Molecular Engineering of High Energy Barrier in Single-Molecule Magnets Based on [Mo^III(CN)₇]⁴⁻ and V(II) Complexes

Abstract

Molecular engineering of high energy barrier U_eff in single-molecule magnets (SMMs) of general composition Mo^III_kV^II_m based on orbitally-degenerate pentagonal-bipyramidal [Mo_III(CN)₇]⁴⁻ complexes with unquenched orbital momentum and high-spin V(II) complexes is discussed. In these SMMs, the barrier originates exclusively from anisotropic Ising-type exchange interactions −J_xy(S_i^xS_j^x + S_i^yS_j^y) − J_zS_i^zS_j^z in the apical cyano-bridged pairs Mo^III–CN–V^II, which produce a double-well energy profile with a doubly degenerate ground spin state ±M_S. It is shown that the spin-reversal barrier U_eff is controlled by anisotropic exchange parameters J_z, J_xy, and the number n of apical Mo^III–CN–V^II groups in a SMM cluster, U_eff ~ 0.5|J_z − J_xy|n; it can reach a value of many hundreds of wavenumbers (up to 741 cm⁻¹). This finding provides a very efficient straightforward strategy for further scaling U_eff to high values (>1000 cm⁻¹) by means of enhancing exchange parameters J_z, J_xy, and increasing the number of [Mo^III(CN)₇]⁴⁻ complexes in a SMM molecule.

1. Introduction

Single-molecule magnets (SMMs) are individual high-spin molecules featuring slow spin relaxation and preserving their magnetic moment below characteristic blocking temperature T_B [1,2,3,4,5,6,7]. The slow relaxation of SMMs is caused by a double-well potential with the two lowest M_S = +S and M_S = −S spin states separated by an energy barrier U_eff, which arises from the combined effect of the easy-axis (Ising-type) magnetic anisotropy (quantified by the axial zero-field splitting parameter D) and high-spin ground state S. The spin-reversal barrier U_eff is expressed by |D|S² or |D|(S² − 1/4) for integer and half-integer spins, respectively. Since the discovery of the first representative (Mn₁₂Ac) in 1993 [1,2], SMMs have attracted increasing attention due to their huge forward-looking application potential in the design of high-density information storage [8], quantum computing [9,10,11,12], and molecular spintronics [13,14]. Currently, development of SMMs is one of the most rapidly growing lines of research in the field of molecular magnetism. However, rapid advance of the SMM-based technology is hampered by a low blocking temperature T_B, which is normally below a few Kelvins [1,2,3,4,5,6,7]. Designing of SMMs with higher T_B and U_eff characteristics has remained a highly challenging task over the past two decades. Earlier efforts toward increasing U_eff and T_B were mainly focused on obtaining large polynuclear transition metal complexes (mostly based on Mn^III ions) with a high ground-state spin S. However, as these studies progressed, it was realized that the barrier U_eff and temperature T_B do not generally rise with increasing spin and nuclearity of a magnetic molecule because the overall magnetic anisotropy D is inversely proportional to the square of the spin, D ~ S⁻² [15,16,17]. In fact, the barrier U_eff = |D|S² is largely independent on spin S; consequently, the barrier U_eff can be increased only by increasing the molecular magnetic anisotropy D. Basically, magnetic anisotropy D originates from a combined effect of spin-orbit coupling and some orbital angular momentum; it may operate as a first-order perturbation or as a second-order perturbation, depending on the nature of the ground state. In SMMs based on high-spin 3d ions, magnetic anisotropy D originates from single-ion contributions resulting from zero field splitting (ZFS) on individual magnetic ions. Being the product of a second-order effect, the ZFS energy is relatively small, typically within few tens of cm⁻¹ or less for spin-only 3d-ions (such as Mn^III, Fe^III, and Ni^II); thus, it leads generally to a rather small magnetic anisotropy D and low barriers U_eff [15,16,17]. Much stronger first-order magnetic anisotropy is produced by magnetic ions with the unquenched orbital momentum, such as lanthanide and actinide ions [18,19,20,21]. Indeed, all lanthanide ions with open 4f-shell (except Gd³⁺) exhibit large unquenched orbital momentum, which, in concert with strong spin-orbit coupling and weak crystal-field splitting of 4f states, produces extremely strong single-ion magnetic anisotropy of Ln³⁺ ions; these features taken together ultimately lead to high U_eff and T_B values of Ln-based SMMs. With the discovery of the first Ln-based SMM, the double decker mononuclear complex [(Pc)₂Tb]⁻ (Pc = phthalocyanine) in 2003 [22], lanthanide complexes have opened a new avenue to high-performance SMMs [23,24,25,26,27,28]. Even mononuclear lanthanide complexes exhibit slow magnetic relaxation at low temperatures with a high barrier U_eff; they are referred to as single-ion magnets (SIMs) [18,19,20,21,22,23,24,25,26,27,28,29]. The last few years have seen impressive progress in designing advanced SIMs with new record blocking temperature, such as T_B = 20 K [30], and spin-reversal barrier U_eff = 1261 cm⁻¹ (1815 K) [31]). Recently, new record SMM characteristics were reported for a mononuclear dysprosium complex [Dy(Cp(t-Bu₃)₂]⁺, U_eff/k_B = 1760 K, T_B = 60 K [32,33]. These values are an order of magnitude higher than those known for SMMs based on polynuclear transition metal complexes, U_eff = 62 cm⁻¹ and T_B = 4.5 K for a Mn₆ complex with S = 12 [34]).

More recently, there has also been a growing interest in mononuclear 3d complexes with unquenched first-order orbital momentum, which is provided by a special less-common coordination of the metal atom. Especially interesting are complexes with a large uniaxial magnetic anisotropy (Ising spin units), which are capable to behave as SIMs with a large energy barrier originating from the first-order spin-orbit splitting of the orbitally degenerate ground state [35,36,37,38]. In recent years, there have been numerous reports on monometallic 3d complexes with SIM behavior [35,36]. Among them, the largest U_eff barrier has been obtained for linear two-coordinated Fe(I) [37] and Co(II) [38] complexes (246 and 413 cm⁻¹, respectively); high energy barriers were also reported for other low-coordinated 3d transition metal complexes [39,40,41,42].

These results vividly show that magnetic centers with unquenched orbital angular momentum provide great opportunities to maximize magnetic anisotropy and to achieve a high barrier and blocking temperature. However, it should be emphasized that the current record-breaking SMM characteristics are based exclusively on single-center magnetic anisotropy of high-spin monometallic complexes, which is already close to the physical limits for 4f and 3d ions. In fact, the maximum feasible barrier U_eff in 4f-SIMs is specified by the total crystal-field splitting energy of the ground J-multiplet of Ln³⁺ ions, which is normally within several hundreds of cm⁻¹ and very rarely reaches 1000 cm⁻¹ or higher [43]. Therefore, the current record barrier around 1250 cm⁻¹ for Dy-based SIMs [32,33] is close to the maximum CF splitting energy for lanthanide ions and thus it is difficult to increase the barrier far beyond this value [44]. The same is also true for mononuclear high-spin 3d complexes with unquenched orbital momentum, in which maximum value of U_eff is controlled by the spin-orbit splitting energy of the ground orbital manifold; this energy is limited by ca. 1000 cm⁻¹ due to weak spin-orbit coupling for 3d electrons.

Alternative strategy toward high-performance SMMs is based on pair-ion contributions to the molecular magnetic anisotropy D rather than on single-ion contributions [45]. These pair-ion contributions are associated with anisotropic exchange interactions produced by low-spin (S = 1/2) orbitally-degenerate 4d and 5d complexes with unquenched orbital angular momentum, such as pentagonal-bipyramidal (PBP) complexes [Mo^III(CN)₇]⁴⁻ [45,46,47] and [Re^IV(CN)₇]³⁻ [46,48,49], and octahedral hexacyano complexes [Ru^III(CN)₆]³⁻ [50] and [Os^III(CN)₆]³⁻ [51]. It is noteworthy that, unlike high-spin 4f and 3d complexes, these complexes exhibit no single-ion magnetic anisotropy in the usual sense, i.e., as the ZFS of the ground-state spin multiplet 2S + 1 (which occurs at S > 1/2). Instead, their single-ion magnetic anisotropy can be seen only in an anisotropic g-tensor and in anisotropic magnetic susceptibility; moreover, it can completely disappear for magnetically isotropic [Ru^III(CN)₆]³⁻ and [Os^III(CN)₆]³⁻ octahedral complexes [50,51]. In this case, the unquenched orbital momentum of low-spin orbitally degenerate 4d and 5d complexes affects the overall magnetic anisotropy through highly anisotropic exchange interactions with other spin carriers [45,46,47,49]. Thus, being incorporated into polynuclear heterometallic spin clusters together with high-spin 3d ions, these 4d/5d complexes form pair-ion 4d/5d–3d anisotropic exchange interactions, which can produce high magnetic anisotropy D and spin reversal barrier U_eff [45,46,47,49]. It is important to note, however, that the anisotropic exchange interactions in themselves do not necessarily lead to a SMM behavior because of the condition for a molecular spin cluster to have a negative magnetic anisotropy with a small transverse component (D < 0, E ~ 0) [52,53]. For this, two additional conditions are needed: (i) the anisotropic exchange interactions must have the Ising-type character, −J_xy(S_i^xS_j^x + S_i^yS_j^y) − J_zS_i^zS_j^z with |J_z| >> |J_xy|; and (ii) the anisotropic exchange parameters J_z, J_xy must be large in absolute value. Actually, among known orbitally-degenerate complexes, only PBP complexes [Mo^III(CN)₇]⁴⁻ and [Re^IV(CN)₇]³⁻ fully meet these conditions [45,46,47,48,49,50,51,52,53,54]. As shown in our previous works, these complexes have a unique property to form uniaxial anisotropic spin coupling H_eff = −J_xy(S_M^xS_Mn^x + S_M^yS_Mn^y) − J_zS_M^zS_Mn^z (M = Mo^III, Re^IV) with high-spin Mn^II ions (S = 5/2) attached in both the apical and equatorial coordination positions; remarkably, the uniaxial symmetry of the spin Hamiltonian H_eff retains even for the actual low symmetry M–CN–Mn exchange-coupled pairs [47,49]. The apical pairs M–CN–Mn exhibit Ising-type anisotropic exchange interactions (|J_z| > |J_xy|), while exchange interaction in the equatorial pairs is nearly isotropic with a small easy-axis (xy) anisotropic component (|J_z| < |J_xy|, |J_z − J_xy| << |J_z|) [47,49]. Ising-type exchange interactions of apical pairs are especially important since they generate uniaxial magnetic anisotropy of the necessary quality (i.e., D < 0, E ~ 0) and form a barrier. Thus, in the Mo^IIIMn^II₂ trinuclear complex composed of central [Mo^III(CN)₇]⁴⁻ complex and two Mn^II ions in the apical positions, Ising-type exchange interactions −J_xy(S_Mo^xS_Mn^x + S_Mo^yS_Mn^y) − J_zS_Mo^zS_Mn^z of two apical Mo^III–CN–Mn^II pairs (with J_z = −34, J_xy = −11 cm⁻¹) form a double-well potential with the energy barrier of 40.5 cm⁻¹ (58.5 K) and blocking temperature T_B = 3.2 K [55]. Quite recently, another related Mo^IIIMn^II₂ SMM complex with very close characteristics (U_eff = 44.9 cm⁻¹, T_B = 2.5 K, J_z = −35.4, J_xy = −11.4 cm⁻¹) was reported [56]. Remarkably, in these systems, the barrier U_eff is controlled exclusively by the anisotropic exchange parameters (namely, by the difference |J_z − J_xy|) and by the number n of the apical exchange-coupled pairs Mo^III–CN–Mn^II, U_eff ≈ |J_z − J_xy|n [47] (Figure 1).

This opens new avenues for a significant increase in the barrier and the blocking temperature both by enhancing exchange interactions and by increasing nuclearity of the SMM cluster [45,46,47]. To further develop this strategy, this article develops theoretical frameworks for obtaining high SMM characteristics in some hypothetical 4d–3d heterometallic polynuclear complexes based on [Mo^III(CN)₇]⁴⁻ heptacyanometallate and V^II ions. In these systems, high U_eff and T_B values could potentially be obtained due to much stronger exchange interactions in Mo^III–CN–V^II cyano-bridged pairs, i.e., J > 200 cm⁻¹ for V^II [57,58,59] vs. J ~ 40 cm⁻¹ for Mn^II [55,56]. It is important to emphasize that this system fully meets the specific requirements for orbitally-degenerate complexes, which are necessary for obtaining a high barrier. In this respect, it is relevant to note that anisotropic exchange interactions were also observed in other heterometallic cyano-bridged transition-metal complexes [53,54]; among 3d-based cyano-bridged complexes, anisotropic exchange was first reported in a {LCu^II–NC–Fe^III(CN)₅} ferromagnetically coupled cluster [60]. However, as is shown below, most of the d-complexes featuring anisotropic exchange are of little interest in increasing SMM characteristics, either due to small exchange parameters or because of low symmetry of the anisotropic spin Hamiltonian. As for lanthanide-based SMMs, 4f-3d anisotropic exchange contributions to the U_eff barrier and hysteresis have recently been shown to be important in heterometallic {Cr^III₂Dy^III₂} SMM clusters [61,62]. At the same time, it is clear that lanthanide ions are generally not good candidates for the efficient implementation of the claimed strategy owing to inherently weak exchange interactions resulting from the core-like nature of 4f electrons.

The aim of this work is to establish main regularities in the variation of the spin-reversal barrier U_eff in heterometallic Mo^III–V^II molecular spin clusters, depending on their size, composition and topology of anisotropic and isotropic exchange linkages. It is of special interest to explore the interplay between anisotropic exchange interactions in the apical Mo^III–CN–V^II pairs and isotropic exchange interactions in the equatorial pairs and to analyze their impact on the overall magnetic anisotropy and barrier. Some specific approaches to engineering high energy barrier in terms of anisotropic exchange interactions are discussed.

2. Results and Discussion

2.1. Electronic Structure of [Mo^III(CN)₇]⁴⁻ Complex

This section discusses the basic mechanism of anisotropic exchange interactions between [Mo^III(CN)₇]⁴⁻ complexes and V^II ions. In fact, this mechanism is similar to that considered earlier for the spin coupling between [Mo^III(CN)₇]⁴⁻ and Mn^II [47] complexes, since in both cases strong exchange anisotropy in the Mo^III–CN–M(3d) pairs is caused by the unquenched angular orbital angular momentum of the PBP complex [Mo^III(CN)₇]⁴⁻. Although the electronic structure and magnetic properties of [Mo^III(CN)₇]⁴⁻ were previously discussed in detail in Ref. [45,47], it is still appropriate to provide here an overview of its main characteristics, which are essential for understanding specific details of the origin of strong exchange anisotropy. In the PBP ligand surrounding, the two lowest degenerate orbitals 4d_xz and 4d_yz are populated by three electrons to produce an orbital doublet ²Φ_xz,yz = ²Φ(M_L = ±1), whose two components correspond to the M_L = ±1 projection of the unquenched angular orbital momentum L on the polar z-axis of the [Mo(CN)₇]⁴⁻ bipyramid. The ²Φ(M_L = ±1) wave functions are represented by two degenerate electronic configurations (d₊₁)²(d₋₁)¹ and (d₋₁)²(d₊₁)¹ with d_±1 = (d_xz ± id_yz)/√2 being complex d orbitals with definite projection of the orbital momentum (m_l = ±1) on the polar axis of the pentagonal bipyramid (Figure 2). Alternatively, the real wave functions are expressed by ²Φ_xz = (d_yz)²(d_xz)¹ and ²Φ_yz = (d_xz)²(d_yz)¹. The orbital doublet ²Φ undergoes spin-orbit splitting (ζ_MoLS) into the ground φ(±1/2) and excited χ(±1/2) Kramers doublets (Figure 2). In the strong pentagonal-bipyramidal ligand field, the φ(±1/2) wave functions are well described by the single-determinant electronic configurations (d₊₁)²(d₋₁)↑ and (d₋₁)²(d₊₁)↓ with unquenched orbital momentum (L_z = ±1, Figure 2) [45,47]. As a result, [Mo(CN)₇]⁴⁻ complex exhibits highly anisotropic g-tensor (such as g_z = 3.89 and g_x = g_y = 1.77 [63].

In distorted [Mo^III(CN)₇]⁴⁻ complexes, the orbital momentum is reduced or quenched due to low-symmetry ligand field splitting δ of the ground orbital doublet ²Φ_±. However, weak or moderate low-symmetry distortions (when δ < ζ_Mo) do not significantly change the overall picture, leaving the orbital angular momentum unquenched. Moreover, these low-symmetry distortions of the [Mo(CN)₇]⁴⁻ bipyramid do not affect the axial symmetry of the anisotropic exchange Hamiltonian H_eff = −J_xy(S_Mo^xS_V^x + S_Mo^yS_V^y) − J_zS_Mo^zS_V^z (see below).

2.2. Anisotropic Spin Coupling Mo^III–CN–V^II

Analysis of anisotropic exchange interactions between [Mo^III(CN)₇]⁴⁻ complexes and V^II ions basically follows the theoretical model developed for Mo^III–CN–Mn^II exchange coupled pairs in Mo^IIIMn^II₂ SMM [47]. With the spin-orbit coupling (SOC) on Mo^III switched off (ζ_Mo = 0), the exchange interaction between Mo^III and V^II is described by an isotropic orbitally dependent spin Hamiltonian H_orb = A + RS_MoS_V, which acts in the space of the ²Φ_±(Mo) × ⁴A₂(V) wave functions of the Mo^III–CN–V^II pair (with × being the anti-symmetrized product and ⁴A₂(V) the ground state of high-spin V^II ion). Here, S_Mo and S_V are spin operators of Mo^III and V^II, while A and R are, respectively, spin-independent and spin-dependent orbital operators acting on the orbital variables only. In general case, the A and R operators are written as a Hermitian 2 × 2 matrices; it is noteworthy that the spin-dependent orbital R matrix is diagonalized by a rotation of the coordinate frame around the polar z-axis [47]. Therefore, without loss of generality, H_orb can be written as

$$H_{\mathrm{orb}}=\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)-\left(\begin{array}{cc}{J}_1& 0\\ 0& J_2\end{array}\right){\mathit{S}}_{\mathrm{M}\mathrm{o}}{\mathit{S}}_{\mathrm{V}},$$

(1)

where J₁ and J₂ are orbital exchange parameters referring to the real wave functions ²Φ_xz = (d_yz)²(d_xz)¹ and ²Φ_yz = (d_xz)²(d_yz)¹. Then, when the SOC on Mo^III is turned on, the isotropic orbitally dependent operator H_orb in Equation (1) transforms into an anisotropic spin Hamiltonian H_eff that describes effective spin coupling between the ground Kramers doublet φ(±1/2) and the true spin S = 3/2 of V^II ions. The spin Hamiltonian H_eff is obtained by projection of the full Hamiltonian A + RS_MoS_V + ζ_MoL_MoS_Mo acting in the space of the ²Φ_±(Mo) × ⁴A₂(V) wave functions onto the restricted space of the φ(m) × ⁴A₂(V) wave functions (where m = ±1/2 is the projection of the fiction spin S = 1/2 of the ground Kramers doublet φ(±1/2)). At this point it is important to emphasize a remarkable property of the pentagonal bipyramidal complex [Mo^III(CN)₇]⁴⁻ that the SOC operator ζ_4dL_MoS_Mo is diagonal within the space of the ²Φ_±(Mo) × ⁴A₂(V) wave functions since it does not mix the two states ²Φ(M_L = +1) and ²Φ(M_L = −1) of Mo^III; thus the ζ_4dL_MoS_Mo operator can be replaces by its z-component ζ_4dL^z_MoS^z_Mo. This implies that the full spin Hamiltonian A + RS_MoS_V + ζ_4dL^z_MoS^z_Mo of the Mo^III–CN–V^II pair commutes with the operator S^z_Mo + S^z_V of the total spin projection on the pentagonal z axis of [Mo^III(CN)₇]⁴⁻. Thus, the projection of the total spin M_S(Mo) + M_S(V) of the Mo^III–CN–V^II pair on the z axis is a good quantum number, while the total spin S_Mo + S_V is not such due to strong first-order spin-orbit coupling on Mo^III. Therefore, regardless of the actual symmetry of the Mo^III–CN–V^II pair (which is generally low), the effective anisotropic spin Hamiltonian H_eff always exhibit uniaxial symmetry, i.e., −J_xy(S_Mo^xS_V^x + S_Mo^yS_V^y) − J_zS_Mo^zS_V^z (see also ref. [47] for more detail).

For the regular (D_5h) or moderately distorted structure of [Mo^III(CN)₇]⁴⁻, H_eff is calculated by equating the matrix elements of H_eff and A + RS_MoS_V in the space of wave functions |m, M_S> = φ(m) × |S_V,M_S>, <m, M_S|H_eff|m′, M_S′> = <m, M_S|A + RS_MoS_V|m′, M_S′>. This results in

$$H_{\mathrm{eff}}=C-J_{xy}(S_{\mathrm{Mo}}^x{S}_{\mathrm{V}}^x+S_{\mathrm{Mo}}^y{S}_{\mathrm{V}}^y)-J_z{S}_{\mathrm{Mo}}^z{S}_{\mathrm{V}}^z,$$

(2)

where J_z = (J₁ + J₂)/2, J_xy = (J₁ − J₂)/2, and |J₁| ≥ |J₂|; here, C = (A₁₁ + A₂₂)/2 is a constant stemming from the spin-independent orbital operator A in Equation (1); it can be neglected in further calculations. It is also important to note that the uniaxial symmetry of H_eff retains even for moderately distorted PBP [Mo^III(CN)₇]⁴⁻ complexes, which frequently occur in many [Mo^III(CN)₇]⁴⁻ based molecular magnets [64,65,66]. Indeed, the SOC operator remains diagonal (ζ_4dL^z_MoS^z_Mo) in the space of wave functions ²Φ_±(Mo) × ⁴A₂(V), provided that the distortions of the PBP structure of [Mo^III(CN)₇]⁴⁻ slightly admix its high-lying excited 4d³ states with M_L = ±2 and M_L = 0 to the ground state ²Φ(M_L = ±1); in fact, due to large energy gap (~20,000 cm⁻¹ >> δ, Figure 1), this holds true even for pronounced departures from the strict (D_5h) PBP geometry of the [Mo^III(CN)₇]⁴⁻ complex. At the same time, without violating the uniaxial character of the spin Hamiltonian H_eff in Equation (2), distortions tend to quench the orbital momentum of Mo^III and reduce the relative anisotropy of the −J_xy(S_Mo^xS_V^x + S_Mo^yS_V^y) − J_zS_Mo^zS_V^z spin Hamiltonian (which is measured by the J_z/J_xy ratio), especially when δ > ζ_Mo.

Thus, the uniaxial symmetry of the anisotropic exchange Hamiltonian H_eff is an intrinsic property of [Mo^III(CN)₇]⁴⁻ complexes, which is extremely useful for creating the uniaxial symmetry of the magnetic anisotropy of SMM clusters (D < 0, E = 0).

2.3. Estimate of Anisotropic Exchange Parameters

It is important to evaluate anisotropic exchange parameters J_z and J_xy in Mo^III–CN–V^II cyano-bridged pairs, especially given their strong impact on the barrier, U_eff ~ |J_z − J_xy| (see Figure 1) [47]. The Mo^III ion has diffuse and high-energy 4d magnetic orbitals, which may provide strong exchange interactions with high-spin 3d metal ions [54]. Especially intense exchange interactions are expected between [Mo^III(CN)₇]⁴⁻ complexes and vanadium(II) ions, which, similarly to Mo^III ions, also tend to form strong exchange interactions due to more diffuse 3d orbitals of early transition metal ions [67]. However, data on the exchange parameters in [Mo^III(CN)₇]⁴⁻–V^II based molecular magnets are scarce or absent [68]. The magnitude of exchange interactions in cyano-bridged pairs Mo^III–CN–M(3d) can be estimated from available experimental data for the related hexacyano complex [Mo^III(CN)₆]³⁻, J = −122 cm⁻¹ [57] and J = −228 cm⁻¹ [58] for Mo^III–CN–V^II cyano-bridged pairs; very large exchange parameters for the Mo^III–CN–V^II linkages in a Prussian blue type structure were also predicted from a broken symmetry DFT study (J ≈ 350 cm⁻¹) [59]. It is noteworthy, however, that these exchange parameters refer to the spin coupling between high-spin octahedral [Mo^III(CN)₆]³⁻ complex (S_Mo = 3/2) and high-spin V^II ions (S_V = 3/2). For the Mo^III–CN–V^II pairs involving low-spin [Mo^III(CN)₇]⁴⁻ PBP complex (S = 1/2), these exchange parameters should be rescaled for a smaller spin value. Approximately, it can be done in terms of the t and U parameters of the superexchange theory, which are, respectively, electron transfer parameter between magnetic t₂-orbitals of Mo and V and metal-to-metal charge-transfer energy. With two active antiferromagnetic pathways 4d_xz–CN–3d_xz and 4d_yz–CN–3d_yz for the high-spin ground state ⁴A₂ of [Mo^III(CN)₆]³⁻, this gives J = −8t²/9U for the high-spin [Mo^III(CN)₆]³⁻ complex; given that J = −228 cm⁻¹ [58], we have t²/U = 256.5 cm⁻¹, which corresponds to the electron transfer parameter of t ~ 3500 cm⁻¹ at a typical charge-transfer energy of U = 50,000 cm⁻¹ (6 eV). In the low-spin [Mo^III(CN)₇]⁴⁻ PBP complex, only one active superexchange pathway lefts for each of two orbital states ²Φ_xz and ²Φ_yz (4d_xz–CN–3d_xz and 4d_yz–CN–3d_yz, respectively), which results in two equal orbital exchange parameters J₁ = J₂ = −4t²/3U for the linear apical Mo^III–CN–V^II pairs and J₁ = 4t²/3U, J₂ ~ 0 for the equatorial pairs (see ref. [47] for more detail). Therefore, in the linear Mo^III–CN–V^II pair, the orbital exchange parameters for the low-spin Mo^III ion are approximately 1.5 times larger than that for the high-spin Mo^III, i.e., 342 cm⁻¹ vs. 228 cm⁻¹. With the equations for the anisotropic exchange parameters, J_z = (J₁ + J₂)/2 and J_xy = (J₁ − J₂)/2, we have antiferromagnetic Ising spin coupling for the apical pair, J_z = 342 cm⁻¹, J_xy = 0, and isotropic spin coupling J_z = J_xy = −171 cm⁻¹ for the equatorial pairs. However, in reality, the pairs are distorted and thus generally J₁ is not equal to J₂ in the apical pairs [45,47]. In further model calculations, we can conventionally assume J₁ = 350, J₂ = 250 cm⁻¹ for the orbital exchange parameters in distorted apical pairs Mo^III–CN–V^II, which corresponds to the anisotropic exchange parameters J_z = −300 and J_xy = −50 cm⁻¹ in Equation (2). Similarly, for equatorial pairs, the isotropic exchange parameter (J = J_z = J_xy) can be roughly rounded to −150 cm⁻¹. More quantitative calculations of the exchange parameters in [Mo^III(CN)₇]⁴⁻ based systems are beyond the scope of this article; they will the subject of a separate study.

2.4. Spin Energy Spectra of Mo^III_kV^II_m Clusters: Engineering of High Energy Barrier

This Section presents results of a comparative study of spin energy diagrams (E vs. M_S) for polynuclear clusters Mo^III_kV^II_m (k = 1–4, m = 1–5) with highly anisotropic magnetic interactions. These clusters are composed of alternating cyano-bridged [Mo^III(CN)₇]⁴⁻ and V(II) complexes (Figure 3, Figure 4, Figure 5 and Figure 6) and involve variable number of apical and equatorial Mo^III–CN–V^II groups featuring, respectively, anisotropic (Ising-type) and isotropic exchange interactions. This study aimed to investigate the influence of the composition, structure and topology of Mo^III_kV^II_m clusters on the spin energy diagrams and, especially, on the height of the barrier U_eff. In this respect, it is especially interesting to investigate the dependence of the barrier on the number n of apical groups with Ising interactions, which are the main source of molecular magnetic anisotropy in Mo^III_kV^II_m clusters. For this purpose, the spin energy spectra of the Mo^III_kV^II_m clusters are calculated in terms of a spin Hamiltonian

$$\widehat{H}=-{\displaystyle \sum _{<ij>}\left(J_{xy}(ij)(S_{\mathrm{M}\mathrm{o}(i)}^x{S}_{\mathrm{V}(j)}^x+S_{\mathrm{M}\mathrm{o}(i)}^y{S}_{\mathrm{V}(j)}^y)+J_z(ij)S_{\mathrm{M}\mathrm{o}(i)}^z{S}_{\mathrm{V}(j)}^z\right)},$$

(3)

where the sum <ij> runs over all Mo(i)–CN–V(j) pairs in the cluster; note that, since the Mo–Mo and V–V exchange interactions are neglected, here i and j indexes numerate, respectively, Mo and V centers. According to the estimates in Section 2.3, the exchange parameters are set to J_xy = −50 and J_z = −300 cm⁻¹ for apical pairs and to J_xy = J_z = −150 cm⁻¹ for equatorial pairs.

Primary calculations were performed for Mo^IIIV^II_m clusters composed of the single central [Mo^III(CN)₇]⁴⁻ complex and several V^II complexes attached in the apical and equatorial positions (Figure 3 and Figure 4). It is important to note that in these clusters the projection of the total spin M_S onto the z-axis (which is parallel to the pentagonal axis of the [Mo^III(CN)₇]⁴⁻ complex) is a good quantum number due to uniaxial symmetry of the spin Hamiltonian (3), which commutes with the S^z operator of the total spin. As a result, each spin quantum state has a definite M_S spin projection on the polar z-axis of [Mo^III(CN)₇]⁴⁻ and thus spin energy spectra can be visualized in terms of the E vs. M_S diagrams. Figure 3 shows spin energy diagrams calculated for Mo^IIIV^II_m clusters with one apical pair and progressively increasing number of equatorial pairs (m = 1–5).

In all cases (Figure 3a–d), the spin energy diagrams show a butterfly-shaped figure with a double-well profile for the lower part of the energy spectrum with doubly degenerate ground energy level represented by two spin quantum states M_S = +S and M_S = −S, which are separated by the barrier U_eff. Note that the spin energy diagrams of Mo^IIIV^II_m clusters differ considerably from the conventional double-well pattern DS² in ordinary SMMs. In this case, similar to the parabolic energy profile DS², the value of the barrier U_eff can be associated with the energy position of the lowest spin state with the smallest spin projection (i.e., M_S = 0 or M_S = ±1/2 for clusters with integer and half-integer spin, respectively, Figure 3 and Figure 4). There is an interesting feature that in the Mo^IIIV^II_m clusters the barrier U_eff is rather insensitive to the number m of the equatorial Mo–CN–V pairs (with some exceptions for the smallest cluster Mo^IIIV^II₂, Figure 3a). This can be explained by the fact that there is the only (constant) source of the magnetic anisotropy coming from the single apical pair, while addition of the equatorial pairs with isotropic exchange interactions add nothing to the overall magnetic anisotropy D and to the barrier U_eff. On the other hand, albeit the barrier varies only slightly with the increasing number of equatorial Mo^III–CN–V^II pairs, the ground-state spin S increases significantly, which improves the overall SMM performance due to larger separation between the two lowest spin states |M_S = ±S> in the M_S scale.

Addition of the second apical group in the Mo^IIIV^II_m clusters increases the barrier approximately twice (from ca. 150 to 300 cm⁻¹, Figure 4a–d), but the overall picture remains the same: the E vs. M_S diagrams retain a butterfly-shaped pattern with a double-well potential, the barrier U_eff varies rather little with increasing number m of the equatorial groups and the ground-state spin M_S increases considerably (from 5/2 to 7, Figure 4). This suggests that each apical group Mo^III–CN–V^II contributes additively to the barrier U_eff.

Next, important information on the variation of the spin energy diagrams and barrier U_eff on the structure of the Mo^III_kV^II_m clusters was obtained from comparative calculations for the series of Mo^III₂V^II₄ clusters with the central Mo^III₂V^II₂ square and two side V^II complexes connected via various apical and equatorial positions to the [Mo^III(CN)₇]⁴⁻ complexes (Figure 5). In these calculations, an idealized structure of Mo^III_kV^II_m clusters is adopted with the parallel orientation of the local pentagonal axes of two [Mo^III(CN)₇]⁴⁻ complexes to ensure uniaxial symmetry of the total spin Hamiltonian in Equation (3). In this case, z-projection M_S of the total spin is good quantum number, so each quantum spin state has a definite value of M_S.

These calculations show that all the Mo^III₂V^II₄ clusters have a double-well potential with the M_S = ±5 ground spin state and a large barrier U_eff, (up to ~500 cm⁻¹, Figure 5b) which correlates directly with the number n of apical groups; in fact, U_eff is roughly proportions to n (Figure 5). Similar regularities were also found for larger Mo^III₄V^II₄ clusters with cubic and ladder structures (Figure 6). Thus, in the ladder cluster Mo^III₄V^II₄ with six apical Mo–CN–V groups (n = 6), the barrier reaches a value of U_eff = 741 cm⁻¹. Consequently, these results indicate the possibility of further increase of the barrier in larger Mo^III_kV^II_m clusters with alternating [Mo^III(CN)₇]⁴⁻ and V^II units, in which the number n of apical groups may be considerably larger.

To establish a more quantitative correlation between U_eff and anisotropic exchange parameters, numerous calculations were performed for various Mo^III_kV^II_m clusters with variable exchange parameters J_z and J_xy for the apical and equatorial Mo^III–CN–V^II groups. Analysis of these results has revealed that the barrier is approximately given by U_eff ~ 0.5|J_z − J_xy|n, provided that the isotropic exchange interactions (i.e., with J_z = J_xy) for the equatorial pairs are strong enough as compared to the anisotropic Ising-type exchange interactions in the apical pairs (J_z > J_xy). The meaning of this condition is that the isotropic exchange interactions of equatorial pairs integrate the local magnetic anisotropies of the apical pairs into the overall magnetic anisotropy of the Mo^III_kV^II_m cluster, and thus they must be strong enough to carry out this function. Fortunately, for the specific cyano-bridged Mo^III_kV^II_m clusters, this condition is well satisfied due to the relation between anisotropic exchange parameters (J_z, J_xy) and orbital exchange parameters J₁, J₂ involved in the orbitally-dependent spin Hamiltonian (2), J_z = (J₁ + J₂)/2 and J_xy = (J₁ − J₂)/2.

These data indicate the possibility of the direct scaling of the barrier U_eff in molecular cyano-bridged clusters based on alternating [Mo^III(CN)₇]⁴⁻ and V^II complexes by means of increasing the cluster size and proper organization of its structure and composition. In principle, from the ratio U_eff ~ 0.5|J_z − J_xy|n with |J_z − J_xy| = 200–300 cm⁻¹, one can expect to obtain a barrier above 1000 cm⁻¹ in large Mo^III_kV^II_m clusters. The main conditions for obtaining a high barrier are a large number n of apical groups and large absolute values of the anisotropic exchange parameters J_z and J_xy. However, it is important to note that, in the large polynuclear Mo^III_kV^II_m clusters with a fixed number of apical groups n, the maximum barrier U_eff could be obtained only if there are a sufficient number of equatorial groups in the cluster. Thus, the equatorial Mo^III–CN–V^II groups carry out an important function in the formation of a high barrier: although their isotropic exchange interactions do not directly contribute to the magnetic anisotropy of the Mo^III_kV^II_m cluster, they ensure the overall magnetic connectivity of the molecular spin cluster and thus help to convert the local Ising-type exchange anisotropy −J_xy(S_i^xS_j^x + S_i^yS_j^y) − J_zS_i^zS_j^z of individual apical pairs Mo^III–CN–V^II into the value of the barrier U_eff.

A unique feature of the PBP [Mo^III(CN)₇]⁴⁻ complexes as molecular building blocks lies in the fact that their uniaxial anisotropic exchange interactions −J_xy(S_i^xS_j^x + S_i^yS_j^y) − J_zS_i^zS_j^z automatically create a molecular magnetic anisotropy D with a purely axial symmetry, without undesirable transverse component (E = 0), which is very important for obtaining a high barrier U_eff. In other words, the symmetry of the molecular magnetic anisotropy is higher than the geometric symmetry of the molecule. This simplifies drastically the problem of molecular design of nanomagnets, since it eliminates the need to obtain the axial geometric symmetry of a SMM cluster during its chemical synthesis.

However, it is important to note that this study does not fully address the problem of designing high-temperature SMMs—its goal is to demonstrate theoretically a great potential of the new approach in engineering high energy barriers U_eff, which exploits highly anisotropic exchange interactions of PBP [Mo^III(CN)₇]⁴⁻ complexes. Of course, many complicated features of these systems (such as distortions, non-collinear orientation of magnetic axes, specific molecular building blocks and structures, synthetic approaches, etc.) remain so far beyond the limits of this theoretical model. Thus, the paper discusses some new general principles for constructing a high barrier rather than specific recommendations for the synthesis of advanced SMM. Practical development of this strategy will be based on new PBP 4d and 5d complexes with unquenched orbital momentum (instead of [Mo^III(CN)₇]⁴⁻ and [Re^IV(CN)₇]³⁻ heptacyanide complexes), in which the pentagonal coordination of the metal ion in the equatorial plane is enforced by a five-membered chelate ring of the planar pentadentate macrocyclic ligands (see our recent paper [69]); this work is underway.

3. Method and Computation Details

Spin energy spectra of Mo^III_kV^II_m clusters were calculated using specially designed computational approaches and routines based of anisotropic spin Hamiltonian in Equation (3). More specifically, a Fortran routine first calculates the set of matrix elements of the spin exchange Hamiltonian (3) in the full basis set of multicenter spin functions generated by the product of single-ion spin functions of Mo(III) and V(II) ions involved in the Mo_kV_m cluster. Next, the energies of spin states and eigenvectors are obtained by diagonalization of the full matrix of the spin Hamiltonian (3); then, the eigenvectors are sorted according their E energies and M_S spin projections to visualize the spin energy diagrams shown in Figure 3, Figure 4, Figure 5 and Figure 6.

4. Conclusions

In summary, anisotropic Ising-type exchange infractions −J_xy(S_i^xS_j^x + S_i^yS_j^y) − J_zS_i^zS_j^z with large exchange parameters J_z and J_xy generated by PBP [Mo^III(CN)₇]⁴⁻ complexes and V^II ions connected via apical positions represent a very efficient tool for constructing high-performance SMMs. It is shown that in mixed 4d–3d heterometallic SMM clusters composed of alternating [Mo^III(CN)₇]⁴⁻ and V^II complexes the spin-reversal barrier is controlled by anisotropic exchange parameters and the number n of apical Mo^III–CN–V^II groups, U_eff ~ 0.5|J_z − J_xy|n. This finding provides a very efficient straightforward strategy for scaling U_eff and T_B values by means of enhancing exchange parameters J and J_xy and increasing the number of PBP [Mo^III(CN)₇]⁴⁻ complexes in a SMM molecule. Some ideas on molecular engineering high energy barrier U_eff and further developments toward high-T SMMs are discussed, which are illustrated in Figure 3, Figure 4, Figure 5 and Figure 6. Practical implementation of this approach will be focused on a new family of PBP 4d and 5d complexes with pentadentate macrocyclic ligands [69].