Andrea De Maio

Simon Lacroix

Deep Bayesian ICP Covariance Estimation

Covariance estimation for the Iterative Closest Point (ICP) point cloud registration algorithm is essential for state estimation and sensor fusion purposes. We argue that a major source of error for ICP is in the input data itself, from the sensor noise to the scene geometry. Benefiting from recent developments in deep learning for point clouds, we propose a data-driven approach to learn an error model for ICP. We estimate covariances modeling data-dependent heteroscedastic aleatoric uncertainty, and epistemic uncertainty using a variational Bayesian approach. The system evaluation is performed on LiDAR odometry on different datasets, highlighting good results in comparison to the state of the art.

I. INTRODUCTION

With the steady increase in 3D sensors availability, point clouds have seen a rapid diffusion and adoption. Their use benefits from the 3D information that they readily express, which allows to easily retrieve geometric properties and yield the possibility to extract complex geometric features to classify object shapes, perform scene segmentation, or assess complex situations. Point clouds are also used to solve motion estimation and self-localization problems in robotics. Scan registration algorithms provide an estimate of the 3D transformation between the positions at which two point clouds are acquired. Such processes are at the heart of LiDAR-based odometry [START_REF] Zhang | LOAM: Lidar Odometry and Mapping in Real-time[END_REF] and SLAM frameworks [START_REF] Mendes | ICP-based pose-graph SLAM[END_REF]. Originally introduced in [START_REF] Besl | A Method for Registration of 3-D Shapes[END_REF], the Iterative Closest Point (ICP) algorithm, in its many variants, gradually became the standard approach to the point-cloud registration problem [START_REF] Pomerleau | A review of point cloud registration algorithms for mobile robotics[END_REF].

In order to qualify the result of ICP, and especially to integrate it within a localization framework, it is essential to estimate its uncertainty. ICP errors stem from various error sources. Among those, the structure of the scene plays an important role: underconstrained situations, such as corridors or mostly flat environments, yield larger errors than geometrically more complex scenes (Fig. 1).

Building on the advances in deep learning techniques dealing with point clouds, we present an approach that estimates data-dependent error models for the ICP process in form of a covariance matrix. Our method learns heteroscedastic aleatoric uncertainty from ICP input data and uses Bayesian posterior approximation to capture the epistemic uncertainty.

The following section introduces the ICP process, reviews the various sources of errors and works relevant to ours: methods that estimate ICP covariance, data-driven approaches that have proven successful to estimate covariances for other estimation processes, and point-cloud based deep learning techniques. Section III formalizes the problem and depicts the learning process. Section IV evaluates our results against the state-of-the-art, showing an accurate correlation between the real error of ICP and our predicted covariance.

II. BACKGROUND AND RELATED WORK

A. Iterative Closest Point process

ICP tackles the problem of aligning a reading point cloud P ∈ R 3×n to a reference point cloud Q ∈ R 3×m , estimating the true rigid transformation between them. Using an initial estimation Ť ∈ SE(3) of the transformation (e.g as provided by a prior on the motion), the rigid transformation T ∈ SE(3) that estimates the true transformation is the one that minimizes the error function e:

T = arg min T (e(T (P), Q)) (1)
where T (C) is the transformation T applied to the point cloud C. The error function is the result of a weighted average distance between appropriately matched point pairs or point to plane associations. The optimization is an iterative process, and there is no guarantee to reach the optimal solution.

B. Sources of error for ICP

There are five main sources of error for ICP described in the literature: wrong convergence, wrong initialization, sensor noise and bias, underconstrained situations, and intrinsic randomness of the process [START_REF] Censi | An accurate closed-form estimate of ICP's covariance[END_REF], [START_REF] Brossard | A new approach to 3D ICP covariance estimation[END_REF]. a) Convergence and initialization: Wrong convergence comes from the iterative nature of ICP, which does not guarantee global convergence and can get stuck in local minima. This is closely related to wrong initialization, which can be a predominant source of error when the initial guess is far from the true solution [START_REF] Brossard | A new approach to 3D ICP covariance estimation[END_REF].

b) Sensor noise and biases: Each sensor naturally suffers changes in its accuracy that depend on several environmental conditions (e.g temperature) [START_REF] Pomerleau | Noise characterization of depth sensors for surface inspections[END_REF]. Also, sensor calibration can introduce distortion [START_REF] Deschaud | IMLS-SLAM: scan-to-model matching based on 3D data[END_REF]. While these phenomena introduce a bias in the estimate [START_REF] Laconte | Lidar measurement bias estimation via return waveform modelling in a context of 3D mapping[END_REF], there is an additional source of sensor noise added to each point independently, which generally is a function of the distance, orientation and physical nature or texture of the perceived material.

c) Underconstrained situations: These occur when the environment does not offer enough information to fully estimate the transformation between the point clouds, mainly by precluding the establishment of good matches between points. While in the planar 2D case such situations can be exhaustively identified, namely the corridor and the circle, in the 3D world this becomes a more challenging task.

d) Intrinsic ICP randomness: Depending on the configuration and implementation, a certain degree of randomness is introduced by subprocesses of ICP, albeit the process itself is deterministic. For example, filtering clouds with outlier rejection and sub-sampling can yield different estimates, even with the same input data.

C. Estimating ICP covariance

Various methods for estimating ICP covariance have been proposed. Monte-Carlo algorithms [START_REF] Buch | Prediction of ICP pose uncertainties using Monte Carlo simulation with synthetic depth images[END_REF] generate the covariance from a large number of samples of scan registration results. Different clouds and initializations are randomly subsampled to obtain a dispersion of the transformation that is used to compute the covariance. Closed-form methods [START_REF] Censi | An accurate closed-form estimate of ICP's covariance[END_REF], [START_REF] Brossard | A new approach to 3D ICP covariance estimation[END_REF], [START_REF] Prakhya | A closed-form estimate of 3D ICP covariance[END_REF], [START_REF] Gelfand | Geometrically stable sampling for the icp algorithm[END_REF] provide algebraic solutions to the ICP covariance problem by linearizing the error functions defined in the ICP algorithm. These methods are potentially illfounded if point reassociations happen at a scale smaller than the actual error [START_REF] Bonnabel | On the covariance of ICP-based scan-matching techniques[END_REF]. Experiments on synthetic data have shown that this is often the case [START_REF] Landry | CELLO-3D: Estimating the Covariance of ICP in the Real World[END_REF]. Early closedform works, as the one in [START_REF] Censi | An accurate closed-form estimate of ICP's covariance[END_REF], have been demonstrated over optimistic [START_REF] Mendes | ICP-based pose-graph SLAM[END_REF]. Recently, [START_REF] Landry | CELLO-3D: Estimating the Covariance of ICP in the Real World[END_REF] proposed a learning approach to the ICP covariance estimation problem. The work estimates a 3D covariance based on preprocessed feature descriptors that are used in the learning scheme. While such descriptors are generic, it is desirable to have a more flexible representation of features impacting on the error of ICP.

It is worth to notice that most of these work tackle different sources of error. The method presented in [START_REF] Brossard | A new approach to 3D ICP covariance estimation[END_REF] shows the need for a proper initialization of ICP and proposes a novel approach to 3D uncertainty of ICP accounting for initialization errors. Censi's pioneering work [START_REF] Censi | An accurate closed-form estimate of ICP's covariance[END_REF] studies the error due to sensor noise and computes the covariance of this error as a function of the error metric minimized by any ICP implementation. The work in [START_REF] Landry | CELLO-3D: Estimating the Covariance of ICP in the Real World[END_REF], closer to our approach, explores a data-driven approach to estimate the covariance focusing on sensor noise and aiming at detecting underconstrained situations.

D. Learning motion estimation error models

The literature in deep learning features a large amount of work in the estimation of uncertainty. The approaches are often tied to the process for which they predict error models. Interesting works in domains similar to ours have been proposed, mainly tackling the Visual Odometry (VO) case. As ICP, VO is a well-known family of techniques which estimate a rigid transformation between two consecutive frames for the monocular or stereovision cases. End-to-end methods perform frame to frame motion estimation [START_REF] Wang | DeepVO: Towards endto-end visual odometry with deep recurrent convolutional neural networks[END_REF] and covariance prediction [START_REF] Wang | End-to-end, sequence-tosequence probabilistic visual odometry through deep neural networks[END_REF]. Both estimates are in this case delegated to neural networks, and do not rely on any geometrical pipeline. The works in [START_REF] Liu | Deep inference for covariance estimation: Learning Gaussian noise models for state estimation[END_REF], [START_REF] De Maio | Simultaneously Learning Corrections and Error Models for Geometry-based Visual Odometry Methods[END_REF] respectively show covariance and joint bias-covariance estimation to the geometric implementation of VO.

E. Deep learning for point clouds

Recently, deep learning methods applied on point clouds have proven successful for several tasks [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF]- [START_REF] Qi | Deep Hough voting for 3D object detection in point clouds[END_REF]. In particular, PointNet [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF] paved the way for neural networks that directly use 3D point clouds for object classification and segmentation purposes, and has been extended towards learning point cloud features in metric space [START_REF] Qi | Pointnet++: Deep hierarchical feature learning on point sets in a metric space[END_REF]. This extension increased the effectiveness of PointNet to learn features on different scales, in the manner of traditional convolutional networks. Other approaches rely on the spherical projection of point clouds, usually coined as range (or depth) imagewhich is, in most cases, the original structure of the sensor data. This is used in the LiDAR odometry techniques [START_REF] Cho | Deeplo: Geometry-aware deep lidar odometry[END_REF]- [START_REF] Li | Lonet: Deep real-time lidar odometry[END_REF] which makes use of the entire scan to estimate the motion between two frames in an end-to-end manner.

III. DATA DRIVEN LEARNING OF ICP UNCERTAINTY

Excluding the process initialization, all error sources of ICP are inherited from the input data. Sensor noise is directly reflected in the data, as well as ICP subprocesses modifying its size, density and structure (i.e randomness due to filtering). Additionally, the scene structure, with its inherent geometry and features, largely impacts the error of ICP (as illustrated in Fig. 1), as it defines the observability of the motion along the different dimensions.

Addressing all these sources in a unified framework is a challenging task. Leveraging recent progresses in deep learning for point clouds, we address the problem of estimating the covariance of ICP that stems from the last three error sources presented in Sec. II-B. Our hypothesis is that a learning scheme that processes the raw input data can capture these error sources and properly estimate ICP covariances.

In this section we describe our approach to estimate ICP covariances and discuss the nature of the learned uncertainties. The optimization problem is contextualized in a probabilistic framework and a description of the learning architecture incorporating different types of uncertainty is provided.

A. Minimization problem

Modeling errors for an estimation process can be tackled via a supervised approach. Consider an estimator, ICP in our case, using sensory data

C i = {P i , Q i+1 } with P i ∈ R ni and Q i+1 ∈ R ni+1 . Its output i
Ti+1 is an estimate of the ground truth transformation i T i+1 . In practice, the point clouds used in the algorithm are the result of several filters [START_REF] Pomerleau | A review of point cloud registration algorithms for mobile robotics[END_REF]. The actual input data is a pair of decimated point

clouds Ci = { Pi , Qi+1 } with Pi ∈ R mi , m i n i and Qi+1 ∈ R mi+1 , m i+1 n i+1 . The error of ICP is computed as e i = i T -1 i+1 • i T i+1 (2)
We represent an error e ∈ SE(3) as a vector ξ ∈ se(3) member of the Lie algebra [START_REF] Sola | A micro Lie theory for state estimation in robotics[END_REF], [START_REF] Barfoot | Associating uncertainty with threedimensional poses for use in estimation problems[END_REF]

ξ ∧ = ρ φ ∧ = φ ∧ ρ 0 T 0 (3)
where the hat operator ∧ turns the vector ξ ∈ R 6 into a matrix ξ ∧ ∈ R 4×4 . The opposite linear map is represented as the vee operator ∨. The exponential map exp(ξ ∧ ) allows to retract an element of the Lie algebra back to the group. Its inverse is the logarithmic map log(T ). Once the error is defined as ξ = log(e) ∨ , the associated uncertainty Σ ∈ R 6×6 can be expressed as

Σ = Σ ρρ Σ ρφ Σ φρ Σ φφ (4)
This uncertainty representation is generally valid for small perturbations that are added to a given pose T = T exp(ξ) where ξ ∼ N (0, Σ) [START_REF] Barfoot | Associating uncertainty with threedimensional poses for use in estimation problems[END_REF]. The equation above matches the error definition in Eq. 2, that is ξ ∈ R 6 can be viewed as the error between the real transformation and its estimate. We define a dataset

D = { Ci , ξ i |∀i ∈ [1, d]}
, where d is the size of the dataset. The goal is to predict the uncertainty of the error vector ξ i along its dimensions. In the data, we noticed that the unbiased error assumption is valid for the vast majority of ICP estimates and the bias is very small otherwise. To keep the uncertainty assumption valid, we encode a bias term for the error vector in the distribution that ensures the error remains close to a zero-mean Gaussian. Therefore, it is possible to estimate the parameters of the Gaussian N (µ, Σ) as arg max

µ 1:d ,Σ 1:d d i=1 p(ξ i |µ i , Σ i ) (5)
To maximize the probability of drawing the correct error given the estimated Gaussian parameters is equivalent to minimize the negative full log-likelihood function [START_REF] De Maio | Simultaneously Learning Corrections and Error Models for Geometry-based Visual Odometry Methods[END_REF]. To ensure a positive definite covariance matrix we use the LDL decomposition as in [START_REF] Liu | Deep inference for covariance estimation: Learning Gaussian noise models for state estimation[END_REF].

B. Uncertainty estimation

ICP covariance can be decomposed into two categories: epistemic and aleatoric [START_REF] Kendall | What uncertainties do we need in bayesian deep learning for computer vision?[END_REF]. Both uncertainties are formulated as probability distributions over different entities.

Aleatoric uncertainty models the uncertainty over the input data to the process. It assumes an observation noise that can be either constant or vary with the input. As discussed in Sec.II-B, we observed that noise levels change with the scene (Fig. 1), hence we are interested in modeling the variation of the noise in correlation with the input data. This is referred to as heteroscedastic aleatoric uncertainty (as opposed to homoscedastic uncertainty for the constant noise case).

Additionally, it is possible to model the variance intrinsic to the learned model, that is how certain of a given prediction the network is. Epistemic uncertainty can, in principle, be reduced if more data is used for training. In our case, it accounts for the network's poor predictions on out-of-data samples such as scene features that were rarely encountered during training. Epistemic uncertainty can be modeled by placing a distribution over the network's weights. Unfortunately, given a prior distribution N W (0, σ) over the weights, it is computationally difficult to analytically infer the posterior p(W |X, Y ), that is the probability of drawing a set of weights given the network input and output. There is a number of proposed methods to approximate this posterior [START_REF] Stephan | Stochastic Gradient Descent as Approximate Bayesian Inference[END_REF], [START_REF] Postels | Sampling-free epistemic uncertainty estimation using approximated variance propagation[END_REF]. The work in [START_REF] Gal | Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning[END_REF] produces an approximation using dropout variational inference. This has proven to be an effective yet easily obtainable Bayesian approximation when dealing with deep complex models.

In our context, From Eq.5, we consider the mean as a noisy error estimate, and the covariance matrix as the aleatoric uncertainty due to the observation noise. To approximate the posterior for a Bayesian neural network estimating the ICP errors we resort to Monte-Carlo dropout. The epistemic uncertainty can be captured by keeping dropout activated at test time. Sampling N times the predicted ICP error vector and ICP covariance yields a distribution [ μ1:N , Σ1:N ] = f W ( C) where f W ( C) is the output of a Bayesian neural network, function of the input point clouds.

The total covariance matrix, integrating aleatoric and epistemic uncertainty is computed as

1 N N n=1 ( μn -μ)( μn -μ) T + 1 N N n=1 Σn (6)
where μ = 1 N N n=1 μn is the mean error vector over the sampled outputs. The first term in Eq. 6 computes the uncertainty over the predicted error vector, which presents a variance proportional to the network confidence over the estimate. The mean vector is not used to correct ICP estimates as the original error is generally noisy. It is rather evaluated in terms of dispersion after having approximated the model posterior.

C. Learning architecture

The improvements in point clouds based network involved a number of estimation processes. Recently, FlowNet3D [START_REF] Liu | FlowNet3D: Learning scene flow in 3D point clouds[END_REF] proposed an approach to tackle 3D scene flow expanding on the building blocks introduced in [START_REF] Qi | Pointnet++: Deep hierarchical feature learning on point sets in a metric space[END_REF]. It provides the 3D flow for each point in a reference point cloud deducing its motion from the comparison with a second, successive, point cloud. Estimating the scene flow is a problem closely related to the scan registration. Intuitively, a network able to approximate such task results suitable for ICP related inferences. In fact, the main difference between the two tasks lies in the generalization needed to shift from a high dimensional flow to the estimate of a rigid transformation in the ICP case. We leverage the main structure of FlowNet3D to tackle the [START_REF] Liu | FlowNet3D: Learning scene flow in 3D point clouds[END_REF].

estimation of ICP errors, modifying few key aspects to adapt it to our problem. To complete the architecture we add a final MLP layer reducing the dimensionality to the number of parameters needed to predict the Gaussian parameters.

First we follow the original regression layer from FlowNet3D reducing the feature space to R 3 × n. The MLP is extended to take as input the feature produced by the regression layer and to return the parameters generating the covariance matrix and predicted error vector. All the MLP (originally Linear-BatchNorm-ReLU) in FlowNet3D are stacked with Dropout modules to allow the estimation of the epistemic covariance matrix. Additionally, this proved to reduce overfitting as expected.

We initially assumed that including the set upconv layer would have not been needed. Intuitively, as in a classical convolutional scheme, the regression task we try to solve can be solved with direct dimensionality reduction, from sensory data to parameters describing uncertainty [START_REF] Liu | Deep inference for covariance estimation: Learning Gaussian noise models for state estimation[END_REF], [START_REF] De Maio | Simultaneously Learning Corrections and Error Models for Geometry-based Visual Odometry Methods[END_REF]. Nonetheless, retaining upsampling layer produced better results in terms of predicted uncertainty and mean. We associate this behavior with the architecture using skip connections to use the same features extracted from the same point sets originally used ICP. This creates a 1:1 matching between the data used by the geometric estimator and the network, leaving less room for outliers. Our network parameterization, with sampling rates and radius, can be found in Table I.

IV. EVALUATION

A. ICP process and dataset

To learn errors produced by a geometric ICP algorithm we selected libpointmatcher [START_REF] Pomerleau | Tracking a Depth Camera: Parameter Exploration for Fast ICP[END_REF]. It is a widely used open-source library offering a chain that filters point clouds, establishes points associations, removes outliers and estimates rigid transformations using an error minimizer. For our setup we parameterized the process similarly to [START_REF] Landry | CELLO-3D: Estimating the Covariance of ICP in the Real World[END_REF]. Cloud random sampling is fixed at 2048 random points. We generate normals, used for the point-to-plane minimizer, using the 10 nearest points before subsampling.

To select the initial transform for ICP, we add an error to the ground truth T . In order not to draw errors from an overoptimistic distribution we sample guesses for the initial transforms from a non-zero mean normal distribution N (N (T , a • T ), b • T ), with a = 0.25, b = 0.2. These figures yield more pessimistic estimates than commonly used aiding sensor (e.g odometry or inertial measurements) posing a challenge to our system.

We select the KITTI odometry dataset [START_REF] Geiger | Are we ready for Autonomous Driving? The KITTI Vision Benchmark Suite[END_REF], using LiDAR scans, which is made of urban and rural road scenes. We train and test using all ground truth tagged sequences but sequence 01, on which ICP fails to register most of the scans due to scenes lacking features. To compare against the state of the art in covariance estimation for ICP, we also present results on the challenging data sets of [START_REF] Pomerleau | Challenging data sets for point cloud registration algorithms[END_REF]. Since for the LiDAR odometry case this dataset is rather small, we refine our KITTI trained model with a few epochs of training, approximately 30 or less for early convergence, on all but one outdoor sequences and validate on the one left out. As expected, features learned on KITTI are ineffective to predict covariances on the apartment and stairs sequences, which are highly structured small indoor environments. Our system learns to model ICP errors with numerically small epistemic uncertainty. For reasons of space, we only show results containing the combined aleatoric+epistemic covariance when moving to unseen datasets, i.e applying models learned on KITTI to the ETH dataset.

B. Single-pair validation

We first assess the quality of the predicted covariances in a single-pair fashion, and then check their consistence on a full trajectory. The metrics we use are based on the relation between the observed errors and the predicted covariance. This is due to the absence of the true distribution which would allow for more classic figures of merit, such as the Kullback-Leibler divergence. While it would be possible to obtain a pseudo-true distribution by random sampling over the point cloud, this would affect the estimated covariance as well, because the network prediction is tied to the ICP configuration, and most importantly to the input data used by ICP.

As in [START_REF] Censi | An accurate closed-form estimate of ICP's covariance[END_REF], [START_REF] Brossard | A new approach to 3D ICP covariance estimation[END_REF], we compute the Normalized Norm Error (NNE) to evaluate the scale of the predicted covariance w.r.t the actual error:

NNE = 1 N N n=1 ||ξ n || 2 2 tr( Σn ) (7)
where tr( Σn ) is the trace of the predicted covariance matrix. The optimal value for NNE is one. Values below and above one respectively represent pessimistic and optimistic uncertainty estimates. n T n+1 ) ∨ , weighted by the predicted covariance Σ n averaged over each consecutive pair in a sequence, is a better metric:

D M = 1 N N n=1 ξ T n Σ -1 n ξ n dim(ξ) (8)
As in the NNE case, the optimal value for a normal distribution is one, with distances below and above one respectively denoting pessimistic and optimistic estimates.

Table III shows the average Mahalanobis distance for the validation sequences. All sequences present values positively close to 1, indicating that the system is able to correctly capture the uncertainty under the Gaussian assumption. For these sequences, the epistemic uncertainty is small and constant, which denotes confidence in the system prediction. Minor variations are encountered for seldom cases: an example is shown in Fig. 2 displaying the epistemic covariance before, during, and after an area corresponding to wide plain scenes shown in Fig. 1. Such spatial features are less frequently encountered during training. Fig. 2: Epistemic and aleatoric 3σ intervals in a rarely seen area lacking sharp geometrical features. The real error computed from the ground truth is represented in black. The zoomed-in area shows epistemic uncertainty increase, which corresponds to a wide plain environment.

C. Trajectory validation

In order to assess the quality of our predicted covariances on trajectories we take a multi-step approach. First, we compose ICP estimates to generate the final pose as 0 Tf =

0 T1 1 T2 • • • f -1
Tf for a sequence with f poses. The ground truth final pose 0 T f is computed similarly. We propagate the covariance associated to each estimate using the 4th order approximation, as in [START_REF] Barfoot | Associating uncertainty with threedimensional poses for use in estimation problems[END_REF], to obtain an uncertainty estimate for the final ICP pose. This way, it is possible to evaluate the Mahalanobis distance (Eq. 8) in a spatially consistent manner. The results are shown in Table IV. They are only slightly optimistic, which is a remarkable outcome given the length of KITTI trajectories (up to more than 2km).

D. Fine tuning on other datasets

We also evaluate the results on the smaller dataset of [START_REF] Pomerleau | Challenging data sets for point cloud registration algorithms[END_REF] and compare to the state of the art. As our model requires a significant amount of data we initially learn on a large dataset. KITTI makes available a considerable number of LiDAR scans with different scenarios, but all correspond to outdoor road environments. To apply our approach to smaller datasets acquired in very different environments, a fine-tuning process is applied using the pre-trained model on the KITTI dataset. This allows to obtain much better results compared to re-training on such a small dataset and shows the importance of epistemic uncertainty as an indicator of the performances of a model on unseen data.

Table V shows the result of our approach on single-pair estimates with the dataset of [START_REF] Pomerleau | Challenging data sets for point cloud registration algorithms[END_REF]. While the environments are very different, the model performs remarkably well in outdoor scenes. Even architectural features, such as in hauptgebaude can be leveraged to produce faithful covariance predictions as, intuitively, the scene presents a structure (flat ground plane, lack of indoor man-made elements) that is geometrically similar to the some scenes encountered in KITTI.

More importantly, finetuning, not only in the expected improved results, but also in the magnitude of the epistemic uncertainty. The improvement brought by considering the model uncertainty is evident when testing the model trained on KITTI directly on an unseen dataset, where, as expected, the network has little confidence in its prediction for new scenarios. In fact, while some outdoor features may recur in the ETH dataset, motion type and scene size are remarkably different when compared to the KITTI dataset. After finetuning, not only the weight of the epistemic covariance is greatly reduced as shown in Fig. 4, but the overall results are closer to the optimal value. Finally, Table VI compares our results against CELLO-3D [14] and Brossard et al. [START_REF] Brossard | A new approach to 3D ICP covariance estimation[END_REF] on aggregated transforms on the same dataset. It is worth noticing that given the strong focus on ICP initialization from Brossard et al., it is more relevant to compare our approach to CELLO-3D, as our work solely focus on the errors introduced directly and indirectly by the data on a purely data-driven approach. CELLO-3D is often over pessimistic while Brossard et al. is generally optimistic. Especially on rotational uncertainty, our method outperforms the two approaches by a significant margin.

V. CONCLUSION

We presented an approach to estimate faithful covariances for the ICP scan registration process. Using a data-driven paradigm we have shown that it is possible to learn uncertainty for the selected algorithm, and to generalize to a certain degree on multiple datasets. Leveraging a neural network architecture conceived to directly process point clouds, we estimate covariances with a close connection to the data used in the ICP algorithm. Standard metrics show the reliability of our approach, and that it generally outperforms the state-of-the-art. Future improvements will concern the network architecture, possibly tailoring it to the scan registration problem. A possible approach is to have the flow embedding layer (and the successive layers) doubled to account for both point clouds motions, whereas in the scene flow the problem is cast on the reference cloud only. Additionally, the role of initialization and the use of the initial guess in the learning process can also be considered.

Fig. 1 :

 1 Fig. 1: Distribution of errors on a series of ICP estimates computed in three different kind of environments: corridorlike (left), wide plain (center), structured (right).

Fig. 3 :

 3 Fig.3: Four propagated covariances in the 2σ interval (95%) on considered ICP trajectories compared to ground truth.

Fig. 4 :

 4 Fig. 4: Epistemic uncertainties normalized by the mean in the direct and finetuned case.

TABLE I :

 I Network architecture parameterization. Set upconv layers are the same as in the updated version of

  Table II shows NNE values computed over test KITTI sequences. The scale of the covariance is well predicted, and is only slightly pessimistic for two validation sequences (05 and 07).

	NNE (KITTI)	05	06	07	09	10
	Translation	0.70 0.61 0.58 0.94 0.90
	Rotation	0.65 0.81 0.55 0.90 1.04

TABLE II :

 II NNE translation and rotation values for the KITTI dataset.The NNE is an immediate measure of the adequacy of the predicted variances, but it does not account for offdiagonal covariances. For this purpose, the Mahalanobis distance between ICP estimates and ground truth ξ n =

	Single-Pair Mah. Dist (KITTI)	05	06	07	09	10
	Translation	0.80 0.83 0.74 0.88 1.03
	Rotation	0.77 0.63 0.70 0.94 0.96

TABLE III :

 III Single-pair Mahalanobis distance in terms of translation and rotation for the KITTI dataset.

	log( n T -1 n+1

TABLE IV :

 IV Trajectory Mahalanobis distance in terms of translation and rotation for the KITTI dataset.

  Table V highlights the impact of the

	Single-Pair Mahalanobis Distance (ETH)	Gazebo winter	Gazebo summer	Mountain	Haupt--gebaude	Wood autumn	Wood summer
	Ours (Direct)	Trans (A) Trans (E+A)	2.05 2.04	2.32 2.12	4.15 3.88	2.68 2.49	4.01 4.19	4.03 3.78
	Ours (Finetuned)	Trans (A) Trans (E+A)	0.64 0.59	1.12 1.06	1.27 1.21	1.04 0.98	1.86 1.70	1.03 0.90
	Ours (Direct)	Rot (A) Rot (E+A)	1.33 1.30	1.93 1.89	3.06 3.03	2.61 2.47	2.10 2.07	2.33 2.19
	Ours (Finetuned)	Rot (A) Rot (E+A)	0.48 0.44	0.53 0.49	0.65 0.65	1.02 0.96	0.54 0.51	0.82 0.71

TABLE V :

 V Single-pair Mahalanobis distance in terms of translation and rotation for the ETH dataset using aleatoric only (A) and combined epistemic-aleatoric (E+A) uncertainty. A comparison between the pre-trained model on KITTI and after finetuning is presented. After finetuning, the epistemic uncertainty is less prominent and its (minor) impact on performances depends on whether the system is being pessimistic or optimistic.

	Trajectory Mahalanobis Distance (ETH)	Gazebo winter	Gazebo summer	Mountain	Haupt--gebaude	Wood autumn	Wood summer
	CELLO-3D [14]	Trans	0.1	0.2	-	0.3	0.1	0.1
	Brossard et al. [6]	Trans	1.8	1.0	1.2	1.8	1.2	1.5
	Ours (Finetuned)	Trans (A)	0.83	1.28	0.87	1.83	0.93	0.69
	CELLO-3D [14]	Rot	0.2	0.2	-	0.2	0.3	0.3
	Brossard et al. [6]	Rot	3.7	2.3	1.2	2.9	4.2	4.7
	Ours (Finetuned)	Rot (A)	0.71	1.03	0.81	1.48	1.69	1.65

TABLE VI :

 VI Trajectory Mahalanobis distance in terms of translation and rotation for the ETH dataset using aleatoric only (A) uncertainty, compared to two state-of-the-art approaches. Epistemic uncertainty is not showed after finetuning.


Sourav Dey

Gopalan Rajaraman
email: rajaraman@chem.iitb.ac.in

Hélène Bolvin
email: bolvin@irsamc.ups-tlse.fr

Analysis of the Magnetic Coupling in a Mn(II)-U(V)-Mn(II) Single Molecule Magnet

{Mn(TPA)I}{UO2(Mesaldien)}{Mn(TPA)I}]I formula (here TPA = tris(2-pyridylmethyl)amine and Mesaldien = N,N'-(2-aminomethyl)diethylenebis(salicylidene imine)) reported by Mazzanti and coworkers (Chatelain et al. Angew. Chem. Int. Ed. 2014, 53, 13434) is so far the best Single Molecule Magnet (SMM) in the {3d-5f} class of molecules exhibiting barrier height of magnetization reversal as high as 81.0 K. In this work, we have employed a combination of ab initio CAS and DFT methods to fully characterize this compound and to extract the relevant spin Hamiltonian parameters. We show that the signs of the magnetic coupling and of the g-factors of the monomers are interconnected. The central magnetic unit [U V O2] + is described by a Kramers Doublet (KD) with negative g-factors, due to a large orbital contribution. The magnetic coupling for the {Mn(II)-U(V)} pair is modeled by an anisotropic exchange Hamiltonian: all components are ferromagnetic in terms of spin moments, the parallel component JZ twice larger as the perpendicular one J ⊥ . The spin density distribution suggests that spin polarization on the U(V) center favors the ferromagnetic coupling. Further, the JZ /J ⊥ ratio, which is related to the barrier height, was found to correlate to the corresponding spin contribution of the g-factors of the U(V) center. This correlation established for the first time offers a direct way to estimate this important ratio from the corresponding g S -values, which can be obtained using traditional ab initio packages and hence has a wider application to other {3d-5f} magnets. It is finally shown that the magnetization barrier height is tuned by the splitting of the [U V O2] + 5f orbitals.

Introduction

Single molecule magnets (SMMs) are of particular interest in the field of molecular magnetism due to their potential application in information storage and quantum computing [1]. The blocking barrier of magnetization reversal (U eff ) and blocking temperature (T B , below which opening of hysteresis is observed) are considered to be figure of merit for the performance of a SMM [2,3]. Recently groundbreaking results of achieving T B as high as liquid nitrogen temperature in [(Cp iPr5 )Dy(Cp * )] + (Cp iPr5 = pentaisopropylcyclopentadienyl, Cp * = pentamethylcyclopentadienyl) reveals that the magnetic anisotropy has reached its axial limit as further finetuning of the Ln-ligand interactions are not realistic [4,5]. One way to circumvent this problem is to employ actinides that have diffused 5f orbitals exhibiting greater metal-ligand covalency -thanks to their extra radial node -compared to the 4f orbitals of the lanthanides [6,7]. But the radioactivity of actinides makes them less explored than lanthanide and transition metal complexes.

Actinide, and in particular uranium complexes have the ability to be superior candidates for SMM due to their high anisotropy over the range of oxidation state and ability to mediate strong magnetic exchange with other metals [START_REF] Van Leusen | Magnetism of Actinide Coordination Compounds[END_REF]. The SMM behavior of U(III) complexes is inherent and well established [START_REF] Rinehart | [END_REF]10,11,12,13,14,6,7,15,16,17,18,19,20,21]. On the other hand, U(V) complexes, though they possess only one unpaired electron, fair better due to a stronger ligand field arising from multiple metal-ligand characteristics with the donor atoms such as U --O [22,23,24,25,17]. The enhanced actinide-ligand covalency is also a disadvantage as even weaker ligands interact strongly yield large transverse anisotropy preventing achieving the Ising type of g-factors that are seen in lanthanides. Due to this factor, mononuclear actinide complexes exhibit stronger QTM, which often overcompensate the ground gained in magnetic anisotropy. However, if a strong exchange coupling is induced, this can substantially quench the tunneling, and this route leads to the birth of a handful of {3d-5f} magnets exhibiting very attractive barrier height/blocking temperatures [22,26,27].

Ab initio calculations based on Complete Active Space (CAS) principles have played an important role in modeling the magnetic anisotropy in lanthanide and actinide complexes. The modeling of one center lanthanide complexes is based on the splitting of the ground J a manifold by the crystal field [28,[START_REF] Ungur | Introduction to the electronic structure, luminescence, and magnetism of lanthanides[END_REF]30], as the splitting of the ground manifold is similar to the thermal energy available at room temperature in most cases. In actinide complexes, the interaction with the ligands is more important and the ground J manifold might not be the good model space [31,32,33]. Even more, the An free ion is not the good starting point for the modeling of the actinyls, which must be considered as the basic unit perturbed by the equatorial ligands [34,24,35,33]. In particular, for the 5f 1 [Np VI O 2 ] 2+ complex, which is isoelectronic to the [U V O 2 ] + cation, the nature of the equatorial ligands completely dictates the magnetic properties.

The coupling between lanthanide centers is often described as an isotropic coupling using Lines model [36,37], where the local properties are determined using ab initio calculations. Combined with a first principle description of the monomers, it allows to determine the value of the magnetic coupling J [38,39,40]. Considering only the couplings between two KDs, the fit can be performed using an Ising Hamiltonian [41]. This approach is valuable to calculate thermodynamic properties where the whole 2J + 1 manifold of the free ion term is populated and estimates J values in good accordance against accurate High-Field EPR spectroscopy for a range of {3d-4f} dimers [42,43]. The exchange coupling between two local J angular momenta may be expressed using the irreducible tensor algebra by a tensor which rank depends on the values of the local J [37]. In the present case, we limited the interaction to its second-order term since the local states are a KD doublet and a spin-only manifold. Furthermore, we neglected the antisymmetric contribution. Consequently, in the following, the exchange coupling is described by the tensor J [44,45]

ĤAB = -Ŝ Ŝ Ŝ A J Ŝ Ŝ Ŝ B (1) = -J X ŜA X ŜB X -J Y ŜA Y ŜB Y -J Z ŜA Z ŜB Z
where S S S A and S S S B are the local pseudo-spin operators. This simple form of Eq. 1 allows to model the interaction with a reduced number of parameters, whose physical interpretation is straighforward. Its applicability will be checked by comparing the response to a magnetic field with the model Hamiltonian to ab initio calculations. This coupling can be determined using EPR [START_REF] Walter | Andersen[END_REF]40,47,48].

There are few ab initio description of this coupling. CAS based methods are widely used to de-scribe the crystal-field levels in mononuclear lanthanide/actinide complexes. For polynuclear systems the calculation of exchange couplings requires large active spaces which renders CAS based calculations extremely demanding. Furthermore, the calculation of exchange couplings requires the inclusion of dynamical correlation by multi-reference configuration interaction, perturbative methods being non reliable in those cases. This has been described using variational methods (DDCI) [START_REF] Gendron | Inorg[END_REF]50], the local-density-fitted configuration-averaged Hartree-Fock (LDF-CAHF) method [51], Density-Matrix Renormalization Group (DMRG) method [52], and Density Functional Theory (DFT) [53].

DFT appears as an alternative tool to compute magnetic exchange coupling and to explore the electronic structure and magnetic properties of actinide-containing molecules, especially when the considered systems are very large [54].

A detailed understanding of the mechanism underlying the anisotropic coupling is of first importance in order to understand the anisotropy of polymetallic complexes. The coupling often reduces the SMM properties by creating desexcitations channels to the magnetization. It was shown that an anisotropic antiferromagnetic coupling should be the source for skyrmions [55]. This work is devoted to the detailed analysis of the anisotropic properties in a trinuclear exchange coupled {3d-5f} SMM [{Mn(TPA)I}{UO 2 (Mesaldien)}-{Mn(TPA)I}]I (denoted [Mn -U -Mn], TPA = tris(2-pyridylmethyl) amine, Mesaldien = N,N-(2aminomethyl)diethylenebis(salicylidene imine)), studied by Mazzanti and coworkers (see Figure 1) [26]. The Mn(II) centers with a high-spin 3d 5 configuration have a half-filled shell: this leads to a simple electronic configuration that makes their description simpler and mostly spin-only non degenerate electronic state, with a negligible zero-field splitting.

The central U(V) unit is highly anisotropic. The electronic and magnetic properties of actinyls are well described by CAS based methods including spin-orbit coupling and a perturbative description of the dynamical correlation [56,57,58,59,60,34,24,61]. In this work, the magnetic monomers are first described using diamagnetic substitution, then the magnetic coupling is computed in the dimers and trimers using Configuration Interaction (CI) and DFT methods. This allows to discuss the spin Hamiltonian of the trimer and the subsequent properties.

Results and discussion

The [Mn -U -Mn] complex was synthesized and characterized in the group of Mazzanti [26]. It consists of two [Mn(TPA)I] + cations bound to the two oxo groups of the [UO 2 (Mesaldien)] anion (see Fig- ure 1). The U(V) atom is heptacoordinate with a slightly distorted pentagonal bipyramid geometry, with the two uranyl oxygen atoms O yl and the five donor atoms of the Mesaldien 2 -ligand in the equatorial plane. The transition metal centers are hexacoordinate, with a slightly distorted octahedral geometry defined by the four nitrogen atoms of the TPA ligand, one oxygen atom from the uranyl(V) group, and a coordinated iodide anion (see the SHAPE analysis in Table S1). The U --O bond length of 1.901 Å denotes a slight lengthening due to charge donation to the Mn(II) atom and the Mn -O yl of 2.055 Å is significantly shorter than in equivalent complexes. The complex is almost linear, with a M -U -M angle of 173.77°, the deviation from linearity is due to intramolecular hydrogen bonds between the protons on the TPA ligand and the oxygen atoms of the Mesaldien 2 -ligand. The magnetic susceptibility was measured and the high temperature χT curve was fitted by an Heisenberg Hamiltonian ĤS = -J (S S S Mn1 S S S U + S S S Mn2 S S S U ) with a ferromagnetic coupling J = 15 cm -1 b and an isotropic g-factor for the U(V) center g U = 1. The complex exhibits a slow relaxation of magnetization in zero field with U eff = 56.3 cm -1 . It is reported to have the highest U eff among all {3d-5f} complexes having a {M-O=U=O-M} (M = 3d metal ions) moiety.

The monomers

The Mn(II) cations have a 3d 5 configuration and their high-spin ground state is an orbitally nondegenerate spin sextet. The zero-field splitting arises by spin-orbit coupling with the excited quartets at more than 20000 cm -1 and is less than 1 cm -1 (see Table S2 and1). It will be neglected in the following and the Mn(II) centers described by a pure spin S = 5/2. The canonical orbitals are shown on Figure S3. They are similar for the two Mn(II) centers, and characteristic for an octahedral symmetry: the three t 2g -like orbitals split by 500 cm -1 , the e g -like orbitals by 2000 cm -1 and the gap between the two groups is 6000 cm -1 .

The uranyl group [U

V O 2 ] + is isoelectronic to the neptunyl [Np VI O 2 ]
2+ with a 5f 1 configuration. The magnetic properties of the latter are very sensitive to the nature of the equatorial ligands [24,61]. The four f δ are non-bonding to the O yl atoms, while the f π and f σ are strongly destabilized, forming anti-bonding orbitals. In the ground state, the single electron lies in the f δ or f ϕ orbitals, and magnetic properties are determined by the composition of the ground KD in terms of those orbitals. The canonical orbitals of the [Zn -U -Zn] mod complex are shown in Figure 2. The two f δ orbitals are the lowest ones with a splitting of 400 cm -1 , followed by the two f ϕ at 2000 and 4000 cm -1 , and finally the two f π orbitals, almost degenerate at 6700 cm -1 . This scheme is usual for actinyls with equatorial ligands: the two former being involved in π interaction with the equatorial ligands, the two next in σ interactions and the two latter in π interaction with the two oxo groups, all those interactions being anti-bonding. The f δ and f ϕ orbitals have four-fold and three-fold symmetries along the U -O yl axis, respectively, while the equatorial ligand is five-fold, but irregular. The two oxygen atoms are closer to the U(V) center than the three nitrogen ones (2.25 vs 2.61 Å). Consequently, f δ2 is slightly more destabilized than f δ1 because it interacts with the oxy-

f δ1 (0) f δ2 (440) f π2 (6750) f π1 (6650) f φ1 (2100)
f φ2 (4120) gen atoms of the Mesaldien ligand. The splitting between f ϕ2 and f ϕ1 arises for the same reasons.

The energies of the U(V) monomer are given in Tables S3 and 2. The simplified [Zn -U -Zn] mod complex gives energies close to the [Zn -U -Zn] complex. The magnetic orbitals in the complex are slightly perturbed as compared to the free [U V O 2 ] + , and are denoted according to the linear parentage: |m l | the value of lZ for the orbitals and |m j | the value of ĵZ for the spinors. The main effect of the equatorial ligands is to lift the degeneracy of the orbitals, the two f δ by 700 cm -1 and the two f ϕ by more than 2000 cm -1 . The states are labeled after the single occupied orbital (spin-orbit free) or spinor (spin-orbit). The ground spin-free state is of 2 ∆ 1 parentage, with the single electron in the lowest f δ1 orbital. The lowest excited state, at 700 cm -1 , is of 2 ∆ 2 parentage, with the single electron in the f δ2 orbital. The next excited states are of 2 Φ parentage. In the [U V O 2 ] + free ion, the spin-orbit coupling splits the 2 ∆ state according to Ω = |M J | the value of ĴZ in ∆ 5/2 and ∆ 3/2 . In our complex, the composition of the ground KD is close to ∆ 3/2 , ensued from the coupling between the two 2 ∆ components, whatever the level of calculation (see Table S6). The next KDs arise from the mixing between 2 ∆ and 2 Φ manifolds. KD2 lies between 600 and 1100 cm -1 , depending on the level of calculation. The main effect of correlation is to lower the energy of 2 Φ 1 from 2000 to 1300 cm -1 . But this does not affect the ground KD, since of Ω = 3/2 linear parentage, and the spin-orbit free 2 Φ free-ion state spits in Φ 5/2 and Φ 7/2 by spin-orbit.

As a consequence, the ground KD can be modeled in the ∆ space, as proposed in Section S2.2 [24].

The key parameter is the splitting by the equatorial ligands of the spin-orbit free 2 ∆ state in 2 ∆ 1 and 2 ∆ 2 . The larger this splitting, the more quenched the orbital contribution. If the two δ orbitals are degenerate, the g-tensor is axial with a negative g Z value. On the opposite, with an extremely large gap, only the 2 ∆ 1 state contributes and as a pure spin doublet, it has an isotopic g tensor equals to g e = 2. The present case is an intermediate regime, with a splitting of 700 cm -1 leading to relative weights of 60% and 30% for 2 ∆ 1 and 2 ∆ 1 and the orbital contribution is partially quenched. Ab initio g-factors are given in Table S5 in ESI. Since the composition of the ground KD in terms of spin-orbit free states barely depends on correlation, so do the g-factors. g A and g A ⊥ equal 1.6 and 0.5. g X and g Y are almost equal, as expected from D 5h symmetry, and will be considered to be equal in the following. The parallel spin and orbital contributions are in good accordance with Eqs. S2, with an orbital contribution about twice the spin one, and with opposite sign, leading to a negative value for g . But Eqs. S2 predict a vanishing orbital contribution to the equatorial g L ⊥ , while the ab initio value is by far non zero, and even larger than the spin one. This might be due to the small weights of the ground KD on the 2 Φ and 2 Π states. The sign of g ⊥ is undetermined, since only the product g X g Y g Z has a physical significance [62,[START_REF] Bolvin | Relativistic methods for calculating Electron Paramagnetic Resonance (EPR) parameters[END_REF]. The sign of g S ⊥ can be assigned by switching off gradually the spin-orbit coupling: at the spin-free limit, all values equal to 2, by continuity g S ⊥ is positive, and consequently, g ⊥ negative. In the following, we will keep this choice, keeping in mind its arbitrariness. The magnetic U(V) center will be modeled by a unique KD, with g-factors g Z = -1.6 and g X = g Y = -0.5.

The Natural Spin Orbitals (NSOs) correspond to the natural orbitals issued from the spin magnetization for a given direction of the external magnetic field as defined in references [24,[START_REF] Gendron | [END_REF][START_REF] Gendron | Complete Active Space Wavefunction-Based Analysis of Magnetization and Electronic Structure[END_REF]. They allow to vizualize the orbitals contributing to the spin density, for a given quantification axis. NSOs with a positive (negative) population participate to the spin density with an α (β) spin for this quantification axis. The NSOs of the ground KD issued from the CASSCF and RASSCF calculations are represented in Figures S4 andS5, respectively and the corresponding spin magnetization densities are shown in Figure 3. As shown in Section S2.2 of the SI, the decomposition of the two Kramers partners in their α and β components depends on the direction of the quantification axis, and consequently, of the applied magnetic field. Along both X and Z directions, f δ1 and f δ2 are NSOs with respective populations 0.6 and 0.3 (see Eqs. S4 and S7). But along Z, f δ1 and f δ2 orbitals appear both with a positive spin density, those contributions are additive and integrate to a large value, leading to an important longitudinal g-factor. Along X, f δ1 and f δ2 orbitals appear with opposite spins, and the spin density alternates with lobes of opposite signs: this integrates to a smaller value than in the Z direction. With both SO-CASSCF and SO-RASSCF, the NSOs correspond well to this model. The increase of the active space with orbitals of the oxo groups in RASSCF allows a better description of the correlation in these bonds, the main effect being the spin polarization [START_REF] Kahn | [END_REF][START_REF] Lloret | [END_REF]68,69]. In the ground spin-orbit free 2 ∆ 1 state, the Mulliken spin densities on the U and O yl atoms are 0.9967 and 0.0005 with CASSCF, the spin density is positive on the oxo groups, since only spin delocalization is introduced. They become 1.0276 and -0.0154 with RASSCF, due to spin polarization. With spin-orbit coupling, there is a small amount of negative spin density, due to the coupling of the ∆ states with the Π ones with opposite spin.

The trimer

The trimer consists of the [UO 2 ] + unit (denoted A) in the center, coupled to two Mn(II) centers (denoted B and C). Monomer A is the [UO 2 ] + unit, restricted to its ground KD, is described by a pseudo-spin S A = 1/2, the tensor g A and its principal values

g A X = g A Y = g A ⊥ and g A Z . ĤA = µ B B B B g A Ŝ Ŝ Ŝ A = µ B g A X B X ŜX + g A Y B Y ŜY + g A Z B Z ŜZ (2)
Monomers B and C are described by a pure spin S B = 5/2 with a negligible zero-field splitting due to the half-shell structure. The spin Hamiltonian reduces to the isotropic Zeeman term

ĤB/C = µ B g e B B B Ŝ Ŝ Ŝ B/C (3)
The coupling between the two magnetic centers is described by the anisotropic coupling Hamiltonian of Eq. 1. By symmetry,

J X = J Y = J ⊥ and ĤAB = -J ⊥ ŜA X ŜB X + ŜA Y ŜB Y -J Z ŜA Z ŜB Z = -J Ŝ Ŝ Ŝ A Ŝ Ŝ Ŝ B + D ŜA Z ŜB Z (4) - 1 2 ŜA X ŜB X - 1 2 ŜA Y ŜB Y with J = (2J ⊥ + J Z )/3 and D = 2/3(J ⊥ -J Z ).
We suppose the principal axes frame (PAF) of J AB and g A to be identical.

We started by studying the dimers, and as a first example, the coupling between two KDs, S A = 1/2 and S B = 1/2 as developped in Section S3.1.1. It appears that the zero-field energies of the dimer are not affected by the reversal of sign of two components, J α and J β . The response to an external magnetic field is unaffected by this change of sign, if one flips the sign of the corresponding g-factors on one of the site: either g A α and g A β , or g B α and g B β . By changing both g A α , g A β , g B α and g B β , one retrieves the original sign for J α and J β . This rule applies whatever the values of S A and S B are. As depicted in Table S7, a ferromagnetic interaction (J > 0) favors the states with parallel pseudo-spins, ++ or --. With positive g factors, it corresponds to parallel magnetic moments. If one of the g is negative, + corresponds to a negative magnetic moment, and the favored state has opposite magnetic moments; this is equivalent to an antiferromagnetic coupling (J < 0) with positive g factors. It comes out that the fitting of the ab initio spectra is not sufficient to provide the sign of the coupling parameters, and is related to the sign of the corresponding g factors.

We then considered the dimers, built by replacing one Mn(II) by a diamagnetic Zn(II). A chemical model [Mn -U -Zn] mod was considered, where the cycles of the ligands are removed, and the coordination sphere of the Zn(II) center simplified (see Figure S2). The dimers were first described by SO-CASSCF, with the 3d and 5f active orbitals. The dynamical correlation is further described, performing a RASSCF calculation which includes the bonding and antibonding σ and π orbitals of the yle bond. Finally, CI calulations were performed using the DDCI method. All results are given and discussed in details in Section S3.2.1 of the Supplementary Material. At the spin-free level, each of the six doublets of the [U V O 2 ] + unit couples with the spin sextet of the Mn(II) center to form a septet and a quintet. For the ∆ and Φ states, the septet lies below the corresponding quintet, which is the a: fit of energies for the [Mn -U -Zn]mod dimer of Table S10 by spin Hamiltonian of Eq. 4. b: fit of the experimental χT = f (T ) curve using g A ⊥ = -0.6 and g A Z = -1.57 and spin Hamiltonian of Eq. 5. c: fit with only J iso using Eq. 9 with g SA X = 0.6; g SA Z = 1.6.

fingerprint of a ferromagnetic coupling. This coupling increases with correlation. With spin-orbit coupling, the 12 low lying states are fitted by the spin Hamiltonian, leading to parameters given in Table S11: J Z is positive, while the sign of J ⊥ is undetermined. The values increase with the level of correlation, and J ⊥ is 2-3 times smaller than J Z . It appears that the exchange coupling is largely reduced by the spin-orbit coupling as compared to the spin-free spectrum, by roughly a factor of 2. A similar effect was observed in the cerocene triple decker [START_REF] Gendron | Inorg[END_REF].

The comparison between the M = f (B) and χT = f (T ) curves calculated from first-principles and deduced from the spin Hamiltonian with the parameters deduced at the same level of calculation, allows to assess that the spin Hamiltonian is able to reproduce the physics, and to assign the sign of the g factors of the central [U V O 2 ] + unit, relatively to the J values. As developped in the Supporting Informations, it comes out that the J and g for the direction have opposite signs. In order to be coherent with the spin-free limit, the J components should be positive, leading to a ferromagnetic coupling in terms of spin densities, and the g Z negative, due to the large orbital contribution, as shown in Section S2.2.

The trimer [Mn -U -Mn] was described at SO-CASSCF level, DFT using a broken-symmetry scheme and by a spin Hamiltonian. The CASSCF energies are given in Tables S12 andS13 of the ESI. At spin-orbit free level, each of the six spin doublets of the [U V O 2 ] + unit couples with the two spin sextets to form one doducaplet, two docaplets, two octets, two sextets, two quartets and two doublets. Except for the Π states, the energy decreases with total spin, which denotes a ferromagnetic coupling. The simplified complex [Mn -U -Mn] mod , where the cycles have been removed from the ligands (cf Figure S2), leads to energies similar to the whole one [Mn -U -Mn], except for the splitting between the two ∆ manifolds which increases by more than 100 cm -1 , as for the dimer (cf Table S8), but not in the monomer (cf Table S3). The SO-CASSCF calcula-tion provides the 36 KDs issued from the coupling between the two spin sextets of the Mn(II) centers and the ground KD of the [U V O 2 ] -unit (cf Table S13). The overall splitting is 72 cm -1 , about twice the whole splitting in the dimer. There is a gap between KD21 and KD22. The g tensors go from axial to more or less equatorial from KD1 to KD21, and then from KD22 to KD36, from more or less equatorial to axial. KD36 has a larger g Z than KD1.

The spin Hamiltonian for the trimer is the following Ĥ = ĤA + ĤB + ĤC + ĤAB + ĤAC + ĤBC (5) where ĤA is given in Eq. 2, ĤB and ĤC in Eq. 3 and ĤAC , ĤAB in Eq. 4. The coupling is considered to be equal with the two Mn(II) centers. ĤBC represents the Mn(II)-Mn(II) coupling and is described by a Heisenberg spin Hamiltonian ĤBC = -j Ŝ Ŝ Ŝ B Ŝ Ŝ Ŝ C . The spectra for different classes of parameters are shown on Figure S8. The SO-CASSCF spectrum of the trimer was fitted by this Hamiltonian (cf Table S14), leading to parameters very close to the dimer, with a positive value for J Z , and undetermined sign for J X = J Y . The fit is improved with a small dissymmetry between J X and J Y which does not affect their average value and by adding a tiny positive magnetic direct coupling between the two Mn(II) centers, j = 0.06 cm -1 . As for the dimer, the response to the magnetic field with the spin Hamiltonian using the g-values of the monomer with different signs was compared to the ab initio curves (cf Figure S12). Again, negative g ⊥ and g Z match better. This corresponds to positive spin contributions (see Section S2.2). Furthermore, the g-factors deduced from spin Hamiltonian compare well with the ab initio values which confirms that the spin Hamiltonian of Eq. 5 is an adequate model to describe the magnetic properties of the [Mn -U -Mn] complex.

The experimental χT curve is represented in Figure 4. The SO-CASSCF curve does not match the experimental one, the peak appears at too low temperatures. This is improved using the DDCI2 parameters, with a larger coupling. The fit proposed by Chatelain et al. [26] works only at high temperature and provides a too high and too narrow peak at low temperatures. We fitted the experimental χT = f (T ) curve with the two J and J ⊥ parameters, the g-factors fixed to their ab initio values, g A ⊥ = -0.6 and g A Z = -1.57. One gets J Z = 56.4 cm -1 and J ⊥ = 30.9 cm -1 , two-three times larger than the DDCI2 values. The experimental M = f (B) curves are shown on Figure S12 and compared to those deduced from the spin Hamiltonian with the different sets of parameters. The set of parameters fitting the χT = f (T ) does not fit the M = f (B) curves, leading to too small values of the magnetization. It is actually the SO-CASSCF curves which is in best accordance. We tried to fit jointly χT = f (T ) and the M = f (B) curves at different temperatures, but unsuccessfully. The axial of χT denotes a maximum, while the transverse component is smaller and increases smoothly with temperature; the two components converge to a plateau, where the contributions of the three centers are additive. The contribution for the U(V) center corresponds to a g-value of 2g Broken-symmetry DFT provides magnetic couplings in good agreement with experimental data in {3d-4f} systems [70,71]. The same approach was performed on the [Mn -U -Mn] complex using the B3LYP/TZVP set up. The calculations are performed without spin-orbit coupling and are modeled according to the broken-symmetry scheme for isotropic magnetic couplings. The high-spin and three broken-symmetry configurations were considered to estimate the three isotropic exchange coupling constants J AB , J AC and j. This leads to J AB = J AC = 49.7 cm -1 and j = 0.5 cm -1 . We remind here that those values are issued from calculations where the spin-orbit coupling is not included. As mentionned previously, the coupling reduces by roughly a factor of 2 with the spin-orbit coupling.

Spin densities of the four configurations are shown in Figure S13 and are compared to SF-RASSCF values for the dimer in Table S15 and Figure S6. In any case, there is a negative spin density on the bridging O yl in the high-spin state. The magnetic orbital of the U(V) center is the same f δ1 as in the ground state of CAS-based calculations. In all configurations, the spin density of the bridging O yl atoms is opposite to the U(V) one; this shows the importance of the spin polarization of those atoms and that the magnetic unit is the [U

V O 2 ] + cation.
Mulliken charges on the oxo oxygens are larger with DFT, which may explain the larger coupling with this method, but the comparison should be performed with care, since the basis sets are different with the two methods.

Mn Mn U O yl O yl = =

The coupling in the dimer is finally described as a ferromagnetic coupling between a center with positive g-factors and an other with negative g-factors.

The sign of the J-parameters denotes which state is favored in terms of pseudo-spins, while the sign of the g-factors denotes the relationship of the magnetic moment to the pseudo-spin. The coupling is found to be ferromagnetic because it was analyzed in terms of spin moments, with positive spin g-factors on the central unit. But, since the orbital contribution on the [U V O 2 ] + unit is opposite to the spin one and dominant, this ferromagnetic coupling leads to opposite moments on the [U V O 2 ] + and Mn(II) units. In the trimer, the moments of the two Mn(II) units are parallel, as a ferrimagnetic coupling scheme. We can summarize the coupling scheme by Figure 5: i) the coupling between the spin densities of the [U V O 2 ] + and Mn(II) magnetic centers is ferromagnetic, in all directions, due to the spin polarization of the oxo bridging groups. ii) the anisotropy of the coupling is related to the anisotropy of the spin contribution of the g of the central magnetic [U V O 2 ] + unit iii) the orbital contribution of the [U V O 2 ] + unit being dominant and opposite to the spin one, the total magnetic moment of [U V O 2 ] + is opposite to the ones of the two Mn(II) units. This is modeled by positive J ⊥ and J Z and negative g A ⊥ and g A Z parameters. iv) the overall magnetic scheme is ferrimagnetic.

The [Mn -U -Mn] complex exhibits a SMM behavior with a relaxation barrier U eff of 81 K and exhibits an open magnetic hysteresis loop at temperatures below 3 K, with a significant coercive field of 1.9 T at 1.8 K. In lanthanide complexes, correlations can be found between the barrier U eff and the energy of the first excited state, which is the sign of an Orbach relaxation process. In actinide complexes, the energy gaps are much larger, and this correlation is not as clear, which suggests that other relaxation pathways occur [72]. Anyhow, a ground state with an anisotropic magnetic moment, and excited states with anisotropic moments and large excitation energies are necessary to the SMM behavior. In Figure 6, the energy of the states is rep-

⊥ . g A ⊥ = -0.5, g A Z = -1.6, J ⊥ = 30.9 cm -1 , JZ = 56.4 cm -1 .
resented as a function of the axial (red) and transverse (blue) g-factors, deduced from the spin Hamiltonian with the coupling parameters fitting the experimental χT . One notices that the perpendicular magnetic component is zero up to 80 cm -1 while the parallel component decreases, leading to an energy barrier close to the experimental U eff value. In Section S4.3 of the Supplementary Material, this scheme is depicted for different sets of parameters. It comes out that the barrier appears when the coupling is close to an Ising Hamiltonian, with J ⊥ as small as possible. The sign of J Z does not influe much and the anisotropy of the g A tensor does not impact this scheme ; the magnetization barrier issued from electronic states arises essentially from the anisotropy of the magnetic coupling.

In the Mn(III) -M(III) -Mn(III)(M=Ru,Os) SMM complexes [73,74], the ground KD is issued from an orbitally degenerate spin-free state; the coupling is orbitally-dependent and consequently anisotropic [75]. In the present case, the ground KD is issued from two spin-free states 2 ∆ 1 and 2 ∆ 2 . These two states have the same coupling with the Mn(II) centers. When degenerate, the KD is the ∆ 3/2 of the [U V O 2 ] + cation, with a pure axial g tensor with both orbital and spin contributions (see Eqs. S2 with B = 0). The spin density of the [U V O 2 ] + unit is represented in Figure 3 (see as well Eqs. S5 and S5 and NSOs in Figure S5); along Z, the contributions of the two δ orbitals are additive, which leads to a non-zero g SA Z , while along X or Y , they have opposite signs and this leads to a zero value for g SA ⊥ . With a splitting of the two δ orbitals, the cancellation is only partial, g SA ⊥ is non-zero, and with a very large gap, the g SA -factors are isotropic. The magnetic interaction between two magnetic centers is based on local spin densities. The spin g-factors are directly related to the spin densities along the corresponding direction ρ S α (r r r ) as

g SA α = 4 ρ S α (r r r )dr r r (6)
The two δ orbitals are of the same symmetry as the corresponding d δ , and consequently have the same interaction with the Mn centers, in accordance with the X ≡ Y symmetry. But the spin density is axially anisotropic. One can expect the coupling interaction in a given direction to depend on the amount of spin density in this direction: if the latter vanishes, the interaction should vanish. In Table 3, the J ⊥ /J Z ratio is worth between 0.3 and 0.38, and 0.55 with the fitted parameters, while in Table S5, g SA X /g SA Z is worth 0.38. It suggests to express the anisotropy of the coupling in terms of the anisotropy of the local g-tensors, as it is usually performed, but in terms of the anisotropy of the spin g-tensors. In Eq. 1, the exchange coupling is described as an anisotropic interaction between pseudo-spins. The pseudo-spins are mathematical operators which allow to describe the anisotropy of the local magnetic moments through the g tensor. Those magnetic moments break down in orbital and spin contributions, as specified by the g L and g S tensors, respectively. Consequently, the spin magnetic moment of a magnetic unit is g S Ŝ Ŝ Ŝ . Since the anisotropy of the coupling arises from the anisotropy of the spin densities, we propose to model the coupling between two magnetic centers A and B with respective spin g tensors g SA and g SB by an isotropic interaction between the anisotropic spin moments

ĤAB = - 1 4 J iso g SA Ŝ Ŝ Ŝ A † g SB Ŝ Ŝ Ŝ B (7)
The factor 1/4 is introduced in order to obtain the usual Heisenberg Hamiltonian in the case of two isotropic spin-only magnetic centers. In the case where the two local g-tensors have the same PAF, Eq. 7 becomes

ĤAB = - 1 4 J iso g SA X g SB X ŜA X ŜB X + g SA Y g SB Y ŜA Y ŜB Y +g SA Z g SB Z ŜA Z ŜB Z (8)
and in the present case

ĤAB = - 1 2 J iso g SA ⊥ ŜA X ŜB X + g SA ⊥ ŜA Y ŜB Y +g SA Z ŜA Z ŜB Z (9)
which is equivalent to Eq. S18 with

J ⊥ = 1 2 J iso g SA ⊥ ; J Z = 1 2 J iso g SA Z and J = 1 2 J iso 2g SA ⊥ + g SA Z .
We applied this approach by fitting the experimental χT curve; as shown in the last column of Table 3, this leads to parameters rather close from the twoparameters fit. This approach allows to reduce the fitting of the experimental curve to one parameter, once the one-center anisotropic magnetic properties are known.

The barrier is related to the anisotropy of the coupling, which depends on the anisotropy of the spin g-tensor. The composition of the ground KD is tuned by the f δ splitting. While this splitting smoothly impacts the axial spin density, it determines the transverse one: the transverse spin density sums up to 0 in the case of no gap and to 2 when the gap becomes important. It shows that this energy gap between the two f δ orbitals plays a key role in the anisotropic properties, not only by introducing a large orbital contribution in the case of no gap, but by quenching the transverse spin contribution, and consequently by quenching the transverse magnetic coupling. The f δ splitting is related to the symmetry of the equatorial ligands of the [U V O 2 ] + unit. The splitting will be the largest with ligand with a four-fold symmetry, one of the f δ being in this case largely anti-bonding.

Conclusions

The large metal-ligand covalency and stronger anisotropy of actinides make it very attractive for SMMs over lanthanides. Furthermore, the strong magnetic exchange substantially quenches the tunnelling, as shown in several cases. Despite these advantages, actinide chemistry is less explored. This is essentially due to a lack of solid theoretical understanding of the design principles and various factors that contribute to a reduction in barrier heights/blocking temperature. This work proposes a detailed analysis of the anisotropic magnetic properties in a {3d-5f-3d} trimer. This analysis is based on CAS based and scalar DFT methods, and spin Hamiltonian. The two Mn(II) centers are half-filled 3d 5 shell with spin-only isotropic magnetic properties. The unpaired electron of the central [U V O 2 ] + unit is mostly borne by the two 5f δ orbitals; the ground KD has an axial g-tensor, with negative values, due to a large orbital contribution, opposite to the spin one. The coupling between the [U V O 2 ] + unit and the two Mn(II) centers is ferromagnetic, due to a large spin polarization of the oxo groups. It is anisotropic, larger in the parallel direction than in the transverse one, and increases with correlation. The sign of the transverse coupling can only be determined relative to the sign of the transverse g-factors. The spin Hamiltonian fits well the 72 magnetic states calculated with SO-CASSCF, which validates the ability of the spin Hamiltonian to describe this system. The two coupling parameters were finally fitted on the χT curves, using the ab initio g-factors of the [U V O 2 ] + unit, leading to values larger than those determined by CAS-based and DFT methods. But, the main contribution to the magnetic moment of the [U V O 2 ] + unit is orbital, which is opposite to the spin one, leading to negative g-factors. It follows that the magnetic moments are opposite on Mn(II) and U(V) centers: the ferromagnetic interaction between the spin mag-netic moments becomes a ferrimagnetic interaction for the total moments, due to the local Hund's rule on the central cation.

The magnetization barrier is deduced from the spin Hamiltonian. Supposing that the relaxation follow an Orbach relaxation scheme, it corresponds to the experimental value [76,77,78,79]. It is related to the anisotropy of the coupling, but is not affected by the anisotropy of the central g-tensor. The ratio of the transverse and axial components of the coupling and of the spin g-factors of the central unit are close. One can expect the anisotropy of the coupling to arise from the anisotropy of the spin density. The transverse spin density sums up to zero when the two f δ orbitals are degenerate. As a result, in order to increase the barrier, one should quench the transverse spin contribution of the [U V O 2 ] + unit, which is tuned by the splitting between the f δ orbitals. Due to the four-fold symmetry of the latter, the splitting is favored by equatorial ligands denoting the same symmetry. As a consequence, one should avoid four-fold equatorial ligands in order to increase the barrier.

The fitting of the magnetic response for complexes with anisotropic properties is difficult, even impractical, due to the large number of parameters. Thanks to the use of the local g-tensors determined from ab initio calculations, only the coupling is fitted. In lanthanide complexes, the coupling is reduced to only one parameter, when the ground J manifold is populated [38,39,40]. This was performed in this work by expressing the coupling Hamiltonian as an isotropic interaction between the spin magnetic moments, which are anisotropic. The new fit, with only one parameter, is in good accordance with the two-parameters fit.

To summarize (i) The g factors of the ground KD of the central anisotropic unit [U V O 2 ] + are negative, due to a large orbital contribution opposite to the spin one. Further, due to strong U-ligand covalency, we show that this anisotropy originates due to splitting within the 5f orbitals, particularly between two 5f orbitals. This allows to offer a design principle to enhance the anisotropy via ligand tuning. (ii) Due to the large orbital contribution opposite to the spin one, even if the coupling is ferromagnetic in terms of spin densities, the interaction in terms of magnetic moments is ferrimagnetic. (iii) The anisotropy of this interaction is related to the anisotropy of the spin density of the U(V) unit. Therefore, the use of local spin contribution to the g-tensor extracted from ab initio calculations allows reducing the number of parameters to be fitted in this anisotropic case to only one parameter. Consequently, the method can be used to analyze the magnetic properties of any other 3d-5f SMM, what we will do in a close future. (iv) The magnetization barrier is related to the anisotropy of the coupling, not from the anisotropy in the g factors as seen in the lanthanide SMMs. Thus for the design of actinide SMMs, one has to focus on enhancing this exchange coupling via suitable ligand and not the individual single-ion anisotropy.

Computational details

WFT calculations

Calculations were performed on the X-ray structures [26] or simplified structures: (see Section S1 for more details) i) the trimers, with three paramagnetic centers, either complete [Mn -U -Mn] (cf Figure 1) or truncated [Mn -U -Mn]mod in which the TPA and Mesaldien ligands are simplified (cf Figure S2). ii) the dimers obtained by diamagnetic substitution: one of the Mn(II) centers is replaced by a diamagnetic Zn(II) cation with a simplified coordination sphere. The Mn(II) and U(V) center are either with the original coordination sphere [Mn -U -Zn] or with simplified ligands [Mn -U -Zn]mod (cf Figure S2). iii) the monomers, with only one paramagnetic center. In the Zn -U -Zn complex, the two Mn(II) centers are replaced by diamagnetic Zn(II) cations with a simplified coordination sphere, and in [Mn -U VI ] complexes, U(V) is replaced by the diamagnetic U(VI) and the other Mn(II) center is removed (cf Figure S1). Our earlier work suggests that such approach has negligible effect on anisotropic properties [50].

WFT calculations were performed using MOLCAS 7.8 program package [80]. Spin-free wave functions were generated by the state average complete active space self-consistent field (SF-CASSCF) method [81]. The DKH (Douglas Kroll Hess) Hamiltonian was used to account of relativistic effects [82] and Cholesky decomposition to reduce the computational cost of calculating two-electron integrals [83]. Dynamical correlation in the U(V) monomer was calculated with the CASPT2 method [84]: the 5p(U) and 5d(U) are included in the correlation space, since it was shown that they may impact the results [61]. In the dimers, dynamical correlation was calculated by Configuration Interaction (CI) calculations performed with the CASDI program [85]: CAS+S and DDCI2 [86]. If S1 (S3) is the space of the orbitals which are doubly occupied (unoccupied) in the CASCI (Complete Active Space Configuration Interaction) calculation and n h (np) the number of allowed holes (particles) in S1 (S3), the CAS+S CI corresponds to n h ≤ 1 and np ≤ 1 and DDCI2 space contains all the configurations satisfying n h + np ≤ 2. CI calculations for the [Mn -U -Zn]mod complex involved 118 electrons in 429 orbitals, with a CAS(6,9) using the orbitals of the septet. CAS+S CI corresponds to n h ≤ 1 and np ≤ 1. The spin-orbit interaction was calculated as a state interaction by RASSI-SO module [87]. Spin-Orbit (SO) integrals are calculated using the AMFI (Atomic Mean-Field Integrals) approximation [88]. SO-CAS+S and SO-DDCI2 results were obtained by dressing the spin-orbit matrix obtained with CASSCF with the corresponding Spin-orbit Free (SF) energies. Both for the state average CASSCF and the spin-orbit state interaction, all magnetic states were considered. Parameters for the zero-field splitting spin Hamiltonian of the Mn(II) centers were computed with the SINGLE ANISO code [89]. g-factors and their spin and orbital contributions were calculated according to ref. [90] and magnetic susceptibility to ref. [91]. Basis sets, active spaces and spin-orbit state interaction are detailed in Section S1.

DFT calculations

DFT calculations were carried out using the G09 suite [START_REF] Frisch | Gaussian 09 Revision A.01[END_REF] using the hybrid B3LYP exchange-correlation functional [START_REF] Becke | [END_REF]. Quadratic convergence was used to get the most stable wave function. A quasi relativistic ECP with 60 core electrons (ECP60MWB) along with ECP60MWB ANO basis set was employed for Uranium as this ECP is well established to give a numerical accuracy [94]. Basis sets of Ahlrichs TZVP quality were used for Mn, O and C atoms and SVP for H [95]. Since these calculations do not include the spin-orbit coupling, the exchange coupling is isotropic and the following Heisenberg Hamiltonian was considered

Ĥ = -J Ŝ Ŝ Ŝ A Ŝ Ŝ Ŝ B -J Ŝ Ŝ Ŝ A Ŝ Ŝ Ŝ C -j Ŝ Ŝ Ŝ B Ŝ Ŝ Ŝ C ( 10
)
where Ŝ Ŝ Ŝ A , Ŝ Ŝ Ŝ A and Ŝ Ŝ Ŝ A are the local spin operators with SA = 1/2 and SB = SC = 5/2. We underline that Ŝ Ŝ Ŝ A acts as a real spin in Eq. 10 while it acts as a pseudo-spin in Eq. 1. For our calculations, four configurations were considered: one high-spin HS = Mn1↑-U↑-Mn2↑ and three broken-symmetry, BS1 = Mn1↓-U↑-Mn2↑, BS2 = Mn1↑-U↑-M2↓, BS3 = Mn1↑-U↓-Mn2↑. Using the broken symmetry approach developed by Noodleman [96], the two coupling parameters J and j were estimated using the non-spin projected formula [97]: EBS1 -EHS = EBS1 -EHS = 3J + 15j and EBS3 -EHS = 6J.

Figure 1 :

 1 Figure 1: X-ray structure of [Mn -U -Mn] = [{Mn(TPA)I} {UO2(Mesaldien)}{Mn(TPA)I}]I complex. Color code U: cyan, Mn: pink; I: aqua, O: red, N: blue, C: gray. H atoms are omitted for clarity.

Figure 2 :

 2 Figure 2: Canonical orbitals and energies (in parentheses, cm -1 ) from SF-CASSCF of [Zn -U -Zn]mod. Two perspectives are shown for the 4 lowest ones. Isovalue: 0.01 au.

Figure 3 :

 3 Figure 3: Spin density of the ground KD of the [Zn -U -Zn]mod complex, for the principal directions of the g tensor, from SO-CASSCF (above) and SO-RASSCF (below). Isovalue = 0.0002 e/bohr -3 .

Figure 4 :

 4 Figure 4: χT = f (T ) experimental, and deduced from Spin Hamiltonian with parameters from CASSCF (red), DDCI2 (green), from ref. [26] (blue) and fit of the χT curve (black; Z component dashed, ⊥ component dotdashed).

Figure 5 :

 5 Figure 5: Magnetic scheme in the [Mn -U -Mn] trimer. Blue: spin, red: orbital.

Figure 6 :

 6 Figure6: Energy (cm -1 ) of the KDs of the trimer as a function of the g-factors; red: gZ , blue g ⊥ . g A ⊥ = -0.5, g A Z = -1.6, J ⊥ = 30.9 cm -1 , JZ = 56.4 cm -1 .

Table 1 :

 1 SO-CASSCF energies and zero-field splitting parameters (cm -1 ) of the Mn(II) monomers.

		[Mn1 -U VI ]	[U VI -Mn2]
		CAS (5,5) CAS (5,10) CAS (5,5) CAS (5,10)
	KD1	0	0.0	0.0	0
	KD2	0.4	0.5	0.4	0.5
	KD3	1.2	1.3	1.3	1.4
	KD4 26701	24685	26278	24228
	KD5 26704	24687	26281	24230
	D a	0.20	0.21	0.21	0.23
	E a	0.02	0.02	0.02	0.02
	a: deduced from the three first KDs with spin Hamiltonian
	ĤS = D S 2 Z -1 3 S(S + 1) + E S 2 X -S 2 Y and S=5/2.

Table 2 :

 2 Energies (cm -1 ) of the U(V) monomers from spin-orbit calculations.

		[Zn -U -Zn]		[Zn -U -Zn]mod
		CAS(1,6)	CAS(1,6)	RAS(13,18)
		SCF	SCF PT2	SCF PT2
	KD1	0	0	0	0	0
	KD2	1124	1007 774	918 610
	KD3	6486	6443 6296 6363 6214
	KD4	9034	8973 8928 8844 8751
	KD5	11658	11595 12036 11906 11378
	KD6	14481	14420 14861 14636 14175

Table 3 :

 3 Spin Hamiltonian parameters (cm -1 ) from the ab initio energies of the [Mn -U -Zn]mod dimer and from the experimental χT = f (T ) curve.fit energies [Mn -U -Zn]mod a fit χT exp

		CASSCF RASSCF CAS+S DDCI2	b	c
	J ⊥	4.7	6.0	9.6	10.0	30.9 22.5
	JZ	13.7	20.2	25.9	26.2	56.4 60.0
	J	7.7	10.7	15.0	15.4	39.4 35.0
	D	-6.0	-9.5	-10.8 -10.8 -17.0 -25.

  2 ⊥ + g 2 /3 = 1.03, a value very close to the g U from Chatelain et al. The ab initio gfactors are consequently in good accordance with the experimental data.

a Note the typographic difference between J the total angular momentum quantum number and J the magnetic coupling parameter.

Acknowledgments S.D. thanks the grant NanoX n°ANR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir" for the financial support during his stay in Toulouse. This work was supported by the ANR under convention N°ANR-17-CE06-0010. G.R. thanks CSIR (01(2980)/19/EMR-II) for funding.

Data availability

The data that support the findings of this study are openly available in ZENODO at https://doi.org/10.5281/zenodo.6659929.