Anisotropic Magneto‐Chiral Dichroism in Lanthanide Complexes

ABSTRACT

In this work, we report the preparation of a chiral Er(III) complex specifically designed to assess the influence of the magnetic anisotropy axes orientation on its magneto‐chiral dichroism (MChD). High resolution MChD measurements performed along the three principal crystallographic axes revealed pronounced anisotropic effects on both MChD signal line shapes and intensities. By combining, for the first time, angle‐resolved torque magnetometry, comprehensive magneto‐optical spectroscopy on oriented single crystals, absorption measurements under magnetic fields up to 30 T and theoretical calculations, we provide clear experimental insight into the anisotropic nature of the MChD in lanthanide complexes. Specifically, we evaluate the extent to which Er moderate anisotropy enables signal persistence across a wide range of crystallographic orientations. These results allow the identification of the key role played by both the orientation and the magnitude of magnetic anisotropy in the MChD response of chiral lanthanide complexes, representing a significant step toward the directional control of magneto‐chiroptical responses in lanthanide‐based molecular materials.

The anisotropic magneto‐chiral dichroism (MChD) of a chiral Er(III) complex is investigated here through a combined experimental and theoretical approach based on high‐resolution MChD spectroscopy, cantilever torque magnetometry, optical absorption under high magnetic fields and ab initio calculations. We provide a detailed description of its magneto‐chiroptical properties including the field‐induced variations driven by magnetic anisotropy.

1. Introduction

Magneto‐chiral dichroism (MChD) represents a non‐reciprocal absorption or emission of light that arises in chiral magnetized systems [1, 2]. It originates from the simultaneous breaking of mirror symmetry by structural chirality and time‐reversal symmetry by the magnetic field B or, at zero field, by the remanent magnetization of the system M [3, 4, 5]. These symmetry arguments make MChD independent of the polarization state of light and fundamentally distinguish MChD from other chiroptical effects such as natural circular dichroism (NCD) or magnetic circular dichroism (MCD) that require circular polarization states of light to be observed [6]. While NCD and MCD represent differential absorption of left and right circularly polarized light, MChD manifests as a differential absorption of unpolarized light that depends on the relative orientation of the magnetic field B and the light propagation wavevector k [7].

MChD is a universal magneto‐optical phenomenon and has been observed from frequencies spanning microwaves to hard x‐rays [8, 9, 10, 11]. The strong MChD responses recently observed in the vis and NIR spectral ranges on chiral transition metal and lanthanide complexes has allowed MChD to evolve towards a fundamental and versatile spectroscopic technique to study chiral magnetic materials [12, 13, 14].

Recent theoretical and experimental studies have established key principles governing MChD intensity [15, 16]. The magnitude of the MChD signal critically depends on specific factors: the system net magnetization, the magnetic‐dipole character of the optical transitions, the magnitude of spin–orbit and vibronic coupling, and the local symmetry at the metal center [16, 17]. In this context, lanthanide‐based chiral complexes offer an ideal platform to observe strong MChD responses, owing to the combination of their sharp intrashell f–f transitions, strong spin–orbit coupling (SOC), and tunable magnetic anisotropy, leading to potential applications in chiral photonics and non‐volatile data storage [18, 19, 20, 21, 22, 23, 24]. Additionally, the fine‐tuning of the coordination environment to accommodate a non‐spherical electron density can lead to magnetic bistability, optically detectable through MChD spectroscopy along the entire vis‐NIR range [25, 26]. Yet, to improve the sensitivity of this optical readout method, intense MChD signals are necessary.

It was recently experimentally validated that Richardson's theory of lanthanide optical activity, which classifies f–f electronic transitions​ in terms of electric‐dipole strength E, rotatory strength R and dissymmetry factor D, used to predict strong NCD and circularly polarized luminescence (CPL) responses in chiral lanthanide complexes [27], also applies to MChD [28]. In this theory, magnetic‐dipole allowed transitions, following the selection rules ΔJ = 0, ±1 (except 0 ↔ 0), with ΔL = 0 and ΔS = 0, are defined as R‐I type transitions. Although parity‐forbidden, they are predicted to be highly MChD‐responsive thanks to their high rotatory strength. For this reason, the ⁴I_13/2 ← ⁴I_15/2 electronic transition of Er^III at λ ∼ 1530 nm leads to stronger ΔA_MChD responses and g _MChD dissymmetry factors in comparison to that of chiral Dy^III complexes, that are highly anisotropic but without accessible magnetic‐dipole allowed transitions in the vis‐NIR spectral range [15, 28]. Given its higher Z value compared to Dy^III, Er^III has a higher SOC constant [29]. Nonetheless, in a comparable ligand environment its axiality is generally smaller than that of Dy^III [30]. Although unfavorable when searching for high‐T _B single‐molecule magnets, this modest anisotropy can be exploited to afford measurable MChD signals for multiple crystallographic orientations whereas the strong magnetic axiality of the Dy^III analogous can restrict the observation of strong MChD responses for magnetic field and light oriented along the magnetic easy axis [25].

The present work intends to provide experimental evidence that supports these structure–property relationships with the final aim of defining design principles to finely tune the magnetic anisotropy and the local symmetry to boost MChD responses. To achieve this goal, we have prepared a pair of enantiopure chiral Er^III complexes featuring a coordination environment that does not provide a trivial orientation and spatial distribution of the lanthanide magnetic anisotropy. We have investigated its MChD response along the three principal crystallographic axes with high‐resolution MChD spectroscopy under magnetic fields up to 9 T. The experimental MChD results were analyzed together with angle‐resolved torque magnetometry, absorption spectroscopy under high magnetic fields and ab initio theoretical calculations performed on both the ground and the first optically excited states. This combined analysis affords a fine interpretation of the MChD results, which represents a fundamental advance with respect to the current state‐of‐the‐art.

2. Results and Discussion

2.1. Synthesis and Structural Characterization

[Er(Ph₃SiO)₂((R,R,R,R)‐L^N6)]PF₆·CH₂Cl₂ (1‐(R,R,R,R)) and [Er(Ph₃SiO)₂((S,S,S,S)‐L^N6)]PF₆·CH₂Cl₂ (1‐(S,S,S,S)) (L^N6 = chiral hexaazamacrocyclic ligand) were synthesized according to the previously described procedure for the Dy^III analogues (see experimental section) [25]. Erbium(III) chloride was reacted with enantiopure (1R,2R)‐ or (1S,2S)‐1,2‐diphenylethylenediamine and 2,6‐diformylpyridine in MeOH, followed by a treatment with triphenylsilanol in the presence of N,N‐dimethylethanolamine and sodium hexafluorophosphate in a mixture of CH₂Cl₂ and water. After separation of the organic layer, slow diffusion of pentane led to the formation of large air stable rhombohedral single crystals of 1‐(R,R,R,R) or 1‐(S,S,S,S), suitable for x‐ray diffraction and MChD measurements.

Single‐crystal x‐ray diffraction reveals that both enantiomers are isostructural to the Dy^III analogues and crystallize in the monoclinic non‐centrosymmetric space group P2₁ (Table S1) [25]. The unit cell contains two complexes, two dichloromethane molecules, and two hexafluorophosphate ions that ensure charge neutrality (Figure S1). The complex is octacoordinated, and SHAPE 2.1 analysis [31, 32] (Table S2) confirms a distorted hexagonal bipyramidal coordination environment around the central Er^III ion. This geometry arises from the flat conformation of the chiral macrocyclic ligand, which provides six nitrogen donor atoms in the equatorial plane, combined with two oxygen donor atoms originating from the triphenylsilanolate coordinating anions in the axial positions (Figure 1a,b). For both enantiomers, the Er–N and Er–O bond lengths are in the 2.622–2.718(3) Å and 2.117–2.120(2) Å ranges, respectively, while the O–Er–O angle is 178.6(1)°. The shortest Er···Er distance is 11.87(3) Å, suggesting negligible intermolecular interaction between paramagnetic centers. Powder x‐ray Diffraction analysis was performed on crushed single crystals of both enantiomers and Le Bail refinement of the diffractograms confirmed the structural phase purity of the samples (Figure S2; Table S3).

FIGURE 1.

Side (a) and top (b) view of the molecular structure of 1‐(S,S,S,S). Color code: orange, Er; red, O; blue, N; grey, C; pale yellow, Si. Hydrogen atoms have been omitted for clarity, except for the ones linked to the asymmetric carbons of the macrocycle (marked with asterisks).

2.2. Magnetic Properties and Magnetic Anisotropy

The magnetic properties of 1 were investigated by direct current (dc) magnetic measurements on randomly oriented powder of crushed single crystals. Since enantiomeric pairs exhibit identical magnetic behavior, only the data of 1‐(S,S,S,S) are described. The product of the molar magnetic susceptibility times the temperature (χ _M T) at 300 K is 11.28 cm³ K mol^{− 1} (Figure S3), in good agreement with the expected value for an isolated Er^III ion (11.48 cm³ K mol^{− 1}, ⁴I_15/2, S = 3/2, L = 6, J = 15/2, g_J = 6/5) [33, 34]. Upon cooling, χ _M T gradually decreases down to ca. 20 K and then drops sharply to 6.8 cm³ K mol^{− 1} at 4 K. Considering the large intermolecular Er···Er separations in the crystal packing, dipolar magnetic interactions are expected to be negligible, and the observed decrease in χ _M T can therefore be attributed to the single‐ion magnetic anisotropy arising from the crystal field (CF) splitting of the ⁴I_15/2 ground spectroscopic term of the Er^III center. Isothermal magnetization measurements up to 5 T in the 2–5 K range show that saturation is not fully achieved, with a maximum value of 5.1 μ _B mol⁻¹ at 5 T (Figure S4). When plotted as a function of B/T, the magnetization curves collected at different temperatures do not superimpose, further highlighting the pronounced magnetic anisotropy and the presence of low‐lying energy levels close to the ground Kramers doublet (KD).

To gain further insights into the magnetic anisotropy of this complex and obtain a good initial basis to fit the experimental data, ab initio calculations at the CASSCF level were performed (see below and Supporting Information for details). The computed CF parameters are reported in Table S4. Calculations predict a composition of the ground KD dominated by ±13/2 M _J states (Table S5). However, as expected for a prolate Er^III ion [35], the axial coordination environment promotes a significant mixing, resulting in a moderately axial g tensor (Table S6), and an easy axis that points approximately towards the nitrogen atom of one of the two pyridyl moiety of the macrocyclic ligand. Additionally, the first and second excited KD lie at relatively low energy (17 and 39 cm⁻¹, Figures S5 and S6), which agrees with the nonsuperimposable nature of the reduced magnetization curves.

Alternate current susceptibility measurements at low temperature (T = 2.0 K) do not show slow magnetic relaxation at zero field within the available frequency range. This is in agreement with the small energy difference between the low‐lying KDs of the ground state experimentally detected (vide infra) and calculated through ab initio methods (vide supra).

To obtain accurate experimental information on the orientation of the magnetic reference frame of the Er^III center with respect to the crystallographic frame, we employed cantilever torque magnetometry (CTM) [36]. This technique has been used to precisely determine the magnetic anisotropy of both weakly [37] and highly [38] anisotropic molecules. Moreover, it allows the disentanglement of noncollinear contributions arising from the crystal packing [39, 40]. CTM experiments at higher temperatures have been used to obtain information on the contribution of thermally populated excited crystal field states in lanthanides [41, 42] and actinides [37] or to achieve outstanding sensitivity on zero field splitting parameters in transition metal complexes [43]. In the present work, however, the low temperature range was selected to isolate the ground state magnetic response and thus obtain a more reliable determination of the orientation of the molecular magnetic frame.

CTM measurements were performed on an oriented single‐crystal of 1‐(S,S,S,S) in the T = 2.5−25 K and B = 0−9 T ranges. Due to the presence of two magnetically inequivalent molecules in the unit cell, which contribute together to the overall magnetic behavior, three distinct experiments were performed by rotating the same single crystal along three orthogonal rotation axes (Table S7). A schematic representation of the three rotations is provided in Figure S7. The first and second rotations probe the magnetic anisotropy within a plane containing the O−Er−O bond, whereas the third rotation explores the plane almost perpendicular to this direction. Experimental results are shown in Figures S8–S10.

CTM data were simulated on the basis of the following Hamiltonian operator (Equation 1):

$$\widehat{H}=\mathrm{\Lambda }\widehat{L}·\widehat{S}+\sum _{k=2,4,6}\sum _{q=-k}^k{B}_k^q{\widehat{O}}_k^Q\left(L\right)+{\mu }_B\mathit{B}\left(g_e\widehat{S}+g_L\widehat{L}\right)$$ (1)

where the three terms account for SOC, CF, and Zeeman interactions, respectively, and ${\widehat{O}}_k^q$ are the extended Stevens operator equivalents acting on the orbital angular momentum L. Although CTM data for lanthanide complexes are commonly simulated using a Hamiltonian restricted to the ground state J multiplet, in the present case a larger basis set was necessary to simulate both the CTM data and the absorption spectra under magnetic field (vide infra). The spin‐orbit coupling constant Λ was fixed to −865.2 cm⁻¹ to reproduce the experimental energy of the ground doublet of the excited ⁴I_13/2 multiplet (for the aqua ion, Λ = −793.6 cm⁻¹) [29]. The CF parameters $B_k^q$ and the orientation of the magnetic reference frame were initially fixed to the values obtained from ab initio calculations obtaining the results reported in Figures S8–S10.

The simulated torque amplitudes closely reproduce the experimental intensities over the entire investigated temperature and magnetic field range; however, noticeable discrepancies are observed in the angular dependence of the curves. This result indicates that the calculations correctly predict the overall magnitude of the magnetic anisotropy, whereas the orientation of the magnetic reference frame with respect to the crystallographic axes (Table S8) is not accurately captured.

To improve the agreement between simulations and experimental data, a fitting procedure was performed. Owing to the absence of symmetry at the single molecule level and the strongly mixed character of the ground KD, symmetry constraints could not be applied to reduce the number of CF parameters. Consequently, the magnitudes of the $B_k^q$ parameters were kept fixed during the fitting procedure, and only the three Euler angles defining the orientation of the magnetic reference frame at the Er^III center were refined (Table S9). The agreement with the experimental data is now very good for the first and second rotations, while minor discrepancies in the signal intensity remain for the third rotation (Figures 2a and S11–S13). These residual differences could likely be further reduced by slight adjustments of the energy and/or composition of the low‐lying KDs. However, given the large number of parameters involved, several sets of parameters could lead to comparable refinements, precluding a physical meaningful fit. Figure 2b shows the magnetic reference frame extracted from the fitting procedure, while Table S10 reports its components with respect to the crystallographic ab'c* frame. Figure S14 compares the fitted magnetic frame with that derived from ab initio calculations, highlighting that the direction of the easy axis is nearly identical. These results confirm that the magnetic easy axis lies in the plane of the macrocyclic ligand along the Er‐N(pyridyl) bond. In contrast, the orientation of the intermediate and hard magnetic axes differs significantly between the two approaches. This indicates that, while theoretical calculations correctly predict the dominant axial anisotropy, they are less accurate in describing the transverse components. Nonetheless, for both approaches, the O─Er─O bond appears to be almost in between the hard and intermediate magnetic axes.

FIGURE 2.

(a) Experimental (dots) and simulated (lines) CTM curves at T = 2.5 K and B = 9 T through rotations 1 (purple), 2 (orange) and 3 (green) on an oriented single crystal of 1‐(S,S,S,S) indexed by single‐crystal x‐ray diffraction analysis. Simulations were obtained from the fitting procedure of the Euler angles, while crystal field parameters were kept fixed to ab initio results. (b) Orientation of the magnetic axes obtained from the fitting procedure with respect to molecular frame (hard: red; intermediate: green; easy: blue) and the crystallographic reference frame. The molecular structure is superimposed to the anisotropic free energy surface calculated at T = 2.5 K and B = 4 T.

2.3. Optical Spectroscopy

Low temperature (T = 4.0 K) absorption spectra were recorded at zero‐field on single crystals of both enantiomers indexed by single‐crystal x‐ray diffractometry in the vis–NIR range (λ = 400–1540 nm) with unpolarized light propagating along the three principal crystallographic axes (a, b, and c). Although these directions are not simply related to the principal magnetic axes (vide supra), they have the advantage of showing a single response from all the molecules in the crystal, due to the symmetry elements of the unit cell (C₂ axis along b which is orthogonal to both a and c).

In the 400–1000 nm spectral range, the tail of an intense intra‐ligand absorption band is observed between 400 and 650 nm, together with six sharp, weak‐to‐medium intensity, absorptions corresponding to Er^III intra‐shell f–f electronic transitions between the ground state energy level (⁴I_15/2 term) and the ⁴F_7/2, ²H_11/2, ⁴S_3/2, ⁴F_9/2, ⁴I_9/2, and ⁴I_11/2 excited states levels (Figure S15a). The f–f electronic transitions in this spectral region do not satisfy the selection rules for magnetic‐dipole allowed transitions (vide supra) and are Laporte‐forbidden, that is, parity‐conserving. They are electric‐dipole‐induced transitions enabled by relaxation of the Laporte selection rule by the coordination environment [44, 45, 46]. In contrast, the ⁴I_13/2 ^{← 4}I_15/2 transition falling in the NIR range (λ ca. 1530 nm) is magnetic‐dipole allowed (|ΔJ| = 1) and is expected to provide a sizeable MChD response [27, 28]. Owing to its magnetic‐dipole character, high‐resolution (< 0.15 nm) optical and MChD spectroscopy measurements and analysis have been done for this electronic transition.

High‐resolution absorption spectroscopy was used to experimentally determine the CF splitting of the first optically excited ⁴I_13/2 manifold (Figure 3a). At 4.0 K, six of the seven expected CF contributions are clearly observed: three well‐separated bands at 1524.0, 1515.9, and 1504.8 nm (6561.6, 6596.8 and 6645.6 cm⁻¹), followed by two closely spaced signals at 1497.0 and 1493.4 nm (6679.9 and 6696.0 cm⁻¹), and a singlet at 1480.0 nm (6756.8 cm⁻¹). The contribution associated to the highest‐energy M _J sublevel is detected at λ = 1399.1 nm (7189.5 cm⁻¹). Thus, it lies outside the spectral range of our high‐resolution grating.

FIGURE 3.

(a) Low temperature and zero field absorption spectra of 1‐(R,R,R,R) along the three crystallographic axes. Along c the strong response at 1524.0 nm decreases the signal‐to‐noise ratio associated to the other CF contributions, thereby hindering the detection of the absorption at 1480.0 nm clearly detected along a and b. (b) Contour plot showing the temperature‐dependent evolution of the absorption spectra of 1‐(R,R,R,R) with the light propagation vector k oriented along the b crystallographic axis. Extracted spectra at 4.0, 10, and 60 K are shown, highlighting the energy separation between the main transitions and the emerging satellite bands. (c) Comparison between the experimental and calculated energy levels of the optical ground state ⁴I_15/2 (only the first three KD are experimentally accessible) and the first optically excited state ⁴I_13/2.

To rationalize the optical properties, ab initio calculations at the CASSCF level were performed for the first optically excited state (⁴I_13/2). The calculated energy splitting of the entire multiplet is shown in Figure S6. The experimentally detected values are in agreement with ab initio calculations (Figure 3c). The deviations of the first and the second excited doublets from the experimental values are 2 and 9 cm⁻¹, respectively. At the same time, the calculated centroid of the ⁴I_13/2 multiplet differs from the experimental value by only 1.7%, indicating that the relative position of the relevant excited manifold is accurately captured by the computational model. Within this validated framework, the ordering of the low‐lying Kramers doublets and their dominant magnetic character serve as the key electronic‐structure components for interpreting optical and MChD spectra.

The orientation of light propagation vector k relative to the unit cell does not impact the transition energies but can affect the relative absorption intensities. Absorption spectra of similar intensity are observed along the a and b crystallographic axes (Figure 3a). Along these directions, the L^N6 macrocycle is effectively screened, which may account for the similar absorption response. In contrast, when k // c, a stronger CF contribution is observed at 1524.0 nm with respect to the other contributions.

Upon increasing the temperature (T = 4.0–290 K), the first excited sublevels of the ⁴I_15/2 ground multiplet become thermally populated, leading to the appearance of satellite peaks adjacent to the low‐temperature absorption bands (Figures 3b and S15b and S16). These are detected at 1528.6, 1520.3, and 1509.2 nm (6541.6, 6577.5, 6626.0 cm⁻¹; ΔE ≈ 19.6(4) cm⁻¹) with a maximum in intensity at around 32 K, and at 1535.4 and 1527.1 nm (6513.0 and 6548.4 cm⁻¹; E ≈ 48.6(4) cm⁻¹) with a maximum in intensity at around 60 K. Accordingly, the first and the second set of absorption maxima is related to the first and the second excited KDs, respectively. These results are in excellent agreement with the theoretical calculations, which place the first two excited M_J sublevels of the ground state at 16.9 and 38.7 cm⁻¹, respectively (Figure 3c, Tables S5 and S6).

To further support theoretical calculations and CTM findings, the optical absorption was studied at T = 4.0 K under magnetic fields up to 30.0 T. At this temperature only the lowest energy KD of the ground spectroscopic level ⁴I_15/2 is populated, simplifying the data analysis. Upon increasing the magnetic field intensity, the six absorption bands within the investigated energy range split, leading, in some cases, to crossings of sublevel components (Figures 4b–d and S17). This splitting is due to the Zeeman effect on both the ground and excited states. Indeed, under these conditions, the magnetic field lifts the degeneracy of both the ground state KD and the first optical excited state KD involved in the transition, leading to four non‐degenerate states named |↓ _g 〉, |↑ _g 〉, |↓ _e 〉 and |↑ _e 〉 (g = ground, e = excited) as shown in Figure 4a.

FIGURE 4.

(a) Schematic representation of the Zeeman effect on the energy levels of the ground and first excited state and representation of the transition and Zeeman energies used in the discussion and the respective equations allowing their determination. (b–d) Contour plot showing the field‐dependent evolution of the absorption spectra of 1‐(R,R,R,R) with the light propagation vector k oriented along each crystallographic axis (see legend). (e–g) Simulated plot of the evolution of the energy of the multiplet as a function of the magnetic field for different crystal orientations (see legend).

As a representative example, the impact of the Zeeman splitting on the absorption spectra will be discussed for the lowest energy transition at 6561.6 cm⁻¹. This absorption band splits into three components, with the lowest energy component that gradually decreases in intensity and become undetectable at ca. 5.0 T (Figure 4a). The component shifting toward lower energy is assigned to the |↓ _e 〉 ← |↑ _g 〉 transition: the Zeeman effect destabilizes |↑ _g 〉 and stabilizes |↓ _e 〉, reducing their energy gap relative to zero field and the intensity gradually decreases upon gradual depopulation of the |↑ _g 〉 state. The second component is assigned to the |↑ _e 〉 ← |↑ _g 〉 and |↓ _e 〉 ← |↓ _g 〉 transitions. Because both transitions imply the simultaneous destabilization and stabilization, respectively, of both ground and excited states, the energy difference between them appears to be smaller than the experimental bandwidth, in particular at low fields. Finally, the third component originates from the |↑ _e 〉 ← |↓ _g 〉 transition. As the magnetic field intensity increases, the two high energy components are shifted toward higher energies. This is associated to a Zeeman splitting (ΔE ∝μ_B gB ) of the ground state higher than that of the first optical excited state. A similar trend is observed for the transitions to the other excited KDs, highlighting that the Zeeman splitting of the ground state KD is higher than that of any other KDs. Because the spectra were recorded along three crystallographic axes, one can in principle access the anisotropy of the g _g (g = ground) tensor. Although the |↑ _g 〉 state is depleted at high fields, it is possible to quantify g _g and its anisotropy by comparing the experimental absorption energies at 0 T and 30 T. The Zeeman splitting of the excited state, ΔE_e , is given by the difference in energy between the two high energy transitions:

$$\mathrm{\Delta }{E}_e=\mathrm{\Delta }{E}_{\left|\uparrow e〉\right.\leftarrow \left|\downarrow g〉\right.}-\mathrm{\Delta }{E}_{\left|\downarrow e〉\right.\leftarrow \left|\downarrow g〉\right.}$$ (2)

These energies can be rewritten as:

$$\mathrm{\Delta }{E}_{\left|\uparrow e〉\right.\leftarrow \left|\downarrow g〉\right.}=\mathrm{\Delta }{E}_0+\frac{\mathrm{\Delta }{E}_g+\mathrm{\Delta }{E}_e}{2}$$ (3)

$$\mathrm{\Delta }{E}_{\left|\downarrow e〉\right.\leftarrow \left|\downarrow g〉\right.}=\mathrm{\Delta }{E}_0+\frac{\mathrm{\Delta }{E}_g-\mathrm{\Delta }{E}_e}{2}$$ (4)

where ΔE ₀ is the energy difference between the two KDs at zero field (Figure 4a). Hence, the Zeeman splitting of the ground state ΔE_g is:

$$\mathrm{\Delta }{E}_g=2\left(\mathrm{\Delta }{E}_{\left|\uparrow e〉\right.\leftarrow \left|\downarrow g〉\right.}-\mathrm{\Delta }{E}_0\right)-\mathrm{\Delta }{E}_e$$ (5)

$$\mathrm{\Delta }{E}_g=2\left(\mathrm{\Delta }{E}_{\left|\downarrow e〉\right.\leftarrow \left|\downarrow g〉\right.}-\mathrm{\Delta }{E}_0\right)+\mathrm{\Delta }{E}_e$$ (6)

The extracted experimental energy values ΔE _{|↑e〉 ← |↓g〉} and ΔE _{|↓e〉 ← |↓g〉} and the calculated ΔE_e  and ΔE_g are summarized in Table 1 and confirm that the Zeeman splitting is maximum along the b axis, intermediate along a and minimum along c. This trend is observed for all absorption bands within this spectral range. To further support this interpretation, simulations of the absorption spectra under magnetic field (see Supporting Information for details) were performed using the CF parameters obtained by ab initio calculations and the magnetic reference frame refined by CTM. The experimental (Figure 4b–d) and simulated spectra (Figure 4e–g) show an overall good agreement. A closer inspection of Figure 4c reveals additional splitting above ca. 15 T, resulting in more lines than the expected 2J + 1 = 14 states. Interestingly, the simulations indicate that this effect can be reproduced by introducing a slight misalignment of 2.5° with respect to the b crystallographic axis (Figure S18). Under such conditions, the two molecules present in the unit cell are no longer equivalent and therefore exhibit distinct spectral responses.

TABLE 1.

Extracted absorption energies of the lowest energy transition, at zero and high field (B = 30 T), and experimental Zeeman splitting of 1‐(R,R,R,R) at T = 4.0 K as a function of the probed crystallographic axis obtained with equations 2 and 5or 6. All energy values are in cm⁻¹.

Orientation	ΔE 0	ΔE |↑e〉 ← |↓g〉	ΔE |↓e〉 ← |↓g〉	ΔEe	ΔEg
k // a	6562.1(5)	6635.3(5)	6585.4(5)	49.9(5)	96.5(5)
k // b	6562.1(5)	6648.3(5)	6585.9(5)	62.4(5)	110(5)
k // c	6562.1(5)	6619.4(5)	6580.7(5)	38.7(5)	75.9(5)

2.4. Magneto‐Chiral Dichroism

Temperature‐dependent MChD measurements were performed in the 4−290 K temperature range with a fast‐ramping (frequency Ω = 0.04 Hz) magnetic field B = ±1.0 T on the same single‐crystals, crystallographic orientations and spectral range of absorption spectroscopy. Several sharp signals are observed in the entire Vis‐NIR range. They correspond to the f–f transitions of the Er^III ion already described whereas no contribution from the chiral ligand is observed (Figure S19). At T = 4.0 K the magnetic‐dipole allowed transition ⁴I_13/2 ← ⁴I_15/2 (|ΔJ| = 1) exhibits a MChD response (ΔA _MChD) one order of magnitude stronger than the transitions in the visible range and g _MChD dissymmetry factors values two order of magnitude higher (Tables S11 and S12). These results are in agreement with Richardson's theory of lanthanide optical activity [27, 28]. The g _MChD dissymmetry factors changes as a function of the probed orientation, with values of ca. 0.2 T⁻¹ along c, ca. 0.4 T⁻¹ along a, and between 0.1 and 1.0 T⁻¹ along b (Table S12).

High‐resolution MChD (HR‐MChD) spectroscopy measurements were carried out at T = 4.0 K and B = ±1.0 T in the 1470–1540 nm spectral range with both the light propagation vector k and the magnetic field B applied along each of the three crystallographic axes. Six sharp and well‐defined MChD signals are observed, originating from the six previously identified transitions within this energy range (Figure 5). Both enantiomers exhibit signals of comparable intensity with opposite sign (Figure S20). The MChD line shapes fall into two categories: absorption‐type and derivative‐like shapes. When the two M _J sublevels of the ground state KD, whose degeneracy is lifted by the applied magnetic field via the Zeeman effect, contribute with the same sign, an absorption‐type profile is observed. When the sublevels contribute with opposite sign, a derivative‐like signal is observed, enabling the resolution of the two transitions originating from the individual M _J sublevels within the KD. This behavior is illustrated by comparing the high‐resolution absorption (B = 0 T) and MChD (B = 1.0 T) spectra at the same temperature (Figure 5). This is particularly evident for the lowest energy transitions (λ < 1500 nm) that show spectral line shapes obtained as the sum of two Gaussian profiles of opposite sign whose energy difference is due to the Zeeman splitting of both the ground and excited states, and the difference in relative intensity from the different population of the two M _J sublevels within the ground state KD under magnetic field. Instead, the higher energy transitions (λ > 1500 nm) show broader spectral profiles that do not allow them to separate well the contributions of the M _J sublevels. Notably, both the sign and the relative intensity of the individual M _J sublevels within each KD exhibit a strong dependence on the orientation of the light propagation vector k with respect to the crystallographic axes, which depends on the composition and g anisotropy of the involved energy states.

FIGURE 5.

Low temperature (T = 4.0 K) high resolution zero field absorption spectra (dotted lines) and MChD spectra of 1‐(R,R,R,R) acquired at B = ±1.0 T (solid lines) for k applied along the three crystallographic axes. Dotted vertical orange lines are a guide to the eyes to visualize the absorptions maxima at zero‐field.

Upon increasing the temperature, the intensity of the signal decreases (Figure S21), indicating that the difference in Boltzmann population between the M_J sublevels of the ground‐state KD split by the magnetic field is responsible for the observed signal (MChD C term) [16, 47]. The same applies to the MChD signal associated to the satellite signals arising from the first two magnetically excited states of the ground state (T = 5–60 K), whose intensity decreases at higher temperatures (T > 60 K). Above 150 K, the signals assume a dispersive line shape and their intensity remains nearly constant up to room temperature, suggesting that the temperature‐independent MChD A term, associated with the zero‐field splitting of both ground and excited states, dominates over the C term, as the difference in thermal population of the M _J sublevels within each KD became similar [6]. This is in agreement with previous studies [12, 19, 25, 48].

Magnetic field dependent HR‐MChD measurements were performed at T = 4.0 K over the same energy range up to 9.0 T. The increase in intensity with increasing field is accompanied by a shift in the energy of the signal maxima, consistent with the Zeeman splitting (Figure S22). A clear splitting of the signal at ca. 1503 nm is observed when k and B are applied along a and b (Figure 6). Stacking the MChD spectra recorded at increasing magnetic fields unambiguously reveals the Zeeman‐induced evolution of the levels’ energies (insets of Figure 6) similarly to what have been shown through absorption measurements under magnetic field (Figure 4).

FIGURE 6.

Field dependent high resolution MChD spectra of 1‐(R,R,R,R) acquired up to B = ±9.0 T at T = 4.0 K with k applied along the three principal crystallographic axes. The insets to the left show MChD stacked spectra, highlighting the Zeeman effect, and the insets to the right represent the integrated area of the entire multiplet (1470–1540 nm range) as a function of the magnetic field, combining the data from low field (up to ±2.0 T) and high field experiments (up to ±9.0 T).

The evolution of the normalized absolute area of the entire multiplet as a function of the magnetic field testifies the anisotropic nature of the MChD signal as differences appear when changing the probed crystallographic orientation, consistent with the magnetic anisotropy of this Er^III complex previously assessed by CTM (Figures 6 and S23). The steepest slope at low field corresponds to the b crystallographic axis, followed by a and then c, which agrees with theoretical calculations and CTM measurements predicting the easy axis being the closest to b, the intermediate axis being the closest to a, and the hard axis being the closest to c (Tables S6 and S10). The difference between a and c is however less pronounced than expected, suggesting an impact of the small angle deviation of k with respect to the easy, intermediate and hard axes of magnetization imposed by the crystal geometry. This orientation‐dependent evolution of the MChD signal's area with magnetic field, is also in line with absorption measurements under high magnetic fields and the orientation‐dependent magnitude of the Zeeman splitting. In both cases we observe some CF components that are more affected by the Zeeman effect than others. For example, the CF component at λ ca. 1515 nm shows a high energy shift along the three directions compared to the lowest energy component at λ ca. 1524 nm that is less affected by the Zeeman effect. For this reason, the MChD spectra at high fields can show spectral overlaps between different CF components that are well separated at low fields (Figure 6).

In contrast to highly axial systems, such as Dy^III‐based complexes, the presence of important transverse anisotropic components enables the MChD signal to remain intense over a wider range of orientations, providing robust spectroscopic responses even away from the principal magnetic axes. This represents an advantage from both an experimental point of view and for practical applications.

3. Conclusion

In conclusion, we have synthesized and investigated a pair of enantiomeric mononuclear Er^III complexes featuring a coordination environment composed of an equatorial hexaazamacrocyclic ligand and triphenylsilanolate axial groups. This coordination environment, which does not ideally accommodate the prolate electron density of the Er^III ground state, gives rise to a complex anisotropic magnetic behavior, with comparable axial and transverse components, and a nontrivial orientation of the magnetic easy axis of the lanthanide center, as determined by CTM.

The determination of the magnetic reference frame and energy level scheme from theoretical calculations alone remains challenging. However, by combining them with CTM and absorption spectroscopy at high fields we have been able to refine the magnetic frame and establish an optimized ab initio computational protocol, leading to an accurate description of the electronic structure of the Er^III ion for its ground and first optical excited state.

Unprecedented high‐resolution MChD measurements on oriented single crystals under magnetic fields up to 9.0 T and absorption measurements up to 30 T have allowed us to provide detailed insights into the magneto‐chiroptical response of this compound, including the optical detection of individual M _J sublevels within the ground and first optical excited KDs and their field‐induced evolution driven by the magnetic anisotropy of the system.

This work establishes MChD as a versatile magneto‐optical probe of the anisotropic behavior of chiral lanthanide complexes, complementary to those based on luminescence [49, 50, 51, 52]. Furthermore, it opens new perspectives for the rational design and characterization of functional lanthanide‐based materials for chiroptical applications, molecular magnetism, and quantum technologies. In particular, further synthetic efforts should target chiral Er^III complexes with a coordination environment adapted to its prolate character, to benefit from their magnetic dipole‐allowed transitions within the optical telecommunication window, enabling exceptionally large g _MChD dissymmetry factors and facilitating optical readout with unpolarized light [12, 25, 26]. This strategy could be of high interest in the development of new low‐weight and versatile molecular optical components for telecommunication applications.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

The authors have cited additional references within the Supporting Information [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. Deposition numbers 2548335 (for 1‐(S,S,S,S)), 2548336 (for 1‐(R,R,R,R)), contain the supporting crystallographic data for this paper. These data are provided free of charge by the joint Cambridge Crystallographic Data Centre.

Supporting File 1: anie73539‐sup‐0001‐SuppMat.pdf.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.