Single-Ion Magnet Et4N[CoII(hfac)3] with Nonuniaxial Anisotropy

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Single-Ion Magnet Et4N[CoII(hfac)3] with Nonuniaxial Anisotropy: Synthesis, Experimental Characterization, and Theoretical Modeling Andrew V. Palii,*,†,‡ Denis V. Korchagin,*,† Elena A. Yureva,† Alexander V. Akimov,† Eugenii Ya. Misochko,† Gennady V. Shilov,† Artem D. Talantsev,† Roman B. Morgunov,† Sergey M. Aldoshin,† and Boris S. Tsukerblat*,§ †

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

S Supporting Information *

ABSTRACT: In this article we report the synthesis and structure of the new Co(II) complex Et4N[CoII(hfac)3] (I) (hfac = hexafluoroacetylacetonate) exhibiting single-ion magnet (SIM) behavior. The performed analysis of the magnetic characteristics based on the complementary experimental techniques such as static and dynamic magnetic measurements, electron paramagnetic resonance spectroscopy in conjunction with the theoretical modeling (parametric Hamiltonian and ab initio calculations) demonstrates that the SIM properties of I arise from the nonuniaxial magnetic anisotropy with strong positive axial and significant rhombic contributions. magnetic field (so-called field-induced SMMs and SIMs) in spite of the fact that they possess positive axial magnetic anisotropy. An important specific feature of such Co(II) compounds is that along with positive axial anisotropy they exhibit strong rhombic anisotropy, which was argued to be a prerequisite of slow magnetic relaxation in this kind of system. The origin of slow relaxation of magnetization in Kramers ions with nonuniaxial anisotropy was recently revealed in ref 7. While the magnetic behavior of the systems with axial magnetic anisotropy is well-studied and, in general, clear enough (at least conceptually), the adequate description of systems possessing nonaxial anisotropy represents a challenging problem. This problem is especially interesting when we are dealing with the high-spin Co(II)-based SIMs whose properties are closely related to different aspects of the orbital degeneracy of the ground term. Since the orbital degeneracy is closely related to the orbital angular momentum that gives rise to the magnetic anisotropy, the study of the orbitally degenerate (or pseudo-degenerate)

1. INTRODUCTION Mononuclear complexes exhibiting slow magnetic relaxation are known as single-ion magnets (SIMs), which represent an important class of single-molecule magnets (SMMs). As well as polynuclear SMMs, the mononuclear complexes exhibiting SIM behavior are currently a focus of research in molecular magnetism due their intriguing physical properties and potential applications in high-density data storage,1 spintronics,2 and quantum computing.3 First SIMs were found among the lanthanide complexes (see ref 4 and refs therein), but also numerous SIMs based on transition-metal ions have been recently reported (see ref 5 and refs therein). Different aspects of of physics and chemistry of SIMs based on Co(II) ions were discussed in detail in recent years.6 For a long time there was a belief that slow magnetic relaxation can be driven only by negative magnetic anisotropy or, alternatively, by existence of an easy axis of magnetization required for the creation of the magnetization reversal barrier. More recently, however, a few examples of mononuclear7−16 and polynuclear17 cobalt(II) complexes have been shown to exhibit slow relaxation of magnetization in applied static (direct current (DC)) © XXXX American Chemical Society

Received: June 20, 2016

A

DOI: 10.1021/acs.inorgchem.6b01473 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry Table 1. Hexa-Coordinated (Quasi-Octahedral) High-Spin Co(II) Complexes Exhibiting SIM Propertiesa compound [Co(dca)2(atz)2]n [Co(oda) (aterpy)] [Co(abpt)2(tcm)2] [Co(btm)2(SCN)2]n [Co(ppad)2]n [Co(AcO)2(py)2(H2O)2] [Co(dca)2(bim)4] [Co(dca)2(bim)2]n [Co(dca)2(bmim)2]n [Co(dmphen)2(NCS)2] [Co(acac)2(H2O)2] [Co(bpy)2(ClAn)]·EtOH

D, cm−1 −7.44 +48(2) +56.6 +76 |69.6| |74.3| |75.8| +98 +57 +64.55

E, cm−1

13 3.70 +6.5

+8.4 17 10.13

|E/D|

gs

0.27(2) 0.065 0.086

2.05 2.461(6) 2.49 2.46

0.086 0.31 0.157

2.49 2.48 2.57 2.78 2.50 2.57, 2.40 2.46

(3/2) κ

λ, cm−1

Δax, cm−1

Ueff, cm−1

ref

1.18(1)

−125(1)

−509(10)

1.48(1) 1 1.18 1.16 1.24 1.225

−147(1) −170.0 −132 −134.1 −134.0 −165.8

−482(5) −279 −416.3 −402.7 −605.8 493 130

5.1 2.9 86.2 31.6 11.37 25.0 5.5−7.7 4.5−9.2 11.5−15.4 17.0 14−17 16.6

18 19 12 13 20 14 15 15 15 16 7 6d

a

Abbreviations: dca = dicyanamide; atz = 2-amino-1,3,5-triazine; oda = oxodiacetate dianion; aterpy = 4′-azido-2,2′:6′,2″-ter-pyridine; abpt = 4-amino-3,5-bis(2-pyridyl)-1,2,4-triazole; tcm = tricyanomethanide anion; btm = bis(1H-1,2,4-triazol-1-yl)methane; ppad = N3-(3-pyridoyl)-3pyridinecarboxamidrazone; AcO = acetate anion; py = pyridyl; bim = 1-benzylimidazole; bmim = 1-benzyl-2-methylimidazole; dmphen = 2,9-dimethyl-1,10-phenanthroline; acac = acetylacetonate; bpy = 2,2′-bipyridyl; ClAn = chloranilate dianion.

(alternating current (AC)) magnetic susceptibility, which is evidence in favor of the existence of SIM properties of the Co(II) complex under the study.

systems can shed light on the origin of the SIM properties in nonaxial SIMs. Among hexa-coordinated (quasi-octahedral) high-spin Co(II) complexes exhibiting SIM properties there are complexes with both positive and negative axial anisotropies.7,12−16,18−20 The main systems are summarized in Table 1 along with the data on magnetic parameters, which will be used in Section 3 for the analysis of the models. The most important features of the magnetic behavior of such complexes arise from the fact that the ground crystal-field term of the octahedrally coordinated high-spin Co(II) ion is the orbital triplet 4T1g. The presence of the orbital degeneracy does not allow, in general, to treat the high-spin Co(II) complexes as simple spin systems (with S = 3/2) even taking into account the anisotropic contributions to the spin-Hamiltonian. In this article we report the synthesis, structural, magnetic, and electron paramagnetic resonance (EPR) characterization and theoretical modeling of the new Co(II) complex Et4N[CoII(hfac)3] (I) (hfac = hexafluoroacetylacetonate) and demonstrate that this system exhibits SIM behavior resulting from strong positive axial and significant rhombic anisotropies. In conformity with the above arguments we use for the description of the DC magnetic properties of I the Griffith Hamiltonian. The determination of the sign of the axial crystal-field parameter is based on the analysis of EPR spectra confirmed also by the quantum-chemical spin-averaged Hartree−Fock (SAHF) calculations. The paper is organized as follows. In Section 2 we describe the experimental data on structure determination, DC magnetic susceptibility, and EPR. Then (Section 3) we give a short overview and critical analysis of the models for high-spin Co(II) complexes and present the theoretical model (including ab initio orbital analysis), which we employ for the treatment of the Et4N[CoII(hfac)3] complex. We will demonstrate that the model is able to provide a unified description of the full set of the experimental data and make a judgment about the reliability of the information provided by each kind of the approaches separately. We show that only the information obtained by the joint analysis of all experimental data in the framework of the physically justified model accompanied by the orbital analysis is able to give an unambiguous explanation of the properties of the novel Co(II) complex Et4N[CoII(hfac)3]. A separate Section 4 is devoted to the measurements and interpretation of the dynamic

2. EXPERIMENTAL SECTION 2.1. Synthesis of Et4N[CoII(hfac)3] (I). All chemicals were purchased from commercial sources and used without further purification. The Fourier transform infrared spectra of microcrystalline powders were recorded on PerkinElmer Spectrum 100 spectrometer. The powder X-ray diffraction (XRD) patterns were recorded at room temperature on an ARL X’TRA X-ray diffractometer. Synthesis of the Et4N[CoII(hfac)3] was based on the known literature procedure.21 Anal. Calcd for C23H23N1O6F18Co1: C, 34.09; H, 2.86; N, 1.73; F, 42.20; Co, 7.27%; Found: C, 33.49; H, 2.79; N, 1.59; F, 39.87; Co, 6.7%. Elemental analysis of this complex was performed by pyrolysis. As a result not all F atoms were burned, and cobalt fluoride was formed, which led to the underestimated percentage of fluorine and cobalt. IR (cm−1): 2966 m, 1644 s, 1555 m, 1526 m, 1507 m, 1484 m, 1397 w, 1343 w, 1258 vs, 1197 s, 1139 vs, 1094 s, 1035 m, 948 w, 866 w, 810 s, 794 s, 763 w, 743 w. The powder XRD measurements showed that sample is a monophase crystalline material (Figure S1b,c). Magnetically diluted sample Et4N[Co0.02Zn0.98(hfac)3] for electron spin resonance investigation was prepared by cocrystallization of complex I and its isostructural Zn analogue22 (Figure S1) with ratio Co/Zn = 1:50. 2.2. Crystal and Molecular Structure. X-ray data for a single crystal of I were collected on a CCD diffractometer Agilent XCalibur with EOS detector (Agilent Technologies UK Ltd., Yarnton, Oxfordshire, England) at 90.0(2) K using graphite-monochromated Mo Kα radiation (λ= 0.710 73 Å). The structure was solved by direct methods and refined against all F2 data (SHELXTL23). All nonhydrogen atoms were refined with anisotropic thermal parameters, and the positions of hydrogen atoms were obtained from difference Fourier syntheses and refined with riding model constraints. The X-ray crystal structure data were deposited with the Cambridge Crystallographic Data Center, with reference code CCDC 1486496. Selected crystallographic parameters and the data collection and refinement statistics are given in Table 2. Compound I is crystallized in the triclinic space group P1̅ (Figure 1). Figure 1a shows the electroneutral structural unit of the compound I consisting of anionic complex tris(1,1,1,5,5,5-hexafluoroacethylacetonato)cobalt(II) and tetraethylammonium cation. Figure 1b shows the ac projection of crystal structure. The bond lengths Co−O in coordination polyhedra CoIIO6 are listed in the caption for Figure 1. The coordination geometry around Co is a fully distorted octahedron with pronounced compression along O(3)−Co(1)−O(6) bonds most probably due to the Jahn−Teller effect, which acts as static in the case under consideration. The equatorial bond lengths are different, B


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Inorganic Chemistry Table 2. Crystal Data and Structure Refinement for Compound I parameters

values

C15H3CoF18O6−, C8H20N1+ (810.35) 90.0(2) triclinic, P1̅ a = 10.3118(4) Å α = 86.395(3)° b = 10.7949(4) Å β = 80.349(3)° c = 15.2491(5) Å γ = 67.848(4)° 1549.9(1) volume, Å3 Z, density (calculated), g/cm3 2, 1.736 absorption coefficient, mm−1 0.701 F(000) 810 crystal size, mm3 0.10 × 0.15 × 0.35 θ range for data collection, deg 2.88−29.07 index ranges −14 ≤ h ≤ 14, −14 ≤ k ≤ 13, −20 ≤ l ≤ 18 reflections collected/unique/[I > 2σ(I)] 14937/8273 [R(int) = 0.027]/6384 completeness to θ = 29.07° 0.999 number of parameters 447 goodness-of-fit on F2 1.023 final R indices [I > 2σ(I)] R1 = 0.0457, wR2 = 0.0962 R indices (all data) R1 = 0.0648, wR2 = 0.1069 largest diff. peak and hole, e·Å−3 1.12 and −0.74 formula (M) temperature, K crystal system, space group unit cell dimensions

Figure 2. X-band EPR powder spectrum of the diluted sample I Et4N[Co0.02Zn0.98(hfac)3] at 4.2K (a) and simulation (b) with the bestfit parameters obtained by the least-squares fitting procedure (see Section 3.3). modeling, which will be discussed in Section 3 in conjunction with the model and magnetic data. 2.4. Magnetic Measurements in Static Field. Magnetic data were obtained with the SQUID magnetometer (MPMS-5XL, Quantum Design). The temperature dependence of the DC magnetic susceptibility was measured on polycrystalline sample in the temperature range 2−300 K with the applied magnetic field of 0.1 T. Diamagnetic corrections were made using the Pascal constants. The magnetization has been measured at T = 2, 3, 4, 5.5, 10, and 20 K. The result of measurements along with the theoretical curves (Section 3) are given in Figure 3. The AC magnetic susceptibility data were obtained at the amplitude of 2.0 Oe and the frequencies of the alternating magnetic field in the range of 1−1400 Hz (see Section 4). At room temperature the χMT is found to be ∼2.78 cm3 K mol−1, which is considerably higher than the spin-only value (1.875 cm3 K mol−1) due to a significant orbital contribution to the magnetic moment (Figure 3). This value is within the range of those observed in sixcoordinate high-spin cobalt(II) complexes with the unquenched angular momentum.24,25 When cooled, the value of χMT decreases slightly in the temperature range from 300 to 150 K (2.78 and 2.55 cm3 K mol−1, respectively), and it decreases sharply below 150 K to reach the value of 1.60 cm3 K mol−1 at 2 K. Apparently, the decrease of χMT in I can be attributed to the thermal depopulation of the excited Kramers doublets of the Co(II) ions. For I at T = 2 K the magnetization as a function of magnetic field is almost saturated at 5 T reaching the value of ∼2.2NAμB

and the averaged value of these lengths Co(1)−Oeq 2.079(2) exceeds the averaged axial bond length Co(1)−Oax 2.050(2) Å of the coordination polyhedron. The bond angles O−Co(1)−O deviate from the ideal values 90 and 180° and lie in the ranges of 86.24(6)−96.76(6) and 173.42(7)−177.59(6)°, respectively. In the crystal packing (Figure 1b), Co(II) atoms are well-shielded from each other with an interatomic separation that is longer than 8 Å, excluding thus significant intermolecular magnetic interactions. 2.3. Electron Paramagnetic Resonance Spectra. The EPR data were measured with a Bruker Elexsys E500 spectrometer working in X-band (9.4715 GHz) and modulated at 100 kHz. Powder diluted samples were placed in a 2 mm diameter quartz tube, which was pumped out and filled with helium to obtain rapid thermal stabilization of the entire sample at low temperatures. An RTI ESRCryo202 continuousflow cryostat refrigerated with liquid helium was used for cooling the samples to 4.2 K. The modulation amplitude and microwave power was low enough to avoid the line broadening and the saturation effects. The EPR spectrum of diluted sample I Et4N[Co0.02Zn0.98(hfac)3] measured at 4.2 K is shown in Figure 2 along with the results of theoretical

Figure 1. Crystal structure of compound I: (a) general view of the structural unit coloring: gray-C, light green-F); (b) ac projection of crystal structure, H atoms are omitted for clarity. Bond lengths Co−O (in Å) in the first coordination sphere: Co(1)−O(1)−2.077(2), Co(1)−O(2)-2.078(2), Co(1)−O(3)−2.053(2), Co(1)−O(4)−2.084(2), Co(1)−O(5)−2.077(2), Co(1)−O(6)−2.047(2). C


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Inorganic Chemistry

isomorphism,34 can be regarded as the term with fictitious orbital angular momentum L = 1 (ML = 0, ML = ±1). The axial crystal field arising from the tetragonal component of the distortion splits the 4T1g term into orbital doublet 4Eg (ML = ±1) and orbital singlet 4A2g (ML = 0). The spin−orbital coupling (SOC) further splits the 4Eg multiplet into four Kramers doublets (first-order effect) and also mixes 4Eg and 4A2g resulting in a splitting of the 4 A2g term into two Kramers doublets (second-order effect). If the symmetry is lower than D4h one must take into account also rhombic component of the crystal field that splits the 4Eg term. Providing positive axial field the tetragonal ground term is 4A2g, the anisotropy in this case is also positive (easy plane anisotropy). In contrast, for negative axial field leading to the ground orbital doublet 4Eg, the system exhibits negative magnetic anisotropy, which means the existence of the easy axis of magnetization. Three types of Hamiltonians are commonly used for the description of the high-spin Co(II) ions in distorted octahedral surroundings. The first one was proposed by Griffith34 and then substantially modified by Figgis.35,36 The Figgis approach explicitly accounts for the unquenched orbital angular momentum through the usage of T−P isomorphism. The Griffith Hamiltonian takes into account the low-symmetry crystal field, the SOC, and Zeeman interaction. It includes the parameters of the axial field Δax, the rhombic field parameter Δrh, the SOC parameter λ, and the orbital reduction factor κ, which actually absorbs the covalence effect and mixing of 4T1g(4F) and 4T1g(4P) terms through the octahedral component of the crystal field. Such kind of Hamiltonian has been applied, for example, to the analysis of the DC magnetic properties of two-dimensional coordination polymers [Co(dca)2(atz)2]n18 (Table 1) and [Co(ppad)2]n20 (Table 1) exhibiting SIM behavior. Along with the Griffith Hamiltonian, which can be referred to as microscopic, the effective pseudospin-1/2 Hamiltonian describing the Zeeman splitting of the ground Kramers doublet is often used (see, e.g., the study of complexes [Co(abpt)2(tcm)2]12 and [Co(dmphen)2(NCS)2]16 (Table 1). Such parametric Hamiltonian contains the principal values gZZ, gXX, and gYY of the effective g-tensor for the ground Kramers doublet. It is to be noted that the effective pseudospin-1/2 Hamiltonian can be in principle derived from the Griffith Hamiltonian, and the g-tensor can be expressed in terms of the parameters of Griffith Hamiltonian. The third kind of Hamiltonian is conventionally accepted for the description of ZFS of the spin S = 3/2 ion in a triaxial crystal field and involves also the isotropic Zeeman term and hyperfine interaction:

Figure 3. Temperature dependence of χMT for I measured at B = 0.1 T (blue ○). (inset) The magnetization vs field for I measured at T = 2, 3, 4, 5.5, 10, and 20 K (symbols). Theoretical curves (solid lines) are calculated with the parameters discussed in Section 3.3. (NA is the Avogadro number), which is significantly lower than the value of 3NAμB corresponding to the pure spin S = 3/2 ground state with g = 2. Below we will show that this is because the ground state of the system is a strongly anisotropic Kramers doublet (effective spin 1/2) rather than the spin S = 3/2.

3. MODELING AND DISCUSSION 3.1. Ab Initio Calculation of the Electronic Structure. At the first step of the discussion it is worthwhile to perform the ab initio calculation of the electronic structure to use the obtained energies in the determination of the parameters of the semiempirical Hamiltonian for the quasi-octahedral Co(II) complex. The orbital analysis was performed with the aid of SAHF calculation26 based on restricted open-shell Hartree−Fock method. Ab initio calculations of zero-field splitting (ZFS) parameters were performed using the state-averaged completeactive-space self-consistent-field (SA-CASSCF) wave functions complemented by N-electron valence second-order perturbation theory (NEVPT2)28,27 using ORCA program.28 The active space of the CASSCF calculations was composed of seven electrons in five d-orbitals of Co atom CAS(7,5). The state-averaged approach was used in which all 10 quartet states and 40 doublets states were averaged with equal weights. The polarized triple-ζquality basis set def2-TZVP was used for cobalt and oxygen atoms, while the def2-SVP basis set was used for C, F, and H atoms.29 The calculations utilized the RI approximation with the decontracted auxiliary def2-TZV/C and def2-SVP/C Coulomb fitting basis sets and the chain-of-spheres (RIJCOSX) approximation to exact exchange as implemented in ORCA.30 The calculations were started with the geometry of the experimentally determined X-ray structure. The ZFS parameters, based on dominant spin−orbit coupling contributions from excited states, were calculated through the quasi-degenerate perturbation theory (QDPT).31 In the framework of QDPT in which the spin−orbit coupling operator is approximated to the Breit− Pauli form (SOMF approximation)32 the effective Hamiltonian theory33 is used. The orbital schemes obtained from the evaluation of the electronic structure will be described in Section 3 in course of the discussion of the magnetic properties and EPR, which require a reliable independent (from the experimental data) estimation of the axial and rhombic crystal-field parameters, which are necessary ingredients of magnetic anisotropy. 3.2. Short Overview and Analysis of the Theoretical Models for High-Spin Co(II) Complexes. The ground crystalfield term of the octahedrally coordinated high-spin Co(II) ion is the orbital triplet 4T1g, which, according to the T−P

⎤ ⎡ 2 1 2 2 Ĥ s = D⎢SẐ − S(S + 1)⎥ + E(SX̂ − SŶ ) + μB ΒgsŜ ⎦ ⎣ 3 s ̂ ̂ + SA hf I (1)

where gs and Ashf are the g-tensor and the tensor of the hyperfine interactions, Ŝ and I ̂ are the electronic and nucleus spin operators, and B is the applied magnetic field. In contrast to the Griffith and pseudospin-1/2 Hamiltonians, the applicability of the spin-Hamiltonian, eq 1, often remains controversial. Indeed, such Hamiltonian can be used only providing that the ground term is 4A2g, which is well-separated from the excited term 4Eg. In this case the axial term in eq 1 can be associated with the secondorder spin−orbital splitting of the 4A2g term into two doublets corresponding to MS = ±1/2 and MS = ±3/2. It is important that in the case under consideration, the MS = ±1/2 doublet proves to be the ground state, corresponding thus to the positive D. At the same time the Hamiltonian, eq 1, with negative D is irrelevant to D


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Inorganic Chemistry high-spin Co(II) complex with the dominant cubic crystal field and relatively small low-symmetry contributions. In fact, in the case of negative axial field the low-lying part of the energy pattern of the Co(II) complex contains four Kramers doublets, which cannot be described by the ZFS spin Hamiltonian, eq 1. Meanwhile, in many of studies listed in Table 1 (with the exception of the case of [Co(dca)2(atz)2]n,18 Table 1), this Hamiltonian has been used for fitting the magnetic DC data. In some cases the use of spin Hamiltonian, eq 1, leads to negative D, which indicates the necessity of the reconsideration of the condition for applicability of the model. Sometimes (e.g., in the study of [Co(ppad)2]n,20 Table 1) the Griffith Hamiltonian has been used for the description of the temperature dependence of the magnetic susceptibility, while the ZFS spin-Hamiltonian has been applied to describe the field and temperature dependences of the magnetization. In ref 20, such kind of description has resulted in a doubtful conclusion about the negative sign of Δax, while the D parameter has been found to be positive. In other studies (for example, in those of [Co(dmphen)2(NCS)2]16 and [Co(acac)2(H2O)2]7) the correct correspondence of the signs of Δax and D parameters has been found; that is, the positive Δax values has been found to correspond to positive D values. The above arguments show that the Griffith Hamiltonian (or the corresponding pseudospin-1/2 Hamiltonian) represents the most suitable tool for the treatment of the magnetic data. Although the Griffith Hamiltonian formalism is the adequate tool, the theoretical description of the magnetic susceptibility for the powder samples usually faces the problem of excessive flexibility. In particular, in the case under consideration a satisfactory fit can be achieved with both positive and negative values of the axial field parameter, and the magnetic behavior is quite insensitive to the rhombic crystal-field parameter (see discussion in Section 3.3). This shows that the use of complementary experimental techniques, like EPR, giving direct access to the effective g-tensor for the ground Kramers doublet, as well as quantum-chemical evaluation of the crystal-field parameters, could be crucial to provide a reliable description of the magnetic anisotropy of the Co(II) complex in powder. 3.3. Preliminary Discussion of Static Magnetic Properties. To describe the DC magnetic properties we will use the following Griffith Hamiltonian that explicitly takes into account the unquenched orbital angular momentum of the Co(II) ion:24,25,34 ⎤ ⎡ 3 1 Ĥ = − κλLŜ ̂ + Δax ⎢L̂ Z 2 − L(L + 1)⎥ ⎦ ⎣ 2 3 ⎛ 3 ⎞ + Δrh (L̂ X 2 − L̂ Y 2) + μB B⎜geS ̂ − κL̂⎟ ⎝ 2 ⎠

D4h and represents the main contribution to the overall crystalfield splitting. The structural data allow also a rhombic distortion, which is weaker (according to the definition of the crystal-field tensor) than the tetragonal one. When the axial parameter Δax is positive the ground term proves to be the orbital singlet 4A2g, while in the opposite case the orbital doublet 4Eg becomes the ground state. The rhombic crystal-field parameter Δrh describes the splitting of the orbital doublet 4Eg caused by further lowering of the symmetry. It follows from the definition of the lowsymmetry crystal-field tensor that |Δrh| ≤ |Δax|/3. The Zeeman interaction in eq 2 includes both spin and orbital contributions. Factor 3/2 in SOC and Zeeman terms appears because, within the T−P isomorphism concept, the matrix of the orbital angular momentum operator L defined in the 4T1g(4F) basis differs by this factor from the matrix of L defined in pure atomic 4P basis. The sign of the axial field parameter Δax plays crucial role in the magnetic behavior of the Co(II) complex, since it determines the sign of the magnetic anisotropy of the system. Thus, providing Δax < 0 the system exhibits negative magnetic anisotropy, which corresponds to the existence of an easy axis of the magnetization, while in the case of positive Δax the anisotropy is also positive, which means that the system possesses an easy plane of magnetization.24 Of course, in the presence of a rhombic field this classification is approximate and relates only to the sign of a stronger axial field. The terms “easy axis” and “easy plane” will be used just in this symbolic meaning. The main problem in fitting χMT and M versus T data for the powder sample is that the averaged magnetic data are only slightly sensitive to the influence of the sign of Δax. In fact, the numerical experiments show that very similar magnetic curves (visually, almost indistinguishable) can be obtained with both positive and negative values of Δax and slightly different remaining parameters. This becomes evident from comparison of the two χMT versus T curves calculated with positive and negative values of Δax (see Supporting Information, Figure S.2). One can check that both sets of parameters used in plots in Figure S.2 (Supporting Information) reproduce quite well the experimental χMT versus T dependences. Moreover, it can be seen that these two curves are hardly distinguishable, so it is difficult to make a reliable conclusion about the sign of Δax from the fit of the magnetic data. In fact, in both cases the agreement between the calculated and experimental curves is quite satisfactory, and the slight difference in the agreement criteria in both cases can be deceptive. An additional complication in extracting the crystal-field parameters from fitting of the magnetic data arises from the fact that χMT versus T curves prove to be almost independent of the value of the rhombic field parameter Δrh. As a result, the usage of the Hamiltonian, eq 2, in which Δax and Δrh are regarded as fitting parameters can give a multitude of sets of the best fit parameters describing DC magnetic experiments. Therefore, without additional independent information about the sign of the parameter Δax, neither the adequacy of a model nor the correctness of these parameters can be tested. However, the analysis of the Co−O bond lengths cannot be regarded as a trustworthy evidence of the sign of the parameter Δax because of large rhombicity of the oxygen surrounding of the Co ion. Summarizing these observations, one can state that the use of spectroscopic techniques, like EPR providing a direct access to the sign of the magnetic anisotropy (incorporated in the principal values of g-tensor for ground Kramers doublet of the Co(II) ion) seems to be quite useful as well as the quantum−chemical evaluation of the main components Δax and Δrh of the low-symmetry

(2)

This Hamiltonian operates within the basis of the ground cubic 4T1 term of the Co(II) ion, which can be associated with the fictitious orbital angular momentum L = 1 and spin S = 3/2. The first term in eq 2 is the SOC with λ being the SOC parameter, and L̂ is the orbital angular momentum operator. The orbital reduction factor κ describes the reduction of the orbital angular momentum caused by the two factors: delocalization of the unpaired electrons toward the ligands and the admixture of the excited 4T1g(4P) term to the ground term 4T1g(4F) by the cubic component of the crystal field. The second term models the splitting of the cubic orbital triplet 4T1 into orbital singlet 4A2g (ML = 0, where ML is the projection of the orbital angular momentum) and orbital doublet 4Eg (ML = ±1) due to the tetragonal crystal field. This field lowers the symmetry from Oh to E


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Inorganic Chemistry crystal-field tensor. Keeping in mind the above-mentioned arguments, we use in the analysis of the DC magnetic properties (susceptibility and magnetization) the ab initio values of the parameters Δax and Δrh. The only parameter that we consider as an adjustable one is the orbital reduction factor κ. We will demonstrate (Section 3.3) that with the only fitting parameter one can reproduce quite well the DC magnetic data. Irrespective of the magnetic data this set of parameters proves to be fully compatible with the EPR data, like g-tensor derived from the simulation of the EPR spectra. This kind of combined approach eliminates the difficulties associated with the ambiguity of the interpretation of the magnetic properties within only a semiempirical Hamiltonian. 3.4. Modeling of Electron Paramagnetic Resonance Spectra. The X-band EPR spectrum of the diluted sample of Et4N[Co0.02Zn0.98(hfac)3] was measured at 4.2 K (Figure 2). Taking into account that the ground Kramers doublet of the Co(II) ion is rather well-separated from the excited states, one can simulate the EPR spectra in the X-band using the pseudospin-1/2 Hamiltonian for the effective spin τ = 1/2 related to the ground Kramers doublet having the effective g-tensor with the principal values gZZ, gXX, and gYY. Along with the Zeeman interaction the triaxial pseudospin-1/2 Hamiltonian also includes the hyperfine interaction with the I = 7/2 cobalt nucleus that is described by the hyperfine A-tensor with principal values AZZ, AXX, and AYY. This Hamiltonian is given by

Figure 4. Dependences of the principal values of the g-tensor for the ground Kramers doublet of the Co(II) complex on the parameter Δax calculated with κ = 0.72, Δrh ≈ 90.34 cm−1. Vertical section (dashed line) corresponds to Δax = 428.29 cm−1.

the presence of first-order orbital angular momentum ML = ±1 in the ground state. Meanwhile, the rhombic anisotropy is more pronounced in the case of positive Δax. This argumentation suggests that the sign of the parameter Δax is positive. To confirm this conclusion we turn to the SAHF calculations of the orbital energies performed for complex I. The energy scheme for the three lowest-bonding orbitals (which are mainly of 3d character) is shown in Figure 5, where

Ĥ eff = μB (HZgZZτẐ + HXgXX τX̂ + HY gYY τŶ ) + τẐ A ZZ IẐ + τX̂ AXX IX̂ + τŶ AYY IŶ

(3)

where τ̂Z, τ̂X, and τ̂Y are the components of the pseudospin-1/2 operator. The Hamiltonian, eq 3, is referred to the coordinate frame associated with the principal axes of g-tensor, and it is assumed that the main axes of g- and A-tensors coincide. The parameters were refined by comparison of the spectra calculated through the exact diagonalization of the spin-Hamiltonian by using EasySpin37 with the experimental ones. Least-square procedure gives the following values for the components of g and A:

Figure 5. Orbital diagram of the orbital energies calculated for complex I obtained by using SAHF approach. (left) Orbital energies in idealized tetragonal (D4h) complex. (right) Ab initio orbital energies calculated with the distances determined from X-ray data.

the energy gap between the two lowest orbitals is denoted by Δrh, and the gap between the center of gravity of these two orbital energies and the second excited orbital is denoted by Δax. For a relatively weak low-symmetry crystal field (compared to both cubic crystal-field splitting and intraionic Coulomb interaction) it seems reasonable to attribute these gaps to the values of the crystal-field parameters Δax and Δrh involved in the Griffith Hamiltonian, eq 2. At this stage it is important to emphasize that parameter Δax so far defined should be positive. To clarify this crucial point one should correlate the one-electron (orbital) picture provided by the ab initio calculations with the multielectron schemes (terms) of Co(II) for which the disposition of the energies explicitly indicate the sign of Δax. In Figure 6a we show the orbital schemes (microstates, or visualized Slater determinants) for the ground 5 2 cubic crystal-field term 4T1g(t2g eg ) along with the terms 4 4 4 3 2 A2g(egb2ga1gb1g) and Eg(egb2ga1gb1g), which appear as the result of the splitting of the 4T1g(t52ge2g) term caused by the tetragonal distortion of the octahedron. In these schemes the splitting of the triply degenerate t2g orbital is taken to be consistent with the orbital energies diagram in Figure 5. This means that the ground orbital in the system of D4h symmetry is double degenerate comprising dXZ and dYZ tetragonal eg orbitals, while the first excited dXY(b2g) orbital is

gZZ = 2.502, gXX = 4.251, gYY = 5.467, A ZZ = 67.8 MHz, AXX = 166.0 MHz, AYY = 473.8 MHz

The values of the components of g obtained from simulation of EPR spectra show that we are dealing with pronounced triaxial anisotropy. To rationalize the EPR data let us examine the dependences of the values gZZ, gXX, and gYY on the axial crystal-field parameter Δax calculated with the aid of eigenvalues of the Hamiltonian, eq 2. Here and in all subsequent numerical estimations we use the freeion value of the SOC parameter, which is set to λ = −180 cm−1. Then we use κ = 0.72 obtained by fitting the magnetic properties of complex I (see Section 3.5) and Δrh = 90.34 cm−1, which is obtained from ab initio data (SAHF calculations, see subsequent Discussion). The evaluated dependences are shown in Figure 4. The central part of the plots (dashed lines in Figure 4) should be excluded, because this part corresponds to the invalid range of crystal-field parameters where the inequality |Δrh| ≤ |Δax|/3 is not satisfied. From Figure 4 one can see that for negative values of Δax the inequality gZZ > gXX, gYY holds (Z is easy axis of the magnetization), while for positive Δax the axis Z is the hard axis of magnetization (XY is the easy plane). One can see also that the axial magnetic anisotropy is more pronounced for Δax < 0 due to F


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Figure 6. Tetragonal splitting of the cubic term 4T1g(t52ge2g) of the octahedral (Oh) complex under lowering of the symmetry to D4h. (a) Orbital schemes showing the splitting of the 4T1g(t52ge2g) into tetragonal orbital singlet 4 A2g(e g4b 2ga 1gb 1g) (ground) and 4 Eg(e3gb22ga1gb1g) (excited) states that corresponds to the one-electron splitting in Figure 5. (b) Splitting of the 4T1g (L = 1) term by axial and rhombic crystal fields.

nondegenerate. The populations of these orbitals (depicted in Figure 6a) give rise to the ground tetragonal 4A2g(e4gb2ga1gb1g) term corresponding to ML = 0, while the tetragonal term 4 Eg(e3gb22ga1gb1g) with ML = ±1 is higher in energy by the value of Δax (Figure 6b). Since the situation when the orbital singlet is a ground state corresponds to the positive axial crystal-field parameter, one can conclude that Δax > 0 in complex I in agree ment with the analysis of g-tensor. We assume that the values Δax = 428.29 cm−1 and Δrh = 90.34 cm−1 determined in this way can be used as reasonable estimates for the crystal-field parameters. The configurational mixing of 4T1g(t52ge2g) and 4T1g(t42ge3g) terms does not disprove the validity of this conclusion provided that the splittings caused by the low-symmetry components of the crystal field are not too large, which seems to be rightly in the case under consideration. As the sign of the axial field is of primary importance for justification of the mechanism of SIM behavior, it is important to exclude an ambiguous interpretation of the results. To this end we will plot the evaluated principal values of the g-tensor as functions of the rhombic field for the cases of negative and positive values of Δax (Figure 7). One can see that, for Δax > 0 and Δrh = 90.34 cm−1 (Figure 7a), the principal values of the g-tensor are quite close to those observed in EPR. Since the only adjustable parameter is involved, the agreement can be referred to as satisfactory. In contrast, providing Δax < 0 (Figure 7b), the experimentally observed g-factors cannot be reproduced by variation of the rhombic field. This observation confirms the conclusion that we are dealing with the case of nonuniaxial anisotropy with strong positive axial contribution. 3.5. Modeling of Magnetic Characteristics. Using the eigenvalues Ei(By) (γ = X, Y, Z) of the Hamiltonian one can evaluate the principal values χyy of the molar magnetic susceptibility tensor and the averaged value χM for the powder sample by using the standard definitions: χγγ = NAkBT

Figure 7. Calculated (solid lines) dependences of the principal values of g-tensor upon the rhombic field parameters for positive axial field Δax = 428.29 cm−1 (a) and negative axial field Δax = −421.7 cm−1, κ = 0.69 (b). Horizontal dashed lines indicate observed g-factors.

Using also the eigenvalues Ei(B,θ,φ) in magnetic field B whose orientation with respect to the reference frame is defined by the polar angles θ,φ one can calculate the magnetization M(H,θ,φ) at the arbitrary direction of the field and the magnetization M for the powder sample as follows: M(B , θ , φ) = NAkBT M=

π

∫0 ∫0

2π

M(B , θ , φ)sin(θ )dθ dφ (5)

In the evaluation of the magnetic properties we use the calculated crystal-field parameters Δax = 428.29 cm−1 and Δrh = 90.34 cm−1.24 The only adjustable parameter we use for the description of the DC magnetic data is the orbital reduction factor κ that is assumed to be constrained within the limits typical for the high-spin quasi-octahedral Co(II) complexes (from 0.6 to 0.9).24,25 The best-fit value of κ is found to be κ = 0.72. The theoretical χMT versus T and M versus B curves obtained with this set of parameters are shown in Figure 3. It is seen that the theory quite satisfactory reproduces the observed magnetic behavior of complex I (especially keeping in mind that we used the only fitting parameter) with the agreement criteria 1.5% and 5.9% for χMT versus T and M versus B, respectively.

1 ∂ ln{∑ exp[−Ei(Bγ )/kBT ]} Bγ Bγ i

χM = (χZZ + χXX + χYY )/3

1 4π

∂ ln{∑ exp[−Ei(B , θ , φ)/kBT ]} ∂B i

(4) G


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regarded to as a generalization of the Griffith diagram34 to the case of triaxial symmetry. The vertical section marked by dashed red line in Figure 9 corresponds to the found axial crystal-field value Δax = 428.29 cm−1. In principle, providing positive Δax the two lowest in energy Kramers doublets can be viewed as a result of ZFS that is described by the spin Hamiltonian 2 2 2 1 D⎡⎣SẐ − 2 S(S + 1)⎤⎦ + E(SX̂ − SŶ ). At the same time it is seen from Figure 9 that the usage of this Hamiltonian providing weak axial field (as in the present case) requires some precaution. Indeed for Δax = 428.29 cm−1 the two lowest Kramers doublets cannot be regarded as well-isolated from the second excited Kramers doublet, whose thermal population at room temperature is not negligible. Note that the second excited Kramers doublet cannot be reproduced by the spin Hamiltonian. For this reason the usage of the Griffith Hamiltonian, eq 2, which explicitly takes into account the unquenched orbital angular momentum, seems to be unavoidable in the case under study to correctly describe the magnetic behavior of the system in the whole temperature range. Nevertheless, the Hamiltonian, eq 1, with positive D is expected to provide a correct description of the low-temperature magnetic anisotropy of the complex, and so it seems to be instructive to obtain the parameters of this Hamiltonian from the analysis of EPR spectra. We used the procedure described in detail in Supporting Information of ref 7. It provides the following parameters for S = 3/2: gs = [2.448, 2.444, 2.556], D = +117.8 cm−1, E/D = 0.0853, Ashf = [|95.6|, |211.3|, |69.28|] MHz. It is notable that the procedure proposed in ref 7 allows estimation of the parameter D, which cannot be determined directly from the X-band EPR. These data allow us to compare our results with the results of previous studies as well as with the results of quantum− chemical calculations. SA-CASSCF/NEVPT2 calculations based on the effective Hamiltonian theory33 also showed a large positive D value (D ≈ +121.2 cm−1) with a significant rhombicity (E/D ≈ 0.12). These values are quite close to those obtained from EPR spectra. It is to be emphasized that the widely used QDPT approach gives in this case inappropriate values, both in magnitude and in sign (D = −194.6 cm−1 and E/D = 0.26), perhaps because of the strong magnetic anisotropy.

Figure 8. Temperature dependences of χZZT, χXXT, and χYYT calculated with Δax = 428.29 cm−1, Δrh = 90.34 cm−1, and κ = 0.72.

For the illustration of the magnetic anisotropy in Figure 8 we give the evaluated temperature dependences of the principal values χγγ (γ = X, Y, Z) of the magnetic susceptibility tensor in the form χγγT versus T. The curves in Figure 8 are evaluated with the parameters Δax = 428.29 cm−1, Δrh = 90.34 cm−1, and κ = 0.72. It is seen that χZZT < χXXT, χYYT in compliance with the easy plane anisotropy, and the difference between χXXT and χYYT is rather large due to strong rhombic contribution. Note that the set of values Δax = 428.29 cm−1, Δrh = 90.34 cm−1, and κ = 0.72 corresponds to the vertical section (dashed line) in Figure 4 that allows us to evaluate the principal values of g-tensor. The calculated values gZZ ≈ 2.51, gXX ≈ 4.01, and gYY ≈ 5.32 prove to be quite close to those (gZZ = 2.502, gXX = 4.251, and gYY = 5.467) obtained from simulation of EPR spectra. Note that the low-temperature magnetization curves and, in particular, the saturation values of M are quite different from the values expected for the S = 3/2 ground state and can be naturally interpreted in terms of strongly anisotropic Kramers doublet. A simple estimation of the staturation value for the lowtemperature magnetic moment can be made using the average values of the anisotropic g-factor defined (roughly) as g ̅ = (gXX 2 + gYY 2 + gZZ 2)/3 . This estimation gives ge̅ xp = 4.25 from EPR data, and estimated value g ̅ = 4.4 satisfactorily explains the saturation value for Msat ≈ 2.2NAμB. These are the remarkable results, which show that the consideration so far given is selfconsistent and can be thought of as reliable. In fact, the DC magnetic properties and EPR spectra are reproduced quite well with the only determined set of parameters. 3.6. Further Discussion of the Model. To conclude the discussion of the determined set of parameters let us examine the dependence of the energy levels of the Co(II) ion on the parameter Δax (Figure 9). Such kind of energy pattern can be

4. DYNAMIC SUSCEPTIBILITY The frequency dependence of the components of AC susceptibility for the complex I was measured in a static magnetic field of 0.1 T (Figure 10) in the temperature range of 1.8−3.3 K. At low temperatures the two maxima can be clearly seen in the frequency dependences, and therefore, the experimental data were fitted by a two-component Debye model. A detailed description of the two-step relaxation processes can be found in refs 38−41. The obtained parameters are given in Supporting Information (Table S1). The first (and most intensive) maximum is at ∼40 Hz (1.8 K), and with increasing temperature it shifts to ∼450 Hz at 3.3 K. The second (lower) maximum at 1.8 K is located at ∼450 Hz, and with increase of temperature it also shifts to higher frequencies, which prove to be out of the SQUID frequency band (νmax > 1400 Hz) at 2.6 K. For this reason at temperatures higher than 3.0 K the values of τ2 (Supporting Information, Table S1, Figure S6) were obtained with low accuracy, and for this reason we do not analyze them here. It is seen from the Arrhenius plots of ln(τ) versus 1/T dependences that for both relaxation processes these dependencies are not strictly linear (see Supporting Information, Figures S1 and S2 for τ1 and τ2, respectively). Therefore, in addition to Orbach

Figure 9. Dependences of the energy levels of the Co(II) ion on the parameter Δax calculated with κ = 0.72 and Δrh = 90.34 cm−1. Vertical section (red dashed line) corresponds to Δax = 428.29 cm−1 obtained from the ab initio calculations. The value κ = 0.72 is the best-fit value obtained by fitting the magnetic properties of complex I using the found values of the crystal-field parameters. The central part of the plot (dashed lines) corresponds to the area of the forbidden values of Δax. H


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Figure 10. Frequency dependence of the in-phase (a) and out-of-phase (b) AC susceptibility, Argand (Cole−Cole) plot (c) of complex I under 0.1 T DC field from 1.8 to 3.3 K in increments of 0.1 K. (○) Experimental data; solid lines indicate fit data within the two-component Debye model with parameters listed in Table S1 (Supporting Information).

lies out of the SQUID frequency band. It is notable that Orbach and Raman-type processes produce quite similar contributions to the relaxation time τ1 (Figure 11), while the direct one-phonon

processes, which are responsible for the jumps over the barrier for spin reversal, other relaxation mechanisms should be taken into account as well. Equation 6 includes four relaxation mechanisms driven by the interaction of the coordinated Co(II) ion with phonons. ⎛U ⎞ B1 τ −1 = τ0−1exp⎜ eff ⎟ + CT n + AB2 T + 1 + B2 B 2 ⎝ kBT ⎠

(6)

The first, second, and third terms in the right part of eq 6 describe contributions of Orbach, Raman, and direct spinphonon processes to the overall relaxation, while the last term is the contribution of quantum tunneling of magnetization. From comparison of the dependenc τ−1(T) with the experimental ones one can find the following sets for the parameters C, n, A, B1, B2: Figure 11. Temperature dependence of the inverse relaxation time τ1−1 for complex I under 0.1 T DC field. Red line shows fit to the experimental data (circles) according to eq 6 with the parameters indicated in the text. The contributions of Orbach, Raman, and direct relaxation processes are represented separately by solid color lines.

Ueff = 20.6(5) K; τ0 = 6.4(9) × 10−7 s; C = 2.2(2) K−6 s−1 (n = 6); A = 9.6(3) × 103 T−2 K−1 s−1 for τ1

and

contribution is negligibly small as expected for the transitions between Zeeman sublevels in a Kramers doublet. The frequency dependences of the AC susceptibility components for the complex I were analyzed in different external DC fields of 0, 0.02, 0.05, and 0.1 T at T = 1.8 K (Supporting Information, Figure S3). At zero DC field no AC signal was detected; the explanation for this phenomenon (that seems to be common for all Kramers ions) was recently proposed in ref 7. Even in a weak DC field (0.01 T) the AC signal can be observed, and in all fields the frequency dependence exhibits two maxima. Consequently, the experimental data were described with the aid of the two-component Debye model; the parameters

Ueff = 18(2) K; τ0 = 9(5) × 10−8 s; A = 1.4(2) × 105 T−2 K−1 s−1 for τ2

While evaluating the τ1 the first three relaxation processes in eq 6 were considered, while in the case of τ2 only Orbach and direct processes were taken into account. It seems to be problematic to evaluate the contribution of the Raman process to τ2, because in the present case such contribution is visible only at temperatures higher than 2.8 K, but for such temperatures the position of the second maximum in the frequency dependence I


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obtained in this way are given in Table S2. The position of the maximum strongly depends on the DC field and varies in the ranges of 27−47 and 180−435 Hz for τ1 and τ2, respectively. For both components the relaxation time reaches the maximal value in the DC field of ∼0.05 T; for higher and lower DC fields both maxima are shifted toward higher frequencies. In the plots of the dependencies τ−1(B2) (Figure S4a for τ1 and S4b for τ2) one can see a nonzero contribution of QTM and direct relaxation pathways, thus showing that the low-temperature processes cannot be ignored.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b01473. Powder XRD patterns, comparison of temperature dependences of DC susceptibility for positive and negative values of the axial crystal field parameter, frequency dependence of in-phase and out-of-phase AC susceptibilities, field dependence of the inverse relaxation times for complex I, natural log plots, tabulated best-fit parameters (PDF)

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REFERENCES

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5. CONCLUDING REMARKS We have reported a new quasi-octahedral Co(II) complex Et4N[CoII(hfac)3] exhibiting SIM behavior. It is demonstrated that this Co(II) complex exhibits triaxial magnetic anisotropy with prevailing positive axial Z-component, so the system can be referred to as nonuniaxial SIM with Z-axis being the hard magnetization axis. This complex proved to be a good example justifying the necessity of applying different complementary experimental techniques supported by the quantum−chemical evaluation of the parameters involved in the effective Hamiltonian. We have demonstrated that only in this way we could get reliable information concerning the sign of the magnetic anisotropy and avoid excessive flexibility of the theory, which can result in multiple sets of the best-fit parameters. Thus, the combined analysis of the magnetic (static and dynamic) and EPR data, as well as the calculated orbital energies, g-factors, and ZFS parameters, allowed us to conclude that the slow magnetic relaxation exhibited by this complex is the consequence of the positive axial and strong rhombic magnetic anisotropies.

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AUTHOR INFORMATION

Corresponding Authors

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the research team of the laboratory for microanalysis at the A. N. Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences. This work was supported by the Russian Foundation for Basic Research (Grant No. 16-33-00464 mol_a) and the Presidium of the Russian Academy of Sciences (Program No. 35). J


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Inorganic Chemistry

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