Spin Switching in Crystals Containing Tetranuclear Fe₂Co₂ Clusters as Structural Units: Interplay of Intra- and Intercluster Interactions

Abstract

A microscopic model has been elaborated for the description of charge transfer-induced spin transitions in crystals containing tetranuclear Fe₂Co₂ clusters as structural units. The model takes into account the energy spectrum of each Fe₂Co₂ cluster, formed by the states arising from its initial configuration, two low-spin Fe^II and two low-spin Co^III, final configuration two low-spin Fe^III, and two high-spin Co^II, as well as the states that originate from four intermediate configurations of the type of low-spin Fe^II, low-spin Co^III, low-spin Fe^III, and high-spin Co^II. Two different types of cooperative interactions are accounted for in the model, namely, the electron–deformational coupling arising as a result of the observed elongation of the cobalt-nitrogen bonds under the low-spin Co^III$\to$ high-spin Co^II transition and the interaction via the field of phonons that originates from the coupling of the Co-ions with the full symmetric displacements of the nearest ligand surrounding, which are modulated by crystalline vibrations. The role of cooperative interactions is discussed in detail. Different types of spin transitions are predicted, including the gradual and abrupt ones as well as those manifesting hysteretic behavior. Within the framework of the developed approach, a qualitative and quantitative explanation of the experimental data on the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O compound recently reported oniere is given.

1. Introduction

The subject of charge-transfer-induced spin transitions (CTISTs) has come a long way in studies. Primarily, the interest in these transitions was concentrated on solids, predominantly on Prussian blue analogs [1,2,3,4,5,6,7,8,9,10,11,12] containing iron and cobalt ions as structural elements of the lattice. However, this type of solids possesses different defects [1,2], which prevent clear observation and detection of the electron transfer inside the Fe-Co pair, which involves its following states: ls-Fe^II − ls-Co^III, ls-Fe^III − hs-Co^II, where the symbols ls- and hs- denote the low-spin and high-spin states of the ions participating in the electron transfer, for all that the initial state of the pair ls-Fe^II − ls-Co^III is diamagnetic, while the final state ls-Fe^III − hs-Co^II is a paramagnetic one.

In the desire to obtain an object of study devoid of the defects, inherent to bulk compounds, and to reveal the role of intra- and intercluster interactions in the occurrence of the electron transfer accompanied by the change of structural, magnetic and spectroscopic characteristics, the researchers switched to the synthesis and experimental study of molecular crystals, the structural element of which is definitely determined and represented by Fe-Co clusters of different nuclearity. Actually, the breakthrough in the area occurred in 2004, when K. Dunbar and her team reported the trigonal bipyramidal pentanuclear molecular [Co₃Fe₂] complex demonstrating CTIST [13]. At present, the phenomenon of CTIST is detected in a series of binuclear, tetranuclear, pentanuclear and octanuclear clusters. The number of such type clusters with different nuclearity is permanently increasing. A description of the observed results in the field of CTIST in molecular systems can be found, for instance, in references [14,15,16,17]. The experimental study revealed that, similar to Prussian blue analogs, the above-mentioned clusters demonstrate both thermally and photoinduced electron transfer, converting a diamagnetic molecular complex into a paramagnetic one. Thus, the systems above listed belong to the class of bistable materials possessing two close-lying states, which can be interconverted by temperature, light, pressure and guest molecules. It is clear that the most interesting for the observation of charge-transfer-induced spin transitions are complexes containing a number of Fe-Co pairs larger than one because in this case when passing from the low-spin state of the complex to its high-spin one a pronounced change of the effective magnetic moment can be detected, that is important both for better understanding of the spin transformation as well as for future practical applications.

Recently the switching properties of the cyanide-bridged Fe/Co square complex {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O (where Tp* = tris(3,5-dimethylpyrazol-1-yl)hydroborate, bpy^Me = 4,4′-dimethyl-2,2′-bipyridine, OTf = trifluoro-methanesulfonate anion and DMF = dimethylformamide), that was already reported in 2011 [18], were examined in [19] with the aid of single-crystal X-ray diffraction and X-ray absorption spectroscopy at both Fe and Co sites. These two applied techniques allowed for the restoration of the complete picture of electron transfer and elucidated the role played by the iron and cobalt ions in this transfer.

Three new cyanide-bridged molecular square complexes {[Co(bpy)₂]₂[Fe(tp′)(CN)₃]₂)}(PF₆)₂, {[Co(dmbpy)₂]₂[Fe(tp′)(CN)₃]₂)}(PF₆)(OTf)·2MeOH, {[Co(dtbbpy)₂]₂[Fe(tp′)(CN)₃]₂)}(PF₆)·2MeOH·3EtOH (tp′ = hydrotris(3-methylpyrazol-1-yl)borate, bpy = 2,2′-bipyridine, dmbpy = 4,4′-dimethyl-2,2′-bipyridine, and dtbbpy = 4,4′-di-tert-butyl-2,2′-bipyridine, OTf = trifluoromethane-sulfonate) were obtained in [20] by systematic ligand substitution. Their tunable behavior arising from intramolecular electron transfer was examined both in solution and in the solid state. It was demonstrated that in the whole temperature range, the solid state complex {[Co(bpy)₂]₂[Fe(tp′)(CN)₃]₂)}(PF₆)₂ [20] is in the high-spin phase with the electronic configuration [(hs-Co^II)₂(ls-Fe^III)₂]. A charge-transfer-induced spin transition under rising temperature is demonstrated by the {[Co(dmbpy)₂]₂[Fe(tp′)(CN)₃]₂)}(PF₆)(OTf)·2MeOH compound [20]. At the same time, the characteristics of the spin transition in the {[Co(dtbbpy)₂]₂[Fe(tp′)(CN)₃]₂)}(PF₆)·2MeOH·3EtOH compound [20] depend on the degree of desolvation of the compound in the solid state. Recently, in [21], on the basis of cyanide-bridged molecular squares, supramolecular hybrids have been prepared. These square grids comprising cyanide-bridged tetranuclear complexes demonstrated multistep redox behavior, multistep spin crossover and single-molecule magnet(SMM)-type behavior.

In spite of the sufficiently vast experimental material on metal complexes exhibiting charge-transfer-induced spin transitions, the theoretical description of this phenomenon based on microscopic models is not presented in a large number of papers. In paper [22], a model was developed to describe the magnetic characteristics and Mössbauer spectra of the cyano-bridged pentanuclear {[Os(CN)₆]₂[Fe(tmphen)₂]₃} cluster exhibiting CTIST. The interplay between the intracluster Co-Fe electron transfer and cooperative interactions in a model crystal containing these pairs as a structural element was analyzed in [23]. The common feature of the models suggested in [22,23] as well as in earlier papers [24,25] is the allowance for different elasticity of intra- and intermolecular spaces and for the electron–deformational interaction as the cooperative one promoting the spin transitions induced by electron transfer between the electronic shells of one and the same ion (called traditionally as spin crossover) or the electronic shells of different ions (described by the notion “charge-transfer-induced spin transitions”, for instance, (ls-Fe^II)(ls-Os^III)→ (hs-Fe^III)(ls-Os^II) and (ls-Fe^II)(ls-Co^III)→ (ls-Fe^III)(hs-Co^II), etc.).

The magneto–electric coupling in binuclear Fe-Co cyanide clusters containing a tunneling electron gives promise since it provides access to a new type of control of the magnetic state by an electric field in spintronic devices. The latter represents an important advance in molecular spintronics and a basis for magnetic molecule-based quantum computing. Thereby, in paper [26], the influence of the electric field on the polarizability, magnetic and spectroscopic characteristics of binuclear Fe-Co cyanide-bridged clusters was examined. The approach, which was applied in paper [27] for examination of charge-transfer-induced spin transitions in crystals, containing a binuclear Fe-Co cluster as a structural unit, accounts for the interaction of the Co- ion with molecular (local) vibrations and the vibrations propagating along the crystal (phonons). For all that, as far as we know, a microscopic model describing the charge-transfer-induced spin transformation in tetranuclear compounds of the type of $M_2{M}_2^{\prime }$ (where $M \mathrm{a}\mathrm{n}\mathrm{d}{M}^{\prime }$ are transition metal ions) recently obtained and characterized [19] has not been discussed yet. The aim of the present paper is to elaborate such a model for elucidation of the origin of spin transitions in a crystal composed of tetranuclear Co₂Fe₂ clusters that takes into account the specifics of the cluster structure and the intracluster electron transfer facilitating these transitions, reveals the origin of intercenter interactions responsible for cooperativity in the crystal, predicts the possible types of spin transitions in these compounds as well as the conditions for observation of their hysteretic magnetic behavior and, finally, provides a reasonable qualitative and quantitative interpretation of the available experimental data on the Co₂Fe₂ compounds.

The model suggested below will take into account the energy spectrum of each Co-Fe pair as well as two types of cooperative interactions, promoting the charge-transfer-induced spin transition, and, namely, the electron-deformational interaction and the coupling of tetranuclear clusters through the field of phonons.

2. The Model

The experimental data on the $\chi T$ product of the tetranuclear square Co₂Fe₂ clusters, reported in [19], evidence electron transfer from the diamagnetic ls- Fe^II ion to the diamagnetic ls-Co^III ion. Under this transfer, as mentioned above, each cobalt-iron pair in this type of cluster finally transforms into the ls- Fe^III−hs-Co^II pair. Thus, in the initial and final states of the whole square cluster, its configurations can be denoted as 2 ls-Fe^II − 2ls- Co^III, 2 ls-Fe^III − 2hs- Co^II (Figure 1), respectively. For all that a tetranuclear square cluster also possesses four other configurations of the type of ls-Fe^II − ls- Co^III− ls-Fe^III − hs- Co^II, in which in one Co-Fe pair the electron transfer has occurred, while the other pair remains still diamagnetic. The schematic illustration of these configurations is given in Figure 1. The charge-transfer-induced spin transition that takes place in the compounds under examination is accompanied by a noticeable elongation of the Co-ligand bonds. For instance, in the {[Co(dmbpy)₂]₂[Fe(tp′)(CN)₃]₂)}(PF₆)(OTf)·2MeOH compound [19], which shows thermally induced CTIST, the average Co-N distance undergoes an increase from 1.938 to 2.107 Å, confirming thus intracluster electron transfer. The conversion of ls-Fe^II ions, which are in the mixed nitrogen-carbon surrounding, creating a strong crystal field, into ls-Fe^III ions, occurs without any noticeable elongation of the iron-ligand bonds (average distances are 1.9595 Å and 1.959 Å for 80 K and 240 K, respectively [19]) because the transition from the ls-Fe^II state to the ls-Fe^III one is accompanied only by the decrease of the number of electrons in the t₂ shell by one. This means that the contribution of the iron ions to the deformation of the space between the tetranuclear Co₂Fe₂ clusters, which transfers the interaction between the clusters, is negligible as compared with that given by the Co-ions. Thus, for the description of CTIST in tetranuclear Co₂Fe₂ compounds as well as in binuclear CoFe and pentanuclear ones previously examined in our papers [22,23], the electron–deformational interaction is relevant only for the Co-ions. Since under metal–metal electron transfer, the crystal symmetry does not change, and all elements of the molecular and intermolecular volumes remain similar to themselves, the internal molecular ${\epsilon }_1$ and external (intermolecular) ${\epsilon }_2$ full symmetric strains are introduced in the model, bearing in mind that the intermolecular volume is far softer than that of the tetranuclear molecule itself. Otherwise, the transition ls-Co^III $\to$ hs-Co^II, accompanied by the expansion of the electronic shell of the cobalt ion, will not take place. In this case, within the framework of Kanamori’s approach [28] (see also [24,25]), the operator describing the interaction with strain looks as follows:

$$H_{st}={\displaystyle \frac{N{c}_1{\mathsf{\Omega }}_0{\epsilon }_1^2}{2}}+{\displaystyle \frac{N{c}_{2 }({\mathsf{\Omega }-\mathsf{\Omega }}_0){\epsilon }_2^2}{2}}+\sum _n{V}_n^{st}{\epsilon }_1$$

(1)

where ${\mathsf{\Omega }}_0$ and $\mathsf{\Omega }$ are the volumes of the Co₂Fe₂ cluster and the intermolecular space falling per one cluster, and $c_1$ and $c_2$ are the elastic moduli corresponding to the above-mentioned strains.

For the n-th cluster, the matrix of the operator $V_n^{st}$ of interaction with strain, written in the basis of states arising from one configuration 2ls-Co^III−2ls-Fe^II, four configurations of the type of ls-Fe^II−ls-Co^III− ls-Fe^III−hs-Co^II and one configuration 2ls-Fe^III−2hs-Co^II (Figure 1) of the molecule under examination, can be written as follows:

$$V_n^{st}=\left(\begin{array}{c}\begin{array}{cccccc}2v_{ls}& 0& 0& 0& 0& 0\\ 0& v_{ls}+v_{hs}& 0& 0& 0& 0\\ 0& 0& v_{ls}+v_{hs}& 0& 0& 0\\ 0& 0& 0& v_{ls}+v_{hs}& 0& 0\\ 0& 0& 0& 0& v_{ls}+v_{hs}& 0\\ 0& 0& 0& 0& 0& 2v_{hs}\end{array}\end{array}\right)$$

(2)

where $v_{ls},v_{hs}$ represent the constants of interaction of the Co-ion with the totally symmetric strain in the states ls-Co^III and hs-Co^II, respectively. A simple transformation allows us to represent the matrix of the operator $V_n^{st}$ in the following form:

$$V_n^{st}=(v_{hs}-v_{ls})\left(\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}-1\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 1\end{array}\end{array}\end{array}\right){+(v}_{ls}+v_{hs}){\mathit{I}}_{\mathit{n}}=(v_{hs}-v_{ls}){\tau }_n+{(v}_{ls}+v_{hs}){\mathit{I}}_{\mathit{n}}$$

(3)

Here, $\mathit{I}$ is the unit matrix determined in the accepted basis of states. Further, the second term on the right side of Equation (3) is omitted because it leads to the shift of all energy levels of the system to the same value. Applying the presentation of the operator $V_n^{st}$ in the form given by Equation (3), minimizing Hamiltonian (1) over the strain ${\epsilon }_1$ and supposing that for a uniform crystal compression (or extension) the following approximate relation holds [24,25]

$${\epsilon }_2\approx {\epsilon }_1{\displaystyle \frac{c_1}{c_2}}$$

(4)

one obtains the operator of electron-deformational interaction for the whole crystal in the following form:

$$H_{st}=-{\displaystyle \frac{1}{2N}}{J}_1{\sum }_{n,m}{\tau }_n{\tau }_m,$$

(5)

where the parameter $J_1$ of electron–deformational interaction is determined as

$$J_1={\displaystyle \frac{{c_2 (v_{hs}-v_{ls})}^2}{c_1(c_2{\mathsf{\Omega }}_0+c_1\left(\mathsf{\Omega }-{\mathsf{\Omega }}_0\right))}},$$

(6)

N denotes the number of tetranuclear square clusters in the crystal.

The coupling of the two cobalt ions with the vibrations of the nearest surrounding is taken into account in the model as well. Bearing in mind that the dominating localization effect is produced by the interaction of cobalt ions with the full symmetric breathing modes of the nearest surroundings of the cobalt ions, and this interaction along with the electron-deformational one is also responsible for the observed elongation of the cobalt-ligand bonds when passing from the ls-Co^III state to the hs-Co^II one, in the Hamiltonian the following term, written in the accepted above basis of electronic states, is introduced as follows:

$$H_{e\upsilon }^n=\left(\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}{V_{ls}q}_{1n}+{V_{ls}q}_{2n}\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ {V_{ls}q}_{1n}+{V_{hs}q}_{2n}\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ {V_{hs}q}_{1n}+{V_{ls}q}_{2n}\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}{V_{hs}q}_{1n}+{V_{ls}q}_{2n}\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ {V_{ls}q}_{1n}+{V_{hs}q}_{2n}\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ {V_{hs}q}_{1n}+{V_{hs}q}_{2n}\end{array}\end{array}\end{array}\end{array}\right),$$

(7)

here $q_{1n}$ and $q_{2n}$ are the full symmetric vibrational coordinates of the local surroundings of the Co1 and Co2 ions in the tetranuclear n-th Co₂Fe₂ complex, $V_{ls}$ and $V_{hs}$ are the matrix elements of the operator of electron–vibrational interaction in the ls-Co^III and hs-Co^II states. The interaction of the iron ions with this type of vibration is not taken into account because the effect produced by this interaction is much smaller than that for Co-ions. In fact, during the transition, the configuration of the Co-ions changes from $t_2^6$ to $t_2^5{e}^2$, while for iron ions from $t_2^6$ to $t_2^5.$ Introducing the collective modes of the cobalt ions $Q_{1n}$ and $Q_{2n}$, which are connected with the local coordinates $q_{1n}$ and $q_{2n}$ by the relations

$$Q_{1n}={\displaystyle \frac{q_{1n}-q_{2n}}{\sqrt{2}}},Q_{2n}={\displaystyle \frac{q_{1n}+q_{2n}}{\sqrt{2}}},$$

(8)

the operator $H_{ev}^n$ can be rewritten in the form

$$H_{e\upsilon }^n=\upsilon (Q_{1n}{\sigma }_n+Q_{2n}{\tau }_n)$$

(9)

Here, the term ${\displaystyle \frac{V_{hs}+V_{ls}}{\sqrt{2}}}{Q}_{2n}{\mathit{I}}_n$, which leads to an equal shift of all energy levels, is omitted, the matrix ${\sigma }_n$ looks as follows

$${\sigma }_n=\left(\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ -1\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 1\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}1\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ -1\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}\end{array}\right),$$

(10)

the matrix ${\tau }_n$ is given by Equation (3), $\upsilon ={\displaystyle \frac{V_{hs}-V_{ls}}{\sqrt{2}}}$. The symmetrized coordinates $Q_{1n}$ and $Q_{2n}$ are expressed through the phonon creation $a_{\kappa \nu }^{+}$ and annihilation $a_{\kappa \nu }$ operators by the following relations:

$$\begin{gathered} Q_{1\mathit{n}}={\sum }_{\mathit{\kappa },\nu }{f}_{\mathit{\kappa }\nu }^{-}\left(A_1\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-i\mathit{\kappa }\mathit{n}\right)(a_{\mathit{\kappa }\mathit{\nu }}^{+}+a_{-\kappa \nu }){({\displaystyle \frac{\hslash }{2N{\omega }_{\mathit{\kappa }\mathit{\nu }}}})}^{1/2}, \\ {Q}_{2\mathit{n}}={\sum }_{\mathit{\kappa },\nu }{f}_{\mathit{\kappa }\nu }^{+}\left(A_1\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-i\mathit{\kappa }\mathit{n}\right)(a_{\mathit{\kappa }\mathit{\nu }}^{+}+a_{-\kappa \nu }){({\displaystyle \frac{\hslash }{2N{\omega }_{\mathit{\kappa }\mathit{\nu }}}})}^{1/2}, \end{gathered}$$

(11)

where $\mathit{\kappa }$ is the phonon wave vector, $\nu$ denotes the branch of the phonon mode, and the coefficients $f_{\mathit{\kappa }\nu }^{-}\left(A_1\right)$ and $f_{\mathit{\kappa }\nu }^{+}\left(A_1\right)$ perform the unitary transformation from the symmetrized displacements $Q_{1\mathit{n}}$, $Q_{2\mathit{n}}$ of the nitrogen ligand surroundings of the two Co-ions in the n-th cluster to the crystalline modes, N is the total number of cobalt ions in the tetranuclear Co₂Fe₂ clusters in the crystal, ${\omega }_{\mathit{\kappa }\mathit{\nu }}$ is the frequency of phonons. The relation (11) describing the transformation from molecular vibrations to crystalline ones was obtained with the aid of the procedure suggested in [29,30].

After substitution of Equation (11) into Equation (9), the Hamiltonian describing the electron-phonon interaction of the cobalt ions is obtained in the following form:

$$H_{ev}={\sum }_{\mathit{n},\mathit{\kappa },\nu }{\upsilon }_{\mathit{\kappa }\mathit{\nu }}^{\mathit{n}} \mathrm{e}\mathrm{x}\mathrm{p}\left(-i\mathit{\kappa }\mathit{n}\right)(a_{\mathit{\kappa }\nu }^{+}+a_{-\kappa \nu }),$$

(12)

$${\upsilon }_{\mathit{\kappa }\mathit{\nu }}^{\mathit{n}}={\left({\displaystyle \frac{\hslash }{2N{\omega }_{\mathit{\kappa }\mathit{\nu }}m}}\right)}^{{\displaystyle \frac{1}{2}}}\upsilon ({\sigma }_{\mathit{n}}{f}_{\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)+{\tau }_{\mathit{n}}{f}_{\mathit{\kappa }\nu }^{+}\left(A_1\right)),$$

(13)

where $m$ is the mass of the nitrogen ligand in the nearest cobalt surrounding. As to the ls-Fe^II and ls-Fe^III ions, the difference between the constants of electron-vibrational interaction in these states, in which the $t_2$—shell is occupied by 6 and 5 electrons, respectively, is smaller compared with that for the ls-Co^III and hs-Co^II ions and may lead to insignificant changes in the energies of the Co₂Fe₂ complex. Therefore, when considering CTIST in tetranuclear Co₂Fe₂ compounds, for the first step for the iron ions, the electron–phonon interaction is also omitted along with the electron-deformational one.

The total Hamiltonian of the system also includes the part describing the free phonons:

$$H_{ph}=\sum _{\kappa \nu }{\hslash \omega }_{\kappa \nu }(a_{\mathit{\kappa }\mathit{\nu }}^{+}{a}_{\kappa \nu }+1/2).$$

(14)

Then, the shift transformation is applied to the Hamiltonian $H_{ph}+H_{ev}$, and the operators $a_{\mathit{\kappa }\mathit{\nu }}^{+}$ and $a_{\mathit{\kappa }\mathit{\nu }}$ are replaced by the new ones ${\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}^{+}$ and ${\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}$

$$a_{\mathit{\kappa }\mathit{\nu }}={\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}-{\displaystyle \frac{1}{{\hslash \omega }_{\mathit{\kappa }\nu }}}\sum _{\mathit{n}}{\upsilon }_{\mathit{\kappa }\mathit{\nu }}^{\mathit{n}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-i\mathit{\kappa }\mathit{n}\right), a_{\mathit{\kappa }\mathit{\nu }}^{+}={\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}^{+}-{\displaystyle \frac{1}{{\hslash \omega }_{\mathit{\kappa }\nu }}}\sum _{\mathit{n}}{\upsilon }_{\mathit{\kappa }\mathit{\nu }}^{{\mathit{n}}^{\ast }}\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }\mathit{n}\right).$$

(15)

After this transformation, one obtains the following:

$$\begin{gathered} {\tilde{H}}_{ph}+H_{int}=\sum _{\mathit{\kappa },\mathit{\nu }}\mathit{\hslash }{\omega }_{\mathit{\kappa }\mathit{\nu }}\left({\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}^{+}{\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}+1/2\right)-{\displaystyle \frac{1}{2N}}\sum _{\mathit{n},\mathit{m},\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{m\omega }_{\mathit{\kappa }\nu }^2}}\left[f_{\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)\right.{\sigma }_{\mathit{n}}{\sigma }_{\mathit{m}} \\ \left.{+f}_{\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right){\tau }_{\mathit{n}}{\sigma }_{\mathit{m}}{+f}_{\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right){\sigma }_{\mathit{n}}{\tau }_{\mathit{m}}{+f}_{\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right){\tau }_{\mathit{n}}{\tau }_{\mathit{m}}\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }(\mathit{n}-\mathit{m})\right) \end{gathered}$$

(16)

In its turn, the coefficients $f_{\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right)$ and $f_{-\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)$ can be presented in the form

$$f_{\mathit{\kappa }\nu }^{+}\left(A_1\right)={\displaystyle \frac{f_{\mathit{\kappa }\nu }\left(A_1\right)}{\sqrt{2}}}\left(1+\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }\mathit{R}\right)\right), f_{\mathit{\kappa }\nu }^{-}\left(A_1\right)={\displaystyle \frac{f_{\mathit{\kappa }\nu }\left(A_1\right)}{\sqrt{2}}}(1-\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }\mathit{R}\right)),$$

(17)

where $\mathit{R}$ is the vector connecting the cobalt ions in the cluster, and

$$f_{\mathit{\kappa }\nu }\left(A_1\right)=\sum _{k,\alpha }{e}_{\alpha }(k,\mathit{\kappa }\nu )u_{k,\alpha }^{A_1}\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }{\mathit{R}}_{kL}\right)$$

(18)

are the so-called Van-Vleck coefficients [29,30], here k enumerates the ligands of the n-th cobalt ion, ${\mathit{R}}_{kL}$ is the vector determining the position of the $k$-th ligand in the crystal surrounding of the cobalt ion, $e_{\alpha }\left(k,\kappa \nu \right)$ are the polarization vectors $\alpha =X, Y, Z$. The Van-Vleck coefficients $f_{\mathit{\kappa }\nu }\left(A_1\right)$ perform the unitary transformation from the Cartesian displacements of the ligands, belonging to the first coordination sphere of the Co-ion, to the totally symmetric vibrational mode of its ligand surrounding [29,30] and obey the relation $f_{\mathit{\kappa }\nu }^{\ast }\left(A_1\right)=f_{-\mathit{\kappa }\nu }\left(A_1\right)$.

It is worth noting that the transformed Hamiltonian ${\tilde{H}}_{ph}+H_{int}$ (Equation (16)) does not contain linear in ${\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}^{+}$ and ${\tilde{a}}_{\mathit{\kappa }\mathit{\nu }}$ terms. With the aid of definitions (17), one can prove that in Hamiltonian $H_{int}$ (16), the terms proportional to $f_{\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{-}$ ${\tau }_{\mathit{n}}{\sigma }_{\mathit{m}}$ and $f_{\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right){\sigma }_{\mathit{n}}{\tau }_{\mathit{m}}$ cancel each other out.

Using Equation (18), one easily obtains that the second term in Equation (16), namely, the Hamiltonian $H_{int}$ acquires a simpler form:

$$H_{int}=-{\displaystyle \frac{1}{2}}{\sum }_{\mathit{n},\mathit{m}}{J_2(\mathit{n}-\mathit{m})\sigma }_{\mathit{n}}{\sigma }_{\mathit{m}}-{\displaystyle \frac{1}{2}}{\sum }_{\mathit{n},\mathit{m}}{J_3(\mathit{n}-\mathit{m})\tau }_{\mathit{n}}{\tau }_{\mathit{m}},$$

(19)

where

$$\begin{gathered} J_2\left(\mathit{n}-\mathit{m}\right)={\displaystyle \frac{1}{N}}{\sum }_{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{m\omega }_{\mathit{\kappa }\nu }^2}}{f}_{\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)·\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right), \\ {J}_3\left(\mathit{n}-\mathit{m}\right)={\displaystyle \frac{1}{N}}\sum _{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{m\omega }_{\mathit{\kappa }\nu }^2}}{f}_{\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right)f_{-\mathit{\kappa }\mathit{\nu }}^{+}\left(A_1\right)·\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right). \end{gathered}$$

(20)

After the substitution of expressions (17) for $f_{\mathit{\kappa }\nu }^{+}\left(A_1\right)$ and $f_{\mathit{\kappa }\mathit{\nu }}^{-}\left(A_1\right)$ into (20) and simple transformations the parameters $J_2\left(\mathit{n}-\mathit{m}\right)$ and $J_3\left(\mathit{n}-\mathit{m}\right)$ acquire the following form:

$$\begin{gathered} J_2\left(\mathit{n}-\mathit{m}\right)={\displaystyle \frac{1}{N}}\sum _{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{m\omega }_{\mathit{\kappa }\nu }^2}}{⌈f_{\mathit{\kappa }\mathit{\nu }}\left(A_1\right)⌉}^2\left(1-\mathrm{c}\mathrm{o}\mathrm{s}(\mathit{\kappa }\mathit{R}\right))\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right) \\ {J}_3\left(\mathit{n}-\mathit{m}\right)={\displaystyle \frac{1}{N}}{\sum }_{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{m\omega }_{\mathit{\kappa }\nu }^2}}{⌈f_{\mathit{\kappa }\mathit{\nu }}\left(A_1\right)⌉}^2\left(1+\mathrm{c}\mathrm{o}\mathrm{s}(\mathit{\kappa }\mathit{R}\right))\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right). \end{gathered}$$

(21)

Insofar as the parameters $J_2\left(\mathit{n}-\mathit{m}\right)$ and $J_3\left(\mathit{n}-\mathit{m}\right)$ represent real quantities and $\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{i}\mathsf{\kappa }\left(\mathbf{n}-\mathbf{m}\right)\right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\kappa }\left(\mathbf{n}-\mathbf{m}\right)\right)+i\mathrm{s}\mathrm{i}\mathrm{n} \left(\mathsf{\kappa }\left(\mathbf{n}-\mathbf{m}\right)\right)$, the expressions in (21) can be transformed to the following forms:

$$\begin{gathered} J_2\left(\mathit{n}-\mathit{m}\right)={\displaystyle \frac{1}{N}}{\sum }_{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{m\omega }_{\mathit{\kappa }\nu }^2}}{⌈f_{\mathit{\kappa }\mathit{\nu }}\left(A_1\right)⌉}^2\left(1-\mathrm{c}\mathrm{o}\mathrm{s}(\mathit{\kappa }\mathit{R}\right))\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right), \\ {J}_3\left(\mathit{n}-\mathit{m}\right)={\displaystyle \frac{1}{N}}{\sum }_{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{m\omega }_{\mathit{\kappa }\nu }^2}}{⌈f_{\mathit{\kappa }\mathit{\nu }}\left(A_1\right)⌉}^2\left(1+\mathrm{c}\mathrm{o}\mathrm{s}(\mathit{\kappa }\mathit{R}\right))\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right). \end{gathered}$$

(22)

Since the electron transfer facilitates the redistribution of the electronic density inside each tetranuclear Co₂Fe₂ cluster, the intercluster dipole–dipole interaction is also taken into consideration in the model. The operator of dipole–dipole interaction in the general form can be written as follows:

$$V_{dd}={\displaystyle \frac{1}{2}}\sum _{\mathit{n},\mathit{m}}{R}_{\mathit{n}\mathit{m}}^{-3}\left({\mathit{d}}_{\mathit{n}}{\mathit{d}}_{\mathit{m}}-{\displaystyle \frac{3({\mathit{d}}_{\mathit{n}}{\mathit{R}}_{\mathit{n}\mathit{m}})·({\mathit{d}}_{\mathit{m}}{\mathit{R}}_{\mathit{n}\mathit{m}})}{R_{\mathit{n}\mathit{m}}^2}}\right),$$

(23)

where ${\mathit{d}}_{\mathit{n}}$ is the dipole moment of the tetranuclear cluster, ${\mathit{R}}_{\mathit{n}\mathit{m}}={\mathit{R}}_{\mathit{n}}-$ ${\mathit{R}}_{\mathit{m}}$, $R_{\mathit{n}\mathit{m}}$ is the distance between the n-th and m-th clusters in the crystal, the vector ${\mathit{R}}_{\mathit{n}}$ determines the center of the n-th tetramer. The dipole moment appears in the states arising from four intermediate configurations of the type of ls-Fe^II − ls-Co^III − ls-Fe^III − hs-Co^II (Figure 1). In this case, the vector of the dipole moment lies in the plane of the tetranuclear square, and it is directed from the hs-Co^II ion to the ls-Fe^III one. This allows us to further account for only the components $d_n^x$ and $d_n^y$ of the cluster dipole moment. In the initial ls-Fe^II − ls-Co^III − ls-Fe^II − ls-Co^III and final ls-Fe^III − hs-Co^II − ls-Fe^III − hs-Co^II configurations, the tetranuclear cluster does not possess a dipole moment. The matrices of the components $d_n^x \mathrm{a}\mathrm{n}\mathrm{d} d_n^y$ of the dipole moment of the tetranuclear square cluster in the basis of states belonging to the six above-mentioned configurations have the form:

$$d_n^x=\left(\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ d_0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}{-d}_0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}\end{array}\right),d_n^y=\left(\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ {-d}_0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ d_0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}\end{array}\right),$$

(24)

where $d_0=2\mathrm{e}{R}_0$, $R_0$ is the distance between the iron and cobalt ions along the edge of the tetranuclear cluster, i.e., the length of the side of the square.

Then, the Hamiltonian of dipole–dipole interaction is rewritten in the form:

$$V_{dd}=-{\displaystyle \frac{1}{2}}\sum _{\mathit{n},\mathit{m}}\sum _{\alpha ,\beta }{K}_{\alpha ,\beta }\left(\mathit{n}-\mathit{m}\right)d_{\mathit{n}}^{\alpha }{d}_{\mathit{m}}^{\beta },$$

(25)

where $\alpha ,\beta =x,y$ due to the planar shape of the cluster and

$$K_{\alpha ,\beta }\left(\mathit{n}-\mathit{m}\right)=R_{\mathit{n}\mathit{m}}^{-3}\left(-{\delta }_{\alpha ,\beta }+{\displaystyle \frac{3R_{\mathit{n}\mathit{m}}^{\alpha }{R}_{\mathit{n}\mathit{m}}^{\beta }}{R_{\mathit{n}\mathit{m}}^2}}\right).$$

(26)

Finally, the total Hamiltonian of the crystal also includes the Hamiltonian of isolated tetranuclear clusters:

$$H_0={\sum }_n{H}_n^0,$$

(27)

where the matrix of the operator $H_n^0$ looks as follows:

$$H_n^0=\left(\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ ∆\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ ∆\end{array}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}∆\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ ∆\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 0\\ 2∆\end{array}\end{array}\end{array}\end{array}\right),$$

(28)

Here, $∆$ and $2∆$ are the crystal field gaps between the states arising from the ground configuration ls-Fe^II−ls-Co^III− ls-Fe^II−ls-Co^III and excited ls-Fe^II−ls-Co^III− ls-Fe^III−hs-Co^II, ls-Fe^III−hs-Co^II− ls-Fe^III−hs-Co^II configurations (Figure 1) of the tetranuclear Co₂Fe₂ cluster, respectively.

The total Hamiltonian of the system is determined as

$$H=H_0+H_{st}+H_{int}+{\tilde{H}}_{ph}+V_{dd}.$$

(29)

Then, in order to examine different types of ordering facilitated by ferro- or antiferro- type cooperative interactions following the theory of magnetism [31], the crystal is divided into equivalent interpenetrating sublattices A and B. In this case, the nearest surroundings of sublattice A clusters are sublattice B clusters and vice versa. Further on, the interaction between nearest neighbors is taken into account, and the intrasublattice interactions are neglected. Then, the Hamiltonian of cooperative interaction

$$H_{ci}=V_{dd}+H_{st}+H_{int},$$

(30)

including the above-mentioned three parts (Equations (5), (19) and (25)) takes on the form

$$\begin{gathered} H_{ci}=-\frac{1}{2}\sum _{\mathit{n},\mathit{m}}\sum _{\alpha ,\beta } K_{\alpha ,\beta }^{AB}\left(\mathit{n}-\mathit{m}\right)d_{\mathit{n}}^{\alpha A}{d}_{\mathit{m}}^{\beta B} -\frac{J_1^{AB}}{2N}\sum _{\mathit{n},\mathit{m}}{\tau }_{\mathit{n}}^A{\tau }_{\mathit{m}}^B\hspace{1em}\hspace{1em}\hspace{1em} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{2}{\sum }_{\mathit{n},\mathit{m}}{J}_2^{AB}\left(\mathit{n}-\mathit{m}\right){\mathsf{\sigma }}_{\mathit{n}}^A{\mathsf{\sigma }}_{\mathit{m}}^B-\frac{1}{2}{\sum }_{\mathit{n},\mathit{m}}{J}_3^{AB}(\mathit{n}-\mathit{m}){\mathsf{\tau }}_{\mathit{n}}^A{\mathsf{\tau }}_{\mathit{m}}^B. \end{gathered}$$

(31)

The problem of interacting tetranuclear clusters is further solved in the mean-field approximation. In the framework of this approximation in Hamiltonians $H_{ci}$, the following substitutions are made:

$$\begin{gathered} {\tau }_{\mathit{n}}^A{\tau }_{\mathit{m}}^B={\tau }_{\mathit{n}}^A\overline{{\tau }^B}+\overline{{\tau }^A} {\tau }_{\mathit{m}}^B-\overline{{\tau }^A} \overline{{\tau }^B}, {\sigma }_{\mathit{n}}^A{\sigma }_{\mathit{m}}^B = {\sigma }_{\mathit{n}}^A\overline{{\sigma }^B}+\overline{{\sigma }^A} {\sigma }_{\mathit{m}}^B-\overline{{\sigma }^A} \overline{{\sigma }^B}, \\ {d}_{\mathit{n}}^{\alpha A}{d}_{\mathit{m}}^{\beta B}=d_{\mathit{n}}^{\alpha A}\overline{d^{\beta B}}\mathbf{+}\overline{d^{\alpha A}}{d}_{\mathit{m}}^{\beta B}-\overline{d^{\alpha A}}\overline{d^{\beta B}}, \end{gathered}$$

(32)

where the mean values $\overline{{\tau }^C}(C=A,B)$, $\overline{{\sigma }^C}$ and $\overline{d^{\gamma C}}$ ($\gamma =x,y)$ are determined as

$$\overline{{\tau }^C}={\displaystyle \frac{Tr\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{H}{kT}){\tau }_{\mathit{n}}^C\right)}{Tr\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{H}{kT})\right)}}, \overline{{\sigma }^C}={\displaystyle \frac{Tr\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{H}{kT}){\sigma }_n^C\right)}{Tr\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{H}{kT})\right)}}, \overline{d^{\gamma C}}={\displaystyle \frac{Tr\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{H}{kT})d_{\mathit{n}}^{\gamma C}\right)}{Tr\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{H}{kT})\right)}}.$$

(33)

In the accepted approximation, the Hamiltonian of interacting complexes

$${\tilde{H}=H}_0+H_{st}+H_{int}+V_{dd}$$

(34)

decomposes into the sum of single-cluster Hamiltonians $H_n^C$ (C = A,B):

$$\begin{gathered} H_n^A=H_n^{0A}-\left(J_1^{AB}+J_3^{AB}\right)\overline{{\tau }^B}{\tau }_{\mathit{n}}^A-J_2^{AB}\overline{{\sigma }^B}{\sigma }_{\mathit{n}}^A-\left(L_{xx}^{AB}\overline{d^{xB}}+L_{xy}^{AB}\overline{d^{yB}}\right)d_{\mathit{n}}^{xA}-(L_{yy}^{AB}\overline{d^{yB}}+L_{xy}^{AB}\overline{d^{xB}})d_{\mathit{n}}^{yA}, \\ {H}_n^B=H_n^{0B}-\left(J_1^{AB}+J_3^{AB}\right)\overline{{\tau }^A}{\tau }_{\mathit{n}}^B-J_2^{AB}\overline{{\sigma }^A}{\sigma }_{\mathit{n}}^B-\left(L_{xx}^{AB}\overline{d^{xA}}+L_{xy}^{AB}\overline{d^{yA}}\right)d_{\mathit{n}}^{xB}-(L_{yy}^{AB}\overline{d^{yA}}+L_{xy}^{AB}\overline{d^{xA}})d_{\mathit{n}}^{yB}, \end{gathered}$$

(35)

where the parameters $L_{\alpha ,\beta }^{AB}, { J}_2^{AB}$ and $J_3^{AB}$ are structural parameters, dependent on the mutual arrangement of clusters in the crystal lattice, and are determined as

$$L_{\alpha ,\beta }^{AB}={\sum }_m{K}_{\alpha ,\beta }^{AB}\left(\mathit{n}-\mathit{m}\right), J_2^{AB}={\sum }_m{J}_2^{AB}\left(\mathit{n}-\mathit{m}\right), J_3^{AB}={\sum }_m{J}_3^{AB}\left(\mathit{n}-\mathit{m}\right)$$

(36)

The eigenvalues of the Hamiltonians $H_n^A$ and $H_n^B$ in the accepted approximation appear as follows:

$$\begin{array}{l}{E}_1^A=\left(J_1^{AB}+J_3^{AB}\right)\overline{{\tau }^B}, E_2^A=∆+J_2^{AB}\overline{{\sigma }^B}-\left(L_{xx}^{AB}\overline{d^{xB}}+L_{xy}^{AB}\overline{d^{yB}}\right)d_0,\\ E_3^A=∆-J_2^{AB}\overline{{\sigma }^B}+(L_{yy}^{AB}\overline{d^{yB}}+L_{xy}^{AB}\overline{d^{xB}})d_0, E_4^A=∆-J_2^{AB}\overline{{\sigma }^B}+\left(L_{xx}^{AB}\overline{d^{xB}}+L_{xy}^{AB}\overline{d^{yB}}\right)d_0,\\ E_5^A=∆+J_2^{AB}\overline{{\sigma }^B}-(L_{yy}^{AB}\overline{d^{yB}}+L_{xy}^{AB}\overline{d^{xB}})d_0, E_6^A=2∆-\left(J_1^{AB}+J_3^{AB}\right)\overline{{\tau }^B},\\ E_1^B=\left(J_1^{AB}+J_3^{AB}\right)\overline{{\tau }^A}, E_2^B=∆+J_2^{AB}\overline{{\sigma }^A}-\left(L_{xx}^{AB}\overline{d^{xA}}+L_{xy}^{AB}\overline{d^{yA}}\right)d_0,\\ E_3^B=∆-J_2^{AB}\overline{{\sigma }^A}+(L_{yy}^{AB}\overline{d^{yA}}+L_{xy}^{AB}\overline{d^{xA}})d_0, E_4^B=∆-J_2^{AB}\overline{{\sigma }^A}+\left(L_{xx}^{AB}\overline{d^{xA}}+L_{xy}^{AB}\overline{d^{yA}}\right)d_0,\\ E_5^B=∆+J_2^{AB}\overline{{\sigma }^A}-(L_{yy}^{AB}\overline{d^{yA}}+L_{xy}^{AB}\overline{d^{xA}})d_0, E_6^B=2∆-\left(J_1^{AB}+J_3^{AB}\right)\overline{{\tau }^A}.\end{array}$$

(37)

Since the energies $E_1^A$, $E_1^B$, $E_6^A$ and $E_6^B$ depend on the sum of the parameters $J_1^{AB}$ and $J_3^{AB}$, while in the energies of all other levels of the clusters $A$ and $B$ these parameters do not enter, in further consideration, instead of the sum $J_1^{AB}+J_3^{AB}$, it will be considered only one parameter $J_1=J_1^{AB}+J_3^{AB}.$

The above-cited papers on tetranuclear Fe₂Co₂ compounds [18,19,20,21], exhibiting CTIST, show that the nearest surrounding of both hs-Co^II and ls-Fe^III differs from the perfect octahedral one. Meanwhile, the observed $\chi T$ product for the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O compound [19], which serves further on as an example for application of the model under examination, demonstrates that the orbital angular momenta of both Fe- and Co-ions are not quenched. A similar situation takes place in other tetramers exhibiting this phenomenon [18,20,21]. Therefore, further on, the splitting of the orbitally degenerate ground terms ⁴T_1g and ²T_2g of the hs-Co^II and ls-Fe^III ions, respectively, by the spin–orbital interaction is taken into account, and in the first approximation, the effect of the low symmetry crystal field is neglected. The operators of spin–orbital interaction for the hs-Co^II and ls-Fe^III ions introduced in the model look as follows:

$$H_{SO}^{Co}=-{\displaystyle \frac{3}{2}}{\lambda }_1\mathit{L}\mathit{S}, H_{SO}^{Fe}={-\lambda }_2\mathit{l}\mathit{s},$$

(38)

Here, $\mathit{L}$ and $\mathit{S}$ are the operators of the orbital angular momentum and spin for the hs-Co^II ion, while the designations $\mathit{l},\mathit{s}$ correspond to the operators with the same meaning for the ls-Fe^III ion. For brevity, the indices A and B in the operators of the spin and angular momenta are dropped for both iron and cobalt ions. The spin–orbital interaction splits the ⁴T_1g ground term of the hs-Co^II into three groups of levels, namely, into the ground doublet and the excited quadruplet and sextet, with the energy gaps between the excited terms and the ground doublet being $9\left|{\lambda }_1\right|/4$ and $6\left|{\lambda }_1\right|$, respectively [27,32], where ${\lambda }_1$ = −180 cm⁻¹ is the usually accepted value for the spin–orbit coupling parameter of the hs-Co^II ion [32]. The ²T_2g ground term of the ls-Fe^III ion also undergoes spin–orbital splitting, resulting in the ground doublet and the excited quadruplet separated by the energy gap $3\left|{\lambda }_2\right|/2$ with ${\lambda }_2$ = −486 cm⁻¹ ([32], see also [27]). Further on, we take into consideration all energy levels of the tetranuclear clusters of the A and B types obtained in the mean field approximation (37) and corresponding to different configurations of these clusters and add to them the terms arising from the spin–orbital splittings of the ²T_2g and ⁴T_1g multiplets (38) of the ls-Fe^III and hs-Co^II ions described above. Finally, these energy levels are employed for writing down the free crystal energy, the minimization of which over the set of the order parameters $\overline{{\tau }^C}$, $\overline{{\sigma }^C}$, $\overline{d^{xC}}$, $\overline{d^{yC}}$ ($C=A,B)$ leads to the following system of self-consistent equations for the determination of these parameters:

$$\begin{array}{l}\overline{{\tau }^A}={\displaystyle \frac{f_{hs}^2\left(\mathrm{T}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_6^A}{kT}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_1^A}{kT}\right)}{Z_A}}, \overline{{\sigma }^A}={\displaystyle \frac{f_{hs}(\mathrm{T})\left[-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_2^A}{kT}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{ E}_3^A}{kT}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{ E}_4^A}{kT}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{ E}_5^A}{kT}\right)\right] }{Z_A}},\\ \overline{d^{xA}}={\displaystyle \frac{f_{hs}(\mathrm{T}) \left(\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_2^A}{kT}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_4^A}{kT}\right)\right)}{Z_A}}, \overline{d^{yA}}={\displaystyle \frac{f_{hs}(\mathrm{T}) \left(-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_3^A}{kT}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_5^A}{kT}\right)\right)}{Z_A}},\end{array}$$

(39)

The partition function $Z_A$ and the temperature-dependent factor $f_{hs}\left(\mathrm{T}\right)$ are determined by the following expressions:

$$Z_A=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E_1^A}{kT}}\right)+f_{hs}\left(\mathrm{T}\right)\left[\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E_2^A}{kT}}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{{ E}_3^A}{kT}}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{{ E}_4^A}{kT}}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{{ E}_5^A}{kT}}\right)\right]+f_{hs}^2\left(\mathrm{T}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{E_6^A}{kT}}\right),$$

(40)

$$f_{hs}\left(\mathrm{T}\right)=\left(2+4 \exp \left[-{\displaystyle \frac{9\left|{\lambda }_1\right|}{4kT}}\right]+6 \exp \left[-{\displaystyle \frac{6\left|{\lambda }_1\right|}{kT}}\right]\right)\left(2+4 \exp \left[-{\displaystyle \frac{3\left|{\lambda }_2\right|}{2kT}}\right]\right).$$

(41)

The corresponding equations for sublattice B can be obtained from Equations (39) and (40) by substituting $E_i^A$ $\to E_i^B$.

3. Results and Discussion

We start with the estimation of characteristic energies of the interactions involved in the model. First, the characteristic energy of dipole–dipole interaction is roughly estimated. Taking into account that in [19] the mean experimental value of the length of the square side is $R_o$ = 4.9 Å, the mean distance between the tetranuclear square clusters in the crystal is $R_1=13.94 \AA ,$ and the dipole moment of the tetranuclear cluster in the states arising from the intermediate ls-Fe^II − ls-Co^III − ls-Fe^III − hs-Co^II configuration can be roughly evaluated as 2e$R_o$ one obtains that the characteristic energy of dipole–dipole interaction ${\displaystyle \frac{{\left(2e{R}_o\right)}^2}{R_1^3}}$ is of the order of 4.1 × 10³ cm⁻¹. However, the expression for $K_{\alpha ,\beta }^{AB}\left(\mathit{n}-\mathit{m}\right)$ (Equation (26)) contains the term ${\displaystyle \frac{3R_{\mathit{n}\mathit{m}}^{\alpha }{R}_{\mathit{n}\mathit{m}}^{\beta }}{R_{\mathit{n}\mathit{m}}^2}}$, which depends on the spherical angles ${\theta }_{\mathit{n}\mathit{m}}$ and ${\phi }_{\mathit{n}\mathit{m}}$, determining the position of the vector ${\mathit{R}}_{\mathit{n}\mathit{m}}$ that connects tetranuclear clusters labeled by the vectors $\mathit{n}$ and $\mathit{m}$, and may acquire different signs in dependence on the mutual arrangement of these clusters. Therefore, further on in numerical simulations, the magnitude of the parameters $L_{\alpha ,\beta }^{AB} d_0^2$ characterizing the dipole–dipole interaction is varied within the limits −60 cm⁻¹ ÷ 100 cm⁻¹. By definition, the parameter of the electron–deformational interaction $J_1$ is positive (6) and its effect will be examined by varying its value from 0 up to 410 cm⁻¹ [25,33]. As to the parameters $J_2^{AB}$ and $J_3^{AB}$ (see Equations (22) and (36)) containing the alternating function $\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right)$, they can accept in general negative or positive values depending on the crystal structure and the corresponding spectrum of acoustical and optical phonons. To have an idea about the order of magnitude and signs of these parameters, further on, their rough estimation is performed under the assumption that the long-wave acoustic phonons (that can be associated with the intermolecular strain in systems with labile electronic states [24,25], in which the intermolecular space is much softer than that inside the molecules) mainly contribute to the values of $J_2^{AB}$ and $J_3^{AB}$. In fact, the approximation below employed and aimed at this estimation corresponds to that well-known as the Debye model [34], within which the difference in the coordinates and masses of the particles in the unit cell of the crystal is neglected, and the polarization vectors $e_{\alpha }(k,\mathit{\kappa }\nu )$ in Equation (18) are represented as follows [34]:

$$e_{\alpha }\left(k,\mathit{\kappa }\nu \right)=\sqrt{{\displaystyle \frac{m}{M}}} e_{\alpha }\left(\mathit{\kappa }\nu \right),$$

(42)

where $\alpha =x,y,z,$ M is the mass of the crystal, and the symbol “$m$” was defined above. The new vectors $e_{\alpha }(\mathit{\kappa }\nu )$ (42) do not depend anymore on the type $k$ of the vibrating atom. Due to the account of (42), the expression (18) for the coefficients $f_{\mathit{\kappa }\nu }\left(A_1\right)$ takes on the following form:

$$f_{\mathit{\kappa }\nu }\left(A_1\right)=\sqrt{{\displaystyle \frac{m}{M}}} {\breve{f}}_{\mathit{\kappa }\nu }\left(A_1\right).$$

(43)

where the new functions ${\breve{f}}_{\mathit{\kappa }\nu }\left(A_1\right)$ are defined as

$${\breve{f}}_{\mathit{\kappa }\nu }\left(A_1\right)={\sum }_{k,\alpha }{e}_{\alpha }(\mathit{\kappa }\nu )u_{k,\alpha }^{A_1}\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathit{\kappa }{\mathit{R}}_{kL}\right).$$

(44)

Further on, according to the Debye model, the difference between the longitudinal $c_l$ and transversal $c_t$ speeds of sound is neglected ($c_l=c_t=c$), and for the phonons, the dispersion law ${\omega }_{\mathit{\kappa }\mathit{\nu }}\cong c\nu$ is accepted. With these assumptions and also equating $\cos \left(\mathit{\kappa }\mathit{R}\right)$ and $\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathit{\kappa }\left(\mathit{n}-\mathit{m}\right)\right)$ to one in the expression for the parameter $J_3^{AB}$ (see (22) and (36)) one obtains

$$J_3^{AB}=\sum _m{J}_3^{AB}\left(\mathit{n}-\mathit{m}\right)=2\sum _{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{M\omega }_{\mathit{\kappa }\nu }^2}}{⌈{\breve{f}}_{\mathit{\kappa }\nu }\left(A_1\right)⌉}^2$$

(45)

Then, the average of the values ${⌈{\breve{f}}_{\mathit{\kappa }\nu }\left(A_1\right)⌉}^2$ in the Equation (45) over the orientation of the wave vector $\mathit{\kappa }$ and the summation over the three branches of the acoustic vibrations are performed. Retaining the terms up to (k$R$_L)² (here $R_L$ is the mean distance between the cobalt ion and its nearest surrounding) in the decomposition of the averaged coefficients ${⌈f_{\mathit{\kappa }\nu }\left(A_1\right)⌉}^2$ obtained in [35], performing the integration over the spherical angles of the wave vector $\kappa$ and its magnitude in the limits {0, ${\kappa }_m$}, where ${\kappa }_m$ is determined as $(6{\pi }^2/V_0)$^1/3 [34] with $V_0$ being the unit cell volume, one obtains the following expression for the parameter $J_3^{AB}$:

$$J_3^{AB}{\displaystyle \frac{{\upsilon }^2{R}_L^2{\kappa }_m^3}{9c^2\rho {\pi }^2}}={\displaystyle \frac{2{\upsilon }^2{R}_L^2}{3\rho c^2{V}_0}}.$$

(46)

For the estimation of this parameter, first, the constants of electron–vibrational interaction $V_{hs}$ and $V_{ls}$ of the Co-ion with the totally symmetric displacements of the nearest surrounding, which enter in the definition of the parameter $\upsilon$, were obtained by expressing the operator of interaction $V_{A_1}$ of the Co-ions with these displacements through the cubic crystal field operator $V_{cub}$ [36].

$$V_{A_1}=-{\displaystyle \frac{5}{\sqrt{6}}}{\displaystyle \frac{{ V}_{cub}}{R_L}}$$

(47)

Such a presentation allows one to immediately represent the above-defined constants $V_{hs}$ and $V_{ls}$ of interaction of the hs-Co(II) and ls-Co(III) ions with the totally symmetric vibrations through the cubic crystal field parameters ${Dq}^{hs}$ and ${Dq}^{ls}$:

$$V_{hs}={\displaystyle \frac{40{Dq}^{hs}}{\sqrt{6}{R}_{hs}}}, V_{ls}={\displaystyle \frac{120 {Dq}^{ls}}{\sqrt{6}{R}_{ls}}}$$

(48)

Taking for ${Dq}^{hs}$ and ${Dq}^{ls}$ the values 1300 cm⁻¹ and 2240 cm⁻¹ [37], respectively, for $R_{hs}$ and $R_{ls}$, the mean lengths of the cobalt–nitrogen bonds 2.118 Å and 1.927 Å for the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O compound [19], one obtains that the vibronic coupling constants (48) acquire the following numerical values: $V_{hs}$ = 2 × 10⁻⁴ dyn, $V_{ls}$ = 10⁻³ dyn. Due to the account of these values for the parameters $V_{hs}$, $V_{ls}$ and experimental values $\rho$ = 1.507 g/cm³, $c$ ≈ 3 × 10⁵ cm/s, $R_L$ ≈ 2.022 Å and $V_0$ ≈ 2536 ų [19], the performed approximate estimation gives that the parameter $J_3^{AB}$ is of the order of 1.28 × 10³ cm⁻¹.

The procedure for the approximate evaluation of the parameter $J_2^{AB}$ (Equations (22) and (36)) in the framework of the Debye model resembles the above-described for $J_3^{AB}$ with the only difference that, in the expansion of the factor $\left(1-\mathrm{c}\mathrm{o}\mathrm{s}(\mathit{\kappa }\mathit{R}\right)) \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s} \mathrm{i}\mathrm{n}$ this parameter over $\mathit{\kappa }\mathit{R}$, the terms of the order of ${(\mathit{\kappa }\mathit{R})}^2$ should be retained. Due to the account of this expansion and the approximations described above for the parameter $J_2^{AB}$, the following expression is obtained:

$$J_2^{AB}=\sum _{\mathit{m}}{J}_2^{AB}\left(\mathit{n}-\mathit{m}\right)=\sum _{\mathit{\kappa },\mathit{\nu }}{\displaystyle \frac{{\upsilon }^2}{{2M\omega }_{\mathit{\kappa }\nu }^2}}{⌈{\breve{f}}_{\mathit{\kappa }\nu }\left(A_1\right)⌉}^2{(\mathit{\kappa }\mathit{R})}^2.$$

(49)

Performing in (49), the averaging over the angle composed by the vectors $\mathit{\kappa }$ and $\mathit{R}$, i.e., substituting the square of the cosine of this angle by ½, and applying a procedure similar to that employed for the estimation of the parameter $J_3^{AB}$, one obtains for the parameter $J_2^{AB}$ the following approximate expression:

$$J_2^{AB}={\displaystyle \frac{{\upsilon }^2{R}^2{R}_L^2{\kappa }_m^5}{120{\pi }^2\rho c^2}}.$$

(50)

The evaluation of the parameter $J_2^{AB}$ was performed with the aid of Equation (50) and the same numerical values for the parameters $\upsilon ,$ $R_L$, ${\kappa }_m$, $\rho$ and $c$ as in the case of $J_3^{AB}$, the distance $R$ between two Co-ions inside the cluster was taken equal to R = 7.2 Å [19]. As a result for the $J_2^{AB}$ parameter, the value of about 405 cm⁻¹ was obtained. The values of the parameters $J_2^{AB}$ and $J_3^{AB}$ were calculated applying rather rough approximations. Meanwhile, the obtained order of magnitude of these parameters does not contradict qualitatively the experimental data, and, namely, for the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O compound noticeable growth of the $\chi T$ product occurs at temperatures higher than 150 K, the latter indicates that the parameters of cooperative interactions facilitating the spin transformation are in magnitude at least not lower than 100 cm⁻¹.

The second point to be discussed concerns the signs of the parameters $J_2^{AB}$ and $J_3^{AB}$. In the estimation performed above, only the contribution of long-wave acoustic vibrations was taken into account, and the alternating function cos$\left[\mathit{k}(\mathit{n}-{\mathit{n}}^{\prime })\right]$ in the expressions for these parameters was put approximately equal to one. In this approximation, both parameters $J_2^{AB}$ and $J_3^{AB}$ were shown to be positive. However, the function cos$\left[\mathit{k}(\mathit{n}-{\mathit{n}}^{\prime })\right]$ in dependence of the magnitude and direction of the vectors $\mathit{k}$ and $\mathit{n}-{\mathit{n}}^{\prime }$ can accept in general positive and negative values, and lead to positive or negative signs of the parameters $J_2^{AB}$ and $J_3^{AB}$. Therefore, from the very beginning, to have the possibility to examine in the framework of the suggested model the effects of different signs of the parameters of cooperative interaction and, respectively, different types of ordering, the crystal was subdivided into two different sublattices. As to the parameter $J_1^{AB}$ of cooperative electron–deformational interaction, its estimation is similar to that described in detail in papers [24,25] dealing with the valence tautomeric transformation ls-Co^III-cat $\to$ hs-Co^II-sq (where cat and sq are the catecholate and semiquinone ligands) in cobalt compounds and giving a value of the order of 100 cm⁻¹. As it was already indicated, in the present model, the parameter $J_1^{AB}$ enters the cluster energies in combination with the parameter $J_3^{AB}$ characterizing the cooperative interaction via the field of phonons. So, instead of the sum $J_1^{AB}+J_3^{AB}$, it will be considered only one parameter $J_1=J_1^{AB}+J_3^{AB}$. This parameter will be changed in the range of 200–415 cm⁻¹. In the sample calculations presented below, the behavior of the system is analyzed by calculation of the hs-fraction determined as

$$n_{hs}^A={\displaystyle \frac{\frac{f_{hs}(\mathrm{T})}{2}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_2^A}{kT}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{ E}_3^A}{kT}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{ E}_4^A}{kT}\right)+\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{ E}_5^A}{kT}\right)\right]+f_{hs}^2\left(\mathrm{T}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_6^A}{kT}\right) }{Z_A}}$$

(51)

In the subsequent consideration as well as in the figures captions, for simplicity, the superscripts AB in $J_2^{AB}$, $L_{xx}^{AB}$ and $L_{yy}^{AB}$ parameters are omitted. Figure 2 demonstrates the effect of the parameter $J_1$ on the spin transition. It is seen that the increase of this parameter makes the transition more abrupt, and at some value of the $J_1$ parameter, the hysteresis loop appears. The system demonstrates bistability.

In Figure 3, Figure 4 and Figure 5, the effect of J₂ on the hs-fraction is illustrated. When the value of the parameter $J_1$ is negligible, the increase in the parameter J₂ transforms the curve from a gradual one to one that demonstrates a pronounced step (Figure 3). For non-vanishing values of $J_1$, depending on the magnitude of this parameter and the parameter J₂, various types of high-spin fraction temperature behavior can be observed with temperature growth, namely, gradual increase of this fraction, its abrupt increase, as well as the increase accompanied by a hysteresis loop with one (Figure 4) or two (Figure 5) steps. In the case when the characteristic energy of dipole–dipole interaction is vanishing, $\mathrm{t}\mathrm{h}\mathrm{e}$ change of the sign of the parameter J₂ does not lead to the change of the energy spectra of clusters of the type $A$ and $B$ as it is clearly seen from Equation (37). Therefore, the curves for the hs-fraction presented in Figure 3, Figure 4 and Figure 5 coincide for the same magnitudes and different signs of the parameter J₂.

The effect of dipole–dipole interaction was analyzed (Figure 6) under the simplified condition that the characteristic energies of this interaction $L_{xx}{d}_0^2$ and $L_{yy}{d}_0^2$ are equal. From the analysis of Equation (37), it follows that the effect of this interaction is similar to that produced by the cooperative coupling via the field of phonons, characterized by the J₂ parameter. As a result, the dipole–dipole interaction increases (when the parameters are of the same sign) or reduces (in the case of opposite signs of the parameters) the effect of the coupling via the field of phonons.

Finally, the model above developed is applied for the explanation of the experimental data on the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O complex reported in [19]. The curve calculated within the framework of this model and the experimental one [19] are shown in Figure 7. The set of characteristic system parameters that assures the satisfactory agreement of these two curves is listed in the caption for Figure 7. The obtained value of the gap Δ = 180 cm⁻¹ does not contradict the observed data because the noticeable growth in the $\chi T$ curve starts from 125 K, the value of the parameter J₁ accounting for the contributions coming from the cooperative electron–deformational interaction and that arising from the coupling of clusters through the field of phonons is also within the limits established for systems with labile electronic states and our estimations above described. As to the parameters $L_{xx}{d}_0^2$ and $L_{yy}{d}_0^2$, to avoid overparametrization, they were roughly put equal in the simulation, taking into account the quadratic form of the tetranuclear clusters and their mutual arrangement in the crystal. Following the experimental data, which demonstrate the presence at low temperatures of a fraction of tetranuclear clusters which are in the ls-Fe^III − hs-Co^II − ls-Fe^III − hs-Co^II state, we denote this fraction as x_hs and take it into account while comparing the calculated $\chi T$ curve with the experimental one. The best fit value of the x_hs fraction is determined by the low temperature observed magnetic behavior of the compound under examination, when only the ground diamagnetic configuration 1 predominates.

The dipole–dipole interaction must compensate for the interaction through the phonon field, the parameter J₂ of which was estimated in the limit of long-wave acoustic phonons. Simulating the $\chi T$ curve for the parameter J₂, the above estimated value of 400 cm⁻¹ was taken. Here, it should be mentioned that at the present stage of study, the problem of the determination of the characteristic parameters of cooperative interactions of such a complicated system through DFT and ab initio calculations still remains a difficult one. Therefore, the performed estimation of these parameters expressed through characteristic parameters of the crystal, which represent measurable quantities, is reliable and gives at least the correct order of magnitude. At the same time, the developed model that accounts for all relevant interactions governing the pronounced charge-transfer-induced spin transition in a crystal of interacting tetranuclear clusters provides a reasonable explanation of the experimental data on the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O compound [19]. The low temperature decrease of the χT product can be explained by some small antiferromagnetic interactions between paramagnetic ls-Fe^III and hs-Co^II not included in the model, because it does not affect the spin transition that occurs at higher temperatures. The elaborated approach can be applied for the description of charge-transfer-induced spin transitions and in other compounds containing tetranuclear 2Fe-2Co clusters as a structural element, demonstrating charge-transfer-induced spin transitions.

To understand the role of different cluster configurations in the spin transformation demonstrated by the crystal consisting of teranuclear Fe₂Co₂ complexes, the populations of these configurations as functions of temperature have been calculated with the set of the best fit parameters which assure the coincidence of the observed χT product with the calculated one for the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O complex in frames of the suggested model. As can be seen from Figure 8, four intermediate configurations cannot be excluded from consideration when consistently examining the observed phenomenon because their contribution is appreciable.

4. Concluding Remarks

The approach developed in the present paper is aimed at the description of charge-transfer-induced spin transitions in crystals containing tetranuclear Fe₂Co₂ clusters as structural units and accounts for the energy spectrum of each cluster formed by its states arising from different cluster configurations mentioned above. Traditionally, as in the case of other systems with labile electronic states, the suggested model accounts for electron-deformational interaction, the Hamiltonian of which is obtained by minimization of the elastic energy of the crystal with due account of the different elasticity of intra- and intermolecular spaces. The introduction of this interaction in the model is justified by the observed elongation of cobalt–nitrogen bonds. As distinguished from previous models of spin transitions, in the present examination, besides the electron–deformational interaction playing an essential role in the stabilization of the ground state characterized by the high spin value, the reported model accounts for the interaction of the cobalt ions composing the cluster with the full symmetric displacements of the nearest surrounding modulated by the crystalline vibrations. The latter coupling leads to two types of cooperative interactions via the phonon field, with one of them possessing a matrix structure similar to the electron–deformational interaction, and actually, the account of this interaction results only in the redetermination of the characteristic parameter of the former one. The second type of cooperative coupling via the field of phonons, involved in the model and characterized by the parameter $J_2^{AB},$ leads to effects distinguished from those facilitated by the electron–deformational interaction and that via the phonon field characterized by the parameter $J_3^{AB}$. Under certain parameter values, this interaction may facilitate abrupt transitions characterized by a narrow hysteresis loop at temperatures higher than 175 K, as well as step transitions and those combining a hysteresis loop and a distinct step. In general, all parameters of the suggested model can be evaluated with the aid of ab initio and semiempirical methods. In the present work, a rough estimation of the characteristic parameters of the model was performed in order to determine the order of their values. At the same time, the developed model allows for reproducing the observed magnetic susceptibility of the {[(Tp*)Fe(CN)₃]₂[Co(bpy^Me)₂]₂}(OTf)₂·2DMF·H₂O compound and to determine the range of characteristic energies of the interactions involved in the model that allows for describing the experimental results.