Origin of Dissimilar Single-Molecule Magnet Behavior of Three

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Origin of Dissimilar Single-Molecule Magnet Behavior of Three MnII2MoIII Complexes Based on [MoIII(CN)7]4− Heptacyanomolybdate: Interplay of MoIII−CN−MnII Anisotropic Exchange Interactions Vladimir S. Mironov* A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskii prosp. 59, 119333 Moscow, Russia S Supporting Information *

ABSTRACT: The origin of contrasting single-molecule magnet (SMM) behavior of three MnII2MoIII complexes based on [MoIII(CN)7]4− heptacyanomolybdate is analyzed; only the apical Mn2Mo isomer exhibits SMM properties with Ueff = 40.5 cm−1 and TB = 3.2 K, while the two equatorial isomers are simple paramagnets [Qian, K.; et al. J. Am. Chem. Soc. 2013, 135, 13302]. A microscopic theory of anisotropic spin coupling between orbitally degenerate [MoIII(CN)7]4− complexes (pentagonal bipyramid) and bound MnII ions is developed. It is shown that the [MoIII(CN)7]4− complex has a unique property of uniaxial anisotropic spin coupling in the apical and equatorial MoIII−CN− MnII pairs, H̑ eff = −Jxy(SxMoSxMn + SyMoSyMn) − JzSzMoSzMn, regardless of their actual low symmetry. The difference in the SMM behavior originates from a different ratio between the anisotropic exchange parameters Jz and Jxy for the apical and equatorial Mo−CN−Mn groups. In the apical Mn2Mo isomer, an Ising-type anisotropic spin coupling (Jz = −34, Jxy = −11 cm−1) produces a double-well potential of spin states resulting in SMM behavior. Exchange anisotropy of an xy-type (|Jz| < |Jxy|) in the equatorial Mn2Mo isomers results in a single-well potential with no SMM properties. The prospects of anisotropic uniaxial spin coupling in engineering of high Ueff and TB values are discussed.

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MnIII ions.11a A lot of effort has been made in order to increase Ueff and TB. In particular, many studies were focused on obtaining even larger high-spin 3d complexes with the hopes of increasing the barrier, as one might expect from Ueff = |D|S2.6 However, in recent years it has become increasingly apparent that ever-larger spin does not generally increase Ueff and TB values. Moreover, it has been shown theoretically that the barrier Ueff = |D|S2 is largely independent of S because the molecular zero-field splitting (ZFS) parameter D decreases as S−2 with the increasing spin S.12 Consequently, enhancements in magnetic anisotropy D rather than in spin S represent a more efficient strategy for increasing Ueff and TB. Great efforts toward reaching this goal have been undertaken in the preceding years using various synthetic strategies. These studies have resulted in new diverse SMM compounds with enhanced Ueff and TB: polynuclear metal complexes,6 one-dimensional single-chain magnets,13 metal−radical SMMs,14 and SMM molecules including a single magnetic ion.15−17 Comparative analysis of their SMM characteristics indicates that efficient control and optimization of magnetic anisotropy are of crucial importance for a successful strategy.6,12d,18 Basically, magnetic anisotropy implies the presence of some unquenched angular orbital momentum on magnetic centers.

INTRODUCTION The synthesis and study of single-molecule magnets (SMMs) has attracted increasing attention over the past two decades1−6 because of their potential applications in future information technology, such as information storage at the molecular level, quantum computations, and spintronics devices.7−10 SMMs are molecular metal−organic clusters with a high-spin ground state showing a slow magnetization relaxation rate and magnetic hysteresis below a characteristic blocking temperature (TB). This property arises from a magnetic bistability associated with a double-well potential, in which two lowest-lying quantum states +Ms and −Ms are separated by an energy barrier to magnetization reversal (Ueff). The barrier is the result of a combined effect of a large ground-state spin S and uniaxial magnetic anisotropy (with negative zero-field-splitting parameter, D); it is equal to Ueff = |D|S2 and |D|(S2 − 1/4) for integer and half-integer spin, respectively.4 The blocking temperature TB is the most important SMM characteristic; large energy barrier Ueff is also crucially important although it does not correlate directly with TB.1−6 However, all SMMs based on polynuclear 3d complexes possess a low blocking temperature (within a few Kelvin), which is too small for practical applications;4−6 the largest values of TB = 4.5 K and Ueff = 62 cm−1 are reported for a Mn6 complex (S = 12), [MnIII6O2(Et-sao)6(O2CPh(Me)2)2(EtOH)4] (Et-sao: ethyl-salicyladoxime), with six © XXXX American Chemical Society

Received: August 26, 2015

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DOI: 10.1021/acs.inorgchem.5b01975 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry

Figure 1. Molecular structure and contrasting SMM behavior of three MnII2MoIII complexes based on [MoIII(CN)7]4− heptacyanometallate. (a) Structure of the apical isomer [Mn(LN5Me)(H2O)]2[Mo(CN)7] 1. It exhibits a remarkable SMM behavior with Ueff = 40.5 cm−1 and TB = 3.2 K. (b) Equatorial cis-complex [Mn(LN3O2)(H2O)]2[Mo(CN)7] 2. (c) Equatorial trans-complex [Mn(LDAPSC)(H2O)]2[Mo(CN)7] 3. The two equatorial MnII2MoIII complexes 2 and 3 are simple paramagnets with no SMM behavior.44

character of magnetic anisotropy and the weak spin-coupling limit regime for Ln ions caused by the core-like nature of 4f electrons.15−17 The only exception is the lanthanide dimers bridged by the N23− radical, in which the 4f-radical spin coupling is much stronger.20 In contrast to lanthanide and actinide complexes, in most transition-metal complexes orbital momentum is quenched by a strong ligand field. For d-electrons, unquenched first-order orbital momentum can only appear in orbitally degenerate complexes, such as [FeIII(CN)6]3− and CoII octahedral complexes. A number of Co II-based SMMs has been reported,6,24−26 including single-ion SMMs.26 In the past few years, new orbitally degenerate mononuclear high-spin 3d complexes with a less-common coordination were obtained: linear complexes (FeI and FeII,27 CoI and NiI28), trigonal-planar (FeII),29 trigonal-pyramidal (FeII, CoII, NiII),30 trigonalprismatic (CoII),31 and pentagonal-bipyramidal (FeII) complexes.32 They show strong Ising-type magnetic anisotropy with very high ZFS parameters D (up to 246 cm−1),27c associated with the first-order spin−orbit splitting of the ground-state spin multiplet. Some of them behave as single-ion magnets.33 On the other hand, magnetic anisotropy of orbitally degenerate 4d and 5d complexes differs considerably from that of 4f- and 3d-complexes. Due to a very strong ligand field, all orbitally degenerate 4d and 5d complexes have a low spin of S = 1/2 with no ZFS effect, requiring S > 1/2. Examples are given by [MoIII(CN)7]4−34 and [ReIV(CN)7]3−35 pentagonalbipyramidal complexes, [RuIII(CN)6]3−36 and [OsIII(CN)6]3−37 octahedral complexes, and some trigonal-pyramidal Mo and W complexes.38 It is important to note that isolated complexes of this type cannot exhibit single-ion magnet behavior due to the absence of the ZFS anisotropy for the S = 1/2 spin. Their magnetic anisotropy manifests only in anisotropic g-tensor and anisotropic magnetic susceptibility, or may even be absent, as is the case for magnetically isotropic [RuIII(CN)6]3− and [OsIII(CN)6]3− octahedral complexes.36,37 However, spin coupling of these complexes with other spin carriers is often strongly anisotropic due to the presence of unquenched orbital

In most polynuclear transition-metal complexes, magnetic anisotropy is a tensorial sum of the local ZFS anisotropies Di of magnetically coupled high-spin 3d metal ions projected onto the ground-state spin multiplet 2S + 1.4 Particularly, high-spin MnIII ions with a rather large ZFS parameter Di are routinely used as the source of magnetic anisotropy.6 The resulting magnetic anisotropy is generally rather weak since ZFS is only a second-order effect with respect to the spin−orbit coupling.19 Considerably larger magnetic anisotropy could be obtained by incorporation of magnetic centers with unquenched (firstorder) orbital momentum, f-elements (lanthanide and actinides), and some orbitally degenerate transition-metal complexes. In this respect, lanthanide ions look especially attractive. Because of a strong spin−orbit coupling and very small crystal-field splitting energy, all lanthanide ions with open 4f-shell (except Gd3+) possess large and unquenched orbital momentum L resulting in extremely strong single-ion magnetic anisotropy.15 This is especially true for heavy lanthanide ions (Tb3+, Dy3+, Er3+), which have a large magnetic moment due to a large value of the total angular momentum, J = L + S. The field of lanthanide-based SMMs has developed very rapidly in the past decade. Up to now, a large number of Ln-SMMs have been reported.15−17 Among them are SMMs with the largest effective energy barriers Ueff and blocking temperatures. Thus, the record blocking temperatures were reported for dinuclear lanthanide complexes, in which two anisotropic Ln ions (DyIII and TbIII) are bridged by the N23− radical resulting in a strong spin coupling.20 Especially intense studies have been undertaken for mononuclear Ln complexes with slow magnetic relaxation, which represent a novel class of SMMs, referred to as single-ion magnets.21 These complexes exhibit very high barriers (up to 650 cm−1)22 and slow magnetic relaxation at higher temperatures. Recently several mononuclear actinidebased compounds were reported to show slow magnetic relaxation.23,24 However, despite large Ueff barriers, in most lanthanide-based SMMs, the quantum tunneling of magnetization is very fast, leading to a very small coercitivity in the hysteresis loop. This is related to a prevailing single-ion B


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Inorganic Chemistry momentum and strong spin−orbit coupling.39−41 Anisotropic Ising-type spin couplings can serve as an alternative and efficient source of the overall magnetic anisotropy of a highspin molecule.39,40 In contrast to 3d- and 4f-based SMMs, in this type of 4d- and 5d-based SMMs the barrier Ueff is controlled by anisotropic exchange parameters Jij rather than by single-ion ZFS energy Di of individual magnetic centers;39,40 most notably, the exchange parameters are much easier to increase than the ZFS anisotropy. From this viewpoint, 4d and 5d complexes are especially suitable because high energy and diffuse 4d and 5d magnetic orbitals provide stronger exchange coupling to neighboring magnetic ions. Hence, incorporation of orbitally degenerate 4d and 5d complexes opens up new and exciting possibilities for designing improved SMMs.39−41 Early theoretical analysis indicated that anisotropic Ising-type spin coupling in mixed-metal polynuclear complexes based on [MoIII(CN)7]4−, [ReIV(CN)7]3−, and [OsIII(CN)6]3− complex anions and high-spin 3d ions may result in SMMs with high Ueff and TB.40 More recently, several SMMs of this type were reported, a pentanuclear cross-like [PY5Me2)4Mn4Re(CN)7]4+ complex (ReIVMnII4)42 and related ReIVMII4 complexes (MII = CoII, NiII, and CuII),43 linear trinuclear complexes [Mn2(5Brsalen)2(MeOH)2M(CN)6] (M = Ru and Os),41a and a linear trinuclear [Mn(LN5Me)(H2O)]2[MoCN)7]·6H2O complex (MnII2MoIII).44 It is important to note that the SMM behavior of ReIVMnII4 and MnII2MoIII provides direct evidence of the dominant role of anisotropic spin coupling in the origin of the Ueff barrier. Indeed, despite the metal ions having no (MoIII and ReIV) or very small (MnII) single-ion ZFS anisotropy, these complexes display pronounced SMM properties with the barrier of Ueff = 33 cm−1 in ReMn4 and Ueff = 40.5 cm−1 in Mn2Mo, which are the largest among cyano-bridged SMMs. An especially intriguing situation occurs for MnII2MoIII trinuclear complexes, for which three isomers composed of the same or very similar molecular building blocks were synthesized and magnetically characterized, the apical Mn2Mo isomer 1 and two equatorial isomers 2 and 3, Figure 1.44 There is a drastic contrast between their magnetic properties: only apical Mn2Mo isomer 1 shows a slow magnetic relaxation with high Ueff and TB values, while the equatorial cis- and transisomers 2 and 3 are simple paramagnets with no SMM behavior.44 This difference is even more surprising given the fact that SMM characteristics of a small trinuclear Mn2Mo complex 1 with a moderately low ground-state spin of S = 9/2 (Ueff/kB = 58 K and TB = 3.2 K) are the same as those of the much larger Mn12Ac complex (Ueff/kB = 65 K and TB = 3 K) with 12 magnetic ions and S = 10.4,6 Basically, these experimental results support the main idea that anisotropic pair-spin coupling could be even more efficient in designing high energy barriers Ueff than the single-ion ZFS anisotropy;39,40,45 they also indicate that Ueff and TB are very sensitive to the interplay between anisotropic spin couplings in a SMM cluster. This paper presents a detailed theoretical analysis of magnetic and SMM properties of three Mn2Mo isomer complexes 1−3 (Figure 1). A microscopic model of anisotropic spin coupling in the apical and equatorial MoIII−CN−MnII pairs in the three Mn2Mo complexes is developed, and important magneto-structural correlations are established. These results indicate that the difference in the SMM behavior between three Mn2Mo complexes originates from a different character of anisotropic spin coupling for the apical and equatorial MoIII−CN−MnII pairs, which has an Ising-type

anisotropy for the apical pairs and an xy-type anisotropy for the equatorial pairs. In the apical Mn2Mo complex 1, the spinreversal barrier Ueff originates exclusively from Ising-type anisotropic spin coupling. This theoretical approach provides complete agreement between experimental and simulated magnetic and SMM characteristics of these complexes. New possibilities in engineering high Ueff barriers in terms of anisotropic spin coupling of orbitally degenerate 4d and 5d complexes are discussed.

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THEORY

This section presents results of theoretical study of anisotropic magnetic interactions in the three MnII2MoIII isomer complexes 1−3 (Figure 1). First, a general theory of anisotropic spin coupling between orbitally degenerate [MoIII(CN)7]4− complexes (S = 1/2) and highspin MnII ions (S = 5/2) is developed. Previous analytical superexchange calculations for [MoIII(CN)7]4−−MnII complexes with idealized linear structure indicated that the spin coupling is described by a pure Ising-type spin Hamiltonian −JzSzMnSzMo for the apical MoIII− CN−MnII pairs and by a uniaxial anisotropic spin coupling −Jxy(SxMnSxMo + SyMnSyMo) − JzSzMnSzMo for the equatorial pairs.39 Herein this theoretical analysis is generalized and extended to distorted MoIII−CN−MnII pairs and supported by numerical calculations. This approach is quite similar to that recently used to describe anisotropic spin coupling between orbitally degenerate [ReIV(CN)7]3− complexes (5d counterparts of [MoIII(CN)7]4−) and high-spin MnIII ions in the [MnIII(acacen)]3[ReIV(CN)7] molecular magnet.46 Similar to the case of the rhenium compound, these calculations result in a uniaxial anisotropic spin Hamiltonian −Jxy(SxMnSxMo + SyMnSyMo) − JzSzMnSzMo for all MoIII−CN−MnII pairs in the three Mn2Mo complexes, regardless of their connection mode (i.e., apical or equatorial) and actual lowsymmetry structure (Figure 1). Then, anisotropic exchange parameters are numerically calculated in terms of a many-electron superexchange theory for the bent apical and equatorial MoIII−CN−MnII pairs with the aim to establish some magneto-structural correlations. Electronic Structure of [Mo(CN)7]4−. Electronic structure and magnetic properties of isolated [MoIII(CN)7]4− complexes have been already discussed in refs 39 and 47. The ground state of the [MoIII(CN)7] 4− complex with the regular (D5h) pentagonalbipyramidal structure is orbitally degenerate. In the absence of the spin−orbit effect, the ground state is represented by a low-spin orbital doublet 2Φxz,yz = 2Φ(ML = ±1) with two components corresponding to the ML = ±1 projection of the unquenched angular orbital momentum L on the polar z-axis of the [Mo(CN)7]4− bipyramid. The 2 Φ± wave functions are described by two degenerate electronic configurations (d+1)2(d−1)1and (d−1)2(d+1)1, where d±1 = (dxz ± idyz)/ √2 are complex d orbitals with definite values of the projection of the orbital momentum (ml = ± 1) on the z-axis of the pentagonal bipyramid (Figure 2); they can also be represented by real electronic configurations 2Φxz = (dyz)2(dxz)1 and 2Φyz = (dxz)2(dyz)1. Spin−orbit coupling (SOC) of MoIII splits the orbital doublet 2Φ± into the ground φ(±1/2) and excited χ(±1/2) Kramers doublets (Figure 2). The ground-state wave functions φ(±1/2) have a nearly pure singledeterminant structure with electronic configurations (d+1)2(d−1)↑and (d−1)2(d+1)↓;39 admixture of excited 4d3 ligand-field (LF) states is small due to a large energy separation between the ground state 2Φ± and excited LF states (around 20 000 cm−1).39,47 Due to unquenched orbital momentum (⟨Lz⟩ = ±1), the ground Kramers doublet φ(±1/2) has an anisotropic Ising-like g-tensor (such as gz = 3.89 and gx = gy = 1.7734b). In distorted [MoIII(CN)7]4− complexes, the ground orbital doublet 2 Φ± undergoes the energy splitting δ tending to reduce or quench the orbital momentum and related magnetic anisotropy. The value of the orbital momentum ⟨Lz⟩ depends on the ratio between the SOC energy ζ4d and the orbital splitting energy δ. In moderately distorted [MoIII(CN)7]4− complexes, when δ < ζ4d, the electronic structure changes rather insignificantly: the orbital momentum remains largely unquenched, and the ground-state wave functions are close to those in C


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Inorganic Chemistry

/2) and the S = 5/2 spin of MnII. The operator H̑ eff is obtained by projection of Horb (eq 2) onto the space of wave functions |m, MS⟩ = φ(m) × |SMn, MS⟩, where m = ±1/2 is a projection of the fiction spin SMo(eff) on the polar z-axis of the bipyramid, and |SMn, MS⟩ is a 6A1 ground-state wave function of MnII with the spin projection MS. Formally, H̑ eff is calculated in first-order perturbation theory by equating the matrix elements of Heff and A + RSMoSMn in the space of wave functions |m, MS⟩, ⟨m, MS|H̑ eff|m′, MS′⟩ = ⟨m, MS|A + RSMoSMn| m′, MS′⟩. The resulting set of matrix elements corresponds to an effective anisotropic spin Hamiltonian 1

x x y y z z H̑eff = C − Jxy (SMo SMn + SMo SMn ) − Jz SMo SMn

where Jz = (J1 + J2)/2, Jxy = (J1 − J2)/2, and |J1| ≥ |J2|; here Jz and Jxy have the same sign. The former term C = (A11 + A22)/2 is a constant stemming from the spin-independent orbital operator A in eq 1; it can safely be omitted in further calculations (see Supporting Information for more detail). These results show that the effective anisotropic spin Hamiltonian H̑ eff for Mo−CN−Mn pairs has uniaxial symmetry with the z-axis parallel to the polar axis of the [Mo(CN)7]4− bipyramid. Remarkably, the uniaxial symmetry of H̑ eff is independent of the specific connection mode of bound MnII ions (apical or equatorial) and the C−N−Mn bonding angle. Although the low-symmetry spin terms, the rhombic term Jrh(SxMoSxMn − SyMoSyMn) and antisymmetric Dzyaloshinskii−Moriya exchange G[SMo × SMn], are symmetry-allowed for bent Mo−CN−Mn groups, they are absent in the first-order perturbation spin Hamiltonian H̑ eff (eq 3); they are also absent in the second-order perturbation spin Hamiltonian resulting from mixing of the ground φ(±1/2) and excited χ(±1/2) Kramers doublets (Figure 2) by exchange interaction (eq 1). The underlying reason is that the 2 Φ(ML = +1) and 2Φ(ML = −1) orbital wave functions with definite projection of the orbital momentum ML = ±1 are not mixed by the SOC operator ζ4dLSMo; thus, only the z-component of SOC, ζ4dLzSzMo, is active in the splitting of the orbital doublet 2Φ(ML = ±1) into two Kramers doublets φ(±1/2) and χ(±1/2), Figures 2 and 3. Consequently, within the space of wave functions 2Φ±(Mo) × 6 A1(Mn; MS), the total Hamiltonian A + RSMoSMn + ζ4dLSMo can be replaced by A + RSMoSMn + ζ4dLzSzMo, in which the largest term ζ4dLzSzMo has axial symmetry. In concert with isotropic spin coupling RSMoSMn, this axial symmetry is translated into the uniaxial symmetry of the resulting effective anisotropic spin Hamiltonian Heff (eq 3) acting within the restricted space of wave functions |m, MS⟩ = φ(m) × | SMn, MS⟩ (Figure S1). The axial symmetry breaking of Heff can occur in higher-order perturbation theory due to mixing of the ML = 0 and ML = ±2 components to the ground-state wave functions 2Φ(ML = ±1) that leads to the appearance of the off-diagonal matrix elements of the SOC operator breaking the axial symmetry of its z-component ζ4dLzSzMo. This mixing (caused by intrasite Coulomb repulsion, second-order SOC mixing and/or low-symmetry LF component) is rather weak since these components are represented by high-lying LF states of [MoIII(CN)7]4− comprising xy, x2 − y2, and z2 4d orbitals (Figure 2). As a result, for an undistorted or slightly distorted [MoIII(CN)7]4− bipyramid the low-symmetry spin terms Jrh and G are generally small (typically, less than 1 cm−1, a result from preliminary calculations). In this context, it is relevant to note that the presence of the xy-component (ML = ±2) in the wave functions of the groundstate orbital triplet 2 T 2g (5d 5 ) of the orbitally degenerate [OsIII(CN)6]3− complex breaks the uniaxial symmetry of Heff and results in a three-axis spin Hamiltonian Heff = −JxSxMnSxOs − JySyMnSyOs − JzSzMnSzOs describing anisotropic spin coupling in bent OsIII−CN−MnIII groups in MnIII2OsIII SMM complex.41a In other words, the symmetry of H̑ eff is higher than the geometry symmetry of the Mo−CN−Mn groups (Figure S1 in Supporting Information). This unique feature attributes to orbitally degenerate [Mo(CN)7]4− complexes with the regular or moderately distorted pentagonal-bipyramidal structure. It is also noteworthy that the uniaxial symmetry of H̑ eff is independent of the specific mechanism of Mo−Mn spin-coupling (i.e., superexchange, direct exchange, or spinpolarization mechanism). In fact, it is based exclusively on the

Figure 2. Electronic structure of [MoIII(CN)7]4− complex with the perfect pentagonal structure (D5h). The energy positions of 4d orbitals of [MoIII(CN)7]4− are indicated. Three 4d electrons of MoIII occupy the two lowest degenerate orbitals 4dxz and 4dyz to produce a low-spin orbital doublet 2Φxz,yz = 2Φ(ML = ±1). SOC (ζ4d) on Mo splits the orbital doublet 2Φ(ML = ±1) into the ground Kramers doublet φ(±1/2) and excited Kramers doublet χ(±1/2). Electronic structure of their two components is shown. the undistorted [MoIII(CN)7]4− complex (D5h). Notably, the [MoIII(CN)7]4− bipyramid in the three Mn2Mo complexes 1−3 is considerably less distorted (Figure 2) than in most of the MoIII−MnIIbased polymer molecular magnets which, despite more pronounced distortions in the [MoIII(CN)7]4− units, display very strong magnetic anisotropy.48 The orbital energy splitting δ estimated from angularoverlap calculations for distorted [MoIII(CN)7]4− complexes in 1−3 is less than the SOC energy ζ4d (400−700 cm−1).39 Hence, the regular D5h geometry of [Mo(CN)7]4− is a reasonable approximation in the theoretical analysis of anisotropic spin coupling in complexes 1−3. Origin of Anisotropic Spin Coupling in Mo−CN−Mn Pairs. A Microscopy Theory. The general computational scheme for the spin Hamiltonian describing anisotropic spin coupling in MoIII−CN−MnII pairs is essentially the same as that recently employed for ReIV−CN− MnIII pairs in the [ReIV(CN)7][MnIII(acacen)]3 molecular magnet comprising isoelectronic [ReIV(CN)7]3− complexes.46 First, for Mo− CN−Mn pairs with neglected SOC on MoIII, isotropic orbitally dependent spin Hamiltonian Horb is written in general form

H̑orb = A + RSMoSMn

(1)

where SMo and SMn are spin operators of Mo and Mn; A and R are, respectively, spin-independent and spin-dependent orbital operators acting on the orbital variables only. In the space of the 2Φ±(Mo) × 6 A1(Mn) wave functions ( × being the antisymmetrized product) the A and R orbital operators are written as a Hermitian 2 × 2 matrix. It is important to note that, for the regular or slightly distorted [Mo(CN)7]4− pentagonal bipyramid, the orbital R matrix is diagonalized by a canonical transformation of two wave functions 2 Φ+ and 2Φ− corresponding to a rotation of the coordinate frame (xyz) around the z-axis (see Figure S1 in Supporting Information). This results in

⎛ A11 A12 ⎞ ⎛ J1 0 ⎞ ⎟⎟SMoSMn ⎟ − ⎜⎜ H̑orb = ⎜ ⎝ A 21 A 22 ⎠ ⎝ 0 J2 ⎠

(3)

(2)

where J1 and J2 are orbital exchange parameters referring to the real wave functions 2Φxz = (dyz)2(dxz)1 and 2Φyz = (dxz)2(dyz)1. With nonzero SOC on MoIII, the orbitally dependent isotropic spin Hamiltonian Horb (2) transforms into effective anisotropic spin Hamiltonian H̑ eff describing spin coupling between the ground Kramers doublet φ(±1/2) (referring to a fiction spin SMo(eff) = D


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Inorganic Chemistry

Figure 3. Orbital splitting energy δ and SOC energy of the ground orbital doublet 2Φ(ML = ±1) of distorted pentagonal-pyramidal [Mo(CN)7]4− complexes in compounds 1−3. In the regular pentagonal symmetry (D5h) of [Mo(CN)7]4−, the ground orbital doublet 2Φ(ML = ±1) is strictly degenerate. Distortions split the 2 Φ(ML = ±1) doublet into two orbital components by the energy δ. SOC of Mo further splits the orbital doublet 2Φ± into two magnetically anisotropic Kramers doublets, the ground φ(±1/2) and excited χ(±1/2) Kramers doublets. The orbital momentum remains unquenched when the orbital splitting δ is smaller than the spin−orbit splitting energy ζ4d. Real structure of the [Mo(CN)7]4− units in three Mn2Mo complexes 1−3 is shown. Selected Mo−C distances (blue numbers, in Å) and bonds angles of distorted [Mo(CN)7]4− anions are indicated.

Figure 4. Two principal superexchange pathways in the apical MoIII− CN−MnII exchange-coupled pairs in 1, 4dxz(Mo)−3dxz(Mn) and 4dxz(Mo)−3dxz(Mn), resulting, respectively, in the antiferromagnetic J1 and J2 orbital exchange parameters.

transformational properties of the orbital wave functions 2Φ± and a special electronic structure of wave functions (d+1)2(d−1)↑and (d−1)2(d+1)↓ of the ground Kramers doublet φ(±1/2) of [Mo(CN)7]4−. The relation between the anisotropic exchange parameters Jz and Jxy in eq 3differs considerably for the apical and equatorial positions of attached MnII ions. For the apical positions, there are two main superexchange pathways between the half-filled 4d(Mo) and 3d(Mn) magnetic orbitals, 4dxz(Mo)−3dxz(Mn) and 4dyz(Mo)−3dyz(Mn), resulting in two antiferromagnetic orbital exchange parameters, J1 < 0 and J2 < 0 (Figure 4). According to the relations Jz = (J1 + J2)/2 and Jxy = (J1 − J2)/2 in eq 3, this leads to an antiferromagnetic Ising-type spin coupling with Jz < 0, Jxy < 0, and |Jz| > |Jxy| (since |J1+J2| > |J1−J2| for the same sign of J1 and J2). In particular, there is J1 = J2 for a linear apical Mo−CN−Mn group, because the two superexchange pathways shown in Figure 4 are symmetry-related. In this case eq 3 yields a pure antiferromagnetic Ising-type spin coupling −JzSzMoSzMn, just as has been previously predicted from analytical superexchange calculations for the apical MoIII−CN−MnII group with an idealized high-symmetry geometry.39 These results indicate that distortions in the apical Mo−CN−Mn groups lead only to the appearance of a nonzero Jxy parameter in eq 3 with the retaining uniaxial symmetry of H̑ eff. On the contrary, in the equatorial groups the J1 and J2 orbital exchange parameters differ considerably in magnitude and sign due to contrasting behavior of 4d(Mo)−3d(Mn) superexchange pathways. The larger J1 parameter is related to an antiferromagnetic 4dxz(Mo)− 3dxz(Mn) pathway between two half-filled magnetic orbitals, while the smaller parameter J2 is ferromagnetic due to orthogonality of the 4dyz(Mo) magnetic orbital with respect to all 3d(Mn) orbitals (Figure 5). Thus, in the equatorial groups the anisotropic spin coupling −Jxy(SxMoSxMn + SyMoSyMn) − JzSzMoSzMn reveals xy-character with |Jz| < |Jxy|,

as resulted from the relations Jz = (J1 + J2)/2 and Jxy = (J1 − J2)/2 with |J1| > |J2|, J1 < 0, and J2 > 0. In the following, orbital and anisotropic exchange parameters are analyzed more quantitatively with numerical superexchange calculations (see below, Figures 6−9). Calculation of Anisotropic Exchange Parameters. In order to provide more insight into the mechanism of anisotropic spin coupling in Mn2Mo complexes 1−344 and in other [Mo(CN)7]4−−MnII-based molecular magnets,48 herein anisotropic exchange parameters of MoIII−CN−MnII pairs are evaluated numerically in terms of a manyelectron superexchange theory similar to that described in ref 49. In particular, these calculations are used to study the dependence of the orbital (J1, J2,) and anisotropic (Jz, Jxy) exchange parameters on the C− N−Mn bonding angle in the apical and equatorial pairs. Calculations are performed for Mo−CN−Mn pairs with a symmetrized structure of Mo and Mn centers: [Mo(CN)7]4− is regarded as a regular pentagonal bipyramid (D5h), and a regular octahedral MnN6 coordination is assumed for MnII centers.50 Principal atomic distances are set to the average atomic distances in 1−3 (Figure 1). More details of these superexchange calculations for MoIII−CN−MnII pairs are reported in the Supporting Information (see also ref 46). Calculated variation of the orbital parameters J1 and J2 in (eq 2) and anisotropic parameters Jz, Jxy in (eq 3) (which are related by Jz = (J1 + J2)/2 and Jxy = (J1 − J2)/2) is shown in Figures 6, 8, and 9. These results provide evidence for a strong impact of distortions in the Mo−CN−Mn bridging groups on the exchange parameters. However, exchange parameters vary differently in the apical and equatorial pairs due to a dissimilar behavior of superexchange pathways upon distortions. Thus, for the apical pairs pure Ising-type exchange anisotropy (with Jz < 0 and Jxy = 0) occurs only for a linear E


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Inorganic Chemistry

Figure 5. Two principal superexchange pathways in the equatorial MoIII−CN−MnII exchange-coupled pairs in the equatorial Mn2Mo isomers 2 and 3 related to the orbital parameters J1 and J2. The 4dxz(Mo)−3dxz(Mn) superexchange pathway connecting two halffilled nonorthogonal magnetic orbitals results in larger antiferromagnetic exchange parameter J1. The 4dyz(Mo)−3dyz(Mn) superexchange pathway incorporates the 4dyz(Mo) magnetic orbital, which is orthogonal with respect to all 3d(Mn) magnetic orbitals and thereby results in smaller ferromagnetic exchange parameter J2.

Figure 6. Dependence of exchange parameters on the bending angle 180° − θ for the apical Mo−CN−Mn pair: (a) variation of orbital parameters J1 and J2 in the orbitally dependent spin Hamiltonian Horb (eq 2), and (b) variation of anisotropic exchange parameters Jz, Jxy in the effective anisotropic spin Hamiltonian Heff (eq 3). Spin quantization axes x, y, and z are indicated. The MnN6 unit rotates as a whole in the vertical zx plane around the marked nitrogen atom. With the increasing bending angle, both the exchange parameters Jz, Jxy (anisotropic and antiferromagnetic) increase in absolute value, but the difference Jz − Jxy = J2 is nearly constant.

Mo−CN−Mn group, when θ = 180° and J1 = J2 (Figure 6), as has previously been predicted.39 Then, the absolute value of the J1 orbital exchange parameter increases considerably as the bending angle 180° − θ increases from zero to 45° (Figure 6a). This is caused by opening of a new superexchange pathway of mixed σπ-type, which becomes very efficient at large bending angles (180° − θ > 30°) due to progressively increasing nonorthogonality between 4dxz(Mo) (π-type) and 3dz2(Mn) (σ-type) magnetic orbitals (Figure 7). By contrast, the antiferromagnetic J2 parameter remains nearly constant (Figure 6a) owing to minor changes in the 4dyz(Mo)− 3dyz(Mn) superexchange pathway in the bent Mo−C−N−Mn group (Figure 4). As a result, the anisotropic exchange parameters Jz and Jxy considerably increase (in the absolute value) in strongly bent groups. In the whole range of variation of the bending angle 180° − θ from 0 to 45°, the spin-coupling anisotropy retains an antiferromagnetic Isingtype character with |Jz| > |Jxy|, but the relative degree of the exchange anisotropy (Jz/Jxy) considerably reduces. Note that the difference Jz− Jxy = J2 is nearly constant since J2 changes very little (Figure 6a); this fact plays an important role because the barrier Ueff in the apical Mn2Mo complex 1 is determined by Ueff ≈ 2|Jz−Jxy| = 2|J2| (see below Figure 12 and comments). This implies that the barrier in 1 is insensitive to bending of the apical C−N−Mn groups. In the equatorial pairs, there are two types of bending distortions in the C−N−Mn bridges corresponding to rotations of the MnN6 unit in the vertical plane (polar angle θ) and in the equatorial xy plane (azimuthal angle φ), Figures 8 and 9. Variation of the J1 and J2 parameters with the increasing polar bending angle 180° − θ is similar to that for the apical pair; i.e., J1 rapidly increases while J2 remains small and nearly constant. The reason for a considerable increase of J1 is the same as that in apical pairs, i.e., opening of an efficient superexchange σπ-pathway due to increasing nonorthogonality

between 4dxz(Mo) (π-type) and 3dz2(Mn) (σ-type) magnetic orbitals, Figure 7. However, in contrast to the case of the apical pair, now J2 is weakly ferromagnetic (J2 ∼ +2 cm−1) due to orthogonality between the 4dyz(Mo) orbital and all 3d(Mn) magnetic orbitals, Figure 5. Interestingly, J2 stays ferromagnetic even at large distortions (Figure 8). Hence, as can be seen from eq 3 for J1 < 0, J2 > 0, and |J1| ≫ |J2|, in the equatorial Mo−CN−Mn pairs spin coupling is close to the ordinary isotropic exchange with a minor anisotropic xy-component, Jz ≈ Jxy, |Jz | < |Jxy|, and |Jz−Jxy| ≪ |Jz|, Figure 8b. On the other hand, calculations for the equatorial Mo−CN−Mn pairs with azimuthal angular distortions reveal a rather weak variation of exchange parameters with the bending angle 180° − φ, Figure 9. This is due to the fact that, in contrast to polar angular distortions 180° − θ (Figures 6−8), bending of the bridging groups in the xy plane does not open new efficient superexchange pathways between 4d(Mo) and 3d(Mn) magnetic orbitals.

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RESULTS Origin of the Barrier Ueff in the Apical Mn2Mo Complex 1. On the basis of the developed theoretical background, the magnetic behavior of three Mn2Mo complexes 1−3 is analyzed in terms of a spin Hamiltonian F


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Inorganic Chemistry

Figure 7. 4dxz(Mo)−CN−3dz2(Mn) superexchange pathway in (a) linear Mn−CN−Mn apical group and (b) bent Mo−CN−Mn group. In the linear group, this pathway is inactive due to orthogonality between the 4dxz(Mo) and 3dz2(Mn) half-filled magnetic orbitals. In the bent group, the 4dxz(Mo)−CN−3dz2(Mn) antiferromagnetic superexchange pathway of mixed σπ-type opens up due to nonzero σ-type overlap between 3dz2(Mn) and 2px(N) orbitals. At large bending angles of the C−N−Mn group (180° − θ ∼ 30° in Mn2Mo 1), this superexchange pathway becomes very efficient resulting in a considerable increase of the J1 antiferromagnetic orbital parameter (Figure 6).

Figure 8. Exchange parameters vs bending angle 180° − θ plots for the equatorial Mo−CN−Mn pair: (a) variation of orbital exchange parameters J1 and J2; parameter J2 is weakly ferromagnetic (see Figure 5) and nearly constant; (b) variation of anisotropic exchange parameters Jz and Jxy. The MnN6 unit rotates as a whole in the vertical zx plane around the marked nitrogen atom. The resulting spin coupling is nearly isotropic with a small anisotropic xy-component, |Jxy| > |Jz|. Spin quantization axes x, y, and z are indicated.

Figure 9. Exchange parameters vs azimuthal bending angle 180°− φ dependence calculated for the equatorial Mo−CN−Mn pair: (a) variation of orbital exchange parameters J1 and J2; parameter J2 is weakly ferromagnetic and is nearly constant, J1 is antiferromagnetic; (b) variation of anisotropic exchange parameters Jz and Jxy. The MnN6 unit rotates as a whole in the horizontal xy plane around marked nitrogen atom. Exchange anisotropy has a xy-character, |Jxy| > |Jz|, with small antiferromagnetic exchange parameters. Spin quantization axes x, y, and z are indicated.

y x x y z z Ĥ = − ∑ (Jxy (SMn( i)SMo + SMn(i)SMo) + Jz SMn(i)SMo) i = 1,2

Jz and Jxy parameters is used for the two MnII ions in Mn2Mo, although they are not structurally identical in compounds 1−3. Anisotropic exchange parameters for 1−3 are obtained from a numerical fit of the spin Hamiltonian (eq 4) to the experimental χMT curves (Figures 10 and 15). In these calculations, Jz, Jxy, JMn−Mn, and gMn are allowed to vary, while the components of the anisotropic g-tensor of [Mo(CN)7]4−

− JMn − Mn SMn(1)SMn(2) + μB gMn(SMn(1) + SMn(2))H x y z Hx + gySMo Hy + gz SMo Hz) + μB (gx SMo

(4)

where the small ZFS energy of MnII is neglected (since D < 0.1 cm−1 for a MnIIL5 complex44,51). For simplicity, the same set of G


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Inorganic Chemistry

Figure 10. Calculated and experimental44 temperature dependences of the product of magnetic susceptibility and temperature (χMT) for Mn2Mo 1. Solid red line represents the best fit with the Hamiltonian of eq 4 at Jz = −34, Jxy = −11 cm−1, JMn−Mn = 0, gMn = 2.070, and λ = −0.022 mol cm−3 (see text for detail). Figure 11. Energy levels of spin states of the apical Mn2Mo complex 1 calculated with fitted exchange parameters (Jz = −34, Jxy = −11 cm−1, and JMn−Mn = 0, see Figure 10). A double-well character of a butterfly shaped energy level pattern is clearly seen with the ground state represented by two degenerate spin levels MS = +9/2 and MS = −9/2. Approximately, the spin-reversal barrier Ueff corresponds to the energy position of the first low-lying spin level MS = ±1/2 at 47 cm−1.

are fixed at gx = 1.305, gy = 1.322, and gz = 3.023 obtained from LF calculations (combined with the angular-overlap model) for the actual geometry of the [Mo(CN)7]4− anion in 1. In addition, a small intercluster magnetic coupling was incorporated in terms of a mean-field approach using equation χ = −1 ∑α=x,y,z(χ−1 α (calc) − λ) /3, where χα(calc) represents the susceptibility calculated using the Hamiltonian (eq 4) and λ is the molecular-field constant (see Supporting Information for more detail). For the apical Mn2Mo complex 1, a good fit is obtained at Jz = −34, Jxy = −11 cm−1, JMn−Mn = 0, gMn = 2.070, and λ = −0.022 mol cm−3 for the χMT data above 12 K (Figure 10). Antiferromagnetic and strongly anisotropic Ising-type spin coupling in 1 agrees well with the results of the preceding analysis for the apical Mo−CN−Mn pairs. Qualitatively, fitted exchange parameters are reasonably consistent with those obtained from superexchange calculations for the apical Mn2Mo isomer 1 (Jz = −27 cm−1 and Jxy = −16 cm−1 at the actual bending angle of (180° − θ) ≈ 30° in 1, see Figure 6). It is also noteworthy that the orbital exchange parameters J1 = −45 cm−1 and J2 = −23 cm−1 in Mn2Mo 1 are comparable in magnitude with exchange parameters obtained from broken symmetry density functional theory (DFT) calculations for Mo−CN−Mn fragments in MoIII−MnII polymer compounds.52 Spin energy levels of 1 calculated with these exchange parameters are shown in Figure 11. It is important to note that, although SMo and the total spin STotal of a Mn2Mo complex are not good quantum numbers (since they are coupled with the orbital momentum of the MoIII via SOC), the projection MS of the total spin STotal = SMn(1) + SMo + SMn(2) on the z-axis remains a good quantum number due to the uniaxial symmetry of the spin coupling −Jxy(SxMnSxMo + SyMnSyMo) − JzSzMnSzMo involved in the spin Hamiltonian (eq 4). Hence, because each spin state of Mn2Mo has a definite value of MS, the energy spectrum of spin states is well-described by a conventional E versus MS plot, as is the case for most SMMs (Figure 11). The distribution of energy levels over MS in the apical Mn2Mo complex 1 differs drastically from the conventional DSz2 double-well parabolic energy profile of 3d-ion-based SMMs. However, the butterfly shaped spin energy diagram reveals a distinct double-well character for low-lying spin states

that accounts for the SMM behavior of 1 (Figure 11). The doubly degenerate ground quantum state has a highly anisotropic (Ising-like) character with a large spin projection. It is represented by a well-isolated Kramers doublet with MS = ±9/2; the first excited spin level MS = ±7/2 lies at 14.5 cm−1. By analogy with the giant-spin model |D|(MS2 − 1/4) for SMMs with a half-integer spin, for a double-well potential energy curve of 1 the Ueff barrier can be associated with the energy position of the lower spin level with MS = ±1/2 lying at 47 cm−1 (Figure 11); this value compares reasonably with the experimental barrier Ueff = 40.5 cm−1. It is important to note that the barrier Ueff in 1 is controlled by anisotropic exchange parameters Jz and Jxy rather than by the ZFS anisotropy of MnII ions, which is too small to match the value of Ueff = 40.5 cm−1. Indeed, ZFS on MnII ions in 1 is as small as D = −0.07 cm−1, E ∼ 0;44 these values are consistent with the experimental ZFS parameters D = 0.07 cm−1 and E = 0.008 cm−1 for a related MnIIL5 pentagonal complex.51 Simulation of the energy spectrum with variable Jz and Jxy parameters indicates that the barrier is close to Ueff ≈ 2| Jz − Jxy|, which approximately corresponds to the sum the absolute values of the exchange anisotropy |Jz − Jxy| of two apical Mo−CN−Mn pairs in the Mn2Mo complex 1 (Figure 12). Although the spin energy diagram changes considerably with the increasing Jxy parameter, the barrier remains nearly constant (Figure 12e). At large exchange parameters, the energy profile of low-lying spin states transforms from linear (Figure 12a,b) to the conventional parabolic pattern |D|Sz2 (Figure 12d). It is also remarkable that the barrier Ueff in Mn2Mo 1 is simply equal to the double absolute value of the smaller orbital exchange parameter J2, Ueff ≈ 2|J2| = 2|Jz − Jxy|. Thus, according to superexchange calculations indicating weak variation of J2 H


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Inorganic Chemistry

Figure 12. (a−d) Evolution of the energy spectrum of spin states of the apical isomer Mn2Mo 1 with the increasing Jxy exchange parameter at fixed | Jz − Jxy| = 23 cm−1 (corresponding to Jz = −34 and Jxy = −11 cm−1 in Mn2Mo 1). At large exchange parameters (case d), the energy profile of lowlying spin states is close to the conventional DSz2 parabolic pattern. (e) The Ueff vs Jxy plot. The barrier Ueff is always close to 2|Jz − Jxy|.

(Figure 6a), the barrier Ueff of 1 should be rather insensitive to bending of the C−N−Mn groups. It is noteworthy that a more rigorous estimate of Ueff is a complicated problem that needs further detailed study; particularly, this concerns a deeper understanding of specific processes of the through-barrier quantum tunneling of magnetization (QTM) in SMMs with highly anisotropic spin coupling. In SMMs with integer spin, QTM is related to the tunnel spitting energy Δ−m,m of doubly degenerate spin energy levels ± MS in the double-well potential.4−6 However, for an SMM with half-integer spin the tunnel splitting is not seen in the energy spectrum due to a Kramers degeneracy of energy levels. Qualitatively, this drawback can be avoided by using a trick of considering a modified MnII−MoIII− MnIII complex with integer spin, in which one of the MnII ions (S = 5/2) is formally substituted by a MnIII ion (S = 2) with the same anisotropic exchange parameters (Jz = −34 and Jxy = −11 cm−1). In addition, small ZFS anisotropy of two Mn centers with D = −0.07 cm−1 and E = 0 (the same for MnII and MnIII) is applied as a source of transverse magnetic anisotropy of

Mn2Mo 1. For this purpose, a nonlinear orientation of the local ZFS easy-axes of two MnLN5 complexes is taken into account by means of a rotation of the ZFS tensors Di by the angle of 30° (where Di transforms as RiDiRi−1) corresponding to the C−N− Mn bonding angles in 1. This fictitious replacement changes rather insignificantly the overall character of the double-well energy spectrum, but provides visualization of tunnel splitting energies Δ−m, m of the doubly degenerate ±MS spin states, Figure 13. The resulting Δ−m, m values are very small for the ground and low-lying excited spin levels Figure 13b. In particular, the ground-state tunnel splitting of Δ−4,4 ∼ 4.8 × 10−11 cm−1 (Figure 13b) is comparable to that of Mn12Ac (about Δ−10,10 ∼ 10−11 cm−1 for the MS = ±10 ground spin state4a). In this case, the effective spin-reversal barrier Ueff may be associated with the first excited spin level having a more pronounced tunnel splitting energy, such as Δ−1,1 ∼ 10−1 cm−1 for the lower MS = ± 1 spin state at 38 cm−1. In the parent MnII−MoIII−MnII complex, it corresponds to the lowest MS = ±3/2 spin state lying at 40 cm−1 nearly coinciding with the experimental energy barrier of 40.5 cm−144 (Figure 13a). Given I


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Inorganic Chemistry

Figure 13. Comparison of the energy spectra of low-lying spin states in (a) MnII−MoIII−MnII complex 1 with half-integer spin S = 9/2 and (b) modified MnII−MoIII−MnIII complex with integer spin S = 4, in which one MnII ion is replaced by a MnIII ion. In both cases, calculations are performed with Jz = −34, Jxy = −11 cm−1, and JMn−Mn = 0; small ZFS anisotropy on Mn ions is also involved (see the main text for details). Calculated tunnel splitting energies Δ−m,m (blue numbers, in cm−1) for the ground and excited doubly degenerate spin states ± MS are indicated (see text for detail). In MnII−MoIII−MnIII, the effective energy barrier Ueff can be associated with the energy position of the first excited spin state MS = ±1 with a large tunnel splitting energy (ca. 10−1 cm−1). This spin state corresponds to the lowest MS = ±3/2 level in the parent MnII−MoIII−MnII complex lying at 40 cm−1, which is consistent with the experimental barrier (Ueff = 40.5 cm−1).44

Figure 14. (a) Calculated dependence of spin energy levels of Mn2Mo compound 1 on an external magnetic field B parallel to the z-axis. Critical field values corresponding to the crossover of the ground (MS = +9/2) and first excited spin levels (MS = −7/2) are shown (1.99 and 2.32 T, respectively, marked with blue arrows). (b) Experimental field sweep rate-dependent magnetic hysteresis loops and derivative of the magnetization (dM/dH) versus magnetic field.44 The main resonance field values at 1.83 and 2.25 T are reasonably consistent with the calculated values of 1.99 and 2.32 T.

that Mn2Mo 1 and Mn12Ac have close barriers (Ueff = 40.5 and 45 cm−1, respectively), this explains why these SMMs have the same blocking temperature TB ≈ 3 K despite a large difference in the nuclearity and the ground-state spin. Next, the dependence of spin level energies on an external magnetic field is calculated in terms of the spin Hamiltonian (eq 4), Figure 14. Calculated resonance field values (Figure 14a) are consistent with the positions of the two most intense peaks in Figure 14b (1.99 and 2.32 T calculated vs 1.83 and 2.25 T experimental44). Importantly, these calculations reproduce irregular field positions of the main peaks in

Mn2Mo 1, which differs considerably for the equidistant distribution of resonance peaks typical of the DMS2-type profile of the energy spectrum in the usual SMMs.1−6 It is also noteworthy that a butterfly shaped pattern of the spin state energy spectrum of Mn2Mo 1 (Figure 11) accounts for the origin of a flat minimum in the χMT curve (Figure 10). A considerable increase of the χMT product at low temperatures is due to thermal population of low-lying spin states with large MS values (Figure 11). In the middle energy range between 60 and 100 cm−1, the energy levels are represented by spin states with a small spin projection of MS = ±1/2; as a result, their J


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Inorganic Chemistry

Figure 15. Experimental (○)44 and calculated (red ) temperature dependence of the χMT product for (a) equatorial Mn2Mo complex 2, and (b) equatorial Mn2Mo complex 3. The calculated curves are obtained with the following sets of parameters: Jz = −7.5, Jxy = −9, JMn−Mn = −1.67 cm−1, gMn = 2.058, and λ = −0.016 mol cm−3 for 2 and Jz = 0, Jxy = −4, JMn−Mn = −0.60 cm−1, gMn = 2.120, and λ = −0.028 mol cm−3 for 3.

Figure 16. E vs MS energy diagrams of spin states of (a) Mn2Mo complex 2 and (b) Mn2Mo complex 3. For both equatorial complexes, the energy level profile has a single-well character incompatible with the SMM behavior. The energy levels are calculated at the sets of parameters listed in the captions to Figure 15.

thermal population leads to a local decrease of χMT. Then, the subsequent thermal population of the next excited spin states (lying above 100 cm−1) with progressively increasing spin projection MS gives rise to a steady increase of χMT at higher temperature (Figure 10). A flat minimum in the χMT curve around 80 K approximately corresponds to the waistline of the butterfly shaped figure (around 70 cm−1) in Figure 11. Similar behavior of the χMT curve was also observed in the [MnIII(acacen)]3[ReIV(CN)7] polymer compound based on isoelectronic [ReIV(CN)7]3− complexes exhibiting anisotropic spin coupling.46 Analysis of Magnetic Properties of the Equatorial Mn2Mo Complexes 2 and 3. Magnetic properties of the equatorial Mn2Mo complexes 2 and 3 differ drastically from those of the apical complex 1 (Figure 15). The χMT product of 2 and 3 gradually decreases with the lowering temperature and

then drops below ca. 50 K (Figure 15).44 Simulation of the χMT curves of the equatorial isomers 2 and 3 in terms of the spin Hamiltonian (eq 4) yields quite different sets of exchange parameters, Jz = −7.5, Jxy = −9, JMn−Mn = −1.67 cm−1, gMn = 2.058, and λ = −0.016 mol cm−3 for 2 and Jz = 0, Jxy = −4, JMn−Mn = −0.60 cm−1, gMn = 2.120, and λ = −0.028 mol cm−3 for 3. These data clearly indicate an xy character of the exchange anisotropy with |Jz| < |Jxy|, just as expected from the microscopic theory for the equatorial Mo−CN−Mn pairs (see Figures 5, 8, and 9). Again, the values of the exchange parameters Jz, Jxy and the difference Jz − Jxy = J2 > 0 are fairly consistent with the results of superexchange calculations for the bent equatorial pairs Mo−CN−Mn, Figures 8 and 9. The appearance of a more pronounced JMn−Mn exchange parameter in 2 and 3 is seemingly due to a more efficient antiferromagnetic superexchange pathway between two Mn K


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Inorganic Chemistry

Figure 17. On the efficiency of anisotropic spin coupling in enhancing Ueff and TB values: comparison of SMM characteristics of Mn12Ac,1−3 Mn6,11a Mn3,11b and Mn2Mo44 complexes. In Mn12Ac, Mn6, and Mn3, the barrier Ueff is due to single-ion ZFS on MnIII ions, while in Mn2Mo 1 the barrier originates from anisotropic Ising-type spin coupling in MoIII−CN−MnII groups. Small trinuclear Mn2Mo complex 1 with S = 9/2 has nearly the same Ueff and TB values as those of the much larger Mn12Ac complex with S = 10. Two anisotropic Mo−CN−Mn spin couplings in MnII2MoIII result in similar or better Ueff and TB than ZFS energies of three MnIII ions in a ferromagnetic Mn3 cluster. This indicates that single anisotropic Mo−CN− Mn spin coupling produces larger molecular magnetic anisotropy (measured by ∼|Jz−Jxy| = 23 cm−1, Figure 12) than the ZFS energy of a single MnIII ion (|Di|S2 ≈ 15 cm−1).

centers in the equatorial plane across the central [Mo(CN)7]4− complex, especially for the cis isomer 2 with a shorter Mn−Mn distance. Exchange anisotropy of an xy-type in equatorial Mo−CN− Mn pairs results in dramatic changes in the energy spectrum of the spin states, which loses its double-well character in passing from the apical complex 1 to the equatorial isomers 2 and 3. Contrary to a double-well energy profile in 1 (Figure 11), the E versus MS plot for 2 and 3 reveals a single-well pattern with a MS = ±1/2 ground Kramers doublet, which is followed by a dense (within few cm−1) group of numerous low-lying spin states with a low MS projection (Figure 16). In other words, easy-plane exchange anisotropy with |Jz| < |Jxy| results in a positive overall magnetic anisotropy of the equatorial Mn2Mo complexes 2 and 3, that stabilize a low-spin ground state unfavorable for slow magnetic relaxation. Obviously, these features are incompatible with the necessary conditions for a high-spin molecule to be an SMM. Indeed, no SMM behavior is observed in 2 and 3 down to 1.8 K.44

molecules is very sensitive to the specific connection mode of the attached MnII centers. Accordingly, for 2 and 3 in which MnII ions are connected to MoIII via equatorial CN groups, an xy character of the anisotropic MoIII−MnII spin coupling results in no SMM behavior (Figure 16). Highly anisotropic, Ising-type spin coupling for the apical pairs (such as Jz = −34 and Jxy = −11 cm−1 in 1) represents a very important source of the global magnetic anisotropy for the polynuclear magnetic systems. This is clearly evidenced by a comparison of Ueff and TB characteristics of four SMM complexes, Mn12Ac,1−3 Mn6,11a Mn3,11b and Mn2Mo 1,44 Figure 17. Despite considerably smaller nuclearity and groundstate spin, Mn2Mo 1 displays the same or comparable Ueff and TB values as those of Mn12Ac and Mn6 complexes. Moreover, the energy barrier of 40.5 cm−1 represents a new record for all cyanide-bridged SMMs. It is of special interest to compare Mn3 and Mn2Mo. These small trinuclear clusters display comparable S, Ueff, and TB values albeit having quite different origin of the SMM behavior. The triangular ferromagnetic Mn3 cluster has the largest magnetic anisotropy per one MnIII ion among all Mn-based SMMs due to unique combination of nearly parallel alignment of long Jahn−Teller axes of three MnIII ions and ferromagnetic spin coupling.11b In this case the barrier Ueff of Mn3 approaches the limiting value determined by the sum of ZFS energies on three MnIII ions (with |Di|S2 ∼ 15 cm−1), about 40−45 cm−1 (58−65 K). In Mn2Mo the barrier originates from two apical MoIII−CN−MnII anisotropic Isingtype spin couplings resulting in Ueff ≈ 2|Jz−Jxy| = 46 cm−1, Figure 12. Consequently, anisotropic pair-ion contribution from the single apical MoIII−CN−MnII linkage to the overall magnetic anisotropy is larger than the ZFS contribution from a single MnIII ion, |Jz − Jxy| ≈ 23 cm−1 versus |Di|S2 ≈ 15 cm−1. This is evidence that anisotropic Ising-like MoIII−CN−MnII spin couplings may be considerably more efficient in producing molecular magnetic anisotropy of a SMM cluster than the single-ion ZFS energy of MnIII ions. More quantitatively, according to the developed model, the magnetization reversal barrier Ueff for a molecule based on the Mo−CN−Mn units is directly proportional to the difference |Jz − Jxy| = |J2| and to the number of anisotropic pair-ion spin couplings of the apical Mo−CN−Mn groups (Figure 12). This is exceptionally important since it provides a straightforward

■

DISCUSSION The developed theoretical model provides unambiguous evidence for the fact that the SMM behavior of the apical complex Mn2Mo 1 originates exclusively from anisotropic spin coupling between the MoIII and MnII spin centers, which is strongly dependent on the positions of the CN groups in the pentagonal-bipyramid geometry, just as suggested by earlier theoretical predictions39 and supported by present calculations. In fact, this theoretical analysis explains in a natural way the collection of the experimental data for 1−3 and, particularly, the origin of high Ueff and TB values in the apical Mn2Mo complex 1 and the absence of SMM properties for the equatorial Mn2Mo isomers 2 and 3. Calculations in terms of the anisotropic spin Hamiltonian (eq 4) indicate that the energy profile of low-lying states for 1 is significantly different from the well-known double parabolic potential well where the ground state S is split by the easy-axis ZFS |D|Sz2, Figure 11. This gives rise to irregular intervals of the critical magnetic fields required for the QTM events of 1, as is evidenced by both the experimental data44 and present theoretical calculations, which are consistent with each other (Figure 14). The character of the anisotropic magnetic coupling in [Mo(CN)7]4−-based spin L


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Inorganic Chemistry

Figure 18. Comparison of the Ueff barriers in (a) MnII2MoIII complex 1 based on orbitally degenerate [MoIII(CN)7]4− heptacyanomolybdate (pentagonal bipyramid) and (b) MnIII2OsIII complex based on [OsIII(CN)6]3− orbitally degenerate hexacyanoosmate (octahedron). In the MnIII2OsIII complex, the symmetry of the anisotropic spin Hamiltonian lowers from uniaxial to three-axis one upon bending of the C−N−Mn groups.41a This drastically reduces the barrier Ueff in MnIII2OsIII SMM, which is determined by the energy position ΔE01 of the first excited spin level. By contrast, in Mn2Mo the barrier Ueff is considerably larger than ΔE01 due to retaining uniaxial symmetry of the anisotropic spin Hamiltonian upon the cyanide group bending.

with the spin Hamiltonian (eq 4) for a hypothetical apical VII− MoIII−VII complex (typically, with Jz = −150 and Jxy = −50 cm−1) yield Ueff ≈ 100 cm−1. The fact that the local z-axis of the x x y y anisotropic spin Hamiltonian −Jxy(SMn SMo + SMn SMo ) − JzSzMnSzMo for all Mo−CN−Mn pairs is always parallel to the unique pentagonal axis of the [MoIII(CN)7]4− bipyramid is crucially important for obtaining large molecular magnetic isotropy of uniaxial symmetry required for design of improved SMMs. Indeed, this implies that the local magnetic uniaxial anisotropies coming from separate Mo−CN−Mo groups are additively summarized into the overall molecular magnetic anisotropy of uniaxial symmetry with no or small transversal component. In this context, it is also relevant to mention a similar problem of parallel orientation of the local ZFS easyaxes of individual high-spin 3d ions (such as MnIII), which is important for obtaining large barriers in ordinary SMMs, such as Mn12, Mn6, or Mn3.4−6,11 Importantly, strong exchange anisotropy itself does not inevitably lead to high energy barriers; for this, a uniaxial anisotropy of spin coupling is required as a necessary condition to minimize or suppress the unwanted transversal component of the molecular magnetic anisotropy. This can be wellillustrated by a comparison of SMM characteristics of the MnII2MoIII complex 1 and structurally related cyano-bridged trinuclear complexes MnIII2RuIII and MnIII2OsIII based on orbitally degenerate [Ru III (CN) 6 ] 3− and [Os III (CN) 6 ]3− octahedral complexes featuring highly anisotropic RuIII−CN− MnIII and OsIII−CN−MnIII spin couplings.41a Although the MnIII2RuIII, MnIII2OsIII, and MnII2MoIII compounds have similar trinuclear cyano-bridged structure, ground-state spin, and comparable anisotropic exchange parameters, the energy barriers of MnIII2RuIII and MnIII2OsIII are ∼12−14 cm−1, which are only one-third of the barrier of 1. The underlying

strategy for tuning and controlling the energy barriers of the SMMs. Indeed, because the regular or slightly distorted [Mo(CN)7]4− pentagonal complex invariably produces a uniaxial Ising-type anisotropic spin coupling (with |Jz| > |Jxy|) for apical pairs (regardless of the bending angles of the bridging CN groups and the local symmetry of attached high-spin 3d units, Figure 6), the only other ingredient needed to obtain high barriers is a large value of the exchange parameter J2. In contrast to the single-ion ZFS anisotropy Di of high-spin 3d ions, the exchange parameter J can be enlarged to very high values by using appropriate attached high-spin ions. Thus, the closely related [MoIII(CN)6]3− octahedral complex exhibits very high exchange parameters with some selected high-spin ions, such as J = −226 cm−1 for MoIII−CN−MoIII53 and J = −122 cm−154 or even J = −228 cm−1 for MoIII−CN−VII;55 even larger exchange parameters for MoIII−CN−VII linkages (J ≈ 350 cm−1) in a Prussian blue type structure were obtained from broken symmetry DFT calculations.56 In passing from the highspin [MoIII(CN)6]3− octahedral complex (S = 3/2) to a lowspin [MoIII(CN)7]4− pentagonal complex (S = 1/2) these exchange parameters would further increase (approximately, by a factor of 1.5). A considerable increase in exchange parameters for VII−MoIII compounds is also evidenced by a comparison of the experimental magnetic data on isostructural heptacyanomolybdenum-based polymer compounds MII2[MoIII(CN)7](pyrimidine)2 (MII = MnII and VII), which indicate that the magnetic ordering temperature TC increases from 47 K for the Mn−Mo compound to 110 K for the V−Mo compound.57 Accordingly, given that two apical contributions MoIII−CN− MnII with |Jz − Jxy| = 23 cm−1 yield an energy barrier of 40.5 cm−1 in Mn2Mo 1, considerably larger barriers are expected for the apical MoIII−CN−VII pairs, for which the difference Jz − Jxy = J2 can reach a value on the order of 100 cm−1. Calculations M


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Inorganic Chemistry

Figure 19. Comparison of the Ueff barriers and spin energy diagrams of (a) MnII2MoIII complex 1 and (b) hypothetical MnII4MoIII2 double complex composed of two linear MnII−MoIII−MoII units coupled via two equatorial cyanide bridges with parallel orientation of the local pentagonal axes of two [Mo(CN)7]4− bipyramids (∥z). Calculations for the double complex Mn4Mo2 are performed with the same anisotropic exchange parameters Jz = −34, Jxy = −11 cm−1 for the four apical MoIII−CN−MnII groups as those for Mn2Mo 1, while Jz = Jxy = −10 cm−1 is applied for the two equatorial bridging groups; in the latter case, the small anisotropic xy-component is neglected (see Figure 8). The energy profile of the double-well potential for Mn4Mo2 is indicated by the light blue curve; its top corresponds to the barrier of Ueff ≈ 78 cm−1, which is approximately twice larger than Ueff = 40.5 cm−1 for Mn2Mo 1.

parameters Jz = −34 and Jxy = −11 cm−1 for a hypothetical MnII4MoIII2 double complex composed of two coupled MnII− MoIII−MnII units 1 comprising a total of four apical Mo−CN− Mn linkages result in a barrier of Ueff ≈ 78 cm−1, which is approximately twice larger than Ueff = 40.5 cm−1 for Mn2Mo 1 (Figure 19); besides, the double Mn4Mo2 complex has a twice larger ground-state spin (MS = ±9 vs ±9/2 in 1) favorable for slow magnetic relaxation. These results provide evidence that the barrier Ueff is roughly proportional to the number of the apical MoIII−CN−MnII groups in an extended SMM molecule. Consequently, the Ueff barrier may be scaled up to considerably larger values by increasing nuclearity of [MoIII(CN)7]4−-based SMMs with a specially organized molecular structure that provides parallel orientation of the local pentagonal z-axes of [MoIII(CN)7]4− complexes and contains both apical and equatorial MoIII−CN− MnII linkages (see Figure 19b); an extended 3D structure of this type has been recently described for the [MnIII(acacen)]3[ReIV(CN)7] polymer molecular magnet.46 Importantly, in such SMM clusters apical and equatorial MoIII−CN−MnII exchange interactions should act in concert with each other as they play different and complementary roles: the apical Ising-type anisotropic exchange interactions produce local uniaxial pairion anisotropies (each approximately measured by |Jz − Jxy| = | J2|, Figure 12) while more isotropic equatorial exchange interactions combine these local pair-ion anisotropies into the overall uniaxial magnetic anisotropy of an SMM molecule. For this, apical and equatorial exchange interactions should be properly balanced and tuned. In particular, preliminary calculations indicate that equatorial exchange interactions

reason behind such a strong difference in the Ueff value is tied to the difference in how bending of C−N−Mn groups affects the symmetry of the anisotropic spin Hamiltonian. In MnIII2RuIII/OsIII SMMs, the symmetry of the spin Hamiltonian lowers from uniaxial to a three-axis one, whereas in Mn2Mo spin Hamiltonian retains its uniaxial symmetry upon cyanide ligand bending. The three-axis character of the anisotropic spin Hamiltonian, −JxSxMnSxOs − JySyMnSyOs − JzSzMnSzOs (with Jx < 0, Jy > 0, and Jz < 0, such as Jx = −18, Jy = +35, and Jz = −33 cm−1 in MnIII2OsIII),41a drastically reduced Ueff in MnIII2RuIII/OsIII SMMs; in fact, Ueff is limited by the energy position ΔE01 of the first excited spin level in MnIII2RuIII/OsIII (ΔE01 ≈ 12−15 cm−1).41a In sharp contrast, due to uniaxial symmetry of the spin Hamiltonian, compound MnII2MoIII 1 has a barrier of Ueff = 40.5 cm−1, which is much larger than the energy of the first excited spin level (ΔE01 ≈ 15 cm−1, Figure 18). Consequently, orbitally degenerate, pentagonal-bipyramidal complex [MoIII(CN)7]4− is a fascinating molecular building block that, together with attached high-spin 3d ions, produces high-quality uniaxial pair-ion contributions to the overall magnetic anisotropy, which are all summarized with keeping their local easy-axes parallel to the unique pentagonal z-axis of the bipyramid. Each pair-ion contribution is approximately measured by the difference |Jz − Jxy| = |J2|, which is determined by the value of the smaller orbital exchange parameter J2. This opens up new exciting possibilities for increasing the energy barrier Ueff by means of enhancement of anisotropic exchange parameters and by increasing the number of apical cyanobridged groups exhibiting Ising-type anisotropic spin couplings. In particular, preliminary calculations with the same exchange N


Article

Inorganic Chemistry

Xin-Yi Wang for providing experimental magnetic data on Mn2Mo compounds and helpful discussions.

should be strong enough to provide efficient magnetic coupling between neighboring apical MnII−MoIII−MnII units (Figure 19b). In this respect, polynuclear MoIII−M(3d) heterometallic cyano-bridged complexes with M(3d) = VII, CrII, CrIII, or FeII high-spin 3d ions featuring enhanced Mo−CN−M(3d) exchange interactions are especially interesting. In this case, combined cumulative effect of numerous Ising-type spin couplings of apical MoIII−CN−M(3d) groups with parallel easy-axes and with large exchange parameters J1,2 may potentially provide very large Ueff and TB values.

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CONCLUSION AND PERSPECTIVES In summary, theoretical analysis shows that the spin-reversal barrier Ueff and SMM behavior of MnII2MoIII 1 originate exclusively from anisotropic Ising-type spin coupling in two MoIII−CN−MnII apical groups. High efficiency of anisotropic spin coupling in obtaining high barriers is demonstrated by the fact that SMM characteristics of Mn2Mo 1 (Ueff = 58 K and TB = 3.2 K) are close to the seminal Mn12−acetate compound (Ueff = 65 K and TB = 3 K) despite much smaller ground-state spin and nuclearity. By contrast, the absence of SMM behavior for the equatorial MnII2MoIII isomers 2 and 3 is due to xy-character of the exchange anisotropy in the equatorial cyano-bridged groups. Microscopic calculations indicate that the pentagonalbipyramidal [MoIII(CN)7]4− complex has a unique property of uniaxial anisotropic spin coupling in both the apical and x x equatorial MoIII−CN−MnII pairs, H̑ eff = −Jxy(SMo SMn + SyMoSyMn) − JzSzMoSzMn, regardless of their actual low symmetry. Thanks to these features, the pentagonal-bipyramidal [MoIII(CN)7]4− complex represents a very efficient molecular building block producing high-quality, strong magnetic anisotropy of uniaxial symmetry for polynuclear SMM clusters, especially in concert with specially selected high-spin 3d ions. Due to the direct relationship between the energy barrier and the strength of the anisotropic magnetic coupling, future efforts should be focused on extending this system to the earlier 3d metal centers involving stronger magnetic interactions. Mn2Mo complex 1 together with the ReMn4 complex37 are the first representatives of a peculiar class of SMMs with no or very small single-ion ZFS ansotropy on individual metal ions, in which the barrier is governed solely by pair-ion anisotropic spin couplings. These experimental and theoretical results taken together symbolize an impressive start for implementing a new, highly efficient strategy to reach very high TB SMMs.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.5b01975. Details of magnetic calculations (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported in part by RFBR (Grant 15-0307904-a). Tne author is grateful to Prof. K. R. Dunbar and Dr. O


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