

35

it has been proved that the model of cooperative SCO based on the

Hyzneakov’s approach reproduces all observed types of spin transformations in crystals containing one metal ion in the elementary cell. Moreover, as distinguished from the previous phenomenological models of SCO, the parameters of cooperative interaction in this model can be expressed through the intrinsic crystal parameters. In the present paper we generalize the model suggested in Ref.

35

to the case of binuclear SCO compounds and demonstrate the

possibility to describe in the frame of this model all observed spin transformations in these compounds.

2. Theory 2.1. Crystal Hamiltonian A crystal is considered to be built of binuclear clusters consisting of SCO ions. The energy spectrum of each ion in a cluster consists of the − and ℎ − levels with the multiplicities gls and g hs . An isolated cluster as a whole possesses 4 states: − , ℎ − , − ℎ, ℎ − ℎ. The crystal Hamiltonian is presented as

H = H 0 + H int + H ph ,

(1)

where H 0 is the Hamiltonian of non-interacting clusters, H int is the cooperative interaction between the dimers and H ph is the Hamiltonian of free phonons. The Hamiltonian H 0 looks as follows



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  hω H 0 = ∑ ∑  ls n i = a ,b  2  +∑ ∑

n i = a ,b

  xls  ni 

( )

2

−

 hωhs   ∆hs , ls + 2  

∂2

( )

∂ xnlsi

  x hs  ni 

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   + υ xls τˆls 2  ls ni  ni  

( )

2

−

∂2

( )

∂ x

hs ni

(2)

   + υ x hs τˆ hs + ∑ ∑ 2  hs ni  ni n f = hs ,ls  

∑

f ′ = hs ,ls

U nf , f ′ τˆnfa τˆnfb′ ,

here is the vector that labels the clusters in the crystal, the indices and enumerate the SCO ions and in each cluster and their ls- and hs-states, respectively, the vector = + determines the position of the i-th SCO ion in the -th cell. The first two terms of the Hamiltonian involve the free cluster vibrations and the electron-vibrational interaction in the −and ℎ −states, the diagonal matrices τˆnils and τˆnihs of the dimension ( gls + g hs ) × ( gls + g hs ) possess the following nonvanishing matrix elements

ψ nhsi ( µhs ) τ nhsi ψ nhsi ( µhs ) = 1 and

ψ nlsi ( µls ) τ nlsi ψ nlsi ( µls ) = 1 , where µhs = 1, 2,...g hs and µls = 1, 2,...gls numerate the hs and ls states. The last term in Eq.(2) is introduced following the idea of paper

12

and represents the

Coulomb interaction between the ions and composing each dimer and being in the −

, ℎ − , − ℎ or ℎ − ℎ states. The sense of the other notations in Eq.(2) is the following:

xnlsi and xnhsi are the full symmetric displacements of the nearest surrounding of the −th SCO ion in the −th cluster in the − and ℎ − states, respectively, ωls and ωhs are the frequencies of these molecular vibrations, and ∆hs , ls is the crystal field energy gap between the examined states. Finally, υls and υhs are the vibronic coupling constants. It should be mentioned that the difference in the coordinates xnlsi and xnhsi , and frequencies ωls and ωhs arises on the account of quadratic vibronic interaction

38

which is also taken into consideration while recording Hamiltonian,

Eq.(2). The intracluster Heisenberg exchange interaction that couples the hs-states of the ions in the dimer is not taken into account in Eq. (2) since it is of the order of several wavenumbers

14

and in the vicinity of the spin transition its effect can be neglected in comparison with the intercluster interaction. The main idea underlying the derivation of the Hamiltonian H int of cooperative interaction

35

is based, as mentioned above, on the approach suggested in Ref.

36,37

and can be

formulated as follows. Since each SCO ion participates both in molecular and crystal vibrations, the resulting vibration represents a linear combination of them in which each of these vibrations enters with its weight characterized by a dimensionless parameter of the model. As a result, in the interaction Hamiltonian for a single ion besides the coupling of electrons to the molecular vibrational coordinate an additional bilinear term appears that is proportional to and contains 6 ACS Paragon Plus Environment

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the product of crystal normal coordinates and the coordinate of the molecular vibration.35 Generalized to the case of a crystal consisting of binuclear clusters composed from SCO ions, this term looks as follows

H′ = 2

∑

ϕ κnνi qκν exp ( i κ Rni ) =

n ,i = a , b , κν

ϕ κnνi ( aκ+ν + a− κν ) exp ( i κ Rni ) ,

∑

n ,i = a , b , κν

(3)

where Rni is the position vector of the i-th ion in the n-th binuclear cluster, κ is the phonon wave vector, qκν = ( aκ+ν + a− κν )

+ 2 , aκν and a−κν are the phonon creation and annihilation operators,

and the values ϕκnν are expressed in the matrix form: ϕ κnνi ≡ ϕ κnνi ( A1 ) = −λ

 1 1 ls ls τ nhsi xnhsi ωhs2 − ωκ2ν + τ ni xni ωls2 − ωκ2ν  ωls  ωhs

(

h 2 N ωκ ν

)

(



) f ( A ) , 

κν

1

(4)

where ωκ ν is the phonon frequency, the symbol ν numerates the phonon modes and the VanVleck coefficients 39-41

f κν ( A1 ) =

∑ eα (k , κν ) ukAα exp(i κ Rk )

(5)

1

k, α

perform the unitary transformation from the symmetry adapted totally symmetric displacements

A1 of the ligand surroundings of the metal ions to the normal coordinates of the phonons.41 Only the coupling of SCO ions to fully symmetric vibrations is taken into account since under the SCO transformation the crystal symmetry does not change. The standard shift transformation for the phonon creation and annihilation operators

42,43

applied to Hamiltonian H ph + H ' does not

change the form of the Hamiltonian of free phonons H ph , at the same time it leads to the cooperative interaction between the SCO ions described by the term H int in Eq. (1), whose final form is defined by Eq. (20) in Ref.35. In order to adapt the Hamiltonian H int derived in Ref.

35

to the molecular crystal

composed of dimeric SCO units, we will make a reasonable assumption that the long-wave acoustic phonons produce a dominant contribution to the overall cooperative effect. Within this approximation the strength of each interionic interaction (independently of either this interaction is an intra- or interdimer one) involved in Hamiltonian H int becomes independent of the distance between the two SCO ions (i.e., it represents an infinite-range interaction). Let us focus on the elastic interactions of an arbitrary chosen SCO center with all other SCO centers in the crystal. Within the adopted approximation one can neglect the only elastic interaction of this SCO center with its nearest neighbor center belonging to the same dimer (intradimer interaction), as 7 ACS Paragon Plus Environment


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compared with the infinite number of interionic interdimer interactions. Thus, we arrive at the following Hamiltonian H int that represents a generalization of the Hamiltonian deduced in Ref. 35 to the actual case of a molecular crystal composed of binuclear units: H int = −

1 ∑ 2 n, n'

∑

i , j = a ,b

 J hs , hs ( n i − n′j ) τ nhsi xnhsiτ nhs′ j xnhs′ j + J ls , ls ( n i − n′j ) τ nlsi xnlsiτ nls′ j xnls′ j 

n ≠ n'

(6)

+ 2 J hs , ls ( n i − n′j ) τ nhsi xnhsiτ nls′ j xnls′ j  ,

where ≠ since the coupling of the ions inside each cluster is excluded. In this case, the parameters J f , f ( n i − n′j ) look as follows: '

J f , f ′ ( n i − n′j ) =

πλ 2 h R f R f ′

(

)

16 ω 2f ω 2f ′ − 8 ω 2f + ω 2f ′ ωM2 + 5 ωM4   6 c2 N ω f ω f ′ 

(7)

In the accepted long wave approximation the parameter J f , f ( n i − n′j ) does not depend on the '

indices n i and n′j . Nevertheless, we retain these indices in the notation of the parameter in order to make Eq.(7) suitable for the subsequent use of the mean field approximation (see Section 2.2). A remark needs be made concerning the character of the molecular vibrations involved in Hamiltonian, Eq.(6). They represent the same breathing vibrations of the nearest ligand surroundings of SCO ions as in our approach

35

reported for the case of mononuclear SCO

complexes but not the vibrations of the entire dimeric units. The use of the same kind of molecular vibrations as in Ref.

35

becomes possible due to the above described neglect of the

intracluster cooperative interactions as compared with the intercluster ones. In the framework of this approximation, the cluster specifics are fully taken into account by the last term of Eq. (2). The parameters of intercluster interaction thus depend on the geometry of the nearest surrounding of the SCO ion, on the spectrum of molecular and crystal vibrations as well as on the degree of mixing of the molecular and crystal vibrations expressed by the value of the parameter .

2.2. Mean field approximation To reduce the problem of interacting clusters to the one cluster problem, further on the mean field approximation is applied. Within the framework of this approximation, the crystal Hamiltonian is written in the form:


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   hωls  ls H% = ∑ H% n = ∑ ∑  xn i 2  n n i = a ,b   

( )

+∑ ∑

n i = a ,b

+∑ n

∑

2

−

∂2

  ∂2 hωhs  hs 2  xn i −  ∆hs , ls + 2  ∂ xnhsi    ∑ U nf , f ′ τˆnfa τˆnfb′ ,

( )

( )

∂ xnlsi

( )

   + υ − J ls , ls τ ls x ls − J hs , ls τ hs x hs x ls τˆls ls ni ni 2    

(

)

   + υ − J hs , hs τ hs x hs − J hs , ls τ ls x ls x hs τˆ hs hs ni ni 2    

(

)

(8)

f = hs ,ls f ′= hs ,ls

where the mean values of the products of the electronic matrices and molecular vibrational coordinates

τ ls xls and τ hs x hs play the role of the order parameters and the mean field

parameters J

f , f′

J f , f′ = ∑

take on the form:

∑ ∑

J f , f ′ (ni − n′j ) =

n′≠ n i =a , b j =a , b

πλ2 h R f R f ′ (N − 1) [16 ω 2f ω 2f ′ − 8 (ω 2f + ω 2f ′ )ωM2 + 5 ωM4 ] 2 6 c N ωfωf′

πλ2 h R f R f ′ ≈ [16 ω 2f ω 2f ′ − 8 (ω 2f + ω 2f ′ )ωM2 + 5 ωM4 ] 2 6 c ωfωf′

(9)

here the small value 1/N is neglected as compared with 1. It is worth noting, that with exception of the parameter λ which in principle can be found from the solution of the dynamic problem for all vibrations of the crystal including the molecular ones, all other parameters involved in Eq. (9) are available from the experimental data. The next step in the transformation of the Hamiltonian, Eq. (8), is the exclusion of terms linear in the coordinates

xnils = z nils − xls , xnihs = z nihs − xhs ,

(10)

where xls =

1 hωls

(υ

ls

)

− J ls , ls τ ls x ls − J hs , ls τ hs x hs , xhs =

1 hω hs

(υ

hs

)

− J hs , hs τ hs x hs − J hs , ls τ ls x ls .

(11)

After this transformation the Hamiltonian for a single dimer, Eq.(8), takes on the following form:



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   hωls  ls zn i   2  

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  hωls 2  ls  − − xls τˆn i ∑ ls 2  2 i = a ,b ∂ zn i        hωhs  hs 2 ∂ 2  hωhs 2  hs  zni − xhs τˆni +  ∆hs , ls + − 2 hs  2  2 z ∂   ni     + ∑ ∑ U f , f ′ τˆnaf τˆnbf ′ ,

( )

H% n =

∂2

2

( )

( )

(12)

( )

f = hs ,ls f ′ = hs ,ls

where it has been accepted that U nf , f ′ ≡ U f , f ′ because all dimers in the crystal are equivalent. In this case, the eigenvalues of the Hamiltonian H% n look as follows:

( (τ

) )=∆

ls ,ls ls ls hs hs Edim = hωls ( p + p′ + 1) − , p , p′ τ x , τ x

hs , hs Edim , p , p′

ls

x ls , τ hs x hs

hs , hs , ls ,ls

( 1 − (υ 2 hω

)

∆hs ,hs , ls ,ls 2

hωls

(

υls − J ls , ls τ ls xls − J hs , ls τ hs x hs

)

2

,

+ hωhs ( p + p′ + 1) −

ls , hs ls ls hs hs Edim = , p , p′ τ x , τ x

1

1 hωhs

(υ

hs , hs hs hs τ x − J hs , ls τ ls xls hs − J

)

2

,

− δ + hωls ( p + 1 2 ) + hωhs ( p′ + 1 2 )

ls , ls ls ls τ x − J hs , ls τ hs x hs ls − J

ls

)

2

−

(13)

(

1 υhs − J hs , hs τ hs x hs − J hs , ls τ ls xls 2hωhs

)

2

,

where the following notations are introduced

∆ hs ,hs ,ls ,ls = 2 ∆ hs , ls + U hs ,hs − U ls ,ls ,

δ=

(

)

1 U hs , hs + U ls , ls − U hs , ls . 2

(14)

The eigenvalues of Hamiltonian H% n represent the electronic levels and the adjacent levels of the harmonic oscillators with different frequencies for the hs and ls-states, where p and p ′ are the vibrational quantum numbers determining the energies of the two-dimensional harmonic oscillator. The energies of the electronic levels are counted from the energy of Coulomb repulsion U ls , ls of two iron(II) ions being in the ls-states. Since all dimers are equivalent, the index n numerating the dimers is omitted in Eq. (13). It should be also mentioned that after subtracting of the appropriate free oscillator terms hωls ( p + p′ + 1) , hω hs ( p + p′ + 1) and

(

)

(

ls ,ls ls ls hs hs hs , hs ls ls hs hs hωls ( p + 1 2 ) + hωhs ( p′ + 1 2 ) from the energies Edim , Edim , p , p′ τ x , τ x , p , p′ τ x , τ x

(

)

),

ls , hs ls ls hs hs one obtains the minima of the adiabatic potential surfaces corresponding to Edim , p , p′ τ x , τ x

the − , ℎ − ℎ and − ℎ states of the cluster. 10 ACS Paragon Plus Environment

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The corresponding wave functions of the binuclear cluster look as follows: p

p

χ pf pf′′ (µ af , µ bf ′ ) = ψ af (µ af ) Φ (z af )ψ bf ′ (µ bf ′ ) Φ ′ (zbf ′ ),

(15)

where ψ af (µ af ) and ψ bf ′ (µ bf ′ ) are the electronic wave-functions of ions a and b ( f = hs , ls and

( ) and Φ (z ) are the harmonic oscillator functions corresponding to the f′ b

p

p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


f ′ = hs ,ls ) , Φ zaf

′

equilibrium positions − x f and − x f ′ (see Eq. (12)) of the ligands in the states f and f ′ , respectively. The order parameters are obtained by statistical averaging of the operators τ f x f :





H% n  k BT

τ f x f = Tr τ f x f exp  − 

  

  H% Tr exp  − n  k BT 

With the aid of Eq. (36) from Ref.

35

  

(16)

the matrix elements of the operators τ als xals and τ ahs xahs can

be expressed as follows:

χ lsp lsp′ (µlsa , µlsb ) τ als xals χ lsp lsp′ (µlsa , µlsb )

(

)

(

)

a b = χ lsp phs′ µlsa , µ hsb τ als xals χ lshs =− p p′ µ ls , µ hs

(υ

1 hωls

ls

)

− J ls , ls τ ls x ls − J hs , ls τ hs x hs ,

(17)

χ hsp phs′ (µ hsa , µ hsb )τ ahs xahs χ phsphs′ (µ hsa , µ hsb )

(

)

(

)

ls a b hs hs hsls a b = χ hs =− p p′ µ hs , µ ls τ a x a χ p p′ µ hs , µ ls

1 hω hs

(υ

hs

− J hs , hs τ hs x hs − J hs , ls τ ls x ls

)

Substituting Eqs. (13) and (17) into Eq. (16) we obtain the following system of transcendental equations for the order parameters τ ls xls = −

1

(

Z τ x ,τ x ls

 2  ( gls )  ×  4sinh 2  hωls    2 k BT +

ls

hs

hs

)

hωls

ls

− J ls , ls τ ls x ls − J hs , ls τ hs x hs

 1 exp  υls − J ls , ls τ ls x ls − J hs , ls τ hs x hs k T h ω  ls  B  

(

gls g hs  hωls   hωhs 4sinh   sinh   2 k BT   2 k BT

(

(υ

)

)  2

(18)

 1  ∆hs ,hs , ls ,ls  exp  δ −  2    k BT   

1 + υls − J ls , ls τ ls x ls − J hs , ls τ hs x hs 2k BT hωls

)

2

(

1 + υhs − J hs , hs τ hs x hs − J hs , ls τ ls xls 2k BT hωhs


)

2

    ,   


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

τ hs x hs = −

(

Z τ x ,τ x  2  ( g hs )  ×  4sinh 2  hωhs    2 k BT

ls

hs

hs

) hω

hs

− J hs , hs τ hs x hs − J hs , ls τ ls xls

)

hs

 ∆ hs ,hs , ls ,ls 1 υhs − J hs , hs τ hs x hs − J hs , ls τ ls xls exp  − + ω k T k T h   B B hs  

(

gls g hs  hωls   hωhs 4 sinh   sinh  2 k T  B   2 k BT

+

(υ

1 ls

Page 12 of 26

)  2

 1  ∆ hs ,hs , ls ,ls  exp  δ −  2    k BT   

(

1 + υls − J ls , ls τ ls xls − J hs , ls τ hs x hs 2k BT hωls

)

2

(

(

1 + υhs − J hs , hs τ hs x hs − J hs , ls τ ls xls 2k BT hωhs

)

2

    ,   

)

where the partition function Z τ ls x ls , τ hs x hs has the following form:

(

)

( gls )

Z τ ls xls , τ hs x hs =

( g hs ) +

+

2

2

 1 exp  υls − J ls , ls τ ls x ls − J hs , ls τ hs x hs ω h k T  ls B  hωls  4 sinh 2    2 k BT 

 1 exp  −  k BT

gls g hs  hωls   hωhs 2 sinh   sinh  2 k T  B   2 k BT +

(

(

)  2

(

 1 hs , hs hs hs τ x − J hs , ls τ ls x ls  ∆ hs ,hs , ls ,ls − hω υhs − J hs   hωhs  4 sinh 2    2 k BT 

)   2

 1  ∆ hs ,hs , ls ,ls  exp  δ −  2    k BT   

1 υls − J ls , ls τ ls x ls − J hs , ls τ hs x hs 2k BT hωls

)

2

+

(19)

(

1 υhs − J hs , hs τ hs x hs − J hs , ls τ ls x ls 2k BT hωhs

)  . 2

While deriving Eqs. (18) and (19), the well-known expression for the partition function of the two-dimensional harmonic oscillator has been employed:  hω f ( p + 1 2 ) + hω f ′ ( p ′ + 1 2 )  1 = ω h k T   hω f ′ f  B   4 sinh   sinh   2 k BT   2 k BT

∑ exp − pp ′

  

.

(20)

In order to examine different types of spin transformations that can occur in crystals comprising binuclear SCO clusters, we first discuss the meaning of the parameters ∆ hs ,hs ,ls ,ls and δ defined by Eq. (14). In the absence of interaction between the SCO ions the energy of the first excited 12 ACS Paragon Plus Environment

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level of the dimer corresponding to the (ls)a-(hs)b and (hs)a-(ls)b states is equal to 1/2 of the energy of the (hs)a-(hs)b level, i.e. the first excited level lies in the middle of the gap between the (ls)a-(ls)b and (hs)a-(hs)b–states as it is illustrated in the left part of Figure 1. The account of the intradimer Coulomb interaction leads to the shift of the level energies. Since the matrix elements of this interaction depend on the states of the interacting ions, these shifts are different for the (ls)a-(ls)b, (hs)a-(hs)b and (ls)a(b)-(hs)b(a) states, and the energy of the first excited level is not anymore equal to the half of the gap between the (hs)a-(hs)b and (ls)a-(ls)b states. The shift of the first excited level from the middle of the ∆ hs ,hs ,ls ,ls - gap is just equal to the parameter δ , as it follows from Eq. (14). Thus, here we have assumed that U hs , ls < U ls , ls < U hs , hs . From this it follows that δ > 0 and the first excited level lies lower than the middle of the gap ∆ hs ,hs ,ls ,ls . For 12

the first time the authors of publication

U hs ,ls ≠ (U hs , hs + U ls , ls ) / 2 , later on in Ref. calculations. In Ref.

12

29

have paid attention to the fact that

this conclusion has been confirmed by DFT

it was supposed that this situation facilitates the observed two-step SCO

behavior.

Figure 1. Scheme illustrating the levels of the binuclear cluster involved in SCO and the meaning of the parameters ∆ hs ,hs , ls ,ls and δ describing the spin transformation. Until now we have only discussed the Coulomb interaction as the intercenter one and have shown that it affects essentially the cluster energies leading to a redetermination of the energy gaps between different cluster states. Besides the intercenter Coulomb interaction there is another electronic mechanism which may affect the energy spectrum of the binuclear cluster. The illustration of this kinetic-type mechanism is given in Figure 2. For the sake of definiteness, 13 ACS Paragon Plus Environment


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Figure 2. Illustration of the kinetic-type mechanism of interion interaction in the binuclear iron(II) cluster stabilizing the ) − ℎ)! level. the two-step electron transfer in an iron(II) binuclear cluster in the (ls)a-(hs)b is examined. At the first step, the electron jumps from the half-filled u-orbital of the hs center b to the empty uorbital of the ls center a giving rise to the excited charge transfer (CT) state with an energy that exceeds by K the energy of the (ls)a-(hs)b state, where K is the intracenter Coulomb repulsion parameter. At the second step, the electron jumps back to the center b thus restoring the ground manifold. Such an electron transfer mixing of the ground- and CT-states leads to the stabilization of the ground level by the value that proves to be (up to the second order of the perturbation theory) proportional to t 2 K , where t is the electron transfer integral. One can see that to some extent this interaction is similar to the well-known kinetic exchange interaction introduced by Anderson.44 The (ls)a-(ls)b and (hs)a-(hs)b states of the dimer are also stabilized by the second order electron transfer processes, but these stabilizations are weaker than that for the first excited level corresponding with configurations (ls)a-(hs)b and (hs)a-(ls)b because the Coulomb repulsion (integral K ) of two electrons occupying the same orbital is always stronger than the inter-orbital Coulomb repulsion. From this it follows that the kinetic-type interaction also leads to the energy of the first excited level lower by " than ∆,, , /2. Thus, ∆,, , and " can be regarded as parameters taking into account both the contributions of intercenter Coulomb interaction and electron transfer.

2.3. The effect of intra- and intercluster interactions on the spin transformation in binuclear clusters 14 ACS Paragon Plus Environment

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In this Section, we discuss different types of spin transformations arising in crystals hs hs composed of binuclear clusters and described by the order parameters τ ls x ls and τ x . We aim

to make clear the mutual influence of the cooperative interaction and internal parameters of the binuclear cluster consisting of two SCO ions. In the problem under examination along with the vibronic parameters %&' , %(' and the crystal field gap ∆hs , ls between the hs- and ls-states characterizing one SCO ion, the intracluster Coulomb and intercluster cooperative interactions are also involved. In order to elucidate distinctly the role of the named interactions in the spin transformation, the characteristic parameter of one of these interactions will be varied, while the parameters of the other interactions will be kept constant. Let us proceed with the case when the parameter ∆hs , ls changes, while the parameters " and remain constant. Later on, instead of the + + − * , where crystal field gap ∆hs , ls it is convenient to introduce the parameter ) = * + * = ∆,, , + ℏ −

υ0./

ℏ123

41 +

523,63 78/ 9 ℏ163 7./

:

(21)

and + * = ℏ −

υ08/

ℏ163

41 +

563,63

ℏ163

9

:

(22)

can be easily obtained from Eq.(13) and represent the electron-vibrational energies of the binuclear cluster at low temperatures, when the excited vibrational levels with ; = ; ≠ 0 are not populated, τ ls xls = −

υls and τ hs x hs = 0 . In fact, the gap ) represents the difference in hωls

energies between the zero vibrational levels of the ground (ls) and upper (hs) electronic states. The influence of the parameter ) is illustrated with Figure 3. The curves presented in Figure 3 were calculated with the parameters υls = 2102 cm−1 , υhs = 196 cm−1 , ωls = 200cm−1 ,

ωhs = 161 cm−1 equal to those for the Fe(II) ion in the [Fe(ptz)6](BF4)2 crystal.35,45,46 For the parameters of cooperative interaction, the values J ls , ls λ2 = 595 cm −1 ,

J hs , ls λ2 = 471 cm −1 obtained in Ref.

35

J hs , hs λ2 = 373 cm −1 ,

for this crystal were used. Here and further on in the

simulation of the properties of SCO systems we keep the mentioned values of the characteristic parameters and suppose that they differ insignificantly for SCO Fe(II) ions in the octahedral nitrogen surrounding in different crystals. The value which is presented in Figure 3 and which appears as is determined as follows:

= , + 0.5, + , )

(23) 15

ACS Paragon Plus Environment


hs , hs hs ,ls where , , , and , are the populations of the levels with the energies Edim , p , p ′ , Edim , p , p′ ls , hs and Edim , p , p ′ , respectively. From Figure 3 it is seen, that the decrease of the gap ) leads to a

crucial change in the type of the spin transition. At ) = 1600 cmAB the spin transition contains two steps and it is not complete, the maximum value of is about 0.8. With decrease of

) the transition becomes steeper, the step is more pronounced, the room temperature value n&' grows, and a hysteresis loop appears on the first step. A further decrease in ) leads to hysteresis loops at the two steps and, finally, at ) = 1250 cmAB a transition with one hysteresis loop takes place, the high temperature branch of the hysteresis loop demonstrates a step.

∆hl = 1250, 1300, 1400, 1600 cm

-1

1.0 0.8 0.6

nhs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 26

δ = 195 cm

0.4

-1

2

λ = 0.016

0.2 0.0 50

100

150

200

250

300

Temperature, K

Figure 3. Thermal behavior of the high-spin fraction calculated with : = 0.016, " = 195 cm-1 and following values of the gap ) : 1250 cm-1 (1),1300 cm-1 (2), 1400 cm-1(3) and 1600 cm-1 (4). It is worth noting that the increase of the parameter characterizing the intercluster interaction and the decrease of the gap ) values lead to similar effects. Indeed, the increase of leads to the transformation of the hs-fraction curve demonstrating two steps into the curve with one hysteresis loop. For intermediate values of a two-step curve with a hysteresis loop in the lower temperature step as well as a two-step curve with hysteresis loops in both steps are possible. It is 16 ACS Paragon Plus Environment

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worth noting that the hs-fraction as a function of demonstrates the presence of steps in both branches of the hysteresis loop for : = 0.0164 . Moreover, the increase of shifts the spin transition to lower temperatures. This result is transparent. The stronger the cooperative interaction the lower is the temperature of the transition. The effect of " is illustrated in Figure 4. Since " characterizes the position of the ls-hs cluster states the decrease of " leads to the higher energy of these states (Figure 1) and as a consequence the range of temperatures wherein the hs-

ls configuration exists becomes narrower. The curves corresponding to different values of " are not symmetric. 2

λ = 0.0164, 0.016, 0.014, 0.012

1.0 0.8

nhs

0.6 0.4 ∆hl = 1300 cm

0.2

δ = 195 cm

-1

-1

0.0 0

50

100

150

200

250

300

Temperature, K

Figure 4. The high-spin fraction calculated for) = 1300 cmAB , " = 195 cmAB and different values of : . δ = 170, 195, 220 cm

-1

1.0 0.8 0.6

nhs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.4 0.2

∆hl = 1300 cm

-1

2

λ = 0.016

0.0 50

100

150

200

250

300

Temperature, K

Figure 5. The population of the hs-level as a function of the parameter " for ) = 1300 GHAB ,

: = 0.016. 17 ACS Paragon Plus Environment


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The wider hysteresis loop is obtained for " = 170 cmAB . The increase of " gives rise to a twostep transition with not equally wide hysteresis loops at each step. It is noteworthy that for

" = 195 cmAB , the low temperature hysteresis loop is much wider than the high temperature one. Finally, at " = 220 cmAB the widths of both hysteresis loops are close in magnitude. Thus, the performed sample calculations clearly demonstrate the ability of the model to reproduce different types of observed spin transformations.

3. Results and Discussion As a first example demonstrating the applicability of our microscopic approach, we will consider the binuclear iron(II) compound [{Fe(bpym)(NCSe)2}2bpym]

14

whose structure is

shown in Figure 6, and the observed temperature induced spin transition near 120 K is evidenced by the χ T vsT curve shown in Figure 7a.14 It follows from the magnetic data reported in Ref.

14

and the extensive studies of the Mössbauer spectra,15,16 that in order to explain the observed spin transformation in the [{Fe(bpym)(NCSe)2}2bpym] compound it is also necessary to introduce in the Hamiltonian of the binuclear cluster, Eq. (8), the following term describing the zero-field splitting of the hs-state of Fe(II)-ions:

Figure 6. Molecular structure of [{Fe(bpym)(NCSe)2}2bpym] compound 14 (the crystallographic data have been taken from CSD). B

JKLM = NOP: − OO + 1)),

(24)

Q


Page 19 of 26

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where N is the zero-field splitting parameter, SZ is the operator of the Z-projection of the spin of the Fe(II) ion. Then for the parameters of cooperative interaction J ls , ls λ 2 , J hs , hs λ 2 , J hs , ls λ 2 as well as for the single ion parameters υls , υhs , ωls and ωhs we use the values known for the [Fe(ptz)6](BF4)2 crystal.35,45,46 We suppose that the difference of these parameters in the [Fe(ptz)6](BF4)2 and [{Fe(bpym)(NCSe)2}2bpym] crystals is not large by the reason explained above in Section 2.3. At the same time, if this difference does exist the error in calculations can be overcome by an appropriate choice of the parameters , " , ) and N which are considered as the fitting ones. In Figure 7 the temperature dependence of the product and the ℎ −fraction calculated with the set of the best-fit parameters are shown; the caption for the Figure includes the used parameter values. Quite a good agreement is obtained between the observed and

Figure 7. χ T vs T (a) and nhs vs T (b) dependences for the [{Fe(bpym)(NCSe)2}2bpym] crystal: Open circles – experimental data,14 solid line – theoretical curves calculated with the best-fit parameters ∆hl = 965 cm-1, δ = 198 cm-1, λ2 = 0.0125, D = 9.2 cm-1, gav= 2.19 obtained using downhill simplex minimization procedure.47 The fit accuracy is 3.4%.

experimental values. The obtained result obviously shows that the low temperature behavior of is governed by the zero-field splitting of the hs-5T2 state of one of the iron(II) ions. Then, in the temperature range 30-110 K, this ion still remains in the hs-state while the other one is in the -state. Further temperature increase leads to the abrupt increase in the population of the hsstate of the second ion. However, this population does not achieve its maximum value. Therefore, the population of the hs-state of the whole cluster is less than 1. Within the framework of the developed model we have also explained the magnetic properties of the first binuclear [{Fe(bt)(NCS)2}2bpym] cluster consisting of SCO ions.12 The structure of this complex is illustrated in Figure 8. As it was mentioned in the Introduction the complex demonstrates a two-step spin transition (Figure 9). The calculations show that the 19 ACS Paragon Plus Environment


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temperature dependence of for this system is nicely reproduced with the following set of parameters: ∆hl = 1500 cm-1, δ = 170 cm-1, λ2 = 0.0152, gav= 2.124. From the calculations it also follows that about 7.25% of the ions are always in the hs-state and do not participate in the spin transformation. It is worth noting that this amount of ions is very close to that experimentally determined in Ref.

12

from the magnetic susceptibility data. Concerning the low-temperature

residual hs-fraction we have assumed that the hs ions are distributed over the dimers in such a way that each dimer can contain not more than one ion of this type. The dimers containing the two hs ions are expected to be in excited states with very high energies due to Coulomb destabilization. So this type of dimers is excluded from our consideration.

Figure 8.Molecular structure of [{Fe(bt)(NCS)2}2bpym] 11 (the crystallographic data have been taken from CSD).

Figure 9. χ T vs T (a) and nhs vs T (b) dependences for the [{Fe(bt)(NCS)2}2bpym] crystal: open circles – experimental data,12 solid line – theoretical curve calculated with the best fit parameters ∆hl = 1500 cm-1, δ = 170 cm-1, λ2 = 0.0152, gav= 2.124 obtained using downhill simplex minimization procedure.47 The fraction of ions that do not participate in the spin transition and that are always in hs-configuration is x = 7.25%. The fit accuracy is 7.1%. 20 ACS Paragon Plus Environment

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4.

Concluding remarks In the present paper we have developed a new microscopic approach to the problem of

SCO in binuclear clusters. In the framework of this approach, there are taken into account the Coulomb interaction of the ions composing the cluster, the interaction of these ions with the local vibrations of the nearest surrounding and the intercluster cooperative interaction that arises from the coupling of the crystalline acoustic modes and local vibrations of each SCO ion in the cluster. The model also allows for different strengths of Coulomb interactions in the pairs of the ions in the ls-ls, hs-ls and hs-hs states. It has been clearly demonstrated that the proposed model describes all types of spin transitions observed in binuclear SCO systems. The role of all intrinsic system interactions in the spin transformation has been elucidated. It is shown that at certain values of the cooperative and Coulomb interactions the decrease of the energy gap between the zero vibrational levels of the ground (ls) and upper (hs) electronic states leads to the transformation of the hs-fraction curve, thus giving rise to two steps into a curve characterized by one wide hysteresis loop. At the same time, for intermediary values of this gap there appear curves with hysteresis loops in a low temperature step or in both steps. In the latter case, the widths of the loops will noticeably differ. The increase of the parameter characterizing the cooperative interaction leads to similar effects. Finally, the change of the parameter " describing to some extent the difference of the Coulomb interaction in the ls-ls, ls-hs and hs-hs states of the cluster is shown to lead to a diversity of spin transitions as well. In the framework of the developed approach we have also succeeded in the interpretation of

the

experimental

data

on

SCO

in

the

[{Fe(bpym)(NCSe)2}2bpym]

and

[{Fe(bt)(NCS)2}2bpym] complexes, demonstrating a spin transition with a plateau and a two-step transition, respectively. However, in spite of the success of the model in reproduction of all observable spin transitions, the model contains parameters that are considered as fitting ones. In principle, the parameter λ can be obtained from the solution of the dynamic problem of lattice vibrations with due account of molecular modes while the parameters of the Coulomb interaction as well as the energy gap between hs and ls states can be found from DFT or ab initio calculations. However, at the present stage of study additional efforts are needed to provide a sufficient level of accuracy in estimation of the intrinsic parameters that will give the possibility to reproduce all experimental data. Work in this direction is required.

Acknowledgement A.P., S.O., O.R., S.D. and S.K. are grateful to the Swiss National Science Foundation (project SCOPES IZ73ZO_152404/1) and the Supreme Council for Science and Technological Development of the Republic of Moldova (project 15.817.02.06F) for financial support. 21 ACS Paragon Plus Environment


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Diversity of Spin Crossover Transitions in Binuclear Compounds

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