J M Vazquez-Nicolas

Erik Zamora

Iván González-Hernández

Rogelio Lozano

Humberto Sossa

PD+SMC Quadrotor Control for Altitude and Crack Recognition Using Deep Learning

Keywords: Deep learning, embedded control system, inspection, quadrotor aircraft, robust altitude control, UAV

Building inspection is a vital task because infrastructure damage puts people at risk or causes economic losses. Thanks to the technological breakthroughs in regard to Unmanned Aerial Vehicles (UAVs) and intelligent systems, there is a real possibility to implement an inspection by means of these technologies. UAVs allow reaching difficult places and, depending on the hardware carried onboard, take data or compute algorithms to understand the environment. This paper proposes a real-time robust altitude control strategy for a quadrotor aircraft, also a convolutional neuronal network for crack recognition is developed. The main idea of this proposal is to lay the background for an autonomous system for the inspection of structures using a UAV. For the robust control, a combination of two control actions, one linear (PD) and another nonlinear (Sliding Mode) is used. The combination of these control actions allows increasing the system's performance. To verify the satisfactory performance of proposed control law, simulations and experimental results with a quadrotor, in the presence of disturbances, are presented. For crack recognition in images, several experiments were carried out validating the proposed model. For CNN training, a database of cracks was built from images taken from the Internet.

INTRODUCTION

Unmanned aerial vehicles (UAVs) have become a helpful tool for performing autonomous tasks in the industrial, commercial and research areas. Given your advantages (managed remotely or autonomously, fly over areas of difficult access, etc.), these robots are an ideal tool for inspection, as the case of structures. In Mexico, as well as in other parts of the world, natural disasters are frequent. On September 19, 2017, the worst earthquake in the story of Mexico, since 1985, was occurred. Many buildings suffered structural damages (see Fig. 1) which put the lives of people at risk.

The need to inspect buildings became evident during this tragic event. The civil protection protocol indicates that an inspector must return to the building and identify the risks of the structure and, if there are any, do not allow the re-entry to the building. In this sense, the inspector runs a risk when re-entering the building to perform the inspection. The difficulty of carrying out the check is that the areas to be inspected are usually difficult to access or there exist a risk for the human inspector. One of the most common failures are cracks, which can put at risk the structure and the integrity of people.

To improve the previous work [START_REF] Vazquez-Nicolas | Towards automatic inspection: crack recognition based on Quadrotor UAVtaken images[END_REF] where a Convolutional Neural Network (CNN) was proposed to detect cracks in images taken by a quadrotor aircraft, the present work is focused on the robust altitude control of the UAV, in presence of external disturbances, and in the increase of the images dataset to re-train the CNN and improve their performance. Our future aim is to build an UAV-based autonomous inspection system for structures such as buildings, tunnels, bridges, etc. However, in this paper, we focus on the control subsystem and the crack recognition subsystem.

The main contribution of this work is an altitude hybrid control for a system based on a quadrotor aircraft, focused towards an autonomous wall inspection in the pres-

RELATED WORK

Unmanned Aerial Vehicles (UAV) have been an important research topic in recent years. In particular, the quadrotor helicopter has been under the focus of extensive research due to their many advantages. The control of these systems was extensively covered in the literature; for instance, PD linear control [START_REF] Sandiwan | Robust proportional-derivative control on SO(3) with disturbance compensation for quadrotor UAV[END_REF][START_REF] Liu | Robust tracking control of a quadrotor helicopter[END_REF][START_REF] Elfeky | Modeling and simulation of quadrotor UAV with tilting rotors[END_REF][START_REF] Rafifandi | Leader-follower formation control of two quadrotor UAVs[END_REF][START_REF] Jeong | Cartesian space control of a quadrotor system based on low cost localization under a vision system[END_REF][START_REF] Zuo | Trajectory tracking control design with commandfiltered compensation for a quadrotor[END_REF] and SMC [START_REF] Barghandan | Improved adaptive fuzzy sliding mode controller for robust fault tolerance of a quadrotor[END_REF][START_REF] López-Gutiérrez | Robust quadrotor control: attitude and altitude real-time results[END_REF][START_REF] Aboudonia | Composite hierarchical anti-disturbance control of a quadrotor UAV in the presence of matched and mismatched disturbances[END_REF][START_REF] Escareño | Trajectory control of a quadrotor subject to 2d wind disturbances[END_REF][START_REF] Liu | Altitude control for variable load quadrotor via learning rate based robust sliding mode controller[END_REF][START_REF] L'afflitto | An introduction to nonlinear robust control for unmanned quadrotor aircraft: how to design control algorithms for quadrotors using sliding mode control and adaptive control techniques [Focus on Education[END_REF] have been commonly used to control quadrotor systems.

In this work a hybrid PD+SMC is used to carry out the altitude control. This approach has been used to control robotic arms and robot teleoperation ( [START_REF] Salinas | P+d plus sliding mode control for bilateral teleoperation of a mobile robot[END_REF][START_REF] Ouyang | PD with sliding mode control for trajectory tracking of robotic system[END_REF][START_REF] Tang | Nonlinear PD sliding mode control for robotic manipulator[END_REF][START_REF] Ouyang | Adaptive PD plus sliding mode control for robotic manipulator[END_REF]) but, to the best of our knowledge, this approach has not been used for a quadrotor UAV.

There are works focused on crack detection in structures, some using computer vision algorithms. The images for the experiments are commonly recollected by humans, in a safe area or with the help of a robotic system. However, in recent years, UAVs have been used to obtain images for inspection tasks, mainly in structures that involve risk for the human or in areas where access is difficult.

Concerning the detection of cracks without UAVs, in [START_REF] Lins | Automatic crack detection and measurement based on image analysis[END_REF] authors used the color characteristics via a detection algorithm and, based on this information, an estimate of the length of the cracks is made. In [START_REF] Shi | Automatic road crack detection using random structured forests[END_REF], the authors im-plement crack detection in images of asphalt, where features extraction is carried out by decision trees and classification is performed by k-nearest neighbors method and support vector machines (SVM). Authors in [START_REF] Talab | Detection crack in image using Otsu method and multiple filtering in image processing techniques[END_REF], use classic computer vision techniques like edge detection (Sobel filter) and thresholding (Otsu algorithm) to locate cracks.

In [START_REF] Phung | Automatic crack detection in built infrastructure using unmanned aerial vehicles[END_REF], [START_REF] Wang | Automatic detection of wind turbine blade surface cracks based on UAV-taken images[END_REF], and [START_REF] Choi | Building crack inspection using small UAV[END_REF], present images which are obtained by means of UAVs and subsequently the detection of cracks was performed. In [START_REF] Phung | Automatic crack detection in built infrastructure using unmanned aerial vehicles[END_REF], the authors use lasers to generate a 3D model of the surface to be inspected for the autonomous navigation system, and the detection of cracks is made by means of the histogram and adaptive thresholds. Authors in [START_REF] Wang | Automatic detection of wind turbine blade surface cracks based on UAV-taken images[END_REF], use classifiers and features extraction to find flaws on a wind generator blade surface. A UAV is used to take pictures of a generator. In [START_REF] Choi | Building crack inspection using small UAV[END_REF] a UAV for building inspection is proposed, implementing a basic edge detection algorithm.

Given that our work uses deep learning approach to accomplish the cracks recognition, it is important to mention works related to applications of UAVs and convolutional neural networks (CNN). In [START_REF] Pan | Power line detection via background noise removal[END_REF], [START_REF] Tang | Fast vehicle detection in UAV images[END_REF], [START_REF] Zhang | Application of deep learning and unmanned aerial vehicle technology in traffic flow monitoring[END_REF], [START_REF] Ma | Hand gesture recognition with convolutional neural networks for the multimodal UAV control[END_REF] and [START_REF] Zhang | Road crack detection using deep convolutional neural network[END_REF] where CNN were used to face problems such as high voltage inspection, vehicular traffic tracking, recognition of gestures, and crack recognition on pavement. Authors in [START_REF] Yang | Deep concrete inspection using unmanned aerial vehicle towards CSSC database[END_REF] presents a crack recognition algorithm in images taken by a UAV using a CNN. In terms of deep learning, our work uses a smaller CNN structure than presented in [START_REF] Yang | Deep concrete inspection using unmanned aerial vehicle towards CSSC database[END_REF] and the database of cracks were taken from damaged buildings.

DYNAMIC MODEL FOR QUADROTOR AIRCRAFT

The mathematical model of the quadrotor aircraft is widely discussed in the literature ( [START_REF] Lozano | Unmanned Aerial Vehicles: Embedded Control[END_REF], [START_REF] Castillo | Real-time stabilization and tracking of a four-rotor mini rotorcraft[END_REF] and [START_REF] Gonzalez | Chattering-free sliding mode altitude control for a quad-rotor aircraft: realtime application[END_REF]) where the vehicle is considered as a rigid body that unfolds in three dimensions, which is subject to a main force and three moments that will be formulated by means of the Euler-Lagrange equations. Fig. 2 shows a free-body diagram of the quadrotor aircraft. The generalized coordinates of the vehicle are given by

q = (x, y, z, ψ, θ , ϕ ) ∈ R 6 , (1)
where we can define ξ = (x, y, z) ∈ R 3 which denotes the position of the center of mass of the vehicle with respect to the inertial frame I and we also define η = (ψ, θ , ϕ ) ∈ R 3 which represents the angles of Euler, ψ is the angle of yaw, θ is the angle of pitch and ϕ is the angle of roll which are used to represent the orientation of the vehicle. We can decompose the equations into translational and rotational displacement. As first step, we obtain the Lagrangian of the aerial vehicle, which is given by the following equation

L(q, q) = T trans + T rot -U, ( 2
)
where

T trans = m 2 ξ T ξ , T rot = 1 2
ω T Iω and U = mgz are the translational kinetic energy, the rotational kinetic energy and the potential energy of the system, respectively. The Euler-Lagrange equation is used to obtain the quadrotor dynamic model

d dt ∂ L ∂ q - ∂ L ∂ q = [ F ξ τ ] = F, (3)
where F ξ is the translational force that is applied to the quadrotor and that is due to the control input that is a function of the force applied by the motors, τ ∈ R 3 refers to the moments of Euler's angles.

The translational force F ξ = R F where R refers to the rotation matrix R(ψ, θ , ϕ ) of the vehicle with respect to a fixed reference axis.

We define the force F = [0 0 u] T and the control input

u = f 1 + f 2 + f 3 + f 4 in such that f i = k i ω i 2 , ∀i = 1, ..., 4, (4)
where k i is the value constant of the i-st engine and ω i is its angular velocity. On the other hand, the moments are defined by:

τ =   τ ψ τ θ τ ϕ   △ =   ∑ 4 i=1 τ M i ( f 2 -f 4 )l ( f 3 -f 1 )l   , ( 5
)
where l is the distance between the engines and the center of gravity of the vehicle, while τ M i represents the moment produced by the engine M i , i = 1, ..., 4 around the center of gravity of the quadrotor aircraft. Defining C(η, η) as a function that represents the associated Coriolis terms to the gyroscopic and centrifugal effects corresponding to η

C(η, η) = J - 1 2 ∂ ∂ η ( ηT J). ( 6
)
We can write the equations of the system as

F ξ = m ξ + mgE z , (7)
J η = -C(η, η) η + τ. ( 8
)
Using a change of the input variables

τ = C(η, η) η + J τ, (9)
where τ = [ τ ψ , τθ , τϕ ] T Finally we obtain the system equations given by

m ẍ = -u sin θ , m ÿ = u cos θ sin ϕ , mz = u cos θ cos ϕ -mg, φ = τϕ , θ = τθ , ψ = τψ . ( 10
)

ALTITUDE CONTROL BASED ON PD+SMC ALGORITHM

A widespread method for altitude control in a quadrotor UAV is a typical PD controller, which is widely used due to their simplicity and acceptable performance. However, for a real-time application where there are disturbances such as wind gusts, it is necessary to use a nonlinear control which provides robustness [START_REF] Escareño | Trajectory control of a quadrotor subject to 2d wind disturbances[END_REF][START_REF] Gonzalez | Chattering-free sliding mode altitude control for a quad-rotor aircraft: realtime application[END_REF]. One well-known control method is the Sliding Mode Controller (SMC). Both PD and SMC control have strengths and weaknesses, these motivate the use of a combination of these techniques.

Control design

The design problem is to enforce the behavior of the states towards the desired trajectories which are known. The following procedure describes how to determine a control law for any of the dynamics of the quadrotor (x, y, z, ψ, θ or ϕ ). In this paper, the control for the z dynamics has been obtaining. Denote the reference trajectories by żd and z d which is velocity and altitude desired, respectively. Afterwards, we define the tracking errors by e z = zz d and ėz = żżd , where z d is the desired altitude. Since altitude control concerns only the displacement in the z-axis can be considered the following model from [START_REF] Aboudonia | Composite hierarchical anti-disturbance control of a quadrotor UAV in the presence of matched and mismatched disturbances[END_REF]

z = 1 m (u cos(ϕ ) cos(θ ) -mg). ( 11
)
To start, the control law u is proposed as follows:

u = m(r + g) cos(ϕ ) cos(θ ) (12)
with

r = -k p tanh(e z ) -k d tanh( ėz ) u 1 -ρ sign(s) u 2 , ( 13
)
Fig. 3. Schematic of the control system.

where k p , k d and ρ are gains to tune the altitude controller while s is the sliding surface of SMC. In order to solve the input saturation problem, we use the approach used in [START_REF] Sun | Transportation control of double-pendulum cranes with a nonlinear quasi-PID scheme: design and experiments[END_REF], adding a tanh function to saturate the control input.

A schematic of the control proposed is presented in Fig. 3. Equation ( 13) has u 1 and u 2 that represents a PD and SMC control actions, respectively. The SMC control scheme introduces a "sliding surface" along which the sliding motion is to take place. This surface is denoted by s and is defined as follows:

s = ( d dt + λ ) e z , s = ėz + λ e z , (14)
where λ > 0 is the slope of the sliding line. If a control law enforces the trajectories in the phase space such that s = 0 in [START_REF] Salinas | P+d plus sliding mode control for bilateral teleoperation of a mobile robot[END_REF], then the errors converge in a finite time to the origin due to

0 = ėz + λ e z , ėz = -λ e z , ( 15
)
where the solution is: e z (t) = e z (0)e -λt . Consider the twodimensional system [START_REF] Khalil | Nonlinear Control[END_REF] ẋ1

= x 2 , ẋ2 = h(x) + p(x)u, ( 16
)
where h(x) and p(x) are unknown locally Lipschitz functions and p(x) ≥ p 0 for all x. The goal is to design a controller that constraints the trajectory to the manifold s = ax 1 +x 2 = 0. Choosing a > 0 guarantees that x(t) tends to zero as t tends to infinity and the rate is controlled by the parameter a. The surface s satisfies the equation

ṡ = a ẋ1 + ẋ2 = ax 2 + h(x) + p(x)u. ( 17
)
Suppose that h and g satisfy

ax 2 + h(x) p(x) ≤ ρ(x), ∀x ∈ R 2 . ( 18
)
This assumption allows us to define a bounded region by ρ(x) to limit the control input of the SMC. Defining the states

x 1 = z x 2 = ż, ( 19
) therefore ẋ1 = ż ẋ2 = z, ( 20
)
so that our system is defined as

ẋ1 = x 2 , ( 21
) ẋ2 = -g h(x) + ( cos ϕ cos θ m ) p(x) u, ( 22
)
under the restriction from ( 18)

ax 2 + (-g) cos ϕ cos θ m ≤ ρ(x), ∀ x ∈ R 2 , ( 23
)
where ρ(x) is the upper bound of the states (altitude and velocity in the z-direction). For the design process, the control inputs (u 1 and u 2 ) are analyzed separately.

Stability analysis

We propose the following Lyapunov function candidate given as

V = 1 2 s 2 V 1 + 1 2 ė2 z + k p ln(cosh(e z )) V 2 . ( 24
)
The Lyapunov function candidate is composed by two parts: SMC control (V 1 ) and PD control (V 2 ) so the stability analysis is made separately.

Stability for V 1

For V 1 function, your derivative can be computed as follows:

V1 = s ṡ, ( 25
) considering a control input ũ ũ = -β (x) sign(s), (26)
thereby

V1 ≤ g(x)|s|ρ(x) -g(x)[ρ(x) + β 0 ] sign(s)s ≤ -g(x)β 0 sign(s)s, (27)
considering the next properties sign 2n (s) = 1 for n = 1, 2, 3 . . . , [START_REF] Gonzalez | Chattering-free sliding mode altitude control for a quad-rotor aircraft: realtime application[END_REF] s = sign(s)abss for s ∈ R,

we obtain

V1 ≤ -g(x)β 0 |s| ≤ -g 0 β 0 |s|, ( 30
)
which implies that the system is asymptotically stable since V1 < 0 for s ̸ = 0 in the region bounded by g 0 β 0 .

Stability for V 2

The time derivative of V 2 is V2 = ėz ëz + k p tanh(e z ) ėz , (

where

ëz = z -zd . ( 32
)
The goal to the control is to let z d = 0 so ëz = z. From ( 11) and ( 13), we obtain

V2 = ėz (-k p tanh(e z ) -k d tanh( ėz )) + k p tanh(e z ) ė = -k p tanh(e z ) ė -k d ė tanh( ėz ) + k p tanh(e z ) ė = -k d ė tanh( ėz ), (33)
which implies that the system is asymptotically stable since V2 < 0 for ė ̸ = 0.

SIMULATION RESULTS

Simulation results of the quad-rotor aircraft for altitude control (z-axis dynamic) are presented below. For simulation purposes, an external disturbance has been introduced to test the robustness of the control scheme proposed.

Wind-gust model

In order to perform a more realistic simulation of the system, wind effects are considered as disturbances. The wind model selected is the Dryden wind-gust. This model is defined as a sum of sinusoidal signals

v ω (t) = v 0 ω + n ∑ i=1 a i sin(Ω i t + φ i ), (34)
where v ω (t) is an estimate of the wind vector, Ω i are randomly selected frequencies, φ i are phase shifts, a i is the amplitude of sinusoids and v 0 ω is the static wind vector. The simulation was developed in the Simulink platform, where the force caused by the wind gust is considered proportional to the velocity given by [START_REF] Zhang | Road crack detection using deep convolutional neural network[END_REF]. This force is shown in Fig. 4.

PD Control with Sliding Mode Compensation

(PD+SMC) Three different control laws were simulated; a) Proportional -Derivative (PD), b)Sliding Mode (SM) and c) PD+SM. The results of these control laws simulation are shown, respectively, in Figs. 567.

Note that the quadrotor aircraft reached the desired altitude using sliding mode control and PD control with Sliding Mode Compensation (PD+SM) while with only a PD control the error increase due to disturbances. Fig. 8 shows that PD+SM control presents the best performance since the error converges to zero, as in SM control, in a fast way. On the other hand, Fig. 9 shows the PD and SM control signals from simulation. Fig. 10 is a zoom from 9 to observe in a better way the signals. Fig. 11. Relationship between convolution window and a neuron.

CONVOLUTIONAL NEURONAL NETWORKS

In this work, a deep learning model called convolutional neuronal network (CNN) is used to determine if there exist cracks in images of buildings. The CNNs has been used to solve, in an effective way, different problems related to artificial vision (mainly related to the image recognition) and currently is an important research topic.

Classical algorithms for object detection are based on two-stage: features extraction and classification. By contrast, the CNNs implement this two-stage in the same process.

Fig. 11 shows the basic idea behind the CNN. In classical artificial networks, the synaptic weights (W i j ) and the bias W 0 determine the output of the network. On the other hand, in a CNN, a neuron and convolution window are equivalent. The parameters of the network (synaptic weights and bias) are the coefficients of the convolution window.

The convolution operation is performed sliding the the window (neuron) over the complete image. This convolution generate feature maps that represent the output of each layer of the network. The output y of one layer of the network is

y = Relu(⟨W, x⟩ + b), ( 35
)
where ⟨•, •⟩ represents the dot product between the tensor formed by the synaptic weights and x is the image on which the convolution operation is performed, b is a polarization value or bias while Relu is the activation function. At first, the synaptic weights W i j are selected randomly and the training process modify them in such a way that the CNN is able to classify the cracks in the images.

The last stage within the network is dense or strongly connected, where the flow of information from the convolutional layers is connected to a conventional neural network that functions as a classifier. Since it is a bi-class problem, the last layer consists of a single neuron that will indicate whether or not there is a crack in the image.

In a CNN, the convolutional layers extracts features of the images to later be classified by a classical neural network.

EXPERIMENTAL RESULTS

This section describes the CNN which has been adopted for classifying cracks in images taken by a quadrotor, it is compared with a Viola-Jones classifier commonly used for this task. Also, our quadrotor aircraft platform is described briefly and real-time experimental results are shown. Several experiments were carried out to stabilize the altitude of our platform. The controller's parameters were tuned by trial and error until obtaining better performance.

Convolutional Neural Network Clasification

In order to perform the process of training the CNN and a Viola-Jones classifier, images of cracks were searched on the Internet, mainly of the earthquake in Mexico City in 2017. Fig. 12 shows some of the images from the database.

It is necessary to create two sets, the positive set, that is, those that contain cracks and the other of negatives, which do not contain cracks. We obtained 350 positive images and 450 negative images for a total of 800 images. All of them were resized to 50x50 pixels. In search of improving the detection results, our database was combined with the Fig. 12. Samples of images with cracks taken from the Internet.

one used in [START_REF] Shi | Automatic road crack detection using random structured forests[END_REF], in such a way that the number of positive images was increased to 450. The developed network consists of four convolutional layers and two dense layers, as well as polling operations after each convolutional layer, as can be seen in Fig. 13. We use Keras [START_REF] Chollet | [END_REF] to create a CNN. Keras is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano [START_REF] Chollet | The Python Deep Learning library[END_REF]. We use a computer with AMD A8 processor and 8Gb of RAM to carry out the training stage, under the Ubuntu OS. A total of 400 positive and 400 negative images were used for training while for validation, 50 positive and 50 negative examples were used. A CNN obtained 95% accuracy in the training stage and 96% in the validation stage.

To implement the detection stage, a sliding window method was used. This method uses a fixed size window and this goes through the entire image in search of cracks.

Fig. 14a and14c shows results from the detection using CNN. In general, the detection is satisfactory, with an acceptable amount of cracks detected and some false positives.

On the other hand, results from Viola-Jones classifier are shown in Fig. 14b and14d. Even though detections are effective with this method, some false positives (red ellipses) detect as cracks the wall corner, which does not happen with CNN, in other words, our CNN can distinguish cracks from other edges in an acceptable way and the results are better than classical Viola-Jones classifier, as shown Fig. 15.

Quadorotor UAV real-time altitude control

As an experimental platform, we have used a quadrotor aircraft with a Pixhawk flight controller and a LIDAR-Lite v3 sensor for the altitude measure. This sensor was selected because it offers a better performance than the barometer. Fig. 16 shows the real-time embedded control architecture that we use. As is well known, the chattering problem (which is due to failures in switching devices and delays) is one of the most common handicaps in the sliding mode control. To reduce the chattering, we use the approach presented in [START_REF] Gonzalez | Chattering-free sliding mode altitude control for a quad-rotor aircraft: realtime application[END_REF] and replace the sign function, that is not continuous, for a continuous saturation function sat, defined as follows:

sat ( s ε ) =      s |s| , |s| ≥ ε, s ε , |s| < ε, ( 36
)
where ε is a small positive constant. Fig. 17 present some experimental results obtained by applying the PD+SMC control algorithm. The arrows in Fig. 17b and17d show the time interval where the SMC action was canceled, which correspond with the vertical dotted line in Fig. 17a and17c. It is evident that the performance with PD+SMC is better than when the PD is working alone. Control gains and error until obtaining the desired behavior. A properly k p and k d selection for a PD controller affects the performance of a control system, for instance, improving damping, reducing maximum overshoot, and reducing rise time and settling time. On the other hand, parameters λ and ρ in the SMC let modify the convergence velocity and sliding surface frequency, respectively. This results could be improved by fine-tuning ρ, λ , k p and k d gains. Fig. 18 show some experiments carried out at CINVESTAV.

CONCLUSIONS

The paper presents simulations and experimental realtime results in the altitude control using a combination of two control actions, one linear (PD) and another nonlinear (Sliding Mode Control) for the future autonomous wall inspection task. The PD+SMC takes the advantages of the PD controller (simplicity and easy design) and the robustness of SMC to disturbances and model uncertainty. To our knowledge no works concerinig hybrid PD+SMC for UAV systems have been reported in literature. The simulations were conducted using the quadrotor aircraft dynamical model and the real-time experiments were performed in an outdoor environment where wind-disturbances were present. To see the flight that was carried out to test the control, a video is available on https://youtu.be/Aph-DgsFRF8. On the other hand, we increase our database of crack images for a CNNbased crack recognition system. The experiments showed that the recognition of the cracks is satisfactory. There are very few databases of images with cracks; it is important to have a greater number of images to be able to develop better applications of automatic detection of this type of anomalies. Generally, CNN has shown good results in object recognition tasks given their ability to train feature extractors and their classification stage.

Fig. 1 .

 1 Fig. 1. Damaged building during an earthquake in Mexico in 2017.

Fig. 2 .

 2 Fig. 2. Free-body diagram of the UAV.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Disturbance simulation from Dryden wind-gust model.

Fig. 7 .Fig. 8 .Fig. 9 .

 789 Fig. 7. Altitude PD+SM control.

Fig. 10 .

 10 Fig. 10. Zoom from Fig. 9.

Fig. 13 .

 13 Fig. 13. Architecture of the proposed network.

Fig. 14 .

 14 Fig. 14. Crack detection on the original images.

Fig. 15 .

 15 Fig. 15. Zoom from Fig. 14.

Fig. 17 .Fig. 18 .

 1718 Fig. 17. Altitude position in real-time experiments.


Anton Ashuiev

Md Florian Allouche

Ashraful Islam

José P Carvalho

Kevin J Sanders

Matthew P Conley

Daniel Klose

Giuseppe Lapadula

Michael Wörle

Dirk Baabe

Md Ashraful Islam

Marc D Walter

Andrew J Pell
email: andrew.pell@ens-lyon.fr

Christophe Copéret
email: ccoperet@ethz.ch

Gunnar Jeschke
email: gjeschke@ethz.ch

Guido Pintacuda
email: guido.pintacuda@ens-lyon.fr

Richard A Andersen

Geometry and Electronic Structure of Yb(III)[CH(SiMe3)2]3 from EPR and solid-state NMR augmented by Computations

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Organometallic compounds, perhydrocarbyls in particular (MR n with M = metal; R = hydrocarbyl), often adopt unusual geometries or bonding patterns due to their low coordination numbers and the associated electronic consequences in metal-carbon bondings. [START_REF] Hartwig | Organotransition Metal Chemistry: From Bonding to Catalysis[END_REF][START_REF] Wilkinson | Comprehensive organometallic chemistry: The synthesis, reactions, and structures of organometallic compounds[END_REF] These patterns continue to engage the broader community in understanding how preferences in geometric structure relate to the often complex electronic structure and reactivity in these compounds. As an example, among diamagnetic compounds, carbons with single bonds that exhibit significant π character to metals show specific deshielded nuclear magnetic resonance (NMR) chemical shifts and associated reactivity, indicating that alkyl groups are often better described as both σ-and π-donors to the metal (Figure 1a). [START_REF] Lukens | A π-donor spectrochemical series for X in (Me 5 C 5 )2TiX, and β-agostic interactions in X= Et and N (Me) Ph[END_REF][START_REF] Gordon | NMR chemical shift analysis decodes olefin oligo-and polymerization activity of d 0 group 4 metal complexes[END_REF] Reduced J C-H coupling constants for these carbons, and associated specific structural distortions, provide evidence for agostic interactions. [START_REF] Brookhart | Carbon-hydrogen-transition metal bonds[END_REF][START_REF] Brookhart | Agostic interactions in transition metal compounds[END_REF] Similar bonding peculiarities are also found in compounds with metalcarbon multiple bonds. The prototypical example is Ta( --CH t Bu)(CH t 2 Bu) 3 , which contains a highly deshielded [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C signal and a very low J C-H coupling constant for the formally sp 2 hybridized Ta=C carbon. These signatures are consistent with an α-CH interaction, better described as an additional π-donation from the α-carbon to the electron deficient d 0 metal centre. [START_REF] Lukens | A π-donor spectrochemical series for X in (Me 5 C 5 )2TiX, and β-agostic interactions in X= Et and N (Me) Ph[END_REF][START_REF] Gordon | NMR chemical shift analysis decodes olefin oligo-and polymerization activity of d 0 group 4 metal complexes[END_REF][START_REF] Gordon | π-Bond Character in Metal-Alkyl Compounds for C-H Activation: How, When, and Why?[END_REF] Peculiarities in geometric structure from X-ray diffraction analysis are also observed in monomeric lanthanide perhydrocarbyl complexes, Ln[CH(SiMe 3 ) 2 ] 3 (Ln = Y, [START_REF] Avent | Synthetic and structural experiments on yttrium, cerium and magnesium trimethylsilylmethyls and their reaction products with nitriles; with a note on two cerium β-diketiminates[END_REF] La, [START_REF] Hitchcock | Synthesis and structural characterisation of the first neutral homoleptic lanthanide metal (III) alkyls:[LnR 3 ][Ln= La or Sm, R= CH (SiMe 3 ) 2[END_REF] Ce, [START_REF] Avent | Synthetic and structural experiments on yttrium, cerium and magnesium trimethylsilylmethyls and their reaction products with nitriles; with a note on two cerium β-diketiminates[END_REF] Sm, [START_REF] Hitchcock | Synthesis and structural characterisation of the first neutral homoleptic lanthanide metal (III) alkyls:[LnR 3 ][Ln= La or Sm, R= CH (SiMe 3 ) 2[END_REF] Lu 10 ) that contain three short Ln ... C γ contacts between one Si-CH 3 group of each of the three proximal -SiMe 3 groups (Fig. 1a). The Ln ... C γ contacts result in lengthening of the Si-CH 3 bonds close to the lanthanide by ∼0.04 Å with respect to the other Si-CH 3 groups, indicating that these proximal Si-CH 3 groups interact with the lanthanide. This interaction helps to satisfy the high coordination numbers required to isolate these f -block homoleptic complexes; a strategy that also facilitates isolation of Yb[C(SiMe 3 ) 3 ] 2 , 11 Ln(AlMe 4 ) 3 , [START_REF] Evans | Inclusion of Al 2 Me 6 in the Crystalline Lattice of the Organometallic Complexes LnAl 3 Me 12[END_REF] and Ln[C(SiHMe 2 ) 3 ] 2 . [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF][START_REF] Yan | A tris(alkyl)yttrium compound containing six β-agostic Si-H interactions[END_REF] Recent studies of Lu[CH(SiMe 3 ) 2 ] 3 10 indicated that the Lu• • • C γ interactions are not a result of γ-C-H agostic interactions (2-electron donation from a C-H bond to an electron deficient metal centre), [START_REF] Brookhart | Agostic interactions in transition metal compounds[END_REF] but are rather due to a three-centre-two-electron (3c-2e) interaction between the Si-Me bond and Lu. In other words, the Lu• • • C γ interaction is best described as a pseudo bridging Lu-Me-Si group (Fig. 1a), analogous to Al-Me-Al observed in dimeric AlMe 3 [START_REF] Longuet-Higgins | The structure of some electron-deficient molecules[END_REF][START_REF] Brookhart | Molecular structure and bonding in hexamethyldialuminum[END_REF] or Ln(AlMe 4 ) 3 . [START_REF] Klooster | Neutron Diffraction Study of [Nd(AlMe 4 ) 3 ]•0.5•Al 2 Me 6 at 100 K: The First Detailed Look at a Bridging Methyl Group with a Trigonal-Bipyramidal Carbon Atom[END_REF] The critical data in that study [START_REF] Conley | The nature of secondary interactions at electrophilic metal sites of molecular and silica-supported organolutetium complexes from solid-state NMR spectroscopy[END_REF] were a combination of high quality X-ray diffraction results, variable temperature 1 H and [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C NMR spectra in solution, as well as J-resolved 1 H- [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C and [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si Magic-Angle Spinning (MAS) solid-state NMR spectra. The experimental results were reliably reproduced using both wave-function based and density functional theory (DFT) models of Lu[CH(SiMe 3 ) 2 ] 3 , and natural bond orbital (NBO) analysis showed that the charges on C α and C γ are negative, while the charges on Si β and H γ are positive. This analysis was consistent with the proposed bridging Lu-Me-Si group to describe the Lu• • • C γ interaction. Extending this analysis using similar NMR methods to other Ln[CH(SiMe 3 ) 2 ] 3 is however challenging, because most lanthanides contain a partially filled f-shell in their +3 oxidation state. The resulting paramagnetism perturbs the magnetic properties of the nearby nuclei, making data acquisition and interpretation complicated at best, and typically impractical in most instances. Similarly, while electron paramagnetic resonance (EPR) could offer a potential alternative for this class of compounds, the signals are typically not detectable at room temperature using routine continuous wave (CW) methods due to the short electronic spin relaxation times and large hyperfine couplings. However, both fields have seen impressive recent progress. In solid-state NMR, fast MAS and broadband adiabatic irradiation sequences allow for measurements on such samples [START_REF] Pell | Paramagnetic NMR in solution and the solid state[END_REF] and in EPR, new EPR pulsed tech-niques [START_REF] Roessler | Principles and applications of EPR spectroscopy in the chemical sciences[END_REF] were developed for f -block metal complexes. [START_REF] Denning | Covalency in the 4f Shell of tris-Cyclopentadienyl Ytterbium (YbCp 3 ) -A Spectroscopic Evaluation[END_REF][START_REF] Formanuik | Actinide covalency measured by pulsed electron paramagnetic resonance spectroscopy[END_REF] In parallel, the recent advances in quantum chemical/DFT methods and dedicated formalisms [START_REF] Kurland | Isotropic NMR shifts in transition metal complexes: the calculation of the Fermi contact and pseudocontact terms[END_REF][START_REF] Moon | Calculation of NMR and EPR Parameters[END_REF][START_REF] Pennanen | Density-functional calculations of relativistic spin-orbit effects on nuclear magnetic shielding in paramagnetic molecules[END_REF][START_REF] Pennanen | Nuclear magnetic resonance chemical shift in an arbitrary electronic spin state[END_REF][START_REF] Van Den Heuvel | NMR chemical shift in an electronic state with arbitrary degeneracy[END_REF][START_REF] Van Den Heuvel | NMR chemical shift as analytical derivative of the Helmholtz free energy[END_REF][START_REF] Vaara | Magnetic couplings in the chemical shift of paramagnetic NMR[END_REF][START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF][START_REF] Lang | Solution of a puzzle: Highlevel quantum-chemical treatment of pseudocontact chemical shifts confirms classic semiempirical theory[END_REF] now allow us to interpret NMR and EPR data in terms of the spatial and electronic structure. [START_REF] Rouf | 1 H chemical shifts in paramagnetic Co(II) pyrazolylborate complexes: A first-principles study[END_REF][START_REF] Pigliapochi | CC BY 4.0 investigation of the effect of spin-orbit coupling on the NMR shifts in paramagnetic solids[END_REF][START_REF] Mondal | Large-scale computation of nuclear magnetic resonance shifts for paramagnetic solids using CP2K[END_REF][START_REF] Mondal | Computation of NMR shifts for paramagnetic solids including zero-field-splitting and beyond-DFT approaches. Application to LiMPO 4 (M= Mn, Fe, Co, Ni) and MPO[END_REF][START_REF] Pyykkönen | Paramagnetic Pyrazolylborate Complexes Tp2M and Tp * 2M: 1 H, 13 C, 11 B, and 14 N NMR Spectra and First-Principles Studies of Chemical Shifts[END_REF][START_REF] Andersen | Remarkable reversal of 13 C-NMR assignment in d 1 , d 2 compared to d 8 , d9 acetylacetonate complexes: analysis and explanation based on solid-state MAS NMR and computations[END_REF][START_REF] Ashuiev | Molecular and supported Ti(III)-alkyls: efficient ethylene polymerization driven by the π-character of metal-carbon bonds and back donation from a singly occupied molecular orbital[END_REF] In this study, we showcase a methodology to assess the detailed electronic structure of a prototypical compound, namely Yb[CH(SiMe 3 ) 2 ] 3 , by combining state-of-the-art EPR, NMR and computation (Figure 1b). We first synthesized and isolated a series of Yb[CH(SiMe

3 ) 2 ] x [O- 2,6-tBu 2 -C 6 H 3 ] 3-x (x = 0, 1, 2, 3
), and determined their structures by single-crystal X-ray diffraction at 100 K along with their basic magnetic properties by variable-temperature measurements of their magnetic susceptibilities. We next investigate the full geometric and electronic structure of Yb[CH(SiMe 3 ) 2 ] 3 based on the most advanced EPR and solid-state NMR, supported by quantum chemical calculations. This approach enables a full characterization of the direct metal-carbon bonding as well as the nature of the secondary Yb• • • C γ interaction at low and room temperature. We discuss how this interaction influences the structure and ligand dynamics in solution and in the solid state, and how the CH(SiMe 3 ) 2 ligand leads to an unusual behavior in the crystal field interactions of the Yb(III) ion.

Results and discussion

Synthesis and X-ray diffraction studies of Yb[CH(SiMe 3 ) 2 ] 3

The syntheses of Yb[CH(SiMe

3 ) 2 ] 3 , Yb[CH(SiMe 3 ) 2 ] 2 [O-2,6-tBu 2 -C 6 H 3 ],
and

Yb[CH(SiMe 3 ) 2 ][O-2,6-tBu 2 -C 6 H 3 ] 2 are

Magnetic susceptibility studies

We performed magnetic susceptibility studies for Yb[CH(SiMe We characterized the Yb(III) alkyl complex Yb[CH(SiMe 3 ) 2 ] 3 using pulse EPR spectroscopy [START_REF] Roessler | Principles and applications of EPR spectroscopy in the chemical sciences[END_REF] that, in combination with computational studies, enables the successful characterization of the spatial and electronic structure of paramagnetic complexes. [START_REF] Denning | Covalency in the 4f Shell of tris-Cyclopentadienyl Ytterbium (YbCp 3 ) -A Spectroscopic Evaluation[END_REF][START_REF] Ashuiev | Molecular and supported Ti(III)-alkyls: efficient ethylene polymerization driven by the π-character of metal-carbon bonds and back donation from a singly occupied molecular orbital[END_REF] The continuous-wave (CW) EPR spectrum of Yb[CH(SiMe 3 ) 2 ] 3 (Fig. 3a, blue) was reasonably reproduced by simulation (Fig. 3a, red) by modeling the Yb ion as an electronic spin S = 1/2, with g tensor principal values of g ⊥ = 0.7316 and g ∥ = 7.5698. The spectrum also exhibits a well- A higher-resolution picture can be provided by pulsed EPR spectroscopy, which has evolved into a robust tool for the determination of geometry and electronic configuration of organometallic complexes of paramagnetic d-and f -block metals. In particular, 2D hyperfine sub-level correlation spectroscopy (HYSCORE) [START_REF] Höfer | Hyperfine sublevel correlation (HYSCORE) spectroscopy: a 2D ESR investigation of the squaric acid radical[END_REF] allows us to detect magnetically active nuclei (e.g. 1 H, [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C, [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si etc.) that are coupled to paramagnetic centres by hyperfine interaction, due to close spatial proximity to these centres. The technique can separate the isotropic (primarily Fermi-contact) and anisotropic (primarily spin-dipolar) parts of the hyperfine interaction tensors that are linked to the electronic properties of the systems (e.g. metal-ligand spin density transfer), and bonding between the nuclei and the paramagnetic centre. The HYSCORE spectrum of Yb[CH(SiMe 3 ) 2 ] 3 (Fig. 3c and Fig. S8) reveals the presence of hyperfine coupling to protons in the CH(SiMe 3 ) 2 alkyl ligands. From the simulation of the HYSCORE spectrum a single 1 H hyperfine tensor was determined to have

A iso ( 1 H) = 3.8 ± 0.3 MHz, T dip ( 1 H) = 4.0 ± 0.5 MHz (A dip = [-4.0 ± 0.5; -4.0 ± 0.5; 8.0 ± 1.0] MHz),
with the tensors principal axes frame (PAF) rotated by 45 • ± 15 • with respect to the z axis of the PAF of the g tensor. This interpretation of the HYSCORE data is consistent with the ELDOR-detected NMR (EDNMR) [START_REF] Schosseler | Pulsed ELDOR Detected NMR[END_REF] spectrum shown in Fig. 3d, from which we can extract a single splitting due to an effective hyperfine coupling to 1 H with A eff ( 1 H) = 13 ± 1 MHz.

Based on previous observations of similar isotropic and dipolar parts of hyperfine coupling tensors for the S = 1/2 Ti(III) alkyl complexes, [START_REF] Ashuiev | Molecular and supported Ti(III)-alkyls: efficient ethylene polymerization driven by the π-character of metal-carbon bonds and back donation from a singly occupied molecular orbital[END_REF][START_REF] Allouche | Low-Coordinated Titanium (III) Alkyl-Molecular and Surface-Complexes: Detailed Structure from Advanced EPR Spectroscopy[END_REF] the observed 1 H hyperfine couplings could be assigned to the hydrogen atoms that are close in space to the Yb(III) centre, such as α-H atoms of CH(SiMe 3 ) 2 ligands. The positions of the α-H atoms within the structure of Yb[CH(SiMe 3 ) 2 ] 3 were further refined using quantum chemical computations (vide infra).

This finding indicates that the transferred spin density is significant only for the closest to the Yb(III) atoms, such as C 1 and H 1 , while it is substantially lower for the rest of the ligands.

The observed isotropic 1 H hyperfine couplings that are even smaller than those observed for d 1 metal alkyls [START_REF] Ashuiev | Molecular and supported Ti(III)-alkyls: efficient ethylene polymerization driven by the π-character of metal-carbon bonds and back donation from a singly occupied molecular orbital[END_REF][START_REF] Allouche | Low-Coordinated Titanium (III) Alkyl-Molecular and Surface-Complexes: Detailed Structure from Advanced EPR Spectroscopy[END_REF] suggest that, in contrast to the case of Cp 3 Yb, [START_REF] Denning | Covalency in the 4f Shell of tris-Cyclopentadienyl Ytterbium (YbCp 3 ) -A Spectroscopic Evaluation[END_REF] there is no significant charge and spin density transfer to the ligands in Yb[CH(SiMe 3 ) 2 ] 3 . This is also consistent with the preferential presence of the unpaired electron in the 4f orbital rather than in the valence 5d orbital, which would result in more significant distribution of the spin density into the ligands of Yb(III). A difficulty of promoting 4f electrons of Yb(III) to the 5d orbital, which effectively prevents the formation of double Yb --C bonds, was previously discussed based on CASSCF calculations. [START_REF] Roos | Bonding Trends in Molecular Compounds of Lanthanides: The Double-Bonded Carbene Cations LnCH + 2 (Ln= Sc, Y, La-Lu)[END_REF] Taking into account the results of magnetic susceptibility measurements (vide supra), we propose that the ground state of the Yb[CH(SiMe 3 ) 2 ] 3 complex has a single electronic configuration of 4f 13 (S = 1/2), close to the one of Yb(III) free ion.

NMR characterization of Yb

[CH(SiMe 3 ) 2 ] 3
Over the last twenty years, the development of instrumentation and spectroscopic methods in NMR allows us to target paramagnetic lanthanide complexes such as Yb[CH(SiMe 3 ) 2 ] 3 .

Specifically, solid-state NMR with fast MAS and broadband adiabatic irradiation sequences [START_REF] Pell | Paramagnetic NMR in solution and the solid state[END_REF] can now be used to obtain data on microcrystalline powders, where the complex dynamics are simplified and only the relevant internal ligand motions are retained. Nonetheless, obtaining high-quality magnetic resonance data on these systems represents only part of the challenge, since the resulting spectra are often very difficult to interpret using standard assignment methods. In this respect, recent progress in wave-function based/DFT methods for the computation of NMR parameters of paramagnetic complexes has also been impressive. For NMR, there have been significant advances both in the development of the formalism of the paramagnetic shifts [START_REF] Kurland | Isotropic NMR shifts in transition metal complexes: the calculation of the Fermi contact and pseudocontact terms[END_REF][START_REF] Moon | Calculation of NMR and EPR Parameters[END_REF][START_REF] Pennanen | Density-functional calculations of relativistic spin-orbit effects on nuclear magnetic shielding in paramagnetic molecules[END_REF][START_REF] Pennanen | Nuclear magnetic resonance chemical shift in an arbitrary electronic spin state[END_REF][START_REF] Van Den Heuvel | NMR chemical shift in an electronic state with arbitrary degeneracy[END_REF][START_REF] Van Den Heuvel | NMR chemical shift as analytical derivative of the Helmholtz free energy[END_REF][START_REF] Vaara | Magnetic couplings in the chemical shift of paramagnetic NMR[END_REF][START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF][START_REF] Lang | Solution of a puzzle: Highlevel quantum-chemical treatment of pseudocontact chemical shifts confirms classic semiempirical theory[END_REF] and the computational implementation. [START_REF] Rouf | 1 H chemical shifts in paramagnetic Co(II) pyrazolylborate complexes: A first-principles study[END_REF][START_REF] Pigliapochi | CC BY 4.0 investigation of the effect of spin-orbit coupling on the NMR shifts in paramagnetic solids[END_REF][START_REF] Mondal | Large-scale computation of nuclear magnetic resonance shifts for paramagnetic solids using CP2K[END_REF][START_REF] Mondal | Computation of NMR shifts for paramagnetic solids including zero-field-splitting and beyond-DFT approaches. Application to LiMPO 4 (M= Mn, Fe, Co, Ni) and MPO[END_REF][START_REF] Pyykkönen | Paramagnetic Pyrazolylborate Complexes Tp2M and Tp * 2M: 1 H, 13 C, 11 B, and 14 N NMR Spectra and First-Principles Studies of Chemical Shifts[END_REF][START_REF] Andersen | Remarkable reversal of 13 C-NMR assignment in d 1 , d 2 compared to d 8 , d9 acetylacetonate complexes: analysis and explanation based on solid-state MAS NMR and computations[END_REF] However, applying the developed methods to lanthanides has been a much more difficult matter.

For the Yb[CH(SiMe 3 ) 2 ] 3 complex, we are able to draw the conclusions regarding its electronic structure and on the positions of 1 H nuclei closest to the metal centre (α-H atoms of [CH(SiMe 3 ) 2 ] ligands, H 1 ) based on the EPR data and quantum chemistry calculations.

However, the weak hyperfine interactions between the unpaired electron and the more distant nuclei prevented us from obtaining any high-resolution information about the rest of the molecule by EPR. This gap could, however, be filled by paramagnetic NMR. [START_REF] Pell | Paramagnetic NMR in solution and the solid state[END_REF] Note that the signals of H1 nuclei, most likely, cannot be observed by NMR, in particular, due to a severe reduction of sensitivity because the large paramagnetic relaxation enhancement expected for these nuclei in close proximity to Yb, or due to the fact that potentially large shifts push them out of the excitation window. Therefore, EPR and NMR spectroscopies could be seen as complementary techniques for the characterization of paramagnetic lanthanide organometallic compounds like Yb[CH(SiMe 3 ) 2 ] 3 , allowing to obtain the information from both internal and external coordination environment of the metal center.

The ligands of the complex are expected to exhibit substantial dynamics, which has the effect of averaging the signals. This was observed in the case of the Lu analogue, where variable temperature 1 H solution NMR data were required to partially freeze out these motions and the individual 1 H resonances were obtained by cooling the sample. However, this approach was not successful for the present Yb complex, where no such resolution was obtained over a temperature range between 285 and 182 K, and no detection was possible beyond that because of increased line-broadening (see SI Chapter 3 & Fig. S7).

We therefore proceeded to examine the Yb[CH(SiMe 3 ) 2 ] 3 complex via solid-state MAS NMR, which has the advantage of eliminating molecular rotational diffusion, so that only the internal ligand dynamics remain. Such spectra relate to the room-temperature crystal structure, thereby providing a direct bridge to the low-temperature XRD data.

One-dimensional solid-state MAS NMR spectra of 1 H, [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C and [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si are shown in Figure 4a-c. Each of these spectra exhibits poorly resolved patterns, due to hyperfine interactions with the unpaired electron, which induce large shifts and shift anisotropies and extremely large inhomogeneous line-broadenings. We see shift anisotropies that are very large for all three nuclei, of the order of hundreds of ppm, in contrast to the Lu analogue, where the 1 H and 13 C diamagnetic chemical shift anisotropies (CSAs) were too small to be measured and for [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si were of the order of tens of ppm. In particular, for the Lu analogue, [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si CSAs were used to characterize the metal-ligand interaction. That approach is not at first sight obviously applicable to the paramagnetic Yb complex, since the interpretation of the shift anisotropy is more complex. [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C MAS spectra of at 14.286 kHz MAS and (c) [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si MAS spectra at 30 kHz MAS, showing two distinct [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si signals. The spinning sidebands are marked with asterisks ( * ). 2D aMAT spectrum: (d) 1 H at 30 kHz, (e) [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C at 20 kHz and (f) [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si at 20 kHz. The indirect dimension projections were deconvolved using asymmetric Gaussian functions to extract the isotropic shifts and quantify shift dispersion. (g)-(i) extracted slices with isotropic shift corresponding to the maxima of the deconvolved projections and labelled with Roman numerals. Each slice was subsequently fitted considering a simple CSA model using the Haeberlen convention 43 (δ iso = 1 3 ( δxx + δyy + δzz ), ∆δ = δzz -δ iso and η = ( δyyδxx )/∆δ, where δ xx , δ xx and are the components of the CSA tensor in the principal axes frame). See SI chapter 1.6 for more details.

(I) (II) (III) (IV) (V) (VI) (I) (II) (III) (IV) (V) (VI)
The first problem to be addressed is the lack of resolution of the individual signals in each 13 https://doi.org/10.26434/chemrxiv-2023-ksfb4-v2 ORCID: https://orcid.org/0000-0002-3597-0889 Content not peer-reviewed by ChemRxiv. License: CC BY 4.0 of the three spectra, due to the overlapping spinning sideband manifolds of neighboring resonances. To solve this problem, we employed the 2D adiabatic magic-angle turning (aMAT) experiment, [START_REF] Clément | Spin-transfer pathways in paramagnetic lithium transition-metal phosphates from combined broadband isotropic solid-state MAS NMR spectroscopy and DFT calculations[END_REF] which correlates each spinning side-band manifold to its isotropic shift, thus removing the overlap, as seen in Figure 4d-f for 1 H, [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C and 29 Si, respectively. In all three cases, the aMAT spectra are able to resolve two resonances with distinct isotropic shifts. For [START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si, the interpretation is relatively straightforward, with the two signals V and VI being due to the two inequivalent Si nuclei (specifically, one due to Si 1 and the other to Si 2 ). By extension, it is reasonable to tentatively interpret each [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C signal (III and IV) as being from the three methyl groups attached to each of the Si (C 2/3/4 and C 5/6/7 ), and each 1 H signal (I and II) to the corresponding group of protons (H 2/3/4 and H 5/6/7 ).

Figure 5: (a) 1 H-13 C correlation spectrum achieved by using the TEDOR experiment 45 at 30 kHz MAS, yielding two distinct correlations of the resonances (III/I) and (IV/II), shown in Fig. 4, which were subsequently used to model the paramagnetic shifts in the following section. The spinning sidebands are marked with asterisks ( * ). (b) 1 H NMR shift distributions considering solely the contributions arising from the presence of the unpaired electron and neglecting long-range PCS effects stemming from parametric centres in neighboring unit cells (δ p ). The shifts were calculated considering the Moon and Patchkovski, [START_REF] Moon | Calculation of NMR and EPR Parameters[END_REF] Vaara et al. [START_REF] Pennanen | Density-functional calculations of relativistic spin-orbit effects on nuclear magnetic shielding in paramagnetic molecules[END_REF][START_REF] Pennanen | Nuclear magnetic resonance chemical shift in an arbitrary electronic spin state[END_REF][START_REF] Vaara | Magnetic couplings in the chemical shift of paramagnetic NMR[END_REF] and van den Heuvel and Soncini 26,27 formalism using the g tensor obtained using multireference perturbation theory (SO-CASPT2) on the DFT optimized geometry (see sections 1.8 and 5.3 of the SI for details) and the experimental 1 H hyperfine tensor. A distribution of relative orientations of the g and A tensors were considered using Zaremba, Conroy, Wolfsberg orientational averaging scheme with 1154 orientations. [START_REF] Zaremba | Good lattice points, discrepancy, and numerical integration[END_REF][START_REF] Conroy | Molecular Schrödinger equation. VIII. A new method for the evaluation of multidimensional integrals[END_REF][START_REF] Cheng | Investigations of a nonrandom numerical method for multidimensional integration[END_REF] The relative orientation of the z axes of the g and A tensors was kept fixed at 45 • , in order to match with the EPR results.

For each of the signals I-VI, a slice extracted from the aMAT spectra was fitted to determine the parameters of the shift tensor (isotropic shift δ, shift anisotropy ∆δ and asymmetry parameter η) (Fig. 4g-i). For each of the three nuclei, within each pair one signal (II, IV and V) possesses a larger isotropic shift and shift anisotropy compared to the second (I, III and VI). We can tentatively assign the former set of three signals to the same SiMe 3 group, and the latter set of signals to the other SiMe 3 group. This interpretation is supported by a transferred-echo double resonance (TEDOR) experiment [START_REF] Kervern | Solid-state NMR of a paramagnetic DIAD-FeII catalyst: Sensitivity, resolution enhancement, and structure-based assignments[END_REF] (Fig. 5a), which displays a first unambiguous correlation between I and III, a second unambiguous correlation between II and IV, and no other correlations. These solid-state NMR spectra and our tentative interpretation indicate that each group of 1 H, [START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] Therefore, we suggest that the observed signals are consistent with the presence of the two distinct types of SiMe 3 groups, associated with Si 1 and Si 2 nuclei. Note that the span (Ω = δ 11δ 33 ) of Si 1 is Ω(Si 1 ) = 676 ppm, and the span of Si 2 is Ω(Si 2 ) = 479 ppm. These values are significantly larger than those previously obtained for Lu[CH(SiMe 3 ) 2 ] 3 (Ω(Si 1 ) = 46.8 ppm, Ω(Si 2 = 30.2 ppm), [START_REF] Conley | The nature of secondary interactions at electrophilic metal sites of molecular and silica-supported organolutetium complexes from solid-state NMR spectroscopy[END_REF] which is expected because that compound is diamagnetic. The pattern, observed for Lu[CH(SiMe 3 ) 2 ] 3 , was the strongest experimental evidence that, combined with the DFT interpretations, showed that Si 1 was interacting with Lu through a Lu-Me-Si 3c-2e interaction. [START_REF] Conley | The nature of secondary interactions at electrophilic metal sites of molecular and silica-supported organolutetium complexes from solid-state NMR spectroscopy[END_REF] Although the Ω values for Yb[CH(SiMe 3 ) 2 ] 3 are significantly larger, a similar pattern emerges; Si 1 has a larger Ω(Si) than Si 2 , suggesting that Si 1 is also interacting with Yb to form a 2c-2e Yb-Me-Si.

Computational studies of Yb[CH(SiMe

3 ) 2 ] 3

Ab initio electronic structure

The molecular geometry obtained by XRD was optimized using DFT using the ORCA package [START_REF] Neese | The ORCA quantum chemistry program package[END_REF] (see Fig. S9, and Tables S7 andS8 for the details of DFT-optimized structure of Yb[CH(SiMe 3 ) 2 ] 3 ). Both the starting and optimized geometries were then used to calculate the electronic energy levels using complete active space (CAS) methods using the MOLCAS package. [START_REF] Fdez | OpenMolcas: From source code to insight[END_REF][START_REF] Aquilante | Modern quantum chemistry with[END_REF] The results, reported in Table 1, are similar, and in the following we confined our discussion on the results obtained on the DFT-optimized structure. The seven inner-core active 4f orbitals are shown in Figure 6 along with the three energetically higher bonding type orbitals of the Yb -C α bonds. Out of the seven active orbitals, the lowest one is non-bonding, followed by two sets of doubly-degenerate orbitals:

one undergoes a slight π anti-bonding interaction with the C α p z orbitals and the other one is non-bonding. The final two orbitals characterize a σ bonding interaction with the C α sp 2 hybrid orbitals. The seven 4f orbitals are very little admixed with the ligand orbitals, yet the ligand field splitting at the orbital level, or at the many-electron level, is relatively large compared to the case of O and N donor ligands (usually 200-300 cm -1 ). This is similar to the case of YbCp 3 , which has also been shown to exhibit a large ligand field splitting. [START_REF] Denning | Covalency in the 4f Shell of tris-Cyclopentadienyl Ytterbium (YbCp 3 ) -A Spectroscopic Evaluation[END_REF] The active 4f orbitals are mostly perturbed by electrostatic interactions, while the Yb valence 6s, 6p and 5d orbitals mostly participate in the bonding interactions with the C α sp 2 hybrid orbitals. As a result, population analysis of the ground spin-free CASSCF state (see Table 2) leads to significant electron occupations in the Yb(III) atomic valence and Rydberg orbitals. S7 andS8), are nicely reproduced within the DFT-optimized structure (Fig. S9).

92% 5f z 3 + 7% 5f y(3x
Next, the SO-CASPT2 g factors, g ⊥ = 0.72 and g ∥ = 7.77 (see Table 4) are also in very good agreement with the experimental values and show a substantial improvement as compared to the SO-CASSCF values (g ⊥ = 0.40, g ∥ = 7.91, respectively) due to the better inclusion of the electron dynamical correlation. The anisotropic orbital and spin contributions to the magnetic moments as a result of the figand field and spin-orbit coupling admixing of the 4f orbitals are also shown. The magnetic moment of the ground KD is significantly anisotropic, as characterized by the prolate m J = ±7/2 state stabilizing the unpaired electron in the bonding active 4f y(3x 2 -y 2 ) and 4f x(x 2 -3y 2 ) orbitals, as shown in Figure 6. In addition, the EPR hyperfine coupling parameters to the central 171 Yb (I = 1/2) nucleus were calculated from first principle methods using a finite nuclear model. The SO-CASSCF calculations are in good agreement with experiments as shown in Table 4. Quantitative evaluation of the hyperfine coupling value in the strong relativistic limit requires a forefront level of computational methods and is still in the developing phase. The Fermi contact (A F C ), spin-dipolar (A SD ) and orbital (A orb ) contributions to the hyperfine tensor in Table 4 indicate a dominant anisotropic nuclear spin -electron orbit coupling which leads to the large Yb(III) hyperfine A ⊥ and A ∥ components. The good agreement between calculated and experimental g tensor values and 171 Yb hyperfine couplings indicates that our computational approach, based on DFT-optimized geometry and CAS methods, describes well the electronic structure of Yb(CH(SiMe 3 ) 2 ) 3 . For comparison the first principal calculated molar magnetic susceptibilities of Yb III (CH(SiMe 3 ) 2 ) 3 are provided in Table 5. The main magnetic axis (χ z ) coincides with the C 3 axis. The calculated isotropic part of the magnetic susceptibility χT at 300 K is 2.30 cm 3 K mol -1

A ⊥,SD A ∥,SD A ⊥,orb A ∥,orb A ⊥,F C A ∥,F C A ⊥ A ∥ SO-CASSCF a
and is comparable to the experimental result of 2.57 cm 3 K mol -1 . The axial anisotropy of the molar magnetic susceptibility ∆χ ax = χ z -(χ x + χ y ) /2 is very large in this complex and dominated by the anisotropic contribution from the ground KD (calculated to be ∆χ KD1 ax =23.44 * 10 -8 m 3 mol -1 ), and is due to the large ligand field splitting and consequent non-homogeneous population in the 4f orbitals.

NBO and QTAIM analyses

The DFT optimized structure (Fig. S9) is in particular notable for the unusually short (ca. A similar effect was previously observed for the [CH(SiMe 3 ) 2 ] 3 ligands of the Lu(III) [START_REF] Conley | The nature of secondary interactions at electrophilic metal sites of molecular and silica-supported organolutetium complexes from solid-state NMR spectroscopy[END_REF] or Ti(III) analogues. [START_REF] Allouche | Low-Coordinated Titanium (III) Alkyl-Molecular and Surface-Complexes: Detailed Structure from Advanced EPR Spectroscopy[END_REF] This interaction is essentially described as the donation of electrons from the filled molecular orbital corresponding to the C-H bond to a metal d-orbital of appropriate symmetry, and has been related to a metal-carbon bond acquiring a π (or alkylidene) character, often called α-H agostic interaction in metal alkyls. [START_REF] Lukens | A π-donor spectrochemical series for X in (Me 5 C 5 )2TiX, and β-agostic interactions in X= Et and N (Me) Ph[END_REF][START_REF] Gordon | NMR chemical shift analysis decodes olefin oligo-and polymerization activity of d 0 group 4 metal complexes[END_REF][START_REF] Gordon | π-Bond Character in Metal-Alkyl Compounds for C-H Activation: How, When, and Why?[END_REF] In order to investigate the possible presence of the π character within the Yb-C bonds, we conducted a Natural Bond Orbital (NBO) analysis, [START_REF] Weinhold | What is NBO analysis and how is it useful?[END_REF] using the program NBO 7.0 54 (see SI Chapter 5.2

for more details). The molecular orbital set for the NBO analysis was generated with DFT, using the PBE0 functional.

Indeed, the NBO analysis revealed a high degree of deviation of the natural hybrid orbitals (NHO) on α-C atoms from the Yb-C axis (θ NHO-C-Yb = 17.8 • for one [CH(SiMe 3 ) 2 ] 3 ligand and 17.9 • for the other two alkyl ligands). The value of such deviation can be used as a quantitative descriptor of the degree of the π character within the metal-carbon bonds (for a pure σ bond, no deviation would be expected). [START_REF] Gordon | NMR chemical shift analysis decodes olefin oligo-and polymerization activity of d 0 group 4 metal complexes[END_REF] In fact, the degree of the π character within the Yb-C bonds is higher than the one observed for Ti(III), Ti(IV) and Zr(IV) neutral alkyl complexes, while being somewhat smaller than for the related cationic complexes. [START_REF] Gordon | NMR chemical shift analysis decodes olefin oligo-and polymerization activity of d 0 group 4 metal complexes[END_REF][START_REF] Ashuiev | Molecular and supported Ti(III)-alkyls: efficient ethylene polymerization driven by the π-character of metal-carbon bonds and back donation from a singly occupied molecular orbital[END_REF] Furthermore, the NBO analysis revealed three low-lying natural orbitals, which have been attributed to the low-lying 5d orbitals, with their occupancies being slightly higher than 5 % (see Tab. S11). The spatial distribution of these orbitals (Fig. 7a) includes, in The obtained significant stabilization energies for these 5d orbitals (see Tab. S9) imply that there are other secondary interactions within the structure of Yb[CH(SiMe 3 ) 2 ] 3 that involve these 5d orbitals. One example of such an interaction is the three-centre -two-electron Yb-Me-Si interaction, previously observed for the Lu[CH(SiMe 3 ) 2 ] 3 analogue. [START_REF] Conley | The nature of secondary interactions at electrophilic metal sites of molecular and silica-supported organolutetium complexes from solid-state NMR spectroscopy[END_REF] Indeed, the NBO analysis reveals the constructive overlap between these 5d orbitals and the bonding orbitals of the Si-C γ bonds (Fig. 7b), thus showing that such interactions are also predicted 22 for the Yb[CH(SiMe 3 ) 2 ] 3 complex.

Next, to further probe the chemical bonding interactions, the DFT computed electron density was analysed by the quantum theory of atoms in molecules (QTAIM) method. The QTAIM approach is based on the topological analysis of the physically observable electron density, and is thus theoretically independent of the computational method the C 1 -Si bonds). The total computed shift tensors are presented both before (in Table 7) and after (in Table S10) taking into account these rotations. Table 7: Calculated and experimental shift parameters for the proximal (denoted by p) and distal (denoted by d) -SiMe 3 group of Yb III (CH(SiMe 3 ) 2 ) 3 in the Haeberlen convention after a three-site fast exchange averaging of the shielding tensors. The shielding tensor is computed according to 2 where the magnetic susceptibility tensor has been computed with the SO-CASPT2 method and the hyperfine tensors are either from the SO-CASSCF or DFT(PBE0) methods. Long-range contributions has been accounted using the semi-classical point dipolar equation according to eq. SI10. It is instructive to examine the breakdown of the shifts into the contributions given in Eq. (1). The FC contribution is negligible in all cases, due to the very limited transfer of the unpaired spin density to these atoms over two, three or four bonds. The intra-molecular PCS is more significant, and is consistently larger for the proximal ligand than for the distal ligand, for all three nuclear species, which is a straightforward consequence of the former being in closer proximity to Yb than the latter. This conclusion is also true in the presence of rotational dynamics, where upon rotation about the C 1 -Si bond, the three methyls on Si 2 are on average closer to the Yb ion than those on Si 1 . The same trend is seen for the more dominant PSO contribution, which for the same reasons is consistently larger by up to one order of magnitude for the proximal ligand. Finally, the long-range PCS is also dominant, due to the large magnetic anisotropy of Yb(III) in this complex. A comparison between the proximal and distal ligands reveals more comple trends than for the intra-molecular PCS, due to the packing of molecules in the crystal lattice.

Overall, the interpretation of the NMR data provides evidence that the same structural distortions, observed in the low-temperature XRD structure and associated with the presence of 3c-2e Yb-Me-Si interactions, persist at the higher temperature of the solid-state NMR measurements (ca. 300 K). Therefore, the Yb[CH(SiMe 3 ) 2 ] 3 complex possesses the Yb• • • C γ interaction, described as a pseudo bridging Yb-Me-Si group and previously observed [START_REF] Conley | The nature of secondary interactions at electrophilic metal sites of molecular and silica-supported organolutetium complexes from solid-state NMR spectroscopy[END_REF] for Lu[CH(SiMe 3 ) 2 ] 3 (see Figure 1), at both low-and room-temperature and irrespective of the presence of the incomplete electronic f -shell.

Conclusions

This study establishes that pulsed EPR and paramagnetic solid-state NMR spectroscopies are highly complementary and able, when coupled with the advanced computational methods, to provide detailed information about the geometry and electronic structure of paramagnetic compounds such as Yb[CH(SiMe 3 ) 2 ] 3 . Besides the specific geometry typically accessible by X-Ray diffraction, this study provides direct spectroscopic evidences for the 3c-2e Yb-Me-Si interaction and for the presence of π-character in the Yb-C bond in this paramagnetic compound, paralleling what is observed for the diamagnetic Lu analog10 and so commonly among the corresponding d 0 organometallics, [START_REF] Lukens | A π-donor spectrochemical series for X in (Me 5 C 5 )2TiX, and β-agostic interactions in X= Et and N (Me) Ph[END_REF][START_REF] Gordon | NMR chemical shift analysis decodes olefin oligo-and polymerization activity of d 0 group 4 metal complexes[END_REF][START_REF] Gordon | π-Bond Character in Metal-Alkyl Compounds for C-H Activation: How, When, and Why?[END_REF] often referred to as an α-H agostic interaction. In fact, the magnetic data for Yb[CH(SiMe 3 ) 2 ] 3 indicates a f 13 ground state configuration with no evidence for a multiconfigurational ground state that is often observed in lanthanide compounds, [START_REF] Denning | Covalency in the 4f Shell of tris-Cyclopentadienyl Ytterbium (YbCp 3 ) -A Spectroscopic Evaluation[END_REF][START_REF] Halbach | Understanding the Multiconfigurational Ground and Excited States in Lanthanide Tetrakis Bipyridine Complexes from Experimental and CASSCF Computational Studies[END_REF][START_REF] Booth | CC BY 4.0 bipyridines and diazabutadienes: Multiconfigurational ground states and open-shell singlet formation[END_REF][START_REF] Booth | Intermediatevalence tautomerism in decamethylytterbocene complexes of methyl-substituted bipyridines[END_REF][58] probably further explaining the observed similarities with the Lu isostructural compound. Yet, while magnetic susceptibility measurements indicate that the magnetic moment is consistent with that of a free Yb(III) ion, CAS-based calculations instead reveal an unusually large crystal-field splitting resulting in a ground KD that is almost completely thermally isolated even at room temperature and which results in a large magnetic anisotropy.

Acknowledgement

We acknowledge Laurent Maron (Université Paul Sabatier, Toulouse) for discussions at the early stage of this project. We acknowledge Maciej Korzynski (ETHZ) for a help with preparation of the manuscript. We acknowledge Elizaveta A. Suturina (University of Bath) and Aleksander Jaworski (Stockholm University) for the advices concerning quantum chemistry calculations in lanthanide complexes. We thank Martin Bröring (Institut für Anorganische und Analytische Chemie) at TU Braunschweig for providing access to the SQUID magnetometer.

Supporting Information Available Supporting Information

The Supporting Information is available free of charge on the ACS Publications website.

Table of Contents Figure

Figure 1 :

 1 Figure 1: (a) Typically occurred peculiarities in structure of organometallic compounds, in particular, Ln[CH(SiMe 3 ) 2 ] 3 . (b) Characterization approach, applied to paramagnetic organometallic lanthanide complexes.

  shown in Fig.2a(see SI Section 1 for detailed synthetic procedures). Similar to the lutetium complexes isolated previously, 10 the alkylation depends on the stoichiometry of the organolithium reagent. Key to successful isolation of these compounds is the use of sufficient volumes of pentane to dissolve the sparingly soluble polymeric Li[CH(SiMe 3 ) 2 ].[START_REF] Tecle | X-ray crystal structure of trimethylsilylmethyllithium[END_REF] Addition of Li[CH(SiMe 3 ) 2 ] as a slurry in smaller volumes of pentane results in poor yields and formation of mixtures of these compounds. All organoytterbium compounds shown in Fig.2aare isolated as crystalline solids from concen-

Figure 2 :

 2 Figure 2: (a) Synthesis scheme of Yb[CH(SiMe 3 ) 2 ] x [O-2,6-tBu 2 -C 6 H 3 ] 3-x (x = 1, 2, 3). (b) Structure of the Yb[CH(SiMe 3 ) 2 ] 3 complex with displacement ellipsoids at 50 % probability level. For clarity the hydrogen atoms and the solvent molecules CH 2 (SiMe 3 ) 2 are omitted. (c) Sketch of relevant bond lengths and angles of Yb[CH(SiMe 3 ) 2 ] 3 . The bond lengths are reported in Å, and the bond angles (in red) are reported in degrees (see also Tab. S2). (d) Newman projection down the C(2)-Si(2) bond with bond lengths and angles (in red) obtained from the crystal structure of Yb[CH(SiMe 3 ) 2 ] 3 . (e) Structure of the Lu[CH(SiMe 3 ) 2 ] 3 complex with displacement ellipsoids at 50 % probability level. For clarity the hydrogen atoms and the solvent molecules CH 2 (SiMe 3 ) 2 are omitted. (f) Sketch of relevant bond lengths and angles of Lu[CH(SiMe 3 ) 2 ] 3 . The bond lengths are reported in Å, and the bond angles (in red) are reported in degrees (see also Tab. S2) (f) Newman projection down the C(2)-Si(2) bond with bond lengths and angles (in red) obtained from the crystal structure of Lu[CH(SiMe 3 ) 2 ] 3 . For comparison panels e)-g) are reproduced from 10 with permission from the American Chemical Society, copyright 2016 Conley et al.10.7

  resolved satellite structure, due to hyperfine coupling to the two spin-active isotopes 171 Yb (spin I = 1/2 and natural abundance of 14.31%) and 173 Yb (spin I = 5/2 and natural abundance of 16.13%) of the lanthanide. From spectral simulation (Fig. 3a, red), the 171,173 Yb hyperfine tensors were determined to be A ⊥ ( 173 Yb) = 221 MHz, A ∥ ( 173 Yb) = 1650 MHz, and accordingly A ⊥ ( 171 Yb) = 842 MHz, A ∥ ( 171 Yb) = 6285 MHz.

Figure 3 :

 3 Figure 3: (a) CW EPR spectrum of Yb[CH(SiMe 3 ) 2 ] 3 (blue) and simulation (red), together with an expansion of the low-field region. The black arrow marks the field position of the HYSCORE measurement. (b) Echo-detected EPR spectrum of Yb[CH(SiMe 3 ) 2 ] 3 . The field position for HYSCORE and EDNMR are marked with a black and red arrow, respectively. (c) HYSCORE spectrum of Yb[CH(SiMe 3 ) 2 ] 3 (blue to yellow, see also Fig. S8) and simulation (red) based on a single 1 H hyperfine coupling A iso ( 1 H) = 3.8 ± 0.3 MHz. Anti-diagonal lines indicate the 1 H and 29 Si nuclear Zeeman frequencies. Spurious peaks at 21.5/2.0 MHz and 23.5/23.5 MHz are electronic artefacts. (d) 1 H region of EDNMR spectrum of Yb[CH(SiMe 3 ) 2 ] 3 (blue), together with simulation (red) of a single effective 1 H coupling with A eff ( 1 H) = 13 ± 1 MHz (gray) and a 1 H matrix peak (gray dashed) at a 1 H nuclear Zeeman frequency of 39.4 MHz.

Figure 4 :

 4 Figure4: NMR spectra of Yb[CH(SiMe 3 ) 2 ] 3 : (a)[START_REF] Hartwig | Organotransition Metal Chemistry: From Bonding to Catalysis[END_REF] H MAS spectra at 30 kHz MAS,(b)[START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C MAS spectra of at 14.286 kHz MAS and (c)[START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si MAS spectra at 30 kHz MAS, showing two distinct[START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si signals. The spinning sidebands are marked with asterisks ( * ). 2D aMAT spectrum: (d) 1 H at 30 kHz, (e)[START_REF] Yan | Lewis Acid-Mediated β-Hydride Abstraction Reactions of Divalent M(C(SiHMe 2 ) 3 ) 2 THF 2 (M = Ca, Yb)[END_REF] C at 20 kHz and (f)[START_REF] Parigi | Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories[END_REF] Si at 20 kHz. The indirect dimension projections were deconvolved using asymmetric Gaussian functions to extract the isotropic shifts and quantify shift dispersion. (g)-(i) extracted slices with isotropic shift corresponding to the maxima of the deconvolved projections and labelled with Roman numerals. Each slice was subsequently fitted considering a simple CSA model using the Haeberlen convention 43 (δ iso = 1 3 ( δxx + δyy + δzz ), ∆δ = δzz -δ iso and η = ( δyyδxx )/∆δ, where δ xx , δ xx and are the components of the CSA tensor in the principal axes frame). See SI chapter 1.6 for more details.

  C and 29 Si resonances represents a single SiMe 3 group in which there are rotations about the C 1 -Si bond and about the Si-C bonds of each of the three methyls. The rate constants describing each rotation are larger than the spread of the individual chemical shifts of the exchanging sites. Note that the predicted shifts of H 1 nuclei (see SI Chapter 5.3 for the details) lie within a range between 230 and 200 ppm (Fig. 5b), and are not observed in any of the 1H NMR spectra (1D, aMAT, TEDOR), as expected.

Figure 6 :

 6 Figure 6: Surface plots of the three Yb -C α bonding orbitals and the seven 4f active orbitals of the Yb III (CH(SiMe 3 ) 2 ) 3 complex (separated by the vertical line). The canonical energies (in cm -1 ) are given in parentheses. The isosurface value is at ± 0.02 a.u. The dominant compositions of the orbitals are also shown.

  60 a : basis set: ANO-RCC-TZVP(Yb, Si and C α )+ ANO-RCC-DZP (C,H) in MOLCAS, b :basis set: SARC-DKH-TZVP(Yb) and IGLO-III (rest) in ORCA, c : basis set: SARC-ZORA-TZVP(Yb) and TZVP (rest) in ORCA, d : excess of combined 4f+5f orbital occupations than the formal 4f electron number. At the spin-free (SF) CASSCF level, the 2 F manifold of the central Yb(III) ion splits functional (see SI Chapter 1.8 for the details). While the precise prediction of isotropic hyperfine couplings for H 1 nuclei is unlikely due to the close proximity of H 1 to Yb, which requires high-level relativistic electronic structure methods for accurate determination of spin density delocalization, the dipolar (anisotropic) part of the hyperfine coupling tensor could still be a reliable descriptor for comparison between experimental and computed hyperfine couplings. The computed hyperfine coupling tensors for all three H 1 nuclei of the [CH(SiMe 3 ) 2 ] ligands, similarly to the experimental ones, appeared to be nearly axial: A iso (1) = -0.26 MHz, A dip (1) = [-4.00 -4.50 8.51] MHz; A iso (2) = -0.15 MHz, A dip (2) = [-3.80 -3.84 7.64] MHz; A iso (3) = -0.21 MHz, A dip (3) = [-4.47 -4.32 8.79] MHz. Furthermore, the calculated dipolar hyperfine tensor values A dip are in a very good agreement with the experimental values (A dip = [-4.0 ± 0.5; -4.0 ± 0.5; 8.0 ± 1.0] MHz). This indicates that the position of the α-H (H 1 ) atoms of the [CH(SiMe 3 ) 2 ] ligands of Yb(CH(SiMe 3 ) 2 ) 3 , including the related Yb-H distances and the unusually short Yb-C-H angles of ca. 93 • (see Tables

  SO-CASSCF in ORCA, Experimental g x,y = 0.73, g z = 7.57 and A ⊥ = 842 MHz , A ∥ = 6285 MHz Computational interpretation of the magnetic susceptibility data

  93 • ) Yb-C-H 1 angles for all the [CH(SiMe 3 ) 2 ] 3 ligands (see Tab. S7). The shortened Yb-C-H angles for the α-H atoms of the [CH(SiMe 3 ) 2 ] 3 ligands might be explained by a simultaneous electrostatic repulsion from the Yb centre and the two Si atoms of the CH(SiMe 3 ) 2 ligand.

Figure 7 :

 7 Figure 7: (a) Natural spin-α orbital, representing one 5d orbital of Yb[CH(SiMe 3 ) 2 ] 3 (red for positive and blue for negative signs of the wavefunction, the same in all panels). The p-type parts of the orbital (two out of three) are marked with arrows. (b) Overlap of the same 5d orbital (wireframe contour) with natural spin-α orbital, corresponding to the bonding orbital of one of Si-C γ bonds (solid contour).

  . The bonding descriptors, namely the electron density (ρ b ), its Laplacian (∇ 2 ρ b ), the potential energy density (V b ) and the kinetic energy density (G b ) at the so-called line critical point (LCP, i.e. the inflection point at the gradient path between two neighboring atoms) are of interest in this kind of analysis. Generally at the LCP, ρ b > 0.20 a.u. with a negative Laplacian indicates a shared electron interaction between two atoms. Accompanied by |V b | /G b > 1 and H b = V b + G b < 0, this typically indicates bound electrons at the LCP, often interpreted as a covalent interaction. For Yb[CH(SiMe 3 ) 2 ] 3 , the C-H bonds yield typical values of closed shell interactions with ρ b > 0.25 and negative Laplacian, while the Si-C bonds give ρ b ≈ 0.11 with a positive Laplacian, which is characteristic of an ionic interaction. By contrast, the values for both the Yb -C α and Yb -C γ (ρ b =0.078 a.u. and 0.016 a.u. respectively, with positive Laplacians, see Tab 6) indicate donor-acceptor type bonding interactions.

  ) 3 complex. The NBO and QTAIM charges vary due to their different ways of partitioning the total electron density onto the individual atoms, but both indicate that a Yb -C γ interaction is caused by the Coulomb attraction between the positively charged Yb and negatively charged C γ atoms.

  ∆δ -542±3 409±3 -183±4 140[+30,-270] 153±1 -56±1 On examining the computed shift tensors, we see that each nuclear species gives two distinct signals, with the shift and shift anisotropy of one species being consistently larger than those of the other species. There are clear quantitative discrepancies with the experimental values, but the general trends are reproduced. We can therefore assign the signals from the Me 3 Si (2) group, with the larger shifts and shift anisotropies, to the proximal (p) ligand, and the signals from Me 3 Si (1) , with the smaller shifts and shift anisotropies, to the distal (d) ligand.

  trated pentane solutions. Yb[CH(SiMe 3 ) 2 ] 3 is blue, Yb[CH(SiMe 3 ) 2 ] 2 [O-2,6-tBu 2 -C 6 H 3 ] is brown, and Yb[CH(SiMe 3 ) 2 ][O-2,6-tBu 2 -C 6 H 3 ] 2 isred. The study below focuses on the structure and spectroscopic features of Yb[CH(SiMe 3 ) 2 ] 3 . X-ray structures of Yb[CH(SiMe 3 ) 2 ] 2 [O-2,6-tBu 2 -C 6 H 3 ] and Yb[CH(SiMe 3 ) 2 ][O-2,6-tBu 2 -C 6 H 3 ] 2 are proview of the Yb[CH(SiMe 3 ) 2 ] 3 complex is shown in Figs. 2b & S2. The relevant bond distances obtained from this structure are shown in Fig. 2c-d (see also Tab. S2). For

	vided in SI Chapter 2.2 (see Figs. S3 -S6), and their NMR spectra are provided in SI
	Chapter 7 (see Figs. S15 -S20).
	is 2.324(3) Å, which is nearly the
	same as the Lu-C α distance in Lu[CH(SiMe 3 ) 2 ] 3 (2.319(3) Å) and the Sm-C α distance in
	Sm[CH(SiMe 3 ) 2 ] 3 (2.33(2) Å).
	Yb[CH(SiMe 3 ) 2 ] 3 contains short Yb• • • C γ contacts at 2.963(3) Å, which is slightly longer
	than Lu• • • C γ (2.936(2) Å) and Sm• • • C γ (2.85(3) Å) distances in Lu[CH(SiMe 3 ) 2 ] 3 and
	Sm[CH(SiMe 3 ) 2 ] 3 , respectively. The Neumann projection shown in Fig. 2d contains distances
	and angles for the proximal Si-CH 3 group interacting with Yb. The C-H bond distances
	and angles are close to those expected for a tetrahedral sp 3 hybridized carbon, again similar
	to results obtained for Lu[CH(SiMe 3 ) 2 ] 3 . These structural data infer that a similar 3c-2e
	Yb-Me-Si interaction is present in Yb[CH(SiMe 3 ) 2 ] 3 under the conditions of low-temperature
	X-ray diffraction studies (i.e. 100 K).

A comparison the previously reported bond distances for Lu[CH(SiMe 3 ) 2 ] 3 10 (Fig. 2e) are also shown in Fig. 2f-g (see also Tab. S2). Yb[CH(SiMe 3 ) 2 ] 3 crystallizes in the P 31 c space group and is isostructural with other isolated Ln[CH(SiMe 3 ) 2 ] 3 . Yb[CH(SiMe 3 ) 2 ] 3 adopts a C 3 symmetric structure with the ytterbium displaced from the plane defined by the three Yb-C bonds by 0.783(3) Å. The Yb-C α (Yb-C 1 ) distance

  3 ) 2 ] 3 , Yb[CH(SiMe 3 ) 2 ] 2 [O-2,6-tBu 2 -C 6 H 3 ],

	Yb[CH(SiMe 3 ) 2 ][O-2,6-tBu 2 -C 6 H 3 ] 2 , and
	Yb[O-2,6-tBu 2 -C 6 H 3 ] 3 (see SI Chapter 6, Figs. S11 -S14 and Tab. S11 for more details).
	All four ytterbium complexes exhibit experimental values of the effective magnetic moment
	at 300 K that are in the range between µ ef f (300 K) = 4.30 (χT = 2.31 cm 3 K mol -1 ) -
	4.53 µ B (χT = 2.57 cm 3 K mol -1 ). In particular, the effective magnetic moment of the
	Yb[CH(SiMe 3 ) 2 ] 3 complex is µ ef f (300 K) = 4.53 µ B . These values are in good agreement
	with the expectation value for the free Yb(III) ion, described by a single 2 F 7/2 term (4f 13 ),
	for which µ ef f (300 K) = 4.54 µ B (χT = 2.58 cm 3 K mol -1 ). Hence, these Yb complexes can
	be described by a single 4f 13 configuration, similar to those of the free Yb(III) ion.
	EPR characterization of Yb[CH(SiMe 3 ) 2 ] 3

Table 1 :

 1 Ab-initio energy levels (in cm -1 ) of the 2 F (at spin-free level) and 2 F 7/2-5/2 (at spin-orbit coupling level) manifolds in Yb III (CH(SiMe 3 ) 2 ) 3 . The spin-orbit energy levels are doubly degenerate.

		E 1 E 2	E 3	E 4	E 5	E 6	E 7
			XRD
	SF-CASSCF	0 144 682 687 1031 1123 1203
	SF-CASPT2	0 205 823 928 1181 1392 1769
	SO-CASSCF	0 525 889 1088 10387 11007 11341
	SO-CASPT2	0 659 1043 1452 10420 11166 11660
			Optimized
	SF-CASSCF	0 168 694 694 1074 1083 1203
	SF-CASPT2	0 383 949 1002 1331 1393 1729
	SF-CASSCF a 0 172 746 748 1161 1172 1297
	SO-CASSCF	0 526 888 1070 10387 11006 11327
	SO-CASPT2	0 689 1068 1386 10428 11189 11613
	SO-CASSCF a 0 565 959 1154 9214 9891 10241

a : in ORCA

  90% 5f y(3x 2 y 2 ) + 7% 5f z 3 + 2% 5f x(x 2 3y 2 ) 6% Yb(s) + 46% C (sp 2 ) 20% Yb(pd) + 49% C (sp 2 ) 19% Yb(pd) + 49% C (sp 2 )

	(85375)	(85330)	(81293)	(1273)	(812)
	(0)	e(88)		e(527)

2 y 2 ) 46% 5f yz 2 + 52% 5f xz 2 46% 5f xz 2 + 52% 5f yz 2 52% 5f xyz + 46% 5f z(x 2 y 2 ) 52% 5f z(x 2 y 2 ) + 46% 5f xyz 98% f x(x 2 3y 2 ) + 2% 5f y(3x 2 y 2 )

Table 2 :

 2 The Mulliken (M) and Löwdin (L) charge populations of the Yb atomic orbitals in Yb III (CH(SiMe 3 ) 2 ) 3 from different calculation methods.

		s populations p populations d populations f excess d
	SF-CASSCF (M) a	0.50(6s+7s)	1.52(6p+7p)	1.22(5d+6d)	0.02
	SF-CASSCF (M) b	0.48(6s+7s)	0.76(6p+7p)	1.13(5d+6d)	0.14
	SF-CASSCF (L) b	0.19(6s+7s)	0.60(6p+7p)	1.87(5d+6d)	0.52
	ZORA-DFT(PBE0)(M) 0.72(6s+7s)	1.06(6p+7p)	1.84(5d+6d)	0.18
	ZORA-DFT(PBE0)(L) 0.32(6s+7s)	0.77(6p+7p)	2.28(5d+6d)	0.

Table 4 :

 4 The ab initio g-factors and hyperfine coupling parameters (in MHz) of the central 171 Yb (I = 1/2) in Yb III (CH(SiMe 3 ) 2 ) 3 . L and S are the orbital and spin contributions, respectively, to the magnetic moments. The spin-dipolar (SD), orbital (orb) and Fermicontact (FC) contributions to the calculated hyperfine values are shown.

		g L x	g L y	g L z	g S x	g S y	g S z	g x	g y	g z	g iso ∆g ax
	SO-CASSCF	0.27	0.27	5.92	0.13	0.13	1.99 0.40 0.40 7.91 2.90 7.51
	SO-CASPT2	0.50	0.45	5.81	0.25	0.24	1.96 0.75 0.69 7.77 3.10 7.05
	SO-CASSCF a 0.27	0.27	5.92	0.13	0.13	1.99 0.38 0.38 7.90 2.89 7.52

Table 5 :

 5 The molar magnetic susceptibilities of Yb III (CH(SiMe 3 ) 2 ) 3 (in 10 -8 m 3 mol -1 ) at 300 K from ab initio calculations. iso χ iso T ∆χ ax SO-CASSCF 2.23 2.26 17.56 0.82 0.83 5.87 3.05 3.09 23.43 9.85 2.37 20.36 SO-CASPT2 1.63 1.91 17.69 0.63 0.73 5.94 2.26 2.63 23.63 9.50 2.29 a 21.19 a : SQUID value = 2.57 in CGS unit

	χ L x	χ L y	χ L z	χ S x	χ S y	χ S z	χ x	χ y	χ z	χ

Table 6 :

 6 Bonding descriptors at the Yb -C α and Yb -C : on Lu III (CH(SiMe 3 ) 2

	(ag) γ	line critical points. The NBO

a

https://doi.org/10.26434/chemrxiv-2023-ksfb4-v2 ORCID: https://orcid.org/0000-0002-3597-0889 Content not peer-reviewed by ChemRxiv. License: CC BY 4.0

into three singlets, of relative energies of 0, 168 and 1203 cm -1 , and two orbital doublets at 694 and ∼1074 cm -1 , following the energy stabilization of the unpaired electron or hole in the trigonal ligand field. Dynamic correlation energy correction at the MS-CASPT2 method further increases the energy splitting by further stabilizing and destabilizing the bonding and anti-bonding type orbitals. At the spin-orbit (SO) level, the four Kramers doublets (KDs) of the ground J = 7/2 manifold are split by the axial ligand field in the order m J = ±7/2 < ±5/2 < ±3/2 < ±1/2. The ligand field splitting is around 1400 cm -1 , seven times larger than the room temperature thermal energy (200 cm -1 ). In fact, the first excited KD characterized by m J = ±5/2 is around 690 cm -1 and is not meaningfully populated at room temperature. The irreducible spherical tensor ligand-field parameters extracted from the model spaces of the ground L (at SF) and J (at SO) manifolds are given in Table 3. The dominant ligand-field parameter is the second-rank axial component B 2 0 , but the components of orders three and six are also significant, as expected from the three-fold symmetry axis. The B 2 0 component is related to the quadrupole moment of the ligand field and its negative value corresponds to an oblate coordinating sphere. The excited J = 5/2 manifold is energetically well separated by more than 10000 cm -1 in accordance with the SO coupling strength (ζ 4f ≈ 2900 cm -1 ) of the Yb 4f electrons. [START_REF] Zhou | Electronic spectra of Cs2NaYbF6 and crystal field analyses of YbX63-(X= F, Cl, Br)[END_REF] Table 3: The irreducible spherical tensor ligand-field components of rank k and order m (B k m ) and strength parameters (S) (in cm -1 ) of Yb III (CH(SiMe 3 ) 2 ) 3 determined from the ab initio ground L (SF) and J (SO) manifolds.

Computational interpretation of the EPR data

Based on the DFT-computed structure of Yb(CH(SiMe 3 ) 2 ) 3 (Fig. S9), we first performed a calculation of hyperfine couplings for the α-H atoms of [CH(SiMe 3 ) 2 ] ligands (H 1 ) with PBE0

Computational interpretation of the NMR data

In order to confirm our tentative interpretation and provide a site-specific assignment, we calculated the NMR shift tensors of all the spin-active nuclei in the complex. The total NMR shift can be expressed as:

where δ orb corresponds to the orbital chemical shift, i.e. the contribution from the orbital motion of the electrons, δ FC is the Fermi contact shift due to the through-bond hyperfine interaction, δ PSO is the paramagnetic spin-orbital shift due to the interaction between the nuclear spin and the electron orbital angular momentum, δ SD and δ LR are the shortrange intra-molecular and long-range inter-molecular pseudo-contact shifts (PCS) due to the through-space hyperfine interactions, respectively. To account for long-range contributions, we calculated through-space magnetic dipolar contributions using the semi-classical pointdipolar equation (eq. SI-10) and a spherical bath of surrounding paramagnetic ions up to 50 Å distant in the crystal structure. The intramolecular contributions to the NMR shielding tensor of each nucleus were calculated according to the following equation,

where σ orb is the diamagnetic orbital contribution calculated using a scalar relativistic ZORA-DFT method employing the PBE0 functional, χ s is the spin-only magnetic susceptibility tensor calculated from the SO-CASPT2 method and the hyperfine tensors are computed either from SO-CASSCF or unrestricted DFT methods. µ B is the Bohr magneton and g e is the free electron g-factor. In accordance with the tentative assignment of the experimental spectra, a three-site fast exchange model was employed to average the shift tensors both within the methyl groups (along the Si-C i bonds) and the Me 3 Si groups (along

Author Information

Richard A. Andersen passed away on June 16, 2019 while the early draft of this manuscript was being written; this manuscript is dedicated to his memory.

Corresponding Authors * andrew.pell@ens-lyon.fr; ccoperet@ethz.ch; gjeschke@ethz.ch; guido.pintacuda@ens-lyon.fr

Author contribution