Theoretical Modeling of the Magnetic Behavior of Thiacalix [4] arene

Article pubs.acs.org/IC

Theoretical Modeling of the Magnetic Behavior of Thiacalix[4]arene Tetranuclear MnII2GdIII2 and CoII2EuIII2 Complexes Sergey M. Aldoshin,*,† Nataliya A. Sanina,† Andrew V. Palii,*,†,‡ and Boris S. Tsukerblat*,§ †

Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region, Russia Institute of Applied Physics, Academy of Sciences of Moldova, Chişinău, Moldova § Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel ‡

S Supporting Information *

ABSTRACT: In view of a wide perspective of 3d−4f complexes in single-molecule magnetism, here we propose an explanation of the magnetic behavior of the two thiacalix[4]arene tetranuclear heterometallic complexes MnII2GdIII2 and CoII2EuIII2. The energy pattern of the MnII2GdIII2 complex evaluated in the framework of the isotropic exchange model exhibits a rotational band of the low-lying spin excitations within which the Landé intervals are affected by the biquadratic spin−spin interactions. The nonmonotonic temperature dependence of the χT product observed for the MnII2GdIII2 complex is attributed to the competitive influence of the ferromagnetic Mn−Gd and antiferromagnetic Mn−Mn exchange interactions, the latter being stronger (J(Mn, Mn) = −1.6 cm−1, Js(Mn, Gd) = 0.8 cm−1, g = 1.97). The model for the CoII2EuIII2 complex includes uniaxial anisotropy of the seven-coordinate CoII ions and an isotropic exchange interaction in the CoII2 pair, while the EuIII ions are diamagnetic in their ground states. Best-fit analysis of χT versus T showed that the anisotropic contribution (arising from a large zero-field splitting in CoII ions) dominates (weakexchange limit) in the CoII2EuIII2 complex (D = 20.5 cm−1, J = −0.4 cm−1, gCo = 2.22). This complex is concluded to exhibit an easy plane of magnetization (arising from the CoII pair). It is shown that the low-lying part of the spectrum can be described by a highly anisotropic effective spin-1/2 Hamiltonian that is deduced for the CoII2 pair in the weak-exchange limit.

1. INTRODUCTION The problem of single-molecule magnets (SMMs) and the attendant matter of the exchange interaction have been focuses of research for more than 25 years1−10 to an appreciable extent because of their promise for the design of memory units of molecular size. During this period a series of new highnuclearity clusters have been synthesized (a survey of the most investigated SMMs is given in the book by Gatteschi et al.1), and related fundamental physical concepts have been developed.1−10 However, despite the enormous progress in the chemistry and physics of SMMs, the final technological goal mentioned above has not been achieved to date. In fact, the relaxation time for the spin reorientation in large magnetic spin clusters proved to be not long enough to ensure efficient nanoscale memory storage units at room temperatures. In particular, it became clear that increasing the size of spin clusters does not increase (as it looks at first glance) the barrier for spin reorientation and, consequently, the relaxation time.11 For this reason, in the last few years the focus of research in the area of SMMs has shifted from “spin-only” clusters (mainly, of transition metal ions), in which the magnetic anisotropy is relatively small, to systems containing highly anisotropic rare earth (RE) ions with unquenched orbital angular momenta. The use of magnetic molecules involving RE ions2 promises to © XXXX American Chemical Society

significantly increase the relaxation time as a result of the presence of the unquenched orbital angular momentum, which is directly related to the height of the barrier for the reorientation of the magnetic moment. In fact, even a single RE ion can produce a pronounced SMM effect,2,12−14 and for this reason such systems have been called single-ion magnets. That is why the study of the exchange interactions and magnetic anisotropy in metal clusters incorporating RE and transition metal ions, the topic of the present article, represents a challenging task. At present there are nearly 400 publications reporting molecular complexes containing in their structures both 3d and 4f ions, and more than 100 of these complexes (mainly those containing TbIII and DyIII ions) have been shown to behave as SMMs (see the review in ref 15 and references therein). The first reported 3d−4f SMMs, [CuIILLnIII(hfac)2]2 (Ln = Tb, Dy),16 had square-planar tetrameric structures with alternating 3d and 4f metal ions, and the first attempt to model 3d−4f SMMs theoretically also relates to these systems.17 Since that time, many other 3d−4f tetramers exhibiting SMM properties have been obtained and characterized (see, e.g., refs 18−21). Received: January 13, 2016

A

DOI: 10.1021/acs.inorgchem.6b00065 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry

to be quite important because a specific advantage of the 3d−4f complexes is just the possibility of preparing families of such complexes in which the properties of solely the 3d metal ion fragments or the exchange coupling between the 3d and 4f metal ions can be checked, for example, by using EuIII ions with nonmagnetic ground states or the isotropic GdIII ions. Although these ions usually do not exhibit SMM behavior, they can act as good test systems whose theoretical description would facilitate the subsequent analysis of more complicated complexes containing orbitally degenerate lanthanide ions (e.g., TbIII and DyIII), which often demonstrate SMM properties. Therefore, in this article we shortly summarize the experimental magnetic data, analyze the energy patterns of the spin levels, and present a theoretical modeling study of the magnetic behavior of two thiacalix[4]arene tetranuclear complexes, MnII2GdIII222 and CoII2EuIII2.23

Particularly, in the search for new 3d−4f-type SMMs researchers have paid attention to calixarene complexes with tetragonal tetranuclear units consisting of two 3d metal ions and two lanthanides.22,23 This choice was inspired by the fact that calixarenes represent available polydentate ligands that possess structural versatility allowing their modification (the substituents, bridging fragments between the phenyl rings, and the number of structural units in the calixarene molecule can be varied). As a result, calixarene-based polynuclear complexes containing 3d metal ions and lanthanide ions can be obtained. The synthesis and experimental characterization of the first 3d−4f heterometallic MnII2GdIII2 tetrameric compound with ptert-butylthiacalix [4]arene (tBuTCA) ligands was reported in 2008.22 More recently, other representatives of the series of thiacalix[4]arene tetramers MII2LnIII2 (M = 3d metal, Ln = lanthanide) have been found and experimentally characterized.23 A general insight on the structure of this kind of system is given in Figure 1, which shows a structural fragment of the

2. MAGNETIC EXCHANGE IN THE MnII2GdIII2 CLUSTER 2.1. Experimental Section: MnII2GdIII2. The experimental data on the temperature dependence of χT (where χ is the molar magnetic susceptibility and T is the absolute temperature) for the thiacalix[4]arene MnII2GdIII2 tetranuclear complex were reported in ref 22. The magnetic behavior of the system is represented in Figure 2. The experimental data22

Figure 1. Fragment of the structure of [Eu 2 Co 2 (O)(tBuTCA)2(DMF)0.5(MeOH)1.5(H2O)2](MeOH)2(DMF)2(H2O)3: (a) general view; (b) central fragment emphasizing the metal core and exchange pathways. Hydrogen atoms are not shown. Coloring: Eu, cyan; Co, violet-blue; S, yellow; O, red; C, gray; N, light blue.

Figure 2. Temperature dependence of χT for the thiacalix[4]arene MnII2GdIII2 cluster. Experimental data22 are shown by red circles, and the theoretical curve calculated with the best-fit parameters (see the text for the parameter values) is shown by the solid black line.

compound [Eu 2 Co 2 (O)( t BuTCA) 2 (DMF) 0.5 (MeOH) 1.5 (H2O)2](MeOH)2(DMF)2(H2O)3, represented by the metal complex, two solvate N,N-dimethylformamide (DMF) molecules, two solvate methanol molecules, and three water molecules.23 The molecule has a sandwichlike structure in which the planar Co2Eu2 core with the central bridging oxygen atom (μ4-O) is sandwiched between two tBuTCA ligands having an antiparallel orientation. Both Co ions are sevencoordinate, namely, they are coordinated by four oxygen atoms of the phenoxyl fragments, two bridging sulfur atoms, and one bridging μ4-O ligand. As distinguished from the Co ions, the Eu ions are nine-coordinate. Compared with the Co ions, one of the two Eu ions is additionally coordinated by one MeOH molecule and one H2O molecule, while the second Eu ion is additionally coordinated by one H2O molecule and DMF and MeOH molecules with 50% occupancy. The Co···Co, Co···Eu, and Eu···Eu metal−metal distances in the metal core are 3.07, 3.31/3.32, and 3.65 Å, respectively. In refs 22 and 23, the thiacalix[4] arene tetramers were characterized experimentally, but theoretical modeling of the magnetic behavior of these compounds has not been reported. Meanwhile, theoretical modeling of this kind of system seems

demonstrate a fast increase of χT versus T from 4.93 cm3 mol−1 K at 2 K to the maximal value of 24.57 cm3 mol−1 K at 42 K and then a slight decrease to 24.09 cm3 mol−1 K at 300 K. 2.2. Results and Discussion: MnII2GdIII2. To model the magnetic properties of the system, we must take into account the fact that because the GdIII ions have a half-filled 4f subshell (4f7) and the MnII ions have a half-filled high-spin 3d subshell (3d5), the constituent ions do not carry orbital angular momentum. Under this condition, the tetramer can be regarded as a pure spin system containing two 7/2 spins (gadolinium ions) and two 5/2 spins (manganese ions) coupled through Heisenberg−Dirac−Van Vleck (HDVV) exchange interactions, as shown by the scheme in Figure 3 along with the corresponding notations for the exchange parameters. To simplify the model and to reduce the number of adjustable exchange parameters, we neglect the rather weak 4f− 4f coupling between the two Gd ions (J(Gd, Gd) = 0) as well as the interactions in the distant Mn−Gd pairs occupying the antipodal sites (Jd(Mn, Gd) = 0). Although the last interaction can influence the energy pattern, it is expected to be relatively small with respect to Js(Mn, Gd) because of the greater distance B


Article

Inorganic Chemistry

One can evaluate the molar magnetic susceptibility of the system with the aid of the standard expression (the Van Vleck formula):30 χ (T ) =

NAg 2μB2 3kBT f (S )

∑S ∑v = 1 S(S + 1)(2S + 1)f (S) exp[− Ev(S)/kBT ] f (S )

∑S ∑v = 1 (2S + 1)f (S) exp[− Ev(S)/kBT ]

Figure 3. Scheme of the exchange pathways in the tetranuclear MnII2GdIII2 cluster. The subscripts “d” and “s” stand for the diagonal and the side of the tetragon, respectively, and the subscripts 1 and 2 enumerate the ions of the same kind.

(2)

where the index v numerates the spin states with the same S (so that the summation is extended over all of the spin levels Ev(S)), NA is the Avogadro number, and kB is the Boltzmann constant. A rather good agreement between the experimental and theoretical χT versus T curves is obtained for the following set of best-fit parameters (exchange integrals and the g factor): J(Mn, Mn) = −1.6 cm−1, Js(Mn, Gd) = 0.8 cm−1, and g = 1.97. A more accurate fit could be achieved if the weak exchange between more-distant Mn and Gd ions as well as the Gd−Gd exchange and the difference in the g factors of Mn and Gd ions were taken into account, but it is reasonable to avoid excessive flexibility of the theory that would result in lost confidence in spite of the existence of formal attributes of accuracy. From the fit parameters one can conclude that the system exhibits a relatively strong antiferromagnetic interaction between MnII ions and weak ferromagnetic MnII−GdIII coupling. By comparing the experimental and theoretical temperature dependences of χT (Figure 2), one can see that the model reproduces quite well the features of the magnetic behavior over a wide temperature range. The results show that the observed nonmonotonic behavior of χT is the result of the competitive influence of the ferromagnetic and antiferromagnetic exchange interactions. Figure 4 represents the energy

between the antipodal ions. Therefore, we retain only the most significant exchange pathways, including the 3d−3d interaction between the two manganese ions (J(Mn, Mn) ≠ 0) and the 3d−4f interaction between the closely spaced Mn and Gd ions situated along the sides of the tetragon (Js(Mn, Gd) ≠ 0). This simplified scheme of the exchange pathways can be described by the following spin Hamiltonian including the above-described HDVV exchange interactions and the isotropic Zeeman interaction (last term): H1̂ = −2J(Mn, Mn)s(Mn ̂ ̂ 1) · s(Mn 2) − 2Js (Mn, Gd) [s(Mn ̂ ̂ 1) + s(Mn ̂ ̂ 1) · s(Gd 2) · s(Gd 2)] + gμB [s(Mn ̂ ̂ ̂ 1) + s(Gd ̂ 1) + s(Mn 2) + s(Gd 2)] · H (1)

where ŝ(Mn1) etc. are the ionic spin operators, μB is the Bohr magneton, and H is the external magnetic field. We also neglect the difference between the local (ionic) g factors and introduce the effective g factor for the cluster (g). The meaning of the remaining notations is clear from Figure 3. The anisotropic terms are not included in the Hamiltonian because all of the constituent metal ions are highly isotropic and the anisotropy of the exchange is also expected to be negligible. For this reason the Z axis of the molecular frame can be directed along the external magnetic field, and therefore, the Zeeman interaction can be presented in the form gμBHSẐ , where SẐ is the operator of the projection of the total spin on the Z axis and H is the magnitude of the magnetic field. The full Hilbert space of spin states, with the dimension [2s(Mn) + 1]2[2s(Gd) + 1]2 = 2304 can be divided into subspaces belonging to the allowed total spins (varying from S = 0 to S = 12). The numbers of multiplets with total spin S, denoted as f(S), are given in Table 1. These numbers indicate the dimensions of the S blocks of the exchange Hamiltonian matrix to be diagonalized. The energy levels were found by using the well-developed irreducible tensor operators technique,24−29 which allows large-scale eigenvalue problems to be solved.

Figure 4. Energies vs total spin for the MnII2GdIII2 tetramer calculated with the best-fit parameters (see the text).

Table 1. Numbers of States f(S) with Different Full Spins S in the MnII2GdIII2 Tetramer S

f(S)

S

f(S)

0 1 2 3 4 5 6

6 16 24 29 31 30 26

7 8 9 10 11 12

21 15 10 6 3 1

levels of the system as a function of the total spin calculated with the set of best-fit parameters. One can see that the system has a spin-singlet (S = 0) ground state, which causes the drop in χT at low temperature (experimentally available T = 2 K), while at T = 0 the expected value of χT is zero. Figure 5 illustrates the classical picture of the spin structure of the MnII2GdIII2 tetramer, in which antiferromagnetic (MnII−MnII) and ferromagnetic (MnII−GdIII) connections provide maximal energy gain in the ground state. Table 2 presents the calculated gaps Δ(S, S − 1) = E(S) − E(S − 1) between the neighboring spin levels in the low-lying part of the energy spectrum (S = 0, C


Article

Inorganic Chemistry

from the Landé rule. In fact, the values Jeff corresponding to the gaps Δ(S, S − 1) are not constant but systematically increase with increasing S. Thus, the values of |Jeff| that are found from the gaps Δ(1,0) and Δ(6,5) are 0.129 and 0.162 cm−1, respectively. To account for this discrepancy, a biquadratic term of the form j(Ŝ1·S2̂ )2, where j is a biquadratic interaction parameter, may be formally added to the bilinear HDVV interaction (eq 3). Taking into account the biquadratic term, one can find a modified rule for the gaps, Δ̃(S, S − 1) = 2|Jeff|S + 4jS3, from which one can find that the parameter j is positive and rather small (j ≈ 4.83 × 10−4 cm−1). Nevertheless, the contribution of the biquadratic term is significant for the highspin region of the spectrum because of its strong spindependence (∝S3), and in this sense the biquadratic exchange cannot be called a negligible perturbation (e.g., the contribution of the biquadratic exchange term to the gap Δ̃(6,5) is almost 25%). The spin dependence of the low-lying excitations with the energies ∝S(S + 1) is known as a rotational band,31−36 which is a generic feature of many antiferromagnetic Heisenberg-type molecular magnets, such as rings with even and odd numbers of spins, {Cu12La8}37 (cuboctahedron, s = 1/2), and the giant Keplerate cluster {Mo72Fe30} (icosidodecahedron, s = 5/2). Our calculations demonstrate the presence of the rotational band in the energy pattern of the MnII2GdIII2 tetramer with a quitespecific spin structure combining ferro- and antiferromagnetic interactions. Along with the low-lying band, which is termed the Landé band (or L band for short),31−36 one can also observe in the energy pattern of MnII2GdIII2 (Figure 4) a second band (less pronounced), which was first discovered in an N = 6, si = 5/2 spin ring and named the E band.32 The excited E band is nearly parallel to the L band, but in contrast to the L band, it starts with S = 1. It was argued32 that the existence of the L band in a spin ring also implies the existence of a further sets of rotational bands, in particular the E band (and also more complicated features). The similarity in the rotational band structures of the MnII2GdIII2 tetramer and spin rings suggests some common physical roots for these regularities. For antiferromagnetic spin rings, the low-lying excitations are associated with the so-called rotation of the Néel vector in the classical treatment of the antiferromagnetic systems.31−35 Following this concept lets us represent the topology of the exchange network of MnII2GdIII2 (in which the terminal GdIII ions are assumed to be magnetically disconnected) as a linear chain of the four spins GdIII, MnII, MnII, and GdIII, as shown in Figure 6a. The system can be viewed as an antiferromagnetic dimer formed by the two spins S1 = S2 = s(Mn) + s(Gd) = 6, whose classical ground state (similar to the Néel state) is shown in Figure 6a. Indeed, the spins s(Mn) + s(Gd) (as well as spins s(Mn) and s(Gd) separately) are large enough to be considered as classical ones. The low-lying excited states of this antiferromagnetic dimer are assumed to correspond to the disposition of the “sublattice spins” s(Mn) + s(Gd) (Figure 6b,c) tilted at some angles (interrelated with the full spins of the system), so that magnetization along the dimer does appear. To quantify this classical model, the exchange interaction between MnII−MnII should be rescaled, namely, Jeff = J(Mn, Mn){s(Mn)/[s(Mn) + s(Gd)]}2 ≡ (25/144)J(Mn, Mn). In this way one obtains Jeff = −1.6·25/144 = −0.28 cm−1, which is twice as large as the corresponding value obtained from the numerical analysis of the L band. This result can be compared with a general conjecture in ref 31 according to which the spin

Figure 5. Classical representation of the spin structure of the MnII2GdIII2 tetramer in the ground state.

Table 2. Energy Gaps for Low-Lying Spin Excitations in the MnII2GdIII2 Tetramer Calculated with the Best-Fit Parameters (See the Text) S

Δ(S, S − 1) (cm−1)

1 2 3 4 5 6

0.257 0.523 0.808 1.125 1.495 1.948

1, ..., 6; see Figure 4). The numerical values of the gaps explain the very fast increase of χT at low temperatures, so that even at T = 2 K all of the levels from S = 0 to S = 6 are well-populated. From Figure 4 one can see that the competitive influence of the ferro- and antiferromagnetic interactions results in a specific dependence of the energy levels on the total spin and that in particular the levels with the maximal spin S = 12 are located approximately in the middle of the range of the full spectrum (∼100 cm−1). The population of this level (and neighboring high-spin levels) ensures the maximum in the χT versus T curve at T ≈ 50 K (Figure 2), and then with increasing temperature the value of χT slightly decreases as a result of depopulation of the high-spin levels in favor of the low-spin ones situated at the top of the energy pattern (Figure 4). A distinctive feature of the energy pattern is the dependence of the series of low-lying levels ε(S) on the total spin of the system. First inspection of Figure 4 shows that this dependence is nearly parabolic, so that approximately ε(S) ≈ ε(0) − JeffS2, where ε(0) is the energy of the ground state so far evaluated. More accurately, this dependence can be described by the eigenvalues of the HDVV Hamiltonian of a dimer: Ĥ eff = −2Jeff S1̂ ·Ŝ 2

(3)

in which the two spins S1 = S2 = 6 are coupled through the antiferromagnetic exchange with the parameter Jeff (Jeff < 0). It is remarkable that the spins S1 and S2 are different from S(Mn) and S(Gd), but the total spin S of the dimer ranges from 0 to 12, just as in the low-lying part of the energy pattern for the genuine MnII2GdIII2 system. Formally, the spins S1 and S2 can be associated with the ferromagnetic MnII−GdIII pairs, which act as infrangible spins S1 = S2 = s(Mn) + s(Gd) = 5/2 + 7/2 = 6. A more detailed discussion of this “bi-spin” model will be given later in this section. The energies are expressed as ε(S) ≈ ε(0) − Jeff S(S + 1)

(4)

where the constant term is omitted and the ground level ε(0) is chosen as the reference point for the energies. The energy gaps in eq 4 obey the Landé interval rule Δ(S, S − 1) = −2JeffS. It can be seen from the numerical data (Table 2) that this Landé rule is satisfactorily fulfilled and that |Jeff| is approximately 0.145 cm−1. At the same time, one can observe a noticeable deviation D


Article

Inorganic Chemistry

Figure 7. Additional illustration of the model of the rotational band in MnII2GdIII2 showing the alternative classical representation of the spin excitations in which the spins in the MnII−GdIII pairs are tilted (see the text): (a) the spins of the MnII ions remain antiparallel, while the spins of the GdIII ions are tilted at the angle α; (b) the spins of the MnII ions are tilted at the angle 2ϑ, and at the same time, the spins of the GdIII ions are tilted at an angle β − ϑ with respect to the MnII spins.

Figure 6. Classical representation of the spin excitations in the MnII2GdIII2 tetramer, which is modeled by a linear chain according to the topology of the exchange network: (a) ground-state configuration, in which the spins of the MnII ions are antiparallel while the spins of the GdIII ions are parallel to the spins of the neighboring MnII ions; (b) spin excitation in the (MnIIGdIII)−(MnIIGdIII) dimer, in which spins of the pairs MnIIGdIII are tilted with respect to one another; (c) representation of the excitation in terms of the effective spins of the dimers.

that in spite of the fact that the exchange interaction in the MnII−GdIII pair is weaker than that in the MnII−MnII pair, the former brings a minor contribution to the energies of the lowlying excitations. This speculation justifies (at least qualitatively) the “bi-spin” model of the rotational band so far adopted. At the same time, the notable deviation from the Landé interval rule, especially in the high-energy pattern of the rotational band, can be attributed to tilting of the MnII and GdIII spins, as shown in Figure 7b. More detailed consideration of rotational bands in systems like MnII2GdIII2 (or, more generally, in ferrimagnetic systems) is beyond the scope of this paper.

dependence of the L bands in Heisenberg polytopes can be expressed as being proportional to −J(D(N, s)/N)S(S + 1), where N is the number of spins and D(N, s) is a specific structural coefficient that depends also on the spins of the ions. This coefficient is chosen to properly fit the gap between the lowest antiferromagnetic level of the system and the highest ferromagnetic one within the L band (see the detailed description in ref 31). One can see that within the model under consideration, the coefficient D(2,6) is equal to 1/2. While the “bi-spin” model is successful in the description of the low-lying levels and fits into the overall concept of the rotational bands,31−36 some additional comments appear to be appropriate. An indicative attribute of the model is the nature of the magnetic excitation, which is assumed to be interrelated only with the tilting of the infrangible spins of the ferromagnetic MnII−GdIII dimers (Figure 6). This seems to contradict the fact that the MnII−GdIII exchange interaction (0.8 cm−1) is weaker than that for the MnII−MnII pair (−1.6 cm−1). Therefore, it would seem that the low-lying magnetic excitations could be interrelated with the tilting of the spins in the MnII−GdIII pairs while the spins in the MnII−MnII pairs remain collinear (or at least at small angles α), as shown in Figure 7a. However, it should be taken into account that the ferromagnetic connection is “more resistant” with respect to variation of the classical angle between spins than the antiferromagnetic one. In terms of quantum spins, this means that the gap Δ(Smax, Smax − 1) separating the neighboring lowlying levels of a ferromagnetic dimer exceeds the gap Δ(1,0) in an antiferromagnetic dimer with the same spins and absolute value of the exchange coupling. As applied to the system under consideration, let us imagine that the excitation of the system occurs through the bending (at angle α) of the two ferromagnetic Gd−Mn connections provided that the MnII spins remain collinear in the MnII−MnII pair, as sketched in Figure 7a. As a result, the energy of the system increases by 2Δ(Smax, Smax − 1) = 4Js(Mn, Gd)[s(Mn) + s(Gd)] ≈ 19.2 cm−1, which is considerably larger than Δ(1,0). One can see

3. EXCHANGE COUPLING AND MAGNETIC ANISOTROPY IN THE CoII2EuIII2 CLUSTER 3.1. Experimental Section: CoII2EuIII2. The thiacalix[4]arene CoII2EuIII2 tetranuclear complex was synthesized and experimentally characterized in ref 23 and is fully isostructural with the MnII2GdIII2 cluster described above (see Figure 1) . The temperature dependence of the effective magnetic moment of the powder sample (see ref 23) is given in Figure 8. The magnetic moment exhibits a slow decrease from the roomtemperature value of 7.82μB to about 6μB at 30 K and then a fast drop to 3.4μB at 2 K.

Figure 8. Temperature dependence of the effective magnetic moment of the powder sample of thiacalix[4]arene CoII2EuIII2 complex. Experimental data23 are shown by red circles, and the theoretical curve (section 3.2) calculated with the best-fit parameters (see the text for the parameter values) is shown by the solid black line. E


Article

Inorganic Chemistry 3.2. Results and Discussion: MnII2GdIII2. To model the magnetic behavior of the system, let us first focus on the EuIII ions. The 7F0 ground state of the EuIII ion is nonmagnetic, but the energy gap between this state and the first excited magnetic state, 7F1, is relatively small (λ ≈ 300 cm−1, where λ is the spin−orbit coupling parameter). Hence, this state and the next excited levels become populated as the temperature is increased, and they make a considerable contribution to the magnetic moment of the EuIII ion. In many cases in the calculation of the temperature dependence of the magnetic moment all seven states 7FJ (J = 0, 1, 2, ..., 6) arising from the spin−orbit splitting of the 7F term are taken into account. The energies of these states (counted from the energy of the ground 7 F0 state) are given by E(J) = λJ(J + 1)/2. Then the magnetic susceptibility of the EuIII ion can be calculated as follows:30

this anisotropy being determined by the nearest ligands surrounding the cobalt ion. Thus, in ref 30 it was shown that the first-order orbital angular momentum is quenched in the case of seven-coordinate CoII ion, and therefore, the anisotropy of the cobalt ions can be well-described by the zero-field splitting (ZFS) spin Hamiltonian. To explain and quantify the experimental observations, the total Hamiltonian for the pair of cobalt ions (including the Zeeman term) can be presented as follows: ⎡ 2 Ĥ 2 = −2J s(Co s (Co1) + sZ 2(Co2) ̂ 1) ·s(Co ̂ 2 ) + D⎢ ⎣Z − 3

∑ J = 0 (2J + 1)χ (J ) exp[−λJ(J + 1)/2kBT ] 6

∑ J = 0 (2J + 1) exp[−λJ(J + 1)/2kBT ] (5)

where χ(J) is the partial contribution to the full susceptibility associated with the level E(J), which contains paramagnetic and diamagnetic terms: χ (J ) =

NAgJ 2μB 2 J(J + 1) 3kBT

+

(7)

II

where s = /2 is the spin of the Co ion, D is the single-ion ZFS parameter (Z is the Co−Co axis), J ≡ J(Co, Co) is the parameter for the HDVV exchange between the cobalt ions, and gCo is the g factor for the cobalt ion. We assume that the Z axis passes through the Co−Co bond, so the main local anisotropy axes are collinear. This seems to be a reasonable approximation, taking into account the fact that the ZFS tensors are mirror-related. In a more rigorous treatment, this question must be considered in a more precise way. In the linear regime (weak magnetic field), the diagonal components of the magnetic susceptibility tensor of the cobalt dimer are calculated using the following expression:

6

χ (Eu) =

⎤ 2 s(s + 1)⎥ + gCoμB [s(Co ̂ 1) + s(Co ̂ 2 )] · H ⎦ 3

2NAμB 2 (gJ − 1)(gJ − 2) 3λ (6)

With the exception of g0 = 5, all of the gJ involved in eq 6 are equal to 3/2. Using eq 5 and the formula μeff (Eu) = 8χ (Eu)T , in which μeff is the effective magnetic moment, one obtains the temperature dependence of μeff shown in Figure 9. The effective magnetic moment vanishes

χγγdim = NAkBT

⎫ ⎧ 1 ∂ ln⎨ ∑ exp[−Ei(Hγ)/kBT ]⎬⎪ ⎪ Hγ ∂Hγ ⎩ i ⎭ ⎪

⎪

(8)

The energy levels of the dimer in the applied magnetic field, Ei(Hγ) (γ = x, y, z), are obtained by the numerical diagonalization of the Hamiltonian Ĥ 2 represented by the 16 × 16 matrix. In a weak magnetic field, the average magnetic susceptibility for the powder sample and the corresponding effective magnetic moment are calculated as χ dim =

1 dim dim dim (χ + χYY + χZZ ), 3 XX

μeffdim =

8χ dim T (9)

From the point of view of the magnetic properties, CoII2EuIII2 tetramer represents a paramagnetic mixture of cobalt dimer and the two isolated EuIII ions. Therefore, effective magnetic moment of the entire cluster can calculated as

Figure 9. Temperature dependence of the effective magnetic moment of the EuIII ion calculated with the aid of eq 6.

μeff =

in the low temperature limit since the ground state is diamagnetic, while at room temperature μeff reaches the value of ≈3.4 μB due to population of the excited magnetic levels. It should be emphasized that the EuIII ions in the CoII2EuIII2 complex can be approximately regarded as magnetically isolated. Indeed, the ground states of these ions are diamagnetic and hence they do not participate in the exchange interaction in the ground manifold. As to the exchange coupling involving the excited states of the EuIII ions, such kind of interaction seems to produce a quite negligible effect at high temperatures when these states are populated. Now let us analyze the seven-coordinate CoII ions. As distinguished from isotropic MnII ions, as a rule CoII ions exhibit quite strong magnetic anisotropy, with the character of

(μeffdim )2 + 2μeff2 (Eu)

the the the be

(10)

The experimental data on the temperature dependence of the effective magnetic moment of the Co(II)2Eu(III)2 tetramer reported in ref 23 are shown in Figure 8. To describe this dependence we vary the following three parameters: J ≡ J(Co, Co), D, and gCo. The set of best-fit parameters was found to be J = −0.4 cm−1, D = 20.5 cm−1, and gCo = 2.22 with the agreement criterion R=

1 N

N

∑ {[μefftheor (Ti) − μeffexptl (Ti)]/μeffexptl (Ti)}2 i=1

≈ 1.8 × 10−2

The theoretical curve is in quite good agreement with the experimental data (Figure 8). One can conclude that the Co− F


Article

Inorganic Chemistry

1 9 (9J − 2 4D2 + 9J 2 ) ≈ −2D + J 2 2 1 E1(1) = ( −2D + J − 2 D2 + 2DJ + 25J 2 ) 2 1 ≈ −2D + J 2 1 7 E2(0) = (J − 2 4D2 + 16DJ + 25J 2 ) ≈ −2D − J 2 2 3 E2(1) = E1(2) = J 2 1 3 E3(1) = ( −2D + J + 2 D2 + 2DJ + 25J 2 ) = J 2 2 9 E2(2) = − J 2 1 9 E3(0) = (J + 2 4D2 + 16DJ + 25J 2 ) ≈ 2D + J 2 2 1 9 E4(0) = (9J + 2 4D2 + 9J 2 ) ≈ 2D + J 2 2 9 E(3) = 2D − J 2

Co exchange coupling is weak and antiferromagnetic while the uniaxial ZFS anisotropy is rather strong and positive, thus corresponding to anisotropy with an easy plane of magnetization (the XY plane). In other words, we are dealing with the case that will be termed the weak-exchange limit, where the isotropic exchange interaction is much weaker than the anisotropic interactions (ZFS in the present case). Finally, the obtained best-fit value for the g factor significantly exceeds the pure-spin value g0 = 2, thus demonstrating a strong orbital contribution. The values of D and g so far determined, as well as the presence of an easy plane of magnetization, show the similarity of the CoII ions in the complex under study to those in previously reported mononuclear seven-coordinate CoII complexes.38 Figure 10 shows the temperature dependences

E1(0) =

(11)

The expressions in the middle part of eq 11 are valid for arbitrary interrelations between the key parameters D and J, while the approximate expressions in the right part of eq 11 are obtained by expanding the exact results in series in terms of the small parameter J/D and retaining only the linear terms. The validity of this approximation (which is used here for the discussion rather than for the analysis of the experimental data) follows from the fact that in the actual case of the CoII2EuIII2 complex the inequality D ≫ |J| is fulfilled quite well. Indeed, the analysis of the experimental data shows that |J|/D = 0.02, so this case can be regarded as the weak-exchange limit. One can see that in the weak-exchange limit the levels can be grouped into three sets located around the energies −2D, 0, and +2D (which we will designated as I, II, III), as shown in the correlation diagram (Figure 11), with each group of levels originating from a certain pair (±M1) ⊗ (±M2). The three pairs of unperturbed levels (±1/2) ⊗ (±1/2), (±1/2) ⊗ (±3/2) ↔ (±3/2) ⊗ (±1/2), and (±3/2) ⊗ (±3/2) (with the energies −2D, 0, and +2D, respectively) belonging to the pair of individual CoII ions are split by the exchange interaction into the doublets labeled by the quantum number |M| as shown in Figure 11. The excitation (±1/2) → (±3/2) belonging to group II has two sites of localization in the dimer with the same unperturbed energies 0, and they are mixed by the exchange interaction (which can be called resonance coupling). The energy levels in eq 11 are represented in ascending order. Group I consists of the three levels: a low-lying singlet (| M| = 0), a doublet (|M| = 1) separated from the ground state by a gap of 5|J|, and an excited singlet 3|J| above the doublet. This structure of the levels allows us to comprehend the origin of the easy plane of magnetization. In fact, the ground state of the system is nonmagnetic, but the magnetization appears as a second-order effect at low field directed in the XY plane (gCoμBH⊥ < 8|J|) as a result of mixing of the ground-state singlet with the excited magnetic levels through the Zeeman interaction. On the contrary, the parallel (along Z) magnetization appears only at nonzero temperatures when the doublet

Figure 10. Temperature dependences of χ∥T, χ⊥T and χT for the CoII2EuIII2 complex calculated with the best fit parameters (see text).

of χ∥T, χ⊥T, and χT for the CoII2EuIII2 complex evaluated with the found set of best-fit parameters. As expected for a system with an easy plane of magnetization, χ⊥T > χ∥T, and both of these values increase with increasing temperature. One can also observe a specific temperature behavior of χ∥T, which exhibits rapid growth at very low T and then an almost flat “shelf” until 20 K. Since the measurements were performed on polycrystalline samples, a direct comparison of the theoretically predicted anisotropic effects with the experimental data cannot currently be made. To better understand the physical properties of the system, let us note that the total spin projection M = m1 + m2 (where m1 and m2 are the spin projections of the Co ions) is the exact quantum number for the field-independent part of the Hamiltonian Ĥ 2, which is axially symmetric and thus commutes with ŜZ = ŝZ(Co1) + ŝZ(Co2). Thus, in the absence of a magnetic field, the energy pattern of the cobalt pair consists of doublets that are enumerated by the quantum number |M| (or ±M), and a simple analysis shows that these doublets can be classified as 0 (4), ±1 (3), ±2 (2), and ±3 (1), where the numbers of states with particular |M| values are indicated in parentheses. These numbers are the ranks of the submatrices of the exchange Hamiltonian to be diagonalized. It is to be noted that along with the axial symmetry, the exchange Hamiltonian possesses the inversion center (related to the parity of the irreducible representation of the point-symmetry group to which the dimer belongs). This allows the energy levels Eα(|M|) (where α enumerates the levels with the same |M|) to be evaluated in algebraic form: G


Article

Inorganic Chemistry

magnetization. On the contrary, for the highest excited manifold, group III, the Ising Hamiltonian is obtained, which corresponds to the easy axis of magnetization. It should be noted that the Hamiltonian for the first excited group, (±1/2) ⊗ (±3/2) ↔ (±3/2) ⊗ (±1/2) (group II), cannot be expressed only in terms of the direct products of the spin-1/2 matrices. In fact, the effective Hamiltonian for this group of levels acts in the expanded basis that corresponds to the two resonance configurations. Along with the diagonal (±1/2) ⊗ (±3/2) and (±3/2) ⊗ (±1/2) blocks, which can be described by the pseudospin-1/2 Hamiltonians, the full Hamiltonian also involves the off-diagonal blocks corresponding to the exchange mixing of the resonance levels with the same total spin projection. We will not discuss this question in more detail. The effective pseudo-spin-1/2 Hamiltonians are widely used for the description of the low-temperature magnetic properties and inelastic neutron scattering spectra of clusters of high-spin Co(II) ions in octahedral or pseudo-octahedral ligand surroundings (see the pioneering works of Lines39 and also the reviews in refs 40−43). All of these studies were focused on Co ions with unquenched (first-order) orbital angular momenta. On the contrary, to our knowledge, the pseudospin Hamiltonians have not been considered for Co clusters, which are well-described within the spin Hamiltonian formalism. Another important difference between the effective Hamiltonian obtained here and those derived for clusters of orbitally degenerate Co ions in the framework of the Lines model 39 is that in the latter case the pseudo-spin- 1 / 2 Hamiltonians cannot be close to the XY limit. Indeed, it was shown that in systems composed of orbitally degenerate Co ions, depending on the symmetry of the ligand field around these ions, the pseudo-spin-1/2 Hamiltonian can vary between the fully isotropic Heisenberg limit and the fully anisotropic Ising one. It follows from the above consideration that the pseudo-spin-1/2 Hamiltonian formalism for the pure-spin Co clusters is valid provided that D ≫ |J|. If this criterion is fulfilled, the effective Hamiltonian operating within the ground Kramers doublet (eq 12) proves to be applicable and can be quite useful as a tool both to understand the magnetic anisotropy and to examine the low-temperature magnetic behavior of nanoscopic spin clusters possessing very large Hilbert spaces as well as extended 1D, 2D, and 3D systems. It is worth mentioning in this context that the case of D ≫ |J| occurs in many clusters containing CoII, NiII, and FeIII ions. Many examples and theoretical considerations can be found in refs 44 and 45. In a more common situation, when the orbital contribution is unquenched, the general effective Hamiltonian46 involving orbital as well as spin matrices must be used.

Figure 11. Correlation diagram for the energy levels of the CoII−CoII dimer in the CoII2EuIII2 complex in the limit of weak isotropic exchange. The groups of levels originating from the certain pairs (±M1) ⊗ (±M2) (see the text) of the individual CoII ions are marked by dashed boxes. In order to illustrate the main features of the pattern, the levels are presented out of the actual scale corresponding to the fitted parameters.

|M| = 1 is populated. The structures of groups II and III are clear from eq 11 and Figure 11. An important peculiarity of the weak-exchange limit under consideration is that in this case the effective Hamiltonians acting within the basis sets belonging to the well-separated groups of levels can be deduced. The effective Hamiltonian operating in the (±M1) ⊗ (±M2) subspace can be obtained by expressing the (±M1) and (±M2) blocks of the true spin s = 3/2 matrices in the original Hamiltonian (eq 7) in terms of pseudospin-1/2 matrices. The details are given in the Supporting Information. The effective Hamiltonian matrix for the ground (±1/2) ⊗ (±1/2) manifold (group I) can be obtained in terms of the pseudo-spin-1/2 matrices τX(Co), τY(Co), and τZ(Co) as follows: H[( ± 1 2 ) ⊗ ( ± 1 2 )] ⎡ = − 8J ⎢τX(Co1)τX(Co2) + τY (Co1)τY (Co2) ⎣ +

⎤ 1 τZ(Co1)τZ(Co2)⎥ ⎦ 4

(12)

It should be noted that although the genuine exchange interaction between the Co ions in the initial exchange Hamiltonian (eq 7) is isotropic, the effective interaction described by eq 12 is highly anisotropic since the effective exchange absorbs the effects of the local anisotropy through the truncation of the basis. In the same way, one can deduce the effective Hamiltonian for the upper (±3/2) ⊗ (±3/2) manifold (group III): H[( ± 3 2 ) ⊗ ( ± 3 2 )] = −18JτZ(Co1)τZ(Co2)

4. CONCLUDING REMARKS We have presented a theoretical treatment of the magnetic properties of two representatives of the 3d−4f tetranuclear clusters, namely, the thiacalix[4]arene complexes MnII2GdIII2 and CoII2EuIII2. We have found that in both complexes the exchange interaction between the 3d metal ions is antiferromagnetic and rather weak. In MnII2GdIII2 this antiferromagnetic interaction proves to be in competition with the ferromagnetic 3d−4f exchange between the Mn and Gd ions. The evaluated energy pattern of the spin states well explains the increase in the χT product as the temperature is decreased from 300 to 42 K. At low temperatures, however, the antiferromagnetic exchange plays an important role, leading to an abrupt decrease in the χT product as the temperature decreases. It was

(13)

The effective Hamiltonians in eqs 12 and 13 reproduce by definition the energy levels in the first-order approximation, i.e., when the exchange mixing among groups I, II, and III is neglected. The pseudo-spin Hamiltonian for the ground manifold contains the XY term as the prevailing contribution along with the Ising term as a smaller contribution, demonstrating thus the presence of the easy plane of H


Inorganic Chemistry

■

concluded that the low-lying energy pattern of the MnII2GdIII2 complex exhibits a rotational L band, which is interpreted in terms of the two antiferromagnetically coupled spins S1 = S2 = s(Mn) + s(Gd). It is remarkable that the biquadratic exchange contributes to the energy gaps in the high-energy range of the rotational band. The two main features of the magnetic properties that distinguish the CoII2EuIII2 complex from the MnII2GdIII2 complex should be mentioned. The first is that the 3d−4f exchange coupling in the CoII2EuIII2 cluster is negligible since the ground state of the EuIII ion is nonmagnetic. As a consequence, the magnetic moment of the CoII2EuIII2 complex monotonically changes with the temperature. The second difference between these two complexes is that, as distinguished from the MnII2GdIII2 tetramer comprising only highly isotropic ions, the CoII2EuIII2 system exhibits strong magnetic anisotropy arising from a large ZFS in the CoII ion. We have demonstrated that this is an easy plane of anisotropy (D > 0), in agreement with what was earlier observed for other seven-coordinate CoII complexes. Finally, we have derived a highly anisotropic effective pseudo-spin-1/2 Hamiltonian with a predominant XY component acting in the space of the low-lying levels of the CoII2EuIII2 system. This Hamiltonian seems to be useful as a tool to treat the magnetic anisotropy or, more generally, the low-temperature magnetic behavior of nanoscopic spin clusters and extended systems. It should be noted that although the examined complexes do not behave as SMMs, the analysis of their magnetic properties provides some important indications for exploiting the magnetic properties of 4f ions and continuing the search for thiacalix[4]arene 3d−4f complexes exhibiting SMM properties, especially those containing TbIII and DyIII ions. The synthetic work toward achieving this challenging aim is in progress.

■

ASSOCIATED CONTENT

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b00065. Derivation of the pseudo-spin-1/2 Hamiltonians for a Co pair in the weak-exchange limit (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (S.A.). *E-mail: [email protected] (A.P.). *E-mail: [email protected] (B.T.). Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Gatteschi, D.; Sessoli, R.; Villain, J. Molecular Nanomagnets; Oxford University Press: New York, 2006. (2) Benelli, C.; Gatteschi, D. Introduction to Molecular Magnetism: From Transition Metals to Lanthanides; Wiley-VCH: Weinheim, Germany, 2015. (3) Molecular Cluster Magnets; Winpenny, R. E. P., Ed.; World Scientific Series in Nanoscience, Vol. 3; World Scientific: Singapore, 2011. (4) Molecular Magnetism: From Molecular Assemblies to the Devices; Coronado, E., Delhaès, P., Gatteschi, D., Miller, J. S., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996. (5) Single-Molecule Magnets and Related Phenomena; Winpenny, R. E. P., Ed.; Springer: Berlin, 2006. (6) Milios, C. J.; Vinslava, A.; Wernsdorfer, W.; Moggach, S.; Parsons, S. S.; Perlepes, S. P.; Christou, G.; Brechin, E. K. J. Am. Chem. Soc. 2007, 129, 2754−2755. (7) Tasiopoulos, A. J.; Vinslava, A.; Wernsdorfer, W.; Abboud, K. A.; Christou, G. Angew. Chem. 2004, 116, 2169−2173. (8) Clemente-Juan, J. M.; Coronado, E.; Gaita-Ariño, A. Chem. Soc. Rev. 2012, 41, 7464−7478. (9) Molecular Magnets: Physics and Applications; Bartolomé, J., Luis, F., Fernández, J. F., Eds.; Springer: Berlin, 2014. (10) Molecular Nanomagnets and Related Phenomena; Gao, S., Ed.; Springer: Berlin, 2015. (11) Waldmann, O. Inorg. Chem. 2007, 46, 10035−10037. (12) AlDamen, M. A.; Clemente-Juan, J. M.; Coronado, E.; MartíGastaldo, C.; Gaita-Ariño. J. Am. Chem. Soc. 2008, 130, 8874−8875. (13) Rinehart, J. D.; Long, J. R. Chem. Sci. 2011, 2, 2078−2085. (14) Woodruff, D. N.; Winpenny, R. E. P.; Layfield, R. A. Chem. Rev. 2013, 113, 5110−5148. (15) Piquer, L. R.; Sañudo, E. C. Dalton Trans. 2015, 44, 8771−8780. (16) Osa, S.; Kido, T.; Matsumoto, N.; Re, N.; Pochaba, A.; Mrozinski, J. J. Am. Chem. Soc. 2004, 126, 420−421. (17) Klokishner, S. I.; Ostrovsky, S. M.; Reu, O. S.; Palii, A. V.; Tregenna-Piggott, P. L. W.; Brock-Nannestad, T.; Bendix, J.; Mutka, H. J. Phys. Chem. C 2009, 113, 8573−8582. (18) Funes, A. V.; Carrella, L.; Rentschler, E.; Alborés, P. Dalton Trans. 2014, 43, 2361−2364. (19) Langley, S. K.; Ungur, L.; Chilton, N. F.; Moubaraki, B.; Chibotaru, L. F.; Murray, K. S. Inorg. Chem. 2014, 53, 4303−4315. (20) Koroteev, P. S.; Efimov, N. N.; Ilyukhin, A. B.; Dobrokhotova, Z. V.; Novotortsev, V. M. Inorg. Chim. Acta 2014, 418, 157−162. (21) Mondal, K. C.; Kostakis, G. E.; Lan, Y.; Wernsdorfer, W.; Anson, C. E.; Powell, A. K. Inorg. Chem. 2011, 50, 11604−116011. (22) Bi, Y.; Li, Y.; Liao, W.; Zhang, H.; Li, D. Inorg. Chem. 2008, 47, 9733−9735. (23) Aldoshin, S. M.; Sanina, N. A.; Solov’eva, S. E.; Antipin, I. S.; Dmitriev, A. I.; Morgunov, R. B.; Korchagin, D. V.; Shilov, G. V.; Utenyshev, A. N.; Bozhenko, K. V. Russ. Chem. Bull. 2014, 63, 1465− 1474. (24) Bencini, A.; Gatteschi, D. EPR of Exchange Coupled Systems; Springer: Berlin, 1990. (25) Tsukerblat, B. S.; Belinskii, M. I. Magnetochemistry and Radiospectroscopy of Exchange Clusters; Academy of Sciences of Moldova: Chişinău, Moldova, 1983. (26) Tsukerblat, B. S.; Belinskii, M. I.; Fainzilberg, V. E. Sov. Sci. Rev., Sect. B. 1987, 9, 337−482. (27) Borrás-Almenar, J. J.; Clemente-Juan, J. M.; Coronado, E.; Tsukerblat, B. S. J. Comput. Chem. 2001, 22, 985−991. (28) Borrás-Almenar, J. J.; Clemente-Juan, J. M.; Coronado, E.; Tsukerblat, B. S. Inorg. Chem. 1999, 38, 6081−6088. (29) Tsukerblat, B. S. Group Theory in Chemistry and Spectroscopy; Dover: Mineola, NY, 2006. (30) Kahn, O. Molecular Magnetism; VCH: New York, 1993. (31) Schnack, J.; Luban, M. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 63, 014418. (32) Waldmann, O. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 65, 024424.

S Supporting Information *

■

Article

ACKNOWLEDGMENTS

We express our gratitude to Prof. R. B. Morgunov, Dr. D. V. Korchagin, and Dr. G. V. Shilov for useful discussions of the crystalline structures and magnetic properties of the thiacalix[4] arene tetranuclear complexes. S.M.A., N.A.S., and A.V.P. are grateful to the Russian Foundation for Basic Research (RFBR Grant 13-03-12418) and to the Federal Agency of Scientific Organizations (State Registration No. 01201361865) for financial support. A.V.P. acknowledges the STCU (Project 5929) for financial support of this work. I


Article

Inorganic Chemistry (33) Schnack, J.; Luban, M.; Modler, R. Europhys. Lett. 2001, 56, 863. (34) Schnalle, R.; Schnack, J. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 104419. (35) Waldmann, O. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 75, 012415. (36) Schnalle, R.; Lauchli, A. M.; Schnack, J. Condens. Matter Phys. 2009, 12, 331−342. (37) Blake, A. J.; Gould, R. O.; Grant, C. M.; Milne, P. E. Y.; Parsons, S.; Winpenny, R. E. P. J. Chem. Soc., Dalton Trans. 1997, 485−496. (38) Chen, L.; Chen, S.-Y.; Sun, Y.-C.; Guo, Y.-M.; Yu, L.; Chen, X.T.; Wang, Z.; Ouyang, Z. W.; Song, Y.; Xue, Z.-L. Dalton Trans. 2015, 44, 11482−11490. (39) Lines, M. E. J. Chem. Phys. 1971, 55, 2977−2984. (40) Palii, A.; Tsukerblat, B.; Clemente-Juan, J. M.; Coronado, E. Int. Rev. Phys. Chem. 2010, 29, 135−230. (41) Palii, A.; Tsukerblat, B.; Klokishner, S.; Dunbar, K.; ClementeJuan, J. M.; Coronado, E. Chem. Soc. Rev. 2011, 40, 3130−3156. (42) Murrie, M. Chem. Soc. Rev. 2010, 39, 1986−1995. (43) Lloret, F.; Julve, M.; Cano, J.; Ruiz-García, R.; Pardo, E. Inorg. Chim. Acta 2008, 361, 3432−3445. (44) Boča, R. Theoretical Foundations of Molecular Magnetism; Elsevier: Amsterdam, 1999. (45) Boča, R. A Handbook of Magnetochemical Formulae; Elsevier: Amsterdam, 2012. (46) Borrás-Almenar, J. J.; Clemente-Juan, J. M.; Coronado, E.; Palii, A. V.; Tsukerblat, B. S. Chem. Phys. 2001, 274, 131−144.

J


Theoretical Modeling of the Magnetic Behavior of Thiacalix [4] arene

Recommend Documents