Dipolar Coupling as a Mechanism for Fine Control of Magnetic States in ErCOT-Alkyl Molecular Magnets

Abstract

The design of molecular magnets has progressed greatly by taking advantage of the ability to impart successive perturbations and control vibronic transitions in 4fⁿ systems through the careful manipulation of the crystal field. Herein, we control the orientation and rigidity of two dinuclear ErCOT-based molecular magnets: the inversion-symmetric bridged [ErCOT(μ-Me)(THF)]₂ (2) and the nearly linear Li[(ErCOT)₂(μ-Me)₃] (3). The conserved anisotropy of the ErCOT synthetic unit facilitates the direction of the arrangement of its magnetic anisotropy for the purposes of generating controlled internal magnetic fields, improving control of the energetics and transition probabilities of the electronic angular momentum states with exchange biasing via dipolar coupling. This control is evidenced through the introduction of a second thermal barrier to relaxation operant at low temperatures that is twice as large in 3 as in 2. This barrier acts to suppress through-barrier relaxation by protecting the ground state from interacting with stray local fields while operating at an energy scale an order of magnitude smaller than the crystal field term. These properties are highlighted when contrasted against the mononuclear structure ErCOT(Bn)(THF)₂ (1), in which quantum tunneling of the magnetization processes dominate, as demonstrated by magnetometry and ab initio computational methods. Furthermore, far-infrared magnetospectroscopy measurements reveal that the increased rigidity imparted by successive removal of solvent ligands when adding bridging methyl groups, along with the increased excited state purity, severely limits local spin–vibrational interactions that facilitate magnetic relaxation, manifesting as longer relaxation times in 3 relative to those in 2 as temperature is increased.

Introduction

The synthesis and study of electron spin materials at the molecular level, or molecular magnetism, is a synthetically driven approach to the daunting goal of bringing comprehensive structure–function design principles to atomic level magnetic interactions. Two milestones in this field, both arising from the analysis of a molecular metal oxide cluster Mn₁₂,¹ were the characterization of an energy barrier to spin relaxation for a 0D system² and the manifestation of the resulting macroscopic quantum effect known as quantum tunneling of magnetization (QTM).³⁻⁵ The study and control of QTM, in particular, have taken on a renewed significance with growing efforts to understand and design systems for initialization, evolution, and readout of quantum coherent and spin polarized states at the molecular level. Indeed, the promise of custom quantum spin interactions and molecular level magnet-like behavior has generated an entirely new and fruitful branch of molecular magnetism focused on discrete molecular systems known as single-molecule magnets (SMMs).⁶⁻¹³

Great strides have been made in the field of producing exceptional SMMs with individual spin centers, often denoted as single-ion magnets (SIMs) to denote their propensity to generate giant anisotropies through mechanisms inherent to the spin orbit interaction of an open shell ion with its local environment. Most often derived from transition metal or lanthanide centers, SIMs have generated drastic and continuous enhancement in mimicking bulk-magnet-like behaviors through careful design of their coordination environments.¹⁴ Continued development has elucidated the importance of using highly symmetric ligand scaffolds^13,15−17 or synthetically controlling static^{15,16,18−23} and dynamic²⁴⁻²⁶ elements of the crystal field environment. Such design principles focus on two main parameters: the control of QTM, which is principally dependent on the probability of transitions between states of opposite moment, as determined by the degree of mixing, and thermal excitation via direct mechanisms, which depends on both the separation between magnetic states and the availability of vibrational modes coupled to these transitions.

While SIMs have pushed the boundaries of controlling spin–lattice relaxation out to technologically relevant temperatures, many quantum technologies require arrays of multiple coupled spins;²⁷⁻³⁰ such coupling strategies work to enhance SMM relaxation behavior by affecting QTM at zero and applied fields as well.^31,32 Several examples exist of molecular systems with exceptional hysteretic behavior and an effective quenching of QTM by coupling highly anisotropic magnetic states, either directly or through a molecular bridge bearing spin density, to yield highly anisotropic “giant spins” akin to the clusters that established SMMs.³³⁻³⁹ One route to building higher nuclearity molecular magnets is that of the “building block” approach, wherein several magnetic units with well-defined and synthetically robust ground states are linked to build systems with conserved anisotropy.⁴⁰⁻⁴⁴ However, compatibility with open-shell bridges is difficult due to the charge-dense and highly penetrating radicals either destroying the anisotropy enforced by the comparatively weak crystal field or introducing large coupling so that magnetic energy levels become close and can easily mix.

One alternative is the exchange biasing imposed by neighboring spins – the dipolar interaction. This can be thought of as a local magnetic field perturbation generated by the anisotropic SIM dipole on its neighbors, effectively coupling populated magnetic states.⁴⁵⁻⁴⁷ Dipolar coupling is well understood within condensed matter physics,^14,48−54 with both short- and long-range interactions described for macroscopic spin dynamics, magnetic domain formation, and phase transitions.⁵⁵⁻⁶¹ While such behavior has been noted in weakly coupled SMM clusters exhibiting relaxation behavior far slower than the local symmetry would seem to justify⁶²⁻⁶⁹ and cited as a point of study in frustrated spin systems⁷⁰ and as a possible mechanism for entangling qubits,^30,71 it is more often in competition with other exchange mechanisms and serves to hinder desired properties.^66,72−74 Recently, we have proposed a perspective that such an interaction can instead behave as an advantageous extension of the current SMM design.⁷⁵ In the common heuristic of the 4f electronic structure involving successive perturbations (Figure 1), the dipolar interaction is a reliable method to provide the missing fine-tuning control of the wave function – weak enough to leave the crystal field approximation intact yet strong enough to generate well-defined restrictions on the wave function mixing that can almost completely reshape magnetic transitions in the quantum regime.^32,72

Figure 1.

Illustrative progression of molecular magnetic perturbations. (Left) Spin–orbit coupling is principally governed by the choice in lanthanide. (Middle) Splitting of spin–orbit states into m_J levels is determined by the crystal field strength as enforced by the local ligand environment (r⃗_lig, θ_lig). (Right) Energy and moment of dipolar coupled states depend on the intermetal interactions as described by r⃗_inter and θ_inter.

Herein, we take advantage of the molecular anisotropic building unit ErCOT (COT = cyclooctatetraenide dianion, COT^2–), a coordinatively unsaturated 4f metal site with an isolated, robust M_J = 15/2 ground state and magnetic anisotropy that is well described by structural parameters.⁷⁵ A series of three alkyl-bound ErCOT complexes (Scheme 1 and Figure 2) highlights the degree of control that the dipolar interaction offers over coupling between isolated spin–orbit coupled states. A mononuclear species serves as a control, describing a system subject only to spin dynamics arising from its single-ion state structure and its interactions with the spin bath as a whole. By introduction of small, relatively rigid methyl ligands, two dinuclear species are isolated, clearly exhibiting ferromagnetic dipolar coupling that acts as a first-order perturbation upon the single-ion states. By deliberately directing the structurally pinned magnetic anisotropy axis at each center, we control the coupled energy landscape responsible for through-barrier relaxation processes operant at low temperatures, as supported by ab initio calculations and a combination of static and dynamic magnetometry. Such control illustrates the accessibility and pronounced results of manipulating this small internuclear perturbation while conserving the energy landscape provided by the crystal field. Furthermore, the introduction of rigid methyl groups displaces solvent ligands with high vibrational degrees of freedom and increases the symmetry in half of the coordination sphere of each metal. Using far-infrared magnetospectroscopy (FIRMS), we investigate the origin and efficiency of vibronic coupling responsible for the high-temperature relaxation processes of note.

Scheme 1. Synthetic Overview for 1, 2, and 3.

Figure 2.

Crystallographically determined structures of 1 (left), 2 (middle), and 3 (right) with ab initio calculated ground-state single-ion (blue) and dipolar-coupled magnetic easy axes (orange). Nonalkyl hydrogens and outer-sphere cations are omitted for clarity. Alkyl hydrogens were located explicitly for 1 and 2; hydrogens positions on bridging methyls in 3 were optimized in ORCA.

Results and Discussion

Synthesis and Structure

All reported complexes were synthesized using standard air- and water-free synthetic techniques (Section S1). Three complexes were isolated: one mononuclear complex ErCOT(Bn)(THF)₂ (1) and two dinuclear complexes: [ErCOT(Me)(THF)]₂ (2) and Li[(ErCOT)₂(Me)₃] (3) (Bn = benzyl anion, ^–CH₂C₆H₅; Me = methyl anion, ^–CH₃; and THF = tetrahydrofuran) (Scheme 1). All were obtained through salt metathesis via the addition of an alkali metal salt of the appropriate anion to ErCOT(I)(THF)₂.

Solid-state molecular structures of all compounds were determined by single crystal X-ray diffraction with bulk purity supported by powder X-ray diffraction (Figure 2, Table 1, and Section S2) and a combination of NMR and Fourier transform infrared spectroscopy (Sections S3 and S4, respectively). Metal–carbon bond lengths agree well with previously reported lanthanide alkyl structures.⁷⁶⁻⁹³ Mononuclear 1 features the piano–stool geometry characteristic of the ErCOT moiety with the benzyl anion and THF ligands bound opposite to the COT^2– ring, giving an approximate C_s symmetry. Notably, the benzyl anion is monohaptic, as while the η¹ interaction is well-known,^{76,83,88,90,94−105} lanthanide–benzyl complexes more commonly adopt an η² side-on interaction with the sp² hybridized C–CH₂ bond.^{76,83,88,93,95,96,106−118} The orientation of the benzyl unit can be modified by changing the cosolvent, breaking this symmetry (Figure S4). However, only the structure featured in Figure 2 is further discussed herein. Complex 2 is an inversion symmetric combination of two pseudo-C_s symmetric units — a common dinuclear motif for LnCOT complexes.^119−130 Complex 3 has three identical bridging ligands and is most precisely C_s symmetric due to the incompatibility of the C₈ rotational axis of COT and the approximate C₃ axis between the three bridging methyl ligands. Both 2 and 3 have small internuclear distances [r_Er–Er = 3.5144(5) and 3.2393(9) Å, respectively], with 3 being the third shortest among discrete molecular erbium clusters after a methyl-bridged trinuclear cluster¹³¹ and a pentadienyl-scaffolded dimer.¹³² The r⃗_ErCOT vectors in 2 are strictly parallel, as enforced by crystallographic symmetry, but they are oriented off-axis from the internuclear vector r⃗_int by 22.90(7)°. In 3, however, all three vectors are nearly colinear, deviating by only 0.4–2.3°. As previously demonstrated,⁷⁵ the relative strength of ErCOT dipolar interactions can be predicted by assuming that each SIM moment is pinned to its associated r⃗_ErCOT vector and affecting a dipolar perturbation on the crystal field Hamiltonian. From this model, the parallel noncolinear arrangement (2) and parallel colinear arrangement (3) represent important contrasting experimental cases for understanding strongly anisotropic Ising coupling in the finite case.

Table 1. Selected Distances and Angles for 1, 2, and 3.

	1	2	3a
|r⃗Er–Er|	N/A	3.5144(5) Å	3.2393(9) Å
|r⃗inter|min	8.3298 Å	7.3910(5) Å	7.3117(14) Å
θErCOT–ErCOT	N/A	180.000(19)°	178.2(3)°
θErCOT–ErEr	N/A	155.47(9)°	179.2(2)°

^aAveraged.

Ab Initio Modeling

To model magnetostructural changes engineered into 2 and 3, magnetic energy manifolds for the entire series 1–3 were predicted and evaluated with the SINGLE_ANISO and POLY_ANISO modules of OpenMolcas (Figure 3 and Table 2).¹³³ Typically, lanthanide-based SMMs can be modeled by using crystallographically determined atom coordinates as inputs without further geometry optimization. This has been found to be especially true where a particular metal–ligand pairing, such as the ErCOT unit, provides the majority of the stabilization energy. The addition of small, charge-dense ligands, however, can interfere with this effect by reducing the symmetry restrictions on spin state changes.¹²⁰ In the case of alkyl ligands, we can expect the orientation of the binding to be particularly influential as the orbitals carrying heavy 2p character impart strong locality and directionality on the electron density.^9,134−137 This electron density can then heavily impact the calculated magnetic anisotropy of the single ion states through increased state mixing, affecting both the predicted inter-Kramers dynamics and the expected coupling between them. While hydrogens were located from the crystallographic electron density map in 1 and 2, the degree of disorder in 3 precluded meaningful assignment of methyl protons outside of an idealized geometry. As the calculated natural orbitals follow from input atom positions, optimized hydrogen positions were calculated with the ORCA computational package.^138,139 For the geometry optimization, open-shelled Er³⁺ was replaced with diamagnetic Y³⁺ due to its nearly identical ionic radius, and non-hydrogen atoms were frozen. Crystal packing is not explicitly considered in the gas-phase calculations, but by freezing the position of non-hydrogen atoms, the bulk of their effect on the conformation is assumed to be accounted for. Importantly, the optimized structure predicted significantly increased energy separations between Kramers doublets as well as decreased state mixing relative to the unoptimized structure. This optimized output was also more consistent with the experimental results and was therefore used in further descriptions of the energy landscape of 3. Optimized geometries were also computed and used for magnetic calculations in 1 and 2, and they were found to be largely consistent with the outputs obtained by using crystallographic positions. Thus, their experimentally determined structures were used.

Figure 3.

Predicted primary relaxation pathways, as described by SINGLE_ANISO and POLY_ANISO. (a) In the high-temperature regime, relaxation is predicted to occur primarily via excitation to the KD₁ manifold for 1–3. In the low-temperature regime, only the KD₀ manifold is significantly populated, leading to QTM in 1. (b,c) Low energy excitations in the coupled manifolds with DD₀ and DD₁ at the energy of KD₀ in 2 (b) and 3 (c) due to the exchange bias from dipolar coupling.

Table 2. Ab Initio Calculated Magnetic Parameters.

	1	2	3a
ΔE, KD0→1	58.5 cm–1	65.5 cm–1	66(3) cm–1
gx, KD0	0.011	0.006	0.007(6)
gy, KD0	0.052	0.026	0.03(2)
gz, KD0	17.610	17.559	17.502(2)
gx, KD1	0.106	12.453	11.6(2)
gy, KD1	1.317	6.077	7.2(2)
gz, KD1	14.595	1.241	1.15(4)
ΔE, DD0→1	N/A	2.055 cm–1	3.904 cm–1
gx, DD0	N/A	0.000	0.000
gy, DD0	N/A	0.000	0.000
gz, DD0	N/A	35.118	34.998
gx, DD1	N/A	0.000	0.000
gy, DD1	N/A	0.000	0.000
gz, DD1	N/A	0.000	0.705

^aStructure is optimized and parameters are averaged over both centers.

For each complex, 1–3, an isolated, crystal field split J = manifold was computed for each unique Er³⁺ center (Section S7.1). For mononuclear 1, though manifolds for both crystal structures have been modeled, only the structure featured in Figure 2 is discussed here. For dinuclear complexes 2 and 3, one metal center was replaced with diamagnetic Y³⁺ to limit the calculation to only one open-shell ion. All Er³⁺ centers displayed Kramers doublets ground states (KD₀) predominantly composed of the M_J = ± eigenstate with principal magnetic axes lying nearly coincident with r⃗_ErCOT (Figure 2). While there is minor variation in the composition of KD₀, the consequences of the alkyl coordination are more easily discerned in the first excited states (KD₁). These excited states lie within an energy regime typical of ErCOT complexes bound to charge-dense ligands.^120,140 The percentage of M_J = ± character in the first excited states increases across the series, culminating in 3, where it comprises nearly the entire excited state composition. Thus, even sans interatomic coupling, the first excitation in 3 represents a uniquely useful magnetic manifold for study: a coupling between the most axial Kramers configuration on the periodic table and the most rhombic one (Figure 3a). Here, thermal relaxation behavior is expected to be limited to transitions between relatively pure KD₀ and KD₁, whereas low-temperature relaxation behavior should be dictated by interactions within the lowest energy level at the energy of KD₀. Such a system presents a straightforward two-level system with a highly anisotropic ground state.

Both 2 and 3 bear closely spaced lanthanide ions with parallelly oriented highly anisotropic ground states. Magnetostructurally, this suggests a significant dipolar interaction. Thus, we consider the interactions between the ground states in 2 and 3 as modeled using the POLY_ANISO package in OpenMolcas (Figure 3b,c and Section S7.2).^141−143 To focus on the most important interactions, dipolar coupling was modeled only intramolecularly between single-ion states of KD₀ origin. Both 2 and 3 are predicted to exhibit a coupling of KD₀ states to give new ferromagnetically coupled dipole doublet ground states (DD₀) and antiferromagnetically coupled first excited states (DD₁), consistent with expectations for an end-to-end orientation of anisotropy axes. The calculated energy separation between these states is 2.055 cm^–1 in 2 and 3.904 cm^–1 in 3, representing an increase by a factor of 1.90. The degree of control afforded by the dipolar interaction is immediately apparent. For example, the interaction is small compared to the energy gap between KD₀ and KD₁, allowing the coupled states to be treated as a first-order perturbation on the single-ion manifold while keeping the overall energy landscape intact. These separations are in good accordance with a simplified dipolar Ising model. When considering the M_J = ± eigenstates to be purely axially anisotropic (g_z only), the dipolar interaction can be described with eq 1.

1

Here, μ₁ and μ₂ are moments of individual magnetic centers, which in pseudospin-1/2 systems are 1/2·g_z·μ_B, and r⃗ represents the internuclear vector. When approximating g_z as coincident with r⃗_ErCOT, the coupled doublets are separated by E_dip = 0.0078·(g_z)² cm^–1 and 0.0128·(g_z)² cm^–1 for 2 and 3, respectively. Given the highly axial ground states of both 2 and 3, g_z can be approximated to scale equally in both, and thus, the ratio of energies arising from a purely dipolar interaction (E_dip,3/E_dip,2) is predicted to be ∼1.6. This conforms to the trends observed in our calculated energy separation between DD₀ and DD₁ from POLY_ANISO calculations (vide supra) and with the fitting of the thermal dependence of ac magnetic susceptibility data (vide infra). Furthermore, the intermolecular distance (as measured by the closest Er–Er contacts) is over double the intramolecular distance between magnetic centers, suggesting an approximate 4- to 10-fold decrease in the strength of the dipole interaction between molecules relative to within them.

The foregoing discussion has developed a working model for the magnetic dipolar manifold that is relevant to the behavior governed by transition probabilities between the calculated eigenstates under a perturbing magnetic field.¹⁴⁴ The temperature and field dependence of the magnetic relaxation behavior for 1–3 is limited to, roughly, the energy separation between KD₀ and KD₁, which is still a useful parameter in 2 and 3 due to the relatively small perturbation imparted by dipolar coupling. This limit is set by the nature of the dipolar splitting, which in 2 and 3 further splits this manifold without increasing the overall energy scale. The dipolar interactions generate a highly directional perturbation that is large on the scale of random field fluctuations, making it a highly effective limiter of random QTM mechanisms. These predictions are borne out in our calculations where over-barrier relaxation through the first excited state should be the favored relaxation pathway. At low temperatures, where the population of optical phonon modes with a sufficient energy to drive Orbach relaxation is dramatically decreased,¹⁴⁵ QTM should be rapid for 1. However, in 2 and 3, through-barrier transition probabilities within DD₀ and DD₁ drop to nearly zero compared to those of uncoupled KD₀. Meanwhile, thermally dependent excitations between DD₀ and DD₁ have approximately the same probability as that of the uncoupled QTM mechanism. The extremely slow QTM mechanism is effectively suppressed; therefore, transitions are limited to excitations between DD₀ and DD₁, which are predominantly controlled by the orientation of magnetic axes. The control of this new secondary barrier is then another useful tool toward tuning SMM behavior. More broadly, it can be thought of as a method for the rate control of spin population – a desirable trait for a much broader range of functionality. Furthermore, it represents the capacity of introducing a very small perturbation to dramatically affect relaxation by impacting the transition probabilities between the involved states. To probe the validity of these predictions, calculated properties were compared with physical measurements of spin-based observables.

Magnetometry Studies

Static and dynamic magnetic properties of 1, 2, and 3 were measured in a Quantum Design MPMS3 SQUID magnetometer to probe the effect of tuning the dipolar interactions (Section S5). First, temperature-dependent susceptibility measurements were performed to extract state information from the bulk response (Figure 4). The 300 K experimental χ_MT values measured under a 1000 Oe applied field agree well with the theoretical values for the appropriate number of uncoupled Er³⁺ ions (11.78, 23.03, and 22.86 emu K mol^–1 for 1, 2, and 3, respectively; J = 15/2, g = 6/5 per ion; 11.48 emu K mol^–1 per Er³⁺).¹⁴⁶ At lower temperatures, χ_MT for 1 gradually decreases before decreasing more rapidly near 2 K, consistent with a typical depopulation of states in a SIM with low-lying excited states. For both 2 and 3, χ_MT slightly decreases before rapidly increasing at lower temperatures, confirming the presence of ferromagnetic coupling. In 3, this increase begins at higher temperatures and continues to a higher maximum value than in 2, which coincides with the larger energy gap between DD₀ and DD₁ in 3. This comparison holds under the assumption that the corresponding states have the same moment in both complexes, as corroborated by the output from POLY_ANISO.

Figure 4.

Temperature-dependent ZFC χ_MT measurements for 1 (blue circles), 2 (red triangles), and 3 (green squares) were collected at 1000 Oe. Dashed lines are plotted for theoretical χ_MT values of compounds with one and two Er³⁺ centers at 300 K.

Relaxation times (τ) were extracted from frequency-dependent AC susceptibility measurements by simultaneously fitting the in-phase (χ′) and out-of-phase (χ″) components to a generalized Debye equation (Figure 5).¹⁴⁷ Cole–Cole plots of these fitted data form semicircles, and χ″ curves display one peak each. Thus, only a single relaxation time is described at each temperature for each compound. The τ values were fit to a combination of relaxation terms in eq 2

2

with the first term representing an Orbach process, or a thermally activated pathway involving excited states, and the second term representing Raman, or vibronically coupled, processes. The third term accounts for QTM, whereas the fourth describes the thermal relaxation within the energy regime of KD₀ via excitation from the ferromagnetically coupled ground state (DD₀) to the antiferromagnetically coupled excited state (DD₁) over an energy barrier D_eff.^32,75 Both QTM and inter-DD thermal relaxations are expected to occur at very low temperatures, where sufficient thermal energy is not available for Orbach or Raman mechanisms to be comparatively efficient. QTM is expected to be prevalent in SIMs, and thermal relaxation through D_eff is expected to dominate when relaxation is influenced by highly axial interion coupling. In such a system, weak coupling introduces an energy barrier between the ferromagnetically and antiferromagnetically coupled states, and the suppression of QTM mechanisms limits relaxation to thermal excitations between them with long attempt times (τ_D) dictated by their enhanced anisotropies. Therefore, only one of either the third or fourth terms is included in fits to extracted τ values (Table 3).

Figure 5.

Relaxation behavior for 1@Y (blue circles), 2 (red triangles), and 3 (green squares) as extracted from AC susceptibility versus frequency measurements. Extracted τ values are represented with points, and fits to eq 2 are plotted as curves of their respective colors.

Table 3. Fit Relaxation Parameters for 1@Y, 2, and 3.

	1@Y	2	3
Ueff	34.3(7) cm–1	66(3) cm–1	79(6) cm–1
τ0	1(2) × 10–9 s	2(2) × 10–11 s	2(7) × 10–10 s
C		1(5) × 10–3	3(2) × 10–3
N		8.7(9)	6.9(2)
Deff		1.6(3) cm–1	3.5(1) cm–1
τQTM/τD	2.23(5) × 10–3 s	8(2) × 10–3 s	3.2(3) × 10–3 s
τmax	2.35(14) × 10–3 s	2.61(6) × 10–2 s	3.7(1) × 10–2 s

Under zero applied field, only 2 and 3 display full peaks in χ″ vs ν within the measurable window of 0.1 to 1000 Hz. Curves for 1 display only stacked tails of peaks at frequencies above 1000 Hz. At 2 K, a fit to the incomplete curves places τ near 4.5 × 10^–6 s (Figures S31 and S32 and Table S1). Such rapid relaxation is consistent with fast QTM mechanisms typical for SIMs–especially those with significantly mixed ground states. Dilution into an isostructural diamagnetic Y³⁺ lattice significantly reduces the effect of transient fields caused by nearest-neighbor intermolecular interactions. Thus, a 95:5 mol equivalent Y/Er mixture of 1 (1@Y) was prepared, and full peaks were observable in χ″, allowing the extraction of relaxation times below 6 K. Though fast relaxation processes facilitated by nearest-neighbor interactions are suppressed upon dilution, the fast QTM associated with uncoupled magnetic centers persists.

Both 2 and 3 show SMM behavior below 6 and 7 K, respectively, without dilution. When plotting ln (τ) vs 1/T, both complexes show Arrhenius behavior at their highest temperatures, followed by an attenuation of the temperature dependence of relaxation, culminating in a very weakly temperature-dependent region near 2 K. Notably, neither complex displays a temperature-independent region consistent with QTM being the dominant process. All fit parameters are consistent with ab initio calculations, and the D_eff for 3 is expectedly larger than that for 2. Some notable features when comparing the evolution of τ can be explained using D_eff and τ_D. Notably, 3 has a slightly larger barrier U_eff and τ₀ than 2, and D_eff is approximately twice as large. However, τ_D is faster in 3 than in 2, and thus, there is an intermediate region wherein 2 relaxes more slowly than 3. Fit parameters for 2 and 3 were corroborated by measurements under a small applied field of 800 Oe as well as for diluted samples (Section S5). Under an applied field, the D_eff and τ_D remain largely consistent for both 2 and 3. Diluted samples 2@Y and 3@Y were prepared by cocrystallization of pre-prepared Er³⁺ and Y³⁺ samples of 2 and 3. In both diluted samples, the presence of ferromagnetic coupling is still observed, though metal scrambling is apparent by the much broader peaks in χ″ vs ν resembling two separate overlapping mechanisms. A much higher degree of metal scrambling is noted for 3@Y than in 2@Y, as evidenced by the relative attenuation of the low-temperature increase in χ_MT values. This is a common problem when diluting polynuclear structures;⁷³ we attribute this in our systems to the ability to crystallize 2 at subzero temperatures, but the room-temperature conditions required for 3 can allow for more lability in solution. Regardless, relaxation parameters for 2@Y agree well with the undiluted complex, and while the apparent onset of QTM in 3@Y complicates interpretation, relaxation behavior at low temperatures is still thermally dependent and no attenuation of high-temperature relaxation is observed.

The relaxation of 1 can be juxtaposed against that of 2 and 3 to illustrate the degree of control afforded by dipolar coupling and the resulting consequences for magnetic properties. Primarily, the uncoupled, comparatively weaker moment of 1 is highly susceptible to the complex, low symmetry dipolar environment, which acts to enhance the available QTM processes within individual ions. This same sensitivity to QTM and neighboring spins is not present in 2 or 3. Following the predictions afforded by POLY_ANISO, low-temperature relaxation in these complexes is dominated by the small thermal barrier imparted by intramolecular dipolar coupling, which is directed to enhance the local anisotropy. The probability of QTM is consequently lowered by over 10 orders of magnitude in the case of specific levels determined by the orientation of the anisotropy vector. Importantly, this mechanism is not a result of achieving a well-isolated coupling state, but instead, it is more akin to a targeted exchange bias of the states responsible for magnetic relaxation. While the dipole–dipole interaction between the two ions is weak, it is important to note that it is far larger than the intermolecular dipolar fields between neighboring molecules, making both 2 and 3 relatively inert to fluctuations caused by their nearest neighbors. This is made clear by the lack of significant change in the relaxation dynamics of 2@Y, and in 3@Y, the partial removal of intramolecular coupling seems to hasten relaxation. An alternative interpretation of this result, as might be garnered from the crystal structure of 3, is the removal of a high symmetry bulk dipolar field arising from intermolecular interactions that enhances relaxation.¹⁴⁸ In the absence of further supporting data, we do not explore this beyond speculation, but the possibility invites future investigations of crystal engineering with coupled systems.

Far-Infrared Magnetospectroscopy

To complement the description of the local magnetic landscape that we have crafted, FIRMS was employed to probe vibronically coupled magnetic transitions. While this technique is most often employed in surveying extended magnetic solids and qubit candidates,^{26,29,149−154} recent work has employed FIRMS alongside single-crystal Raman magnetospectroscopy and inelastic neutron scattering experiments to probe zero-field splitting parameters in transition metal complexes^155,156 and Kramers doublet separations in lanthanide SMMs^25,157,158 with energy separations too large to feasibly be observed in electron paramagnetic resonance (>∼ 100 cm^–1) or that do not fulfill the necessary selection rules (ΔM_J = ± 1). While magnetic transitions are often IR-allowed, either by following magnetic/electric dipole selection rules or by relaxing these rules via 4f-5d orbital mixing, these transitions are very weak compared to molecular vibrational and phonon modes.¹⁵⁹ The high difference in moments makes the overlap induced by electromagnetic radiation far poorer for these transitions than for pure vibronic transitions. Magnetic transitions can, however, be isolated from transmission spectra by using the field dependence of the magnetic moment to remove vibrational modes uncoupled from magnetic transitions. The resulting heat map then shows the intensity and energy of a transition as a function of the applied magnetic field. While these spectra cannot be easily deconvoluted into crystal lattice vibration modes (phonons) and local molecular vibrational modes, tentative assignments of these transitions can be made to local vibrations modeled by density functional theory (DFT) calculations performed on optimized structures. Here, Er³⁺ was replaced with the closed-shell Lu³⁺ ion rather than Y³⁺ due to its more similar electronic structure and nuclear mass for calculating vibrational energies, and the entire structure’s geometry was optimized. Due to the density of vibrational modes within this region and the limitations of modeling lanthanide complexes with DFT, these assignments are not meant to pinpoint the atomic origins but rather to suggest the type of mode within the correct energy range to facilitate a dynamic coupling.

FIRMS measurements at 4.5 K were performed from 0 to 17 T for 1, 2, and 3 (Figure 6 and Section S8). Weak, magnetically dependent peaks are visible between 30 and 700 cm^–1 for all three to varying degrees, as discussed below. All transitions’ intensities dramatically increase when in close energy proximity to a coupled vibrational mode, maximizing when the two are resonant. On approach, the magnetic and vibration transitions mix in an observable avoided crossing. Since only states involving KD₀ are expected to be meaningfully populated for all three species at 4.5 K, the magnetic transitions are all assumed to be excitations from KD₀ (or the relevant dipolar splitting manifold for 2 and 3). The high fields used in FIRMS experiments dictate that the Zeeman interaction is now a larger perturbation than the dipolar interaction by an order of magnitude, leading to an energy landscape described by a simplified phenomenological spin–phonon coupling Hamiltonian at the resolution of the crystal field of the individual single-ion spin centers, as described by SINGLE_ANISO (described in Section S9).

Figure 6.

Lowest energy transitions visible in FIRMS spectra for 1 (left), 2 (middle), and 3 (right). White dots represent experimental points, orange curves show full fits as described in the main text and Supporting Information, and dashed and dotted lines represent uncoupled vibrational modes and magnetic transitions extracted from FIRMS data, respectively.

Since relaxation is predominantly measured to occur within KD₀ and KD₁, we focus on transitions below 100 cm^–1. As KD₀ and KD₁ are relatively close together in all three complexes, mixing between them is expected in the high fields in the experiment. Furthermore, the Zeeman perturbation energy in high fields quickly becomes larger than that of dipolar coupling. Thus, for simplicity, magnetovibronic transitions can be described in terms of single-ion states. Within the low energy landscape, two separate instances of one magnetic transition coupled to one vibrational mode were observed for 1, a single magnetic transition coupled to one vibrational mode was observed for 2, and only an exceedingly weak magnetically dependent transition without any resolvable coupled vibrational modes was discernible for 3. The lowest transition for 1 appears at 57(1) cm^–1, consistent with our calculated and experimentally observed barriers. Assuming that the transition originates from the M_J = state, this is also consistent with a transition to the M_J = state (moments μ_KD1,fit = −1.4(8) μ_B, and μ_KD1,calc = ± 1.3 μ_B). The coupled vibrational mode at 59.8(1) cm^–1 most closely resembles a computed librational mode at 60.9 cm^–1 where twisting modes of the THF ligands and rocking of the COT^2– dominate. A second transition is visible in the same region at 67(2) cm^–1 coupled to a vibrational mode at 69.7(3) cm^–1. The excited state moment is μ_KD,fit = −0.3(27) μ_B under the assumption that the same M_J = state is the origin of the transition. While said magnetic transition has a higher energy than predicted for either isolated structure, the similarity both in the coupling strength (spin–phonon coupling constant Λ₁ = 1.6(4) cm^–1 and Λ₂ = 1.4(6) cm^–1) and in the difference in the moment between states implies some congruence between the two transitions.

The fit magnetic transition for 2 at 62(1) cm^–1 also corresponds well to the ab initio value of 65.5 cm^–1. The moment of KD₁ is similarly consistent [μ_fit = −0.9(9) μ_B and μ_calc = ± 0.93 μ_B]. As in 1, the corresponding DFT calculated vibrational mode at 72.1 cm^–1 describes libration involving both THF ligands with moderate twisting and wagging motions, and the transition is coupled more strongly at Λ₁ = 2.9(4) cm^–1. Interestingly, only one of the two COT ligands has a large displacement in its “rocking mode” within the libration. Lastly, the bridging methyl ligands twist about the axis connecting both ligands’ atoms.

The lowest magnetically coupled vibrational mode in 3 displays far weaker intensity than those in both 1 and 2. However, a weak absorption linearly dependent with the field can be resolved. In stark contrast to the other two complexes, the lowest magnetic transition visible in FIRMS for 3 does not appear to couple to any nearby vibrational modes. There is a slight increase in intensity below 1 T near 80 cm^–1 and above 4 T near 100 cm^–1, but the weak signal-to-noise ratio and experimental artifacts preclude meaningful discussion of vibronic coupling in these regions. The weak linearly magnetically dependent transition can be fit to a simple Zeeman perturbation

3

to extract Δ_KD and Δμ_KD. Here, Δ_KD is 78.6(6) cm^–1, and the transition corresponds to an excitation to a state with a moment of −0.6(3) μ_B. The fit barrier is higher than calculated, matching most closely to the first excited state (Δ_1,calc = 66.3 cm^–1), but corresponds well to that measured via magnetometry (U_eff = 79(6) cm^–1). The moment of the excited state likewise resembles that of the first excited state (μ_KD = ± 0.56 μ_B). The closest vibrational modes calculated by DFT lie at 68.29 and 69.69 cm^–1, corresponding to twisting modes of the bridging methyl ligands and slight rocking of the COT rings, respectively. Above the range in which the first transition is resolvable, the first available mode to couple to is at 108.03 cm^–1, which corresponds almost exclusively to twisting motions in the methyl ligands.

While the assignment of local molecular vibrational modes as mediators of the spin–lattice interaction is qualitative, it allows for testable hypotheses regarding molecular design. Specifically, both 1 and 2 display significant motion within the THF ligands. Considering both the large degree of freedom in movement within the ligand and the electron density of the coordinated oxygen atom, such a motion should perturb magnetic states so as to allow for thermally assisted QTM. Decreasing the number of coordinated THF ligands per Er³⁺ center from two in 1 to one in 2 appears to suppress this based on the decreased absorption intensity. Thus, more rigid ligands or, in the case of 3, a total lack of hard Lewis basic solvent ligands may suppress coupling, as has been noted in other discussions.^17,160 The other major contributor to motion—the rocking of the COT^2– ligand—cannot be so easily suppressed. However, introducing bulky substituents, such as the silyl groups that have achieved much synthetic success,^{11,75,129,161,162} could limit the degree of motion available to the scaffold. However, caution must also be taken not to introduce additional entropic vibrational modes within the added functional groups. Additionally, while the increased pseudosymmetry in 3 limits the number of IR active modes, the possibility of Raman-active modes prevents us from drawing clear conclusions relating to it.

While intense magnetic transitions can be seen in the FIRMS heatmaps of all three compounds, the lowest energy transitions show a dramatic difference in the change of their intensities; this merits additional discussion. Alongside the control of internal modes of vibrational degrees of freedom, the spectra we included herein can illustrate the effect of tuning the magnetic state composition as a result of the crystal field. Selection rules dictating intramanifold transitions are not well established, though past discussions have pointed to transitions being magnetic dipole and electric dipole allowed. Such rules dictate that a transition between pure M_J states should be allowed for ΔM_J = ± 1, though this can be relaxed through mixing. Such a case is observed when juxtaposing the heat maps of 1 and 2 against that of 3 in the context of the state composition of their Kramers doublets as modeled by SINGLE-ANISO. The heatmap in 1 reveals a relatively intense magnetically dependent transition in the lowest energy excitation. The corresponding transition in 2 is noticeably weaker, and in 3, the magnetically dependent peaks in the lowest transition are barely resolvable. Such a progression corresponds well to the purity of the first manifold of excited states composed of interactions involving KD₁. While single-ion calculations for 1–3 reveal KD₀ manifolds of almost pure M_J = ± and KD₁ principally composed of M_J = ±, the degree of mixing of the M_J = ± into KD₁ decreases markedly from 1 to 3. In 1, it comprises 13% of KD₁, whereas in 2, it drops to 2%, and in 3, it is less than 1%. Meanwhile, the composition of M_J = ±, to which a transition from M_J = ± would be least allowed, increases from 54 to 81 and 92% in 1, 2, and 3, respectively. This trend is also reflected in the ab initio derived transition probabilities between KD₀ and KD₁, which are nearly an order of magnitude smaller in 2 and 3 than those in 1.

Conclusions

We have presented alkyl-bridged dinuclear SMMs that utilize the placement of internal fields to direct dipolar coupling as an alternative to orbital exchange as a means of fine control over the spin manifold. The dramatic increase in relaxation times at all temperatures observed in the closely bound dinuclear complexes 2 and 3 relative to that in the mononuclear 1 demonstrates how the precision of molecular synthesis offers the chance to transform the oft invoked concept of exchange bias into a design principle for quantum spin architectures. Whereas strong orbital exchange can be used to isolate large spin states, it can also forfeit much of the precise spatial anisotropy garnered by tuning the single-ion crystal field. The use of directed dipolar interactions preserves the overall anisotropy and moment generated by the dominant crystal field interaction while protecting states against random field fluctuation-induced QTM mechanisms. A clear manifestation of this is seen in how the collinear, high pseudosymmetry anisotropy axes of 3 result in exceptional changes in the relaxation dynamics within a temperature regime of nearly equivalent Boltzmann distribution across the manifold of coupled states. Additionally, through the FIRMS technique, the low energy spectrum is monitored to study how the changes in structural rigidity, and selection rules imparted by 1–3 yield very different mixing between the spin and lattice components. Absorptions akin to the magnetic relaxation pathway decrease in intensity across the series in 1–3, with computational results indicating that both the removal of the highly entropic THF ligands and the increased purity of the excited M_J = ± state are likely to play a role.

In summary, 1–3 have been studied comparatively to determine the effects of high-moment, high-anisotropy ions to direct internal dipolar fields and generate a model that describes relaxation behavior in the long time scale regime where it is controlled by the ground single-ion state. Within this regime, relaxation is highly susceptible to random fluctuation factors, unless there are specifically engineered control principles geared to this scale. The results from 1–3 suggest that construction of larger scale architectures assisted by modeling of spin relaxation through Ising-type models offer a natural direction of expansion for single-ion design to higher nuclearities in lanthanide molecular magnetism: it preserves the crystal field design principles carefully crafted over the past decade through the introduction of a much smaller perturbation while allowing for the addition of new ions with a known spin structure based on an additional spin bias (internal Zeeman) Hamiltonian term. As a future direction, it affords an additional mode of observation, modeling, and control with responses to magnetic, crystalline, and electromagnetic fields.

Glossary

Abbreviations

SMM

single-molecule magnet

QTM

quantum tunneling of the magnetization

COT

cyclooctatetraene dianion

THF

tetrahydrofuran

DME

dimethoxyethane

SIM

single-ion magnet

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.3c10412.

• Detailed synthetic procedures, crystallographic parameters and characterization, bulk spectroscopic characterization, experimental magnetic data, computational setup and salient output, FIRMS measurements, and description and derivation of magnetovibronic coupling fitting equations (PDF)

• Representation of salient vibrational modes in 2 included as animation (GIF)

• Representation of salient vibrational modes in 1 included as animation (GIF)

This research was funded through the National Science Foundation Division of Chemistry #2024650. The National High Magnetic Field Laboratory is supported by the National Science Foundation through NSF/DMR-1644779 and the State of Florida.

The authors declare no competing financial interest.