# Cooperative Spin Transitions Triggered by Phonons in Metal Complexes Coupled to Molecular Vibrations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{6}](BF

_{4})

_{2}, binuclear [{Fe(bt)(NCS)

_{2}}

_{2}bpym] and tetranuclear [Fe(tpa){N(CN)

_{2}}]

_{4}·(BF

_{4})

_{4}·(H

_{2}O)

_{2}compounds containing iron ions. The approach is also applied to the description of the charge-transfer-induced spin transition in the [{(Tp)Fe(CN)

_{3}}{Co-(PY

_{5}Me

_{2})}](CF

_{3}SO

_{3}) complex.

## 1. Introduction

_{0.14}Co[Fe(CN)

_{6}]

_{0.71}·4.93H

_{2}O stoichiometry [2]. Then a series of low-dimensional Fe/Co cyanide complexes demonstrating thermally or photo-induced metal to metal electron transfer was reported [21,22,23]. As an example of Fe/Co complexes, the dinuclear cyanide-bridged [{(Tp)Fe(CN)

_{3}}{Co-(PY5Me

_{2})}](CF

_{3}SO

_{3})·2DMF complex is to be mentioned [24].

## 2. Background of the Model

_{3}}{Co-(PY

_{5}Me

_{2})}] (CF

_{3}SO

_{3})·2DMF complex [24] is based on the abovementioned approach recently elaborated in Refs. [27,28,29,30]. The quintessence of this approach consists of the following. The systems demonstrating spin transitions represent molecular crystals. In these crystals, one can distinguish two types of vibrations playing different physical roles in the phenomenon of spin transitions: molecular modes that are localized and confined by the metal ion strongly coupled to its ligand surrounding and phonons that are interrelated with the relatively weak intermolecular interactions and therefore spread out over the crystal lattice. The interaction of the SCO ion with the molecular modes is dominant, because in molecular crystals, the intermolecular interactions are very weak as compared to the intramolecular ones and their thermal modulation can mainly give rise to the coupling with small energy acoustic long-wave vibrations. Thus, one can assume that the molecular vibrations are weakly disturbed by the vibrations arising from the interactions which unite the molecules in the crystal, and the ligands of the iron(II) ions in spin crossover compounds or of the Co-ion in the compound demonstrating CTIST participate both in the molecular and crystalline displacements. The symmetry adapted coordinates of the ligand surrounding can be represented as the normalized linear superposition of the coordinates of molecular (first term) and crystalline (second term) vibrations:

_{1}) mode for the j-th iron ion in

**n**-th SCO complex, the index j = 1, 1−2, 1–4 for mono-, bi- and tetranuclear SCO Fe(II) compounds, for the Co ion in the n-th Fe–Co complex ${\mathit{R}}_{\mathit{n}}\equiv \mathit{n}$ is the position vector of the Co ion, in this case j = 1 and the coordinate ${X}_{{n}_{1}}^{}$ refers to the molecular full-symmetric (A

_{1}) mode of the cobalt ion. ${q}_{\mathbf{\kappa}\nu}$ in Equation (1) represent the dimensionless vibrational coordinates of the phonons with the frequencies ${\omega}_{\nu}\left(\mathbf{\kappa}\right)\equiv {\omega}_{\mathbf{\kappa}\nu}$, $\mathbf{\kappa}$ is the phonon wave vector, the symbol $\nu $ denotes the branch of the phonon mode, and finally, $N$ is the number of ions or clusters in the crystal. The values

**n**, ${\mathit{R}}_{k}$ are the position vectors of these ligands, the coefficients ${u}_{k\text{}\alpha}^{{A}_{1}}$ describe the transformation from the Cartesian displacements $\Delta {R}_{k\alpha}$ to the full-symmetric coordinate, ${e}_{\alpha}\left(k,\mathbf{\kappa}\nu \right)$ are the phonon polarization vectors, $\alpha =X,\text{}Y,\text{}Z$. The dimensionless parameter $\lambda $ characterizes the weight of the phonon-induced displacements in the full vibrational coordinate.

## 3. Mononuclear Spin Crossover Systems

**n**, ${\omega}_{hs}^{}$ and ${\omega}_{ls}^{}$ are the frequencies of molecular vibrations. The elements of the diagonal matrix ${\tau}_{n}^{hs}$ are 1 and 0, while of ${\tau}_{n}^{ls}$ 0 and 1 for the hs and ls states, correspondingly. The meaning of the parameter $\lambda $ is determined by Equation (1).

^{2}, ${f}_{ls}=227\text{}\mathrm{N}{\mathrm{m}}^{-1}=1.14\xb7{10}^{5}\mathrm{c}{\mathrm{m}}^{-1}$/Å

^{2}, ${R}_{ls}$ = 2.0 Å, ${R}_{hs}$ = 2.2 Å, $D{q}_{ls}=2055\text{}{\mathrm{cm}}^{-1},D{q}_{hs}=1176\text{}{\mathrm{cm}}^{-1}$ [39], ${\omega}_{M}=23\text{}{\mathrm{cm}}^{-1}$ [39] $c\approx 2\xb7{10}^{5}\text{}\mathrm{cm}/\mathrm{s}$ [31], and the constants of interaction with the molecular vibrations in the ls and hs states: ${\upsilon}_{ls}=2102\text{}{\mathrm{cm}}^{-1}$, ${\upsilon}_{hs}=196\text{}{\mathrm{cm}}^{-1}$ [27] expressed through the above given crystal field parameters $D{q}_{ls}$ and $D{q}_{hs}$. It should be mentioned that the Debye frequency ${\omega}_{M}=23\text{}{\mathrm{cm}}^{-1}$ of the acoustic lattice vibrations was evaluated in [39] from their parabolic dependence on the wave vector $\kappa $ in the center of the Brillouin zone. This frequency is small because it is interrelated with the relative displacements of heavy metal centers. With the aid of Equation (7), the ratio of the parameters of cooperative interactions was estimated as: ${J}^{ls,ls}$:${J}^{hs,ls}$:${J}^{hs,hs}$ = 1.59:1.26:1. It should also be noticed that the relation between the values of ${J}^{ls,ls}$, ${J}^{hs,ls}$, ${J}^{hs,hs}$ provides a possibility to vary only one of these parameters. The performed examination shows that mononuclear SCO compounds can demonstrate three types of spin transitions: gradual, abrupt and those accompanied by a hysteresis loop. Depending on the internal parameters of the system, these transitions can occur even in the room temperature range.

_{6}](BF

_{4})

_{2}compound. From Figure 2, one can see that the accepted model well describes the observed χT of [Fe(ptz)

_{6}](BF

_{4})

_{2}in the whole temperature interval, namely, the gradual increase of χT in the range 100–129 K, the hysteresis loop between 129 and 135 K and the slow growth above 135 K.

## 4. Binuclear Spin Crossover Systems

_{2}}

_{2}bpym] crystal (Figure 3) containing as a structural unit iron(II) dimers. The theoretical consideration reproduces the observed two-step transition. The experimental curve was described quite well using the following set of parameters: Δ

_{hl}= 1540 cm

^{−1}, δ = 170 cm

^{−1}, ${J}^{ls,ls}$ = 9.08 cm

^{−1}, g

_{av}= 2.124 and the abovementioned ratio between the parameters ${J}^{ls,ls}$, ${J}^{hs,ls}$ and ${J}^{hs,hs}$. Since the observed product $\chi T$ as a function of temperature does not tend to zero at T→0, it was assumed that initially some of the iron ions are in the hs state, the fraction of these ions was determined as y = 7.25% under the assumption that they are uniformly distributed between the binuclear clusters. The inset in Figure 2 shows the populations n

_{0}, n

_{1}and n

_{2}of cluster configurations with 0, 1 and 2 ions in the hs state. It is seen that up to T = 125 K, the main part of ions is in the ls state, the number of dimers with one ion in the hs state is quite low. In the range of temperatures in between 150–200 K the biggest is the number of clusters with one ion in the hs state. It is worth noting that the non-pronounced step in the temperature dependence of the $\chi T$ product is observed, namely, in this range at about 175 K. At temperatures higher than 200 K, binuclear clusters with two hs ions dominate and are responsible for the second observed step in the thermal behavior of the χT product. At the same time, the number of clusters in which only one ion is in the hs state is a bit larger than the initial one at T < 100 K. The suggested model [28] of spin transitions in binuclear systems reproduces all observable types of these transitions and describes quite well the spin transformation in the [{Fe(bt)(NCS)

_{2}}

_{2}bpym] compound.

## 5. Spin Crossover in Tetranuclear Systems

_{i}= ls, hs) of intracluster interactions can be expressed in terms of the parameters ${U}_{ij}^{l}$. Due to intracluster interactions, the effective energy gap between the configurations ls-ls-ls-ls and hs-hs-hs-hs is determined as follows:

^{−1}and 151 cm

^{−1}for the hs and ls configurations, respectively. The new frequencies employed in calculations led to the redetermination of the ratio between the parameters of cooperative interaction as compared with that employed above for mono- and binuclear spin crossover systems: ${J}^{ls,ls}$:${J}^{hs,ls}$:${J}^{hs,hs}$ 3.22:1.79:1. The system of self-consistent equations for the order parameters can be found in Ref. [29].

_{2}}]

_{4}·(BF

_{4})

_{4}·(H

_{2}O)

_{2}compound. From Figure 5, it is seen that the model gives quite a good description of the experimental data with the set of parameters representing a part of the figure caption. The course of the spin transformation in this compound can be easily followed analyzing the temperature dependence of the populations of configurations with different number of iron(II) ions in the hs state. Up to 150 K, the majority of Fe

_{4}clusters are in the ls state, that is confirmed by the plateau observed in the experimental curve in the range 100–160 K then the number of two hs ions situated along the diagonal of the square significantly increases and achieves its maximum value at about 250 K. Namely, this increase provokes the step in the temperature dependence of the effective magnetic moment μ

_{eff}in the range 210–250 K. At temperatures higher than 325 K, clusters with 4 hs ions dominate. At the same time, in this temperature range there exists a non-negligible amount of clusters with two or three hs ions. Therefore, the observed effective magnetic moment does not achieve its maximum value corresponding to four iron(II) ions in the hs state. Here, it is worth noting that in polynuclear spin crossover compounds the interplay between the intra- and intercluster interactions plays a crucial role in the type and temperature of the spin transformation.

## 6. Charge-Transfer-Induced Spin Transition in Binuclear {Fe(μ − CN)Co} Complexes

^{II}−CN−ls-Co

^{III}units into paramagnetic ls-Fe

^{III}–CN–hs-Co

^{II}ones was observed in the Prussian Blue analogue (PBA) of K0.14Co[Fe(CN)

_{6}]

_{0.71}·4.93H

_{2}O stoichiometry [2]. In the present section, it is demonstrated how the approach above, applied to mono-, bi- and tetranuclear spin crossover systems, can be generalized to the case of molecular crystals comprising as a structural unit binuclear cyanide-bridged iron-cobalt systems exhibiting CTIST. As an example, the spin transformation in the [{(Tp)Fe(CN)

_{3}}{Co-(PY5Me

_{2})}](CF

_{3}SO

_{3}) [24] compound is discussed.

^{III}−ls-Fe

^{II}and (II) hs-Co

^{II}−ls-Fe

^{III}. The following states were taken into consideration: the ground non-degenerate state arising from the ls-Co

^{III}−ls-Fe

^{II}configuration and two groups of excited states. The first group of excited states, formed by the direct product of two ground Kramers doublets belonging to the ls-Fe

^{III}and hs-Co

^{II}ions, has the degeneracy ${g}_{hs}^{1}$ = 4 and it is separated from the ground ls-Co

^{III}−ls-Fe

^{II}state by the energy gap ${\mathsf{\Delta}}_{hs}^{1}=\mathsf{\Delta}$. The second group of the cluster excited states arising from the ground Kramers doublet of the ls-Fe

^{III}ion and the quadruplet of the hs-Co

^{II}ion possesses the energy ${\mathsf{\Delta}}_{hs}^{2}=\mathsf{\Delta}+9\left|{\lambda}_{1}\right|/4$ and the degeneracy ${g}_{hs}^{2}$ = 8, where the parameter of spin–orbit coupling ${\lambda}_{1}$ is about −180 cm

^{−1}for the hs-Co

^{II}ion. The magnetic exchange interaction is omitted. Only the ground Kramers doublet of the ls-Fe

^{III}ion is taken into account because the excited quadruplet of this ion is much higher in energy due to large spin–orbital coupling, $\lambda $ = −486 cm

^{−1}[42].

_{3}}{Co-(PY5Me

_{2})}](CF

_{3}SO

_{3}) compound [24] it follows that with temperature rise, the Fe–ligand distances are almost constant in the whole temperature range, while the first coordination sphere of Co ion significantly increases with temperature. From this, it follows that the transformation ls-Co

^{III}$\to $ hs-Co

^{II}assures the cooperativity of the transition. Correspondingly, in the model there are taken into consideration the single state arising from configuration ls-Co

^{III}−ls-Fe

^{II}and 12 above mentioned states of the hs-Co

^{II}–ls-Fe

^{III}configuration. Further on, these configurations will be referred to as the ls and hs ones, respectively. The problem is solved in the mean field approximation (see Ref. [30]).

^{−1}[30], ${\upsilon}_{ls}^{{\mathrm{Co}}^{\mathrm{III}}}$ = 2408 cm

^{−1}[30], $\hslash {\omega}^{ls-{\mathrm{Fe}}^{\mathrm{III}}}$ = 390 cm

^{−1}[43], $\hslash {\omega}^{ls-{\mathrm{Fe}}^{\mathrm{II}}}$ = 200 cm

^{−1}[39], $\hslash {\omega}^{hs-{\mathrm{Co}}^{\mathrm{II}}}$ = 92 cm

^{−1}, $\hslash {\omega}^{ls-{\mathrm{Co}}^{\mathrm{III}}}$ = 108 cm

^{−1}[30]. The procedure of the calculation of the vibronic coupling constants ${\upsilon}_{hs}^{{\mathrm{Co}}^{\mathrm{II}}}$ and ${\upsilon}_{ls}^{{\mathrm{Co}}^{\mathrm{III}}}$ is described in detail in Ref. [30]. The parameters ${J}^{ls,ls},{J}^{hs,hs}$ and ${J}^{hs,ls}$ are evaluated with the use of Equation (7). The Debye frequency is calculated as [44]:

_{0}is the unit cell volume and c is the speed of sound. For Ω

_{0}= 5126.3 Å

^{3}[24] and c = 2 × 10

^{5}cm/s one obtains $\hslash {\omega}_{M}$ = 24 cm

^{−}

^{1}[30]. It is seen that the Debye frequency calculated in this way is practically the same as that presented in [39] and used above for the description of SCO Fe(II) complexes. One easily finds the following ratio between the parameters of cooperative interaction ${J}^{ls,ls}:{J}^{hs,hs}$:${J}^{hs,ls}$ = 1.549:1.245:1. This relation allows us to use only one fitting parameter ${J}^{ls,ls}$ instead of three independent ${J}^{j,l}$ parameters. A weak exchange interaction through the cyanide bridge in the pair $ls-{\mathrm{Fe}}^{\mathrm{III}}$ − $hs-{\mathrm{Co}}^{\mathrm{II}}$ can be neglected.

_{3}}{Co-(PY

_{5}Me

_{2})}](CF

_{3}SO

_{3}) compound [24] is performed. Since at very low temperatures the observed χT product has a non-vanishing value, it means that from the very beginning a part of binuclear complexes ${y}_{hs}$ is in the ls-Fe

^{III}−hs-Co

^{II}state. Therefore, the χT product is calculated as $\chi T={\left(\chi T\right)}_{hs}\left({n}_{hs}\left[1-{y}_{hs}\right]+{y}_{hs}\right)$, where the hs fraction ${n}_{hs}$ = ${Z}_{hs}/Z$ and ${\left(\chi T\right)}_{hs}$ describes the magnetic behavior of the ls-Fe

^{III}−hs-Co

^{II}complex. The exchange interaction of the ls-Fe

^{III}and hs-Co

^{II}ions is neglected; therefore, the magnetic susceptibility of the hs fraction can be presented as a sum of the susceptibilities of the ls-Fe

^{III}and hs-Co

^{II}ions. Here, it should be mentioned that for the correct description of the magnetic behavior of the compound under examination it is necessary to take into consideration the excited energy levels for both the ls-Fe

^{III}and the hs-Co

^{II}ions neglected at the previous stage. For example, the mixing of the ground Kramers doublet of the ls-Fe

^{III}ion with the excited quadruplet by the external magnetic field results in the temperature independent paramagnetism significant for the correct interpretation of the magnetic behavior. The detailed description of this procedure can be found in Ref. [30].

_{3}}{Co-(PY5Me

_{2})}](CF

_{3}SO

_{3}) complex (Figure 6).

## 7. Concluding Remarks

_{6}](BF

_{4})

_{2}, binuclear [{Fe(bt)(NCS)

_{2}}

_{2}bpym] and tetranuclear [Fe(tpa){N(CN)

_{2}}]

_{4}·(BF

_{4})

_{4}·(H

_{2}O)

_{2}compounds based on iron ions. The approach is also applied to the description of the charge-transfer-induced spin transition in the [{(Tp)Fe(CN)

_{3}}{Co-(PY

_{5}Me

_{2})}](CF

_{3}SO

_{3}) complex.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

spin crossover | SCO |

charge-transfer-induced spin transitions | CTIST |

high-spin ion | hs ion |

low-spin ion | ls ion |

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**Figure 1.**Schematic image of molecular SCO crystals: the elastic interactions in the crystal corresponding to intra- and intermolecular vibrations are represented by springs with different rigidity ${k}_{1}$ and ${k}_{2}$.

**Figure 2.**χT vs. T dependence for the [Fe(ptz)

_{6}](BF

_{4})

_{2}crystal: circles—experimental data [31], solid line—theoretical curve calculated with Δ

_{hl}= 460 cm

^{−1}, ${J}^{ls,ls}$ = 4.19 cm

^{−1}and ${g}_{Fe}=2.3$.

**Figure 3.**χT vs. T dependence for the [{Fe(bt)(NCS)

_{2}}

_{2}bpym] crystal: symbols—experimental data [9], solid line—theoretical curve calculated with Δ

_{hl}= 1540 cm

^{−1}, δ = 170 cm

^{−1}, ${J}^{ls,ls}$ = 9.08 cm

^{−1}and g

_{av}= 2.124. The fraction of ions that do not participate in the spin transition and that are always in hs configuration is y = 7.25%. The inset is the thermal variation of the population of different cluster configurations (subscripts indicate the number of Fe ions in hs state).

**Figure 5.**μeff vs. T dependence for the [Fe(tpa){N(CN)

_{2}}]

_{4}·(BF

_{4})

_{4}·(H

_{2}O)

_{2}: symbols—experimental data [17], solid line—theoretical curve calculated with Δ

_{hl}= 5800 cm

^{−1}, δ

_{s}= 280 cm

^{−1}, δ

_{d}= −320 cm

^{−1}, ${J}^{ls,ls}$ = 2.46 cm

^{−1}and g

_{av}= 2.37. The inset shows the thermal variation of the populations of different cluster configurations.

**Figure 6.**μ

_{eff}vs. T dependence for the [{(Tp)Fe(CN)

_{3}}{Co-(PY5Me

_{2})}](CF

_{3}SO

_{3}) complex: symbols—experimental data [24], solid line—theoretical curve calculated with Δ

_{hl}= 410 cm

^{−1}, ${J}^{ls,ls}$ = 1.435 cm

^{−1}and y

_{hs}= 4.0%. The inset is the thermal variation of the population of different complex configurations.

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**MDPI and ACS Style**

Klokishner, S.; Ostrovsky, S.; Palii, A.; Tsukerblat, B.
Cooperative Spin Transitions Triggered by Phonons in Metal Complexes Coupled to Molecular Vibrations. *Magnetochemistry* **2022**, *8*, 24.
https://doi.org/10.3390/magnetochemistry8020024

**AMA Style**

Klokishner S, Ostrovsky S, Palii A, Tsukerblat B.
Cooperative Spin Transitions Triggered by Phonons in Metal Complexes Coupled to Molecular Vibrations. *Magnetochemistry*. 2022; 8(2):24.
https://doi.org/10.3390/magnetochemistry8020024

**Chicago/Turabian Style**

Klokishner, Sophia, Serghei Ostrovsky, Andrew Palii, and Boris Tsukerblat.
2022. "Cooperative Spin Transitions Triggered by Phonons in Metal Complexes Coupled to Molecular Vibrations" *Magnetochemistry* 8, no. 2: 24.
https://doi.org/10.3390/magnetochemistry8020024