# Determination of magnetic anisotropy tensors in actinide complexes using torque magnetometry: a U(IV) case study

Leonardo Tacconi, Adebayo Victor, Chelazzi Laura, Claude Berthon, Hélène Bolvin, Mauro Perfetti

► To cite this version:

Leonardo Tacconi, Adebayo Victor, Chelazzi Laura, Claude Berthon, Hélène Bolvin, et al., Determination of magnetic anisotropy tensors in actinide complexes using torque magnetometry: a U(IV) case study. Journal of the American Ceramic Society, 2025, 2025, 10.1021/jacs.5c16992. cea-05424675

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Distributed under a Creative Commons Attribution 4.0 International License
Determination of magnetic anisotropy tensors in actinide
complexes using torque magnetometry: a U(IV) case study

Leonardo Tacconi$^{a\mathrm{f}}$, Victor Adebayo$^{b\mathrm{f}}$, Laura Chelazzi$^c$, Claude Berthon$^d$, Hélène Bolvin*$^b$, Mauro Perfetti*$^a$

a. Department of Chemistry "Ugo Schiff", University of Florence and INSTM Research Unit, Via della Lastruccia 3-13, 50019 Sesto Fiorentino (FI), Italy.

b. Laboratoire de Chimie et Physique Quantiques, CNRS, Université de Toulouse, 31062 Toulouse, France.

c. Centro di Servizi di Cristallografia Strutturale, CRIST, 50019 Sesto Fiorentino (FI), Italy.

d. CEA, DES, ISEC, DMRC, Université Montpellier, Bagnols-sur-Cèze 30207, France.

* mauro.perfetti@unifi.it; bolvin@irsamc.ups-tlse.fr

£ These authors equally contributed to this work.
**Abstract**

In this study, we employed cantilever torque magnetometry to probe the magnetic anisotropy of a single crystal of U(DOTA)(H₂O) (DOTA = 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid), marking what is, to our knowledge, the first application of this technique to an actinide-based molecular system. Combining cantilever torque magnetometry data with *ab initio* calculations allowed us to determine the magnetic anisotropy tensor of the uranium center. The resulting parameters enabled direct comparison with the isoelectronic lanthanide complex previously reported in the literature, as well as with *ab initio* predictions for the theoretical Es⁴⁺ isostructural analogue. We find that the magnetic anisotropy axes of U(DOTA)(H₂O) (5f²) closely align with those of the Pr³⁺ (4f²) and Dy³⁺ (4f⁹) analogues, and with those predicted for Es(DOTA)(H₂O) (5f⁹). The combination of CTM and electronic-structure calculations was essential to describe the weakly paramagnetic, EPR-silent, non-Kramers ground state of U(DOTA)(H₂O), despite the added complexity of four noncollinear molecules in the unit cell. Overall, these results demonstrate that cantilever torque magnetometry provides a powerful route to characterize the magnetic anisotropy of actinide complexes and offers a valuable experimental benchmark for validating computational approaches to 5f-element magnetism.

**Introduction**

The study of the magnetic properties of actinide complexes is driven by fundamental questions concerning the nature of the 5f orbitals. Actinide complexes offer a unique platform for exploring the interplay between strong spin-orbit coupling, covalency, and low-symmetry ligand fields.¹,² Unlike 4f orbitals in lanthanide ions, 5f orbitals are more spatially extended and participate more actively in bonding³⁻⁵, often leading to complex electronic structures⁶. While this scenario has prompted theoretical speculation that actinides might exhibit enhanced magnetic anisotropy and improved spin dynamics, experimental realizations of competitive molecular structures encompassing actinides remain limited. For example, most reported actinide single molecule magnets display modest blocking temperatures and rapid magnetic relaxation, with a few notable exceptions.⁷⁻¹⁰

However, their characterization is hindered by the challenges associated with radioactivity and the scarcity of techniques capable of probing their magnetic
behavior at very low temperature. Spectroscopic measurements (Magnetic Circular Dichroism, X-rays absorption spectroscopy etc) offer rich insights into the electronic structure<sup>11-15</sup>, while hints on the magnetic anisotropy can be obtained exploiting Electron Paramagnetic Resonance (EPR) selection rules<sup>16,17</sup>, but these same rules can render complexes EPR silent, and temperature-induced peak broadening limits studies across wide temperature ranges. Variable temperature paramagnetic NMR can also offer valuable information<sup>18-20</sup>, but it is usually not measured at cryogenic temperatures and often restricted to solution studies.<sup>21-23</sup> Indeed, most magnetic characterization of actinide complexes in solid state is limited to bulk SQUID magnetometry on powder samples,<sup>13,24-31</sup> which lacks directional sensitivity and obscures anisotropic features due to averaging effects. The amount of information that can be extracted is often insufficient to provide definitive experimental insight into phenomena tied to fine details of eigenstates and eigenvalues – such as quantum tunnelling between states, which strongly depends on state composition<sup>32,33</sup>, or the molecular design required to achieve specific functionalities related to magnetic anisotropy.<sup>34</sup> Ab initio calculations, although frequently employed to address these questions, still require experimental validation. Consequently, a significant gap remains in the experimental literature concerning the direct investigation of magnetic anisotropy in actinide complexes in the solid state.

In this context, Cantilever Torque Magnetometry (CTM) represents a powerful tool for
molecular systems. The technique is known to deliver information about the magnetic
anisotropy axes³⁵, the crystal field (or zero field splitting) parameters³⁶,³⁷, and the
thermal and field evolution of magnetic anisotropy.³⁸,³⁹ However, its usage has been
so far limited to lanthanide and transition metal complexes.⁴⁰,⁴¹ A handful of studies
on actinide intermetallic compounds using CTM are present in the literature, but the
technique is primarily employed to probe critical fields connected to the onset of
superconductivity.⁴²⁻⁴⁵ This gap is probably due to the complexity in interpreting the
torque curves, especially from systems containing noncollinear anisotropy tensors,
and the absence of user-friendly programs to fit the data. Nevertheless, CTM
measurements do not rely on selection rules, offering a route to quasi-spectroscopic
information⁴⁶ particularly valuable for non-Kramers ions that are often EPR-silent⁴⁷,⁴⁸
and tricky to characterize by standard magnetometry alone⁴⁹,⁵⁰.

Here, we present the first CTM measurement on a molecular actinide complex,
demonstrating its capability to reveal the magnetic anisotropy and even fine details
of the electronic structure of 5f elements. Our results show that CTM can uncover
rich information about the magnetic behavior of actinide complexes, offering a
valuable complement to traditional methods and paving the way for a deeper
understanding of 5f-elements magnetochemistry.

**Results and discussion**

**Rationale for the choice of the molecule**

To demonstrate that CTM is a broadly applicable technique capable of addressing
specific gaps in the literature, we selected a uranium-based, X-band EPR-silent
molecule with a weakly magnetic ground state and a crystal packing featuring several
non-collinear anisotropy tensors – conditions that are challenging to investigate using
conventional single-crystal magnetometry. Among candidates meeting these criteria,
the complex U(DOTA)(H₂O) (H₄DOTA = 1,4,7,10-tetraazacyclododecane-1,4,7,10-
tetraacetic acid) proved particularly suitable. Tetravalent uranium (5f²) usually
exhibits a well-isolated (EPR silent) ground state.⁵¹ This molecule possesses a simple
chemical structure (Figure 1a) but crystallizes in the orthorhombic crystal system, with
several symmetry related molecules in the unit cell. While DOTA-based complexes of
lanthanides are extensively studied for their applications in magnetic resonance
imaging⁵²,⁵³, radiotherapy⁵⁴⁻⁵⁶, and luminescence⁵⁷,⁵⁸, their actinide analogues have
received less attention. Nevertheless, the Na[Cf(DOTA)(H₂O)] complex has been
recently reported to be the first californium complex exhibiting slow relaxation of the
magnetization.⁵⁹ In the case of uranium, we noticed that there are only two
publications reporting structures where U binds the unsubstituted DOTA ligand,
focussing on the redox activity of the U center.⁶⁰,⁶¹

**Crystal packing and molecular structure**
<figure><img src="image_2.png" /><figcaption>b)</figcaption></figure>

<figure><img src="image_3.png" /></figure>

Figure 1 – (a) Molecular structure of U(DOTA)(H₂O). (b) Unit cell of the complex showing the four symmetry-related U(DOTA)(H₂O) molecules. Color code: uranium (green), oxygen (red), nitrogen (pale blue), carbon (black). Hydrogen atoms and solvent molecules omitted for clarity.

The compound has been synthesized as reported in the Experimental Section (see SI for further details). A single crystal X-ray diffraction (SC-XRD) experiment on a clear green, block-shaped single crystal revealed that U(DOTA)(H₂O) crystallises in the orthorhombic space group Pca2₁ (n. 29), with Z = 4 (Figure 1b). All the crystallographic data and refinement details are summarized in Table S1. Notably, no proper symmetry elements pass through the uranium centers, indicating a C1 molecular point group symmetry for the complex. The uranium center is coordinated by the DOTA ligand in a κ⁸ binding mode, occupying the central cavity of the macrocycle and binding through four nitrogen atoms and four oxygen atoms. The coordination sphere is completed by an axial water molecule, resembling the coordination environment found in trivalent lanthanide-DOTA complexes.³⁵ However, unlike the tripositive lanthanide analogues, U(DOTA)(H₂O) is overall neutral, due to the +4 oxidation state of the uranium center. Accordingly, no counterions are present in the crystal structure, only water molecules. An analysis of the uranium coordination environment reveals a close resemblance to a spherical capped squared antiprism, with a dihedral angle of 39.13° between the donor atom planes defined by the oxygen and nitrogen atoms. Within the unit cell, there are four molecules (see Figure 1b), related by crystallographic C₂ axes aligned with the orthorhombic main axes (a, b, c). Each molecule may contribute distinctively to the overall magnetic behaviour, with the total magnetic anisotropy of the complex arising as the sum of the individual molecular anisotropies. This poses a significant challenge for determining the molecular magnetic anisotropy. Additionally, low temperature X-band EPR experiments performed on a single crystal along two different crystallographic orientation did not
give any measurable signal (Figure S1), indicating the singlet nature of the ground state.

## Crystal anisotropy

To investigate the magnetic anisotropy of U(DOTA)(H₂O), we performed CTM measurements on an oriented single crystal. Information regarding sample preparation are reported in Supplementary Note 1. Due to the presence of four magnetically inequivalent molecules within the unit cell, we carried out three distinct rotations of the sample, each aligned with a principal crystallographic axis, as illustrated in Figure S2. This choice reduces the number of magnetically inequivalent molecules in each measurement to two as the rotation along a C₂ symmetry axis makes pairs of susceptibility tensors projections in the perpendicular plane equal. To better visualize this, the molecular orientations of the four molecules at the start of each rotation are also depicted in Figure S2.

The high sensitivity of the technique allowed to conduct measurements across a broad temperature (2-260 K) and magnetic field (5-9 T) range, giving access to the ground state as well as to the excited state(s) contributions to the magnetic anisotropy. The torque curves obtained at *B* = 9 T are reported in Figure 2 while all the other temperatures and fields are presented in Figure S3-S5. The zero torque angles, always 0° and 90°, are indicative of vanishing torque when a principal crystallographic axis is oriented along the magnetic field, as imposed by symmetry. Moreover, the values of torque are remarkably small⁴⁰, and the curves shape is sinusoidal, indicating that the system is always within the linear response regime (i.e. the magnetization versus the magnetic field is linear⁴⁰). This hints towards a poorly anisotropic ground state of the molecules.

At low temperature, no measurable signal was detected during the first rotation (Figure 2a), indicating that the crystallographic *ac* plane is nearly magnetically isotropic. In contrast, the phase of the signal obtained in the other two rotations (Figure 2b-c) identifies the *b*-axis as the hard-axis of the crystalline magnetic anisotropy. Moreover, the almost equal signal intensity recorded in rotation 2 and 3 support that the *a* and *c* crystallographic axes are magnetically equivalent. Taken together, these observations suggest that, at low temperature, the U(DOTA)(H₂O) crystal exhibits a perfect easy-plane anisotropy, with *b* being the hard axis of the crystal.
As the temperature increases, the overall signal intensity decreases, consistent with the thermal suppression of magnetic anisotropy. However, the temperature dependence of the signal differs significantly across the three rotations. Notably, the signal associated with the first rotation increases with the temperature (Figure 2a), indicating that the *ac* plane becomes progressively more anisotropic. At T = 260 K and B = 9 T, the signals from the first and third rotations become comparable in magnitude, though both remain smaller than the signal from the second rotation. These results suggest that, at high temperatures, the magnetic anisotropy evolves such that the *a*-axis becomes the easy axis, the *b*-axis remains the hard axis, and the *c*-axis behaves as the intermediate axis. This trend can be reconducted to a marked rhombicity in the *ac* plane.
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

Figure 2 – Experimental (dots) and simulated (solid lines) CTM curves acquired on U(DOTA)(H₂O) at 9 T and different temperatures (dashed lines) during first (a), second (b) and third (c) rotations. Simulations were obtained considering crystalline magnetic anisotropies, as discussed in text.
To extract quantitative information about the crystalline magnetic anisotropy, we performed a fit of the experimental CTM results. The magnetic torque $\vec{\tau}$ is given by the cross product of the magnetization $\vec{M}$ of the sample and the applied magnetic field $\vec{B}$, both expressed in the orthogonal crystallographic abc reference frame. Considering that in the linear regime $\chi = \vec{M}/\vec{B}$, and that in our experimental setup only the component of the torque vector aligned with the rotation unitary vector ($\vec{r}_{abc}$), denoted as $\tau_r$, can be measured, the following equation applies:

$$ \tau_r = (\vec{M}_{abc} \times \vec{B}_{abc}) \cdot \vec{r}_{abc} = [(\chi^{\text{cryst}} \cdot \vec{B}_{abc}) \times \vec{B}_{abc}] \cdot \vec{r}_{abc} \quad \text{Eq. 1} $$

where $\chi^{\text{cryst}}$ is the diagonal 3x3 crystalline magnetic susceptibility tensor expressed in the abc orthogonal reference frame. By defining the susceptibility anisotropies $\Delta\chi_{jk}$, as:

$$ \Delta\chi_{jk} = \chi_{jj} - \chi_{kk} \quad \text{Eq. 2} $$

with $j, k \in \{a, b, c\}$, eq. 1 can be expanded in terms of tensor and vector components, leading to the following compact and general form

$$ \tau_i = \frac{1}{2} \sum_{j,k} \varepsilon_{ijk} \Delta\chi_{jk} B_j B_k r_i \quad \text{Eq. 3} $$

where $i, j, k \in \{a, b, c\}$ and $\varepsilon_{ijk}$ is the Levi-Civita symbol. A full derivation of Equation 3 is provided in Supplementary Note 2. This equation highlights that CTM is extremely sensitive to the differences between the magnetic susceptibility components (i.e. the magnetic anisotropy), rather than their absolute values. This peculiarity makes it complementary with the more widespread Single Crystal Magnetometry.

The simulated curves obtained using this approach are presented in Figure 2, and the extracted anisotropy parameters are shown in Figure 3a. These results are consistent with the qualitative analysis already presented, confirming an easy-plane crystalline magnetic anisotropy, with increasing rhombicity as the temperature rises. A visual depiction of this thermal evolution is shown in Figure 3b.

The torque measurements allowed to unequivocally map the crystal anisotropy. However, this temperature-dependent evolution of the crystal magnetic anisotropy is caused by changes of the molecular magnetic anisotropy. Even though in many cases the torque signal can be used to directly extract single molecule anisotropies$^{37,62,63}$, the present case is exceptionally complicated as it requires disentangling four
contributions belonging to poorly anisotropic tensors. To gain further insight into the
origin of this anisotropy and its thermal behaviour and obtain a reliable starting guess
for the fit of the CTM curves, we performed *ab initio* calculations.

<figure><img src="image_7.png" /><figcaption>Figure 3 – (a) Crystalline magnetic anisotropies $\Delta\chi_{ij}$ extracted from simulations of the experimental CTM measurements performed on U(DOTA)(H$_2$O). Lines connecting the experimental points are a guide to the eye. (b) Visual representation of the thermal evolution of the crystalline magnetic susceptibility tensor. At low temperatures the *ac* crystallographic plane is isotropic, while it becomes more rhombic at higher temperatures.</figcaption></figure>

Molecular anisotropy

*Ab initio* calculations (details in the Experimental Section) enabled us to predict the
principal values of the susceptibility tensor of the complex, and its orientation. Even
though the structure does not possess any real symmetry element, our calculations
predict that the hard axis lies almost exactly along the U–OH$_2$ bond, perpendicular to
the equatorial plane defined by the DOTA ligand architecture. Such result suggests
that the crystal field can be seen as *quasi* tetragonal. This is further corroborated by
the magnitude of the crystal field parameters reported in Table S2. Indeed, only the
three diagonal and two off-diagonal crystal field parameters compatible with
tetragonal symmetry ($\overline{B}_4^4$ and $\overline{B}_4^6$, Wybourne formalism) possess relevant values.
Further comments on the symmetry and predicted electronic structure of
U(DOTA)(H$_2$O) are reported in Supplementary Note 3.

Starting from *ab initio* calculations, it is possible to simulate the experimental
magnetic torque, in a manner analogous to the treatment of crystalline magnetic
anisotropy, with an additional step. Indeed, to simulate experimental measurements,
it is necessary to build the crystallographic tensor $\chi^{cryst}$ as sum of the magnetic susceptibility tensors of the four molecules in the unit cell $\chi_i^{mol}$ ($i = 1, 2, 3, 4$).

This transformation is accomplished using a Euler rotation matrix $R_i$, which relates the molecular and crystallographic frames. The transformed susceptibility tensor is given by the following relation.

$$ \chi^{cryst} = \sum_{i=1}^{4} R_i^T \cdot \chi_i^{mol} \cdot R_i \qquad \text{Eq. 4} $$

Notice that the four Euler matrices corresponding to the four molecules of the unit cell can be derived from each other by applying symmetry operations. Specifically, the molecules are related by three distinct $C_2$ symmetry operations along the three crystallographic axes. The Euler matrices for all four centers obtained from *ab initio* calculations are provided in Table S3. Magnetic torque is then calculated using the transformed tensors, and the total torque is obtained by summing the torque contributions from the four uranium centers.

Initially, experimental CTM curves were simulated considering the rotation matrix and magnetic susceptibility tensors obtained from *ab initio* calculations. Results are shown in Figures S6-S14. *Ab initio* parameters lead to an overall fair agreement between the experiments and simulations, with the most prominent disagreement at low temperatures. In Figures S6-S14 we also reported the contribution of each uranium center to the overall signal. As discussed before, in each rotation we can identify two couples of uranium centers that behave identically, coherently with symmetry constraints previously discussed.

To refine the parameters extracted from *ab initio*, we decided to perform a fitting procedure of both the molecular magnetic susceptibility principal values and rotation matrices $R_i$. More details about the fitting procedure are reported in the Experimental Section. The fitted Euler matrices for all four centers obtained from *ab initio* calculations are provided in Table S4. Using the magnetic susceptibility tensor obtained from the fitting procedure (after selecting one of the four solutions obtained from the fitting of the reference frame, see Figure S24) the agreement between experimental results and simulations becomes excellent, with simulated signals shown in Figures S15-S23. A comparison between the fitted and calculated magnetic susceptibilities is reported in Figure 4a, while Figure 4b presents the superimposed principal axes. A comparison between selected *ab initio* and fitted $\chi$ values are
reported in Table S5. In both cases, the deviations from the *ab initio* values are minimal (2°, 1° and 2° along the x-, y- and z-axes, respectively), probing that the initial calculations were already remarkably accurate and validating the method.

<figure><img src="image_8.png" /><figcaption>Figure 4 – (a) Comparison between the magnetic susceptibilities obtained from *ab initio* calculations (dots) and fitting procedure of the experimental results (solid lines), as discussed in main text. (b) Magnetic reference frame obtained from *ab initio* calculations (x: pink, y: light green, z: cyan) and from the fitting procedure of experimental results (x: red, y: green, z: blue). Atoms color scale: U, green; C, black, O: red, N: pale blue.</figcaption></figure>

The experimental validation of the *ab initio* method allows an experimentally supported discussion of the magnetic anisotropy thermal evolution. Across the entire temperature range (2-260 K) the molecular magnetic anisotropy of U(DOTA)(H₂O) is easy axis (x in Figure 4b). To emphasize the reshaping of the magnetic anisotropy, the magnetic susceptibility anisotropies have been normalized with respect to the largest component, Δχₓₓ, and are presented in Figure 5. Notably, Δχᵧᵧ increases with temperature, indicating an enhancement of the rhombicity within the magnetic yz plane while Δχₓᵧ decreases. This behaviour suggests that thermal effects are crucial in reshaping the magnetic anisotropy of this complex. *Ab initio* calculations provide an insight into this phenomenon. Since the states are singlets (see Supplementary Note 3 and Table S6), the contribution to the magnetization uniquely arises from the coupling between states. The first excited state |1⟩ lies 169 cm⁻¹ above the ground state |0⟩. The matrix element ⟨0|**M**|1⟩ (where **M** is the magnetic moment operator) lies along x and this coupling provides the main contribution to χₓ. The coupling with the second excited state |2⟩ at 427 cm⁻¹ ⟨0|**M**|2⟩ is instead along y. This is the main contribution to χᵧ. This contribution is smaller both because the coupling is slightly
smaller (2.3 and 1.7 µ<sub>B</sub>, respectively) and, more importantly, because the energy gap
is much larger. At 100 K the first excited state begins to be significantly populated,
contributing to reducing susceptibility along *x* (i.e. reducing Δχ<sub>xy</sub>).

<figure><img src="image_9.png" /><figcaption>Figure 5 – Magnetic anisotropies normalized by the Δχ<sub>xz</sub> anisotropy obtained from ab initio calculations (dots) and fitting procedure of the experimental results (solid lines). Increasing the temperature the anisotropy Δχ<sub>yz</sub> increases, pointing towards an increase of the rhombicity of the system.</figcaption></figure>

The information extracted from the torque measurements allow to compare the
magnetic anisotropy of our complex with the literature. Perfetti and co-workers
reported the so-called *f*<sup>n+7</sup> effect<sup>35</sup>, i.e. the orientation of the magnetic anisotropy
tensors of ions in isostructural lanthanide complexes differing by seven *f* electrons
coincides. Such effect was discovered on lanthanide DOTA complexes. The most
immediate, purely experimental, comparison can be done between the magnetic
anisotropy principal axes of U(DOTA)(H<sub>2</sub>O) (5*f*<sup>2</sup>) and the ones of the lanthanide-
based isoelectronic structure, Na[Pr(DOTA)(H<sub>2</sub>O)] (4*f*<sup>2</sup>) and of the lanthanide
differing by 7 *f* electrons, Na[Dy(DOTA)(H<sub>2</sub>O)] (4*f*<sup>9</sup>). Despite the different complex
charge, this comparison is meaningful because the coordination geometry is very
similar (e.g. the torsion angles between the carboxylic O and the closest N is 38.1°,
40.5° and 38.7° for complexes containing Pr<sup>3+</sup>, Dy<sup>3+</sup> and U<sup>4+</sup>, respectively). We notice
that the magnetic principal axes are, also in this case, almost coincident, as visually
reported in Figure 6. Interestingly, the easy axis points close to the closest O-M-O
distance (M = Pr³⁺, Dy³⁺, or U⁴⁺) in all derivatives (horizontal in all the panels of Figure
6), pinpointing that the magnetic anisotropy reference frame in our complex is mostly
determined by electrostatic effects. Notably, this observation is independent from the
degree of bond covalency, typically larger in actinides compared to lanthanides.³,⁶⁴⁻⁶⁶

The experimental validation of the theoretical method prompted the possibility to
theoretically verify the *f* <sup>n+7</sup> effect also in the actinide series. To this aim, we have
performed *ab initio* calculations on the experimentally inaccessible Es<sup>4+</sup> isostructural
analogue of U(DOTA)(H<sub>2</sub>O). The structure on which the calculations were performed
has been obtained by simple substitution of the metal ion inside the complex and
therefore constitutes an approximation. Magnetic susceptibility tensor and energy
level structure are reported in Table S7 and S8, respectively. We find that the reference
frame of Es(DOTA)(H<sub>2</sub>O), calculated at T = 2 K and B = 5 T, is closely aligned with the
one of U(DOTA)(H<sub>2</sub>O) (Figure 6), suggesting that the *f* <sup>n+7</sup> effect could extend to the
actinide series. The easy axis of the structure clearly points, also in this case, along the
shortest O-Es-O direction. The other two directions (intermediate and hard) are
swapped with respect to the other derivatives. However, we note that the hard and
intermediate directions are extremely similar (*g*<sub>hard</sub> = 0.04 and *g*<sub>interm</sub> = 0.1), making
their assignment difficult. Importantly, the anisotropy of the Es-derivative is much
more pronounced compared to the U-based analogue, but this is not surprising, as
observed in the lanthanide series.³⁵ Finally, an interesting comparison can be made
between Es(DOTA)(H<sub>2</sub>O) and its lanthanide isoelectronic analogue,
Na[Dy(DOTA)(H<sub>2</sub>O)]: the complexes stabilize a remarkably similar axial ground
doublet (*g* = [0.04, 0.1, 19.3] and *g* = [0.255, 0.668, 19.238] ³⁵, respectively).
<figure><img src="image_10.png" /><figcaption>Figure 6 – Comparison between the experimentally determined magnetic anisotropy reference frames of **Na[Pr(DOTA)(H₂O)]**, **Na[Dy(DOTA)(H₂O)]** (data obtained from literature³⁵), **U(DOTA)(H₂O)** and **Es(DOTA)(H₂O)** at T = 2 K and B = 5 T. The three molecules were aligned fixing the M-O (M = metal) shortest distance (indicated with an asterisk) along the horizontal axis.</figcaption></figure>

Conclusions

The magnetic anisotropy of a single crystal of **U(DOTA)(H₂O)** was measured using cantilever CTM, applied here for the first time on an actinide complex. While the
crystal magnetic anisotropy was unequivocally mapped with CTM, the synergistic combination of *ab initio* calculations and torque data in a broad range of temperature and fields was key to describe the molecular magnetic anisotropy in detail. We therefore extracted the orientation of the anisotropy tensors, allowing to meaningfully compare the magnetic anisotropy of our complex with the 4f² (Pr³+) and 4f⁹ (Dy³+) isostructural lanthanide complexes. The magnetic anisotropy principal axes of the three complexes almost coincide. *Ab initio* calculations performed on the hypothetical Es⁴⁺ (5f⁹) isostructural analogue suggest that the easy axis of this complex should also coincide with the easy axis direction of the other three complexes, suggesting that the *f*<sup>n+7</sup> effect could be active also in the 5f series. Notably, the chosen complex was intended as a proof of concept for a particularly challenging system – namely a 5f² non-Kramers ion with a weakly magnetic ground state and four magnetically inequivalent molecules in the unit cell. Consequently, we anticipate that uranium complexes with stronger anisotropy and/or crystal structures featuring collinear tensors can be mapped with even greater accuracy. This would provide a valuable route to extract experimentally elusive information and to benchmark *ab initio* methods for actinide complexes

# EXPERIMENTAL SECTION

## Synthesis

The complex (UDOTA)(H₂O) was synthesized in two steps. See Supplementary Note 3 for details and reaction scale. First, the U(DOTA)(DMSO) was prepared by complexation of UCl₄ with DOTA(TEA)₄ in a dimethyl sulfoxide (DMSO) phase in a similar way to that proposed by Kent et al. in 2019.⁶⁰ The UCl₄ reagent was prepared from uranyl nitrate in aqueous solution. After uranyl reduction with Rongalite (HOCH₂SO₂Na), UIV was precipitated with NaOH, washed and recovered with concentrated HCl. The UCl₄ solution was dried at room temperature under nitrogen flow. Commercial H₄DOTA was neutralized by stirring a stoichiometric amount of triethylamine (TEA) in DMSO for 1 hour whereupon a large amount of a white compound precipitates at room temperature. This solid was added to UCl₄ in DMSO (a slight DOTA excess was used) and heated at 60°C for 12 hours. The formation of U(DOTA)(DMSO) was monitored by ¹H NMR spectroscopy. After completion of the reaction, the complex was precipitated by dropwise addition of this DMSO solution in a CHCl₃ phase. The resulting powder was washed three times with THF then dried at room temperature under nitrogen flow.
In a second step, an aqueous solution was prepared by dissolving **U(DOTA)(DMSO)** in a minimum amount of water and **U(DOTA)(H₂O)** complex was subsequently precipitated by addition of THF (tetrahydrofuran). To ensure DMSO removal, this operation was repeated several times. The formation of **U(DOTA)(H₂O)** was confirmed by ¹H NMR spectroscopy (see Figure S27). The final light green solid was redissolved in a minimal amount of water and repeatedly washed with THF. The biphasic mixture was centrifuged for 3 minutes at 3000 rpm, after which the upper THF phase was discarded. This washing-centrifugation cycle was repeated until the THF phase became completely colourless. Slow evaporation of this aqueous solution led to the deposition of crystals.

**Caution!** Natural uranium (mainly the isotope $^{238}$U) is a weak a-emitter (4.5·10⁹ years half-life). Synthesis of U(DOTA)(H₂O) complex has been carried out under fume hoods in laboratory LN1 of the ATALANTE facility. Recrystallization and physical measurements (CTM and Single Crystal X-Ray Diffraction) were performed on the final compound (<1kBq shipping) at the Chemistry Department of the University of Florence.

## Structural characterization

Single crystal X-ray diffraction (SCXRD) data of **U(DOTA)(H₂O)** were collected at 100 K using a Bruker D8 Venture diffractometer equipped with a PHOTON II detector and a microfocus source (Cu-Kα radiation, λ = 1.54184 Å). Frames were collected with the Bruker APEX4 program suite, then integrated and reduced with the Bruker SAINT software. The crystal structure was solved and refined using the Bruker SHELXTL software package. Crystallographic data and refinement parameters are reported in Table S1. SCXRD data have been deposited in the CCDC, the deposition number is 2478981. Molecular plots were produced using the program Mercury⁶⁷ and VESTA⁶⁸.

## Cantilever torque magnetometry measurements

Torque measurements were carried out on a Torque Magnetometry insert of a Quantum Design PPMS. A single-crystal sample of **U(DOTA)(H₂O)** was measured in a wide range of temperatures (2-250 K) and magnetic fields (0-9 T). During each measurement, the crystal was rotated by 180° around an axis perpendicular to the external magnetic field.

## Ab initio calculations
Calculations were performed on the XRD structure using the MOLCAS 7.8 package.⁶⁹
The positions of the hydrogen atoms were optimized using B3LYP KS-DFT.

The spin free complete active space self-consistent field (SF-CASSCF) calculations⁷⁰ were performed with the active space being the seven 5f orbitals and the associated electrons *i.e* CAS (2, 7). In the state-averaged calculations, we used 21 triplets and 27 singlets. Subsequently, the dynamical correlation was added using the multi state complete-active-space perturbation-theory at the second order (MS-CASPT2) method⁷¹ with a level shift of 0.3 a.u. and 63 frozen orbitals. As shown previously⁷² it is important to correlate actinide 5d orbitals. Calculations were done with the ANO-RCC basis sets⁷³, TZP for U, O and N atoms, DZP for C and DZ for H atoms. The scalar relativistic effects accounted for by the means of the Douglas-Kroll-Hess transformation.⁷⁴⁻⁷⁶ In the second phase of the calculations, spin-orbit coupling was introduced via state interactions between the previously calculated spin-free states using the restricted active space state interaction method⁷⁷ (RASSI) leading to the SO-MCASPT2 states. The spin-orbit (SO) integrals are calculated using the atomic mean-field integrals (AMFI) approximation⁷⁸. The susceptibility tensor is calculated according to literature.⁷⁹

Calculations for the (EsDOTA)(H₂O) was done by a simple substitution of the U in the XRD UDOTA.H₂O structure metal with Es. In the state-averaged calculations, we used 21 sextets, 48 quartets and 35 doublets. The dynamical correlation was added with the same level shift as in the UDOTA with the same number of frozen orbitals. The same basis sets were used as in the UDOTA, SO-MCASPT2 calculated from RASSI and the tensor calculated in the same manner as the U.

The crystal-field parameters were calculated based on the AILFT⁸⁰ using the ORCA 5.0.3 quantum chemistry package⁸¹, with SARC-DKH-DEF2-TZVP(-f) basis sets⁸²,⁸³ and AUTOAUX feature⁸⁴ to automatically generate auxiliary basis sets for the resolution of identity approximation (RI-JK)⁸⁵. Scalar relativistic effects were accounted for using the second-order scalar relativistic Douglas Kroll Hess (DKH2) Hamiltonian formalism. The CASSCF calculation was performed with a CAS(2,7) with 21 triplets. Three structures were considered: the **UDOTA** (without its apical water molecule), symmetrized to the C4 point group, the symmetric **U(DOTA)(H₂O)** and the **U(DOTA)(H₂O)** in its XRD structure.

**Fitting procedure**
Cantilever torque magnetometry data were fitted to extract the magnetic susceptibility tensor components as described in text. The fit was performed with a home-made script in MATLAB (R2025a) using a machine-learning approach based on Gaussian process regression (fitrgp) combined with a lower confidence bound (LCB) acquisition function $\mu - 2\sigma$. At each iteration, the model error was quantified as $1 - R^2$, defined as$^{86}$:

$$err = 1 - R^2 = \frac{\sum_{i=1}^{n} (Y_i - S_i)^2}{\sum_{i=1}^{n} (\bar{Y} - Y_i)^2}$$

where the summation runs over the number of points $n$, $Y$ is the experimental data vector, $\bar{Y}$ its mean value, and $S$ the simulated data vector. The parameter space was sampled with 5000 trial points generated by Latin hypercube sampling (lhsdesign), which provides a more uniform and efficient coverage of the multidimensional space than purely random sampling. The model selected the next point to evaluate according to the LCB function. Optimization proceeded until the error dropped below a set threshold of 0.01 or the maximum number of iterations (100) was reached. This machine-learning strategy was employed to efficiently explore the parameters space and avoid stagnation in local minima.

## SUPPORTING INFORMATION

EPR spectra, crystal mounting and information regarding CTM, more CTM curves and fittings, NMR spectra, crystal information, Crystal Field parameters.

## ACKNOWLEDGMENTS

The financial support provided by the MUR - Dipartimenti di Eccellenza 2023 -2027 (DICUS 2.0) (ref. no. B96C1700020008) to the Department of Chemistry “Ugo Schiff” of the University of Florence is acknowledged. Lorenzo Sorace is kindly acknowledged for helping with the acquisition of the EPR spectrum.
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