Jingwen Zhu

Emilie-Laure Zins

Mohammad Esmaïl Alikhani

Characterization of B-H agostic compounds involved in the dehydrogenation of amine-boranes by group 4 metallocenes

and research institutions in France or abroad, or from public or private research centers.

Characterization of B-H agostic compounds involved in the dehydrogenation of amine-boranes by group 4 metallocenes

Introduction

Numerous experimental and theoretical studies have demonstrated the capability of organometallic compounds in catalyzing the dehydrogenation of amine-boranes. [1][2][3][START_REF] Whittell | Advances with Ammonia-Borane: Improved Recycling[END_REF][START_REF] Leitao | Catalysis in service of main group chemistry offers a versatile approach to p-block molecules and materials[END_REF] Of particular importance for actual applications as energy carriers is the detailed reaction mechanism. [START_REF] Waterman | Mechanisms of metal-catalyzed dehydrocoupling reactions[END_REF] Indeed, viable alternative proposal for energy source should satisfy the requirements for sustainable development and green chemistry. As such, care should be taken in step economy and redox economy, which cannot be done without an in-depth understanding of the detailed reaction mechanisms. Based on theoretical as well as experimental investigations, reaction mechanisms were proposed to explain the catalytic cycles leading to the generation of molecular hydrogen from amine boranes. The exact role of the metallic centre M on these reaction mechanisms is still the topic of numerous studies. [START_REF] Rossin | Ammonia -Borane and Amine -Borane Dehydrogenation Mediated by Complex Metal Hydrides[END_REF] At the molecular scale, different non-covalent interactions are involved in these dehydrocoupling reactions: the formation of a 3 centre -2 electrons (3C/2e) intermediate, and Kubas' complexes. [1][2][3][START_REF] Whittell | Advances with Ammonia-Borane: Improved Recycling[END_REF][START_REF] Leitao | Catalysis in service of main group chemistry offers a versatile approach to p-block molecules and materials[END_REF][START_REF] Waterman | Mechanisms of metal-catalyzed dehydrocoupling reactions[END_REF][START_REF] Rossin | Ammonia -Borane and Amine -Borane Dehydrogenation Mediated by Complex Metal Hydrides[END_REF][START_REF] Kubas | Dihydrogen complexes as prototypes for the coordination chemistry of saturated molecules[END_REF] The former species may play a critical role in the reaction, because of the activation of the σ(B-H) bond.

The term "agostic" [START_REF] Brookhart | Agostic interactions in transition metal compounds[END_REF] will be used throughout the present article to describe 3C/2e interaction between a σ(B-H) bond and a metal atom in an organometallic complex. A unique and unequivocal definition of the term agostic would deserve consideration. [START_REF] Clots | Agostic Interactions from a Computational Perspective: One Name, Many Interpretations[END_REF] First proposed by Brookhart and Green in 1983, the term was specifically introduced to describe a 3C/2e interaction involving a H-C bond. [START_REF] Brookhart | Two-electron,three-centre carbon-hydrogen-cobalt bonds in the compounds [Co(Z-C 5 Me 4 R)(Z-C 2 H 4 )(Z-C 2 H 4 -m-H)]BF 4 ,R = Me and Et[END_REF]12 Should the chemical nature of the σ bond involved in the interaction, be considered as a criterion in the use of the "agostic" term? This question was discussed, as was the definition of this interaction. Phenomenological 3C/2e as well as causal (i.e. based on reactivity properties) criteria were applied to use the "agostic" term. Since the phenomenological 3C/2e criterion is used (even tacitly) in almost all the articles dealing with the organometallic-catalyzed dehydrocoupling of amine borane, we will use the term "agostic intermediate" to refer to as the activated species involving a 3C/2e interaction. A thorough understanding of the parameters influencing this interaction may help in choosing either the best catalysts with a given amine-borane compound, or conversely the best N-B (or P-B) containing precursor with a given catalyst, for the generation of molecular hydrogen. The first step toward such an understanding is methodological: a step by step study to analyze the effects of the nature of (1) substituents beared by the ligand, (2) the ligand (amine-versus phosphine-boranes), (3) the metallic centre, and (4) the magnetic properties of the metallic centre, is a needed prerequisite prior to a more systematic computational study. We are convinced that the electron localization function (ELF) [START_REF] Becke | A simple measure of electron localization in atomic and molecular systems[END_REF] and the quantum theory of atoms in molecules (QTAIM) [START_REF] Bader | Atoms in Molecules-A Quantum Theory[END_REF][START_REF] Matta | An Introduction to the Quantum Theory of Atoms in Molecules[END_REF] topological approaches constitute appropriate tools for such a methodological study.

An appropriate electronic and geometric description of the organometallic complex is a prerequisite to any a posteriori topological characterization of the agostic interaction. First discussed by Streuber and Grunenberg in the case of titanium compounds, 16 and by Etienne et al. in the case of niobium compounds, [START_REF] Pantazis | Critical Role of the Correlation Functional in DFT Descriptions of an Agostic Niobium Complex[END_REF] this point is particularly challenging for three main reasons: i) due to the size of the systems, ab initio methods can hardly be carried out in the context of a comparative study; ii) it is a non-covalent intramolecular interaction; and iii) the electronic properties of the d-bloc atoms should be accurately described. Comparison between theoretical studies based on the density functional theory (DFT) and experimental structures definitely suggests that agostic interactions can be described by DFT functionals. [START_REF] Lein | Characterization of agostic interactions in theory and computation[END_REF] Recently, different approaches were implemented in quantum chemistry programs to allow the treatment of dispersion forces in DFT calculations. Such dispersion forces are of fundamental importance in the case of van der Waals interactions. They also may play a prominent role in the case of hydrogen bondings. Among other methods, the D3 empirical correction of Grimme with Becke and Johnson damping was revealed to be particularly adapted to describe intramolecular hydrogen bondings. [START_REF] Grimme | Effects of London dispersion correction in density[END_REF] Would such correction affect the description of agostic interactions, either from a geometric or from a topological point of view?

As far as the electronic description of the metallic centre is concerned, all-electron basis sets can be practically used only in the case of the first-row transition metal elements. For heavier elements, pseudopotentials should be chosen. The further use of relativistic Hamiltonian is needed for an accurate description of topological properties of organometallic compounds. [START_REF] Eickerling | Relativistic effects on the topology of the electron density[END_REF][START_REF] Bučinský | Picture change error in quasirelativistic electron/spin density, Laplacian and bond critical points[END_REF]22 Herein is proposed an estimation of the "minimum level of theory" leading to accurate enough topological description of β-agostic interactions within the framework of the DFT. The accuracy of the levels of theory was evaluated on a model system in a three-step approach:  first, geometric parameters were used to characterize the ability of the level of theory to describe the agostic interaction ,  then, the existence of a trisynaptic basin for the hydrogen atom involved in the agostic interaction was checked using the ELF topological method,  finally, the ρ, 2 ρ, ε and H QTAIM descriptors on the BCP of the agostic bond obtained from DFT calculations carried out at different levels of theory were compared

The influence of the DFT level of theory on the accuracy of the geometric and topological description of the agostic interaction was thus evaluated. A suitable level of theory was then selected for an analysis of a large, representative, set of agostic compounds.

Computational details

A set of nine group 4 metallocenes for which experimental data were reported in the literature, was chosen for this study: Cp 2 TiNH 2 BH 3 , [START_REF] Wolstenholme | Structure and bonding of titanocene amidoborane complexes: a common bonding motif with their β-agostic organometallic counterparts[END_REF] Cp 2 ZrHNH 2 BH 3 and its isomer, [START_REF] Forster | Characterization of β-Agostic isomers in Zirconocene Amidoborane complexes[END_REF] Cp 2 TiNMe 2 BH 3 , [START_REF] Helten | Paramagnetic titanium(III) and zirconium(III) metallocene complexes as precatalysts for the dehydrocoupling/dehydrogenation of amine-borane[END_REF] Cp 2 ZrHNMe 2 BH 3 , [START_REF] Helten | Paramagnetic titanium(III) and zirconium(III) metallocene complexes as precatalysts for the dehydrocoupling/dehydrogenation of amine-borane[END_REF] Cp 2 TiPPh 2 BH 3 , [START_REF] Helten | Paramagnetic titanium(III) and zirconium(III) metallocene complexes as precatalysts for the dehydrocoupling/dehydrogenation of amine-borane[END_REF] and Cp 2 ZrPPh 2 BH 3 . [START_REF] Helten | Paramagnetic titanium(III) and zirconium(III) metallocene complexes as precatalysts for the dehydrocoupling/dehydrogenation of amine-borane[END_REF] All the geometry optimizations were carried out using the Gaussian09 software. 26 The effects of several exchange-correlation functionals and basis sets (including the treatment of relativistic effects) on the geometric and electronic (topological) description of the agostic interaction in the metallocenes were evaluated. This calibration step was carried out on the titanocene aminoborane (compound 1) for which experimental data are available. [START_REF] Wolstenholme | Structure and bonding of titanocene amidoborane complexes: a common bonding motif with their β-agostic organometallic counterparts[END_REF] The Minnesota 27 and range-separated hybrid functionals, 28 dedicated to the study of weak interactions, were considered for the present methodological study. [START_REF] Goerigk | Treating London-Dispersion Effects with the Latest Minnesota Density Functionals: Problems and Possible Solutions[END_REF][START_REF] Zhao | The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals[END_REF][START_REF] Modrzejewski | Range-Separated meta-GGA Functional Designed for Noncovalent Interactions[END_REF] Additionally, the effect of the dispersion-correction within the framework of the Becke-Johnson damping function [START_REF] Grimme | Density functional theory with London dispersion corrections[END_REF][START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF] was evaluated in the present work. [START_REF] Xing | DFT Studies on the Mechanism of Palladium(IV)-Mediated C-H Activation Reactions: Oxidant Effect and Regioselectivity[END_REF][START_REF] Van Der Eide | Isolation of two agostic isomers of an organometallic cation: different structures and colors[END_REF] The following functionals were considered in the present work:

 The local functional M06-L 36 designed for taken into account the exchange-correlation energy ,  The GGA BP86 functional, [START_REF] Perdew | Density-functional approximation for the correlation energy of the inhomogeneous electron gas[END_REF] alone and with the D3 empirical correction of Grimme with Becke and Johnson damping,  The Meta-GGA TPSSTPSS 38 functional, alone and with the D3 empirical correction of Grimme with Becke and Johnson damping,  The hybrid B3LYP, [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF] B3PW91, PBE1PBE, 41,[START_REF] Perdew | Errata: Generalized gradient approximation made simple[END_REF][START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF] M06 and TPSSh 44 functionals, alone and with the D3 empirical correction of Grimme with Becke and Johnson damping ,  The LC-ω-PBE range separated RSH functional, [START_REF] Vydrov | Assessment of a long range corrected hybrid functional[END_REF][START_REF] Vydrov | Importance of short-range versus long-range Hartree-Fock exchange for the performance of hybrid density functionals[END_REF][START_REF] Vydrov | Tests of functionals for systems with fractional electron number[END_REF] alone and with the D3 empirical correction of Grimme with Becke and Johnson damping. All the calculations were carried out with the default value of ω in the Gaussian09 software (ω = 0.4 a.u. -1 ).

Complementarily, the influence of the basis set was studied. In the case of the titanocene compounds, Pople triple ζ basis sets can be used for all the elements, including the transition metal atom. This is not the case for the zirconocene and hafnocene compounds. For these latter cases, the use of pseudopotentials is needed. To determine how these pseudopotentials will affect the description of the agostic interactions, model Ti +III (1) and Ti +IV (1) species were studied using the following basis sets:

 The Pople triple ζ 6-311++G(2d,2p) and 6-311++G(3df,3pd) basis sets,  The Def2-TZVP, Def2-TZVPP and def2-QZVPPD basis sets,  The Pople triple ζ 6-311++G(2d,2p) basis set for the main-element atoms in combination with the SDD pseudopotential for the titanium. This basis set will be further referred to as 6-311++G(2d,2p)+SDD,  The Pople triple ζ 6-311++G(2d,2p) basis set for the main-element atoms in combination with the relativistic LanL2DZ pseudopotential for the titanium. This basis set will be further referred to as 6-311++G(2d,2p)+LanL2DZ,  The DZP [START_REF] Neto | Gaussian basis sets for correlated wave functions, Hydrogen, helium, first-and second-row atoms[END_REF][START_REF] Camiletti | Gaussian Basis Set of Double Zeta Quality for Atoms K Through Kr: Application in DFT Calculations of Molecular Properties[END_REF][START_REF] Barros | Gaussian basis set of double zeta quality for atoms Rb through Xe: application in non-relativistic and relativistic calculations of atomic and molecular properties[END_REF] and TZP [START_REF] Barbieri | Gaussian basis sets of triple and quadruple zeta valence quality for correlated wave functions[END_REF][START_REF] Machado | Gaussian basis set of triple zeta valence quality for the atoms from K to Kr: Application in DFT and CCSD(T) calculations of molecular properties[END_REF][START_REF] Campos | Triple zeta quality basis sets for atoms Rb through Xe: application in CCSD(T) atomic and molecular property calculations[END_REF] basis sets,  The relativistic DZP-DKH [START_REF] Jorge | Contracted Gaussian basis sets for Douglas-Kroll-Hess calculations: Estimating scalar relativistic effects of some atomic and molecular properties[END_REF][START_REF] Neto | All-electron double zeta basis sets for the most fifth-row atoms: Application in DFT spectroscopic constant calculations[END_REF][START_REF] Barros | Gaussian basis set of double zeta quality for atoms Rb through Xe: application in non-relativistic and relativistic calculations of atomic and molecular properties Gaussian basis set of double zeta quality for atoms Rb through Xe: application in non-relativistic and relativistic calculations of atomic and molecular properties[END_REF] and TZP-DKH [START_REF] Barros | Gaussian basis set of double zeta quality for atoms Rb through Xe: application in non-relativistic and relativistic calculations of atomic and molecular properties Gaussian basis set of double zeta quality for atoms Rb through Xe: application in non-relativistic and relativistic calculations of atomic and molecular properties[END_REF][START_REF] Jorge | Contracted Gaussian basis sets for Douglas-Kroll-Hess calculations: Estimating scalar relativistic effects of some atomic and molecular properties[END_REF] basis sets, in combination with the Douglas-Kroll-Hess Hamiltonian, in the context of relativistic calculations, 58  The relativisic RESC approach, in combination with the DZP-DKH basis set.

The nature and the strength of the interactions were further characterized with two topological approaches: the electron localization function (ELF) [START_REF] Becke | A simple measure of electron localization in atomic and molecular systems[END_REF] and the Quantum Theory of Atoms in Molecules (QTAIM). [START_REF] Bader | Atoms in Molecules-A Quantum Theory[END_REF][START_REF] Matta | An Introduction to the Quantum Theory of Atoms in Molecules[END_REF] The ELF streamlines and specifies the chemical bond, in line with the concepts initially introduced by Lewis. 59,[START_REF] Fuentealba | Understanding and using the electron localization function[END_REF] This interpretative tool allows for a partition of space in terms of basins. The number of electrons involved in each basin (i.e. the basin's population), as well as other properties, can easily be obtained by integrating of the property density over the basin volume. The ELF signature of a X-H agostic interaction consists in a protonated basin containing approximately 2 electrons from the X, H and M atoms. Such basins are described as "trisynaptic". The ELF identification of a trisynaptic basin containing approximately two electrons constitutes a necessary and needed proof to the existence of a 3C/2e interaction. In the QTAIM framework, proposed and thoroughly developed by Bader and co-workers, the identification of a bond critical point (BCP) as well as a bond path (BP) between two atoms proves that the two atoms are chemically bonded. In such a case, the chemical bond can be further (quantitatively) characterized by means of the electron density (ρ), the Laplacian of ρ ( 2 ρ), the ellipticity (ε), and the total energy density (H) on the BCP. Other specific (local) descriptors were also introduced in the context of agostic interactions by Joubert et al. 61,[START_REF] Tognetti | Characterizing Agosticity Using the Quantum Theory of Atoms in Molecules: Bond Critical Points and Their Local Properties[END_REF][START_REF] Tognetti | On the Influence of Density Functional Approximations on Some Local Bader's Atoms-in-Molecules Properties[END_REF] . Using the above-mentioned values, the strength of agostic interactions can be accurately evaluated as soon as the interaction is characterized by a BCP. This is generally the case for β agostic interactions, but not necessarily for α agostic interactions. Since the present article focuses on β agostic interactions, the QTAIM tool will be used to estimate the strength of the interaction between the metallic centre and the B-H bond.

All the geometry optimizations and the generation of the wfn files were carried out with the Gaussian 09 Rev D.01 software. The wfn files were further used for the topological analysis. The ELF and QTAIM calculations have been done using the TopMod [START_REF] Noury | Computational tools for the electron localization function topological analysis[END_REF] and AIMAll 65 softwares, respectively. The ELF analyses were carried out using the "fine" grid. As far as the QTAIM calculations are concerned, we should point out the fact that the energy densities are evaluated from the Kohn-Sham first reduced density matrix.

Results and discussion

The Cp 2 TiNH 2 BH 3 compound, for which experimental data are available, was chosen to evaluate the accuracy of the description of the agostic interaction calculated at the different levels of theory. 23

Geometric considerations

Different representative geometric parameters were compared with experimental data. Deviations between experimental and theoretical distance between X and Y atoms were calculated as follows:

where represents the experimental (respectively calculated) distance.

Similarly, deviations between experimental and theoretical angle between X, Y and Z atoms were calculated as follows:

where represents the experimental (respectively calculated) angle.

a. Influence of the functional in combination with the 6-311++G(2d,2p) basis set

To begin with, the effects of the density functionals on representative geometric parameters were studied in the case of the Ti +III (1) compound. The experimental data were taken from Ref. [START_REF] Wolstenholme | Structure and bonding of titanocene amidoborane complexes: a common bonding motif with their β-agostic organometallic counterparts[END_REF] The main results are shown in the Table 1.

The experimental Ti-N and Ti-B distances are 2.153 and 2.520 Å, respectively. The corresponding values are calculated to be in the [2.155 Å; 2.184 Å], and [2.501 Å ; 2.542 Å] ranges, respectively. The δd Ti-N deviation between the experimental and the calculated distance is thus in the [+0.1% ; +1.5%] range, whereas the δd Ti-B deviation is in the [-0.8% ; 0.9%] range. Thus, the calculated and distances are in good agreement with the experimental data.

With all the levels of theories used, the two B-H bonds that are not interacting with the Ti atom are calculated to be identical in length. However, from crystallographic structures, two different lengths are measured for these non-agostic B-H bonds: 1.09 and 1.13 Å. This suggests the existence of additional weak intermolecular interactions either with the solvent during the evaporation, or between molecules in the solid phase.

The corresponding calculated distance ( ) is in the [1.200 Å ; 1.212 Å] range. Thus, the deviation between the experimental and the calculated B-H distance, is in the [8.2% ; 9.2%] range. The deviation observed between the experimental and the calculated B-H distance is much larger than the deviation obtained for the Ti-N and Ti-B distances. Such discrepancy may be due to the uncertainty in the crystallographic data, especially concerning the location of the hydrogen atoms. In the case of the B-H bond involved in the agostic interaction, the experimental distance is 1.21 Å while the corresponding calculated distance is in the [1.282 Å; 1.312 Å] range. Thus, the δ d B-H agost. deviation between the experimental and the calculated B-H distance, is in the [6.0%; 8.5%] range. It is interesting to note that the deviation between the calculated and the experimental distance is similar to the one obtained in the case of the non-agostic B-H bond. To further characterize the 3C/2e interaction, we calculated the difference between the length of the agostic B-H bond and the free B-H bond. This lengthening is of 0.10 Å for the experimental data, and in the range 0.079 -0.100 Å, for the calculated distances. Thus, the δ Δd B-H deviation between the experimental and the calculated difference is in the [-18.0% ; 0%] range, with:

We thus conclude that the lengthening of the B-H bond due to the interaction with the metallic centre is correctly described by all the selected functionals, with a deviation of 0.03 Å depending on the functional used.

In addition with distances, the angles are often used as criteria in the identification of agostic bondings. In the case of σ(C-H) bonds, it is generally considered that a angle in the [90°; 140°] range corresponds to an agostic interaction. Furthermore, a angle in the [110°; 170°] range is generally attributed to an anagostic interaction. [START_REF] Brookhart | Agostic interactions in transition metal compounds[END_REF] The angles reported in the Table 1 show that, in the case of the titanocene amino-borane, the angle is calculated in the [102.2°; 104.0°] range, whatever the functional used. Thus, the geometries calculated with all the chosen functional are consistent with the formation of an agostic interaction between the σ(B-H) bond and the titanium atom. The calculated values for the angle are in the [81.3°; 84.0°] range, whereas the experimental value is 84.5°. This angle is thus systematically calculated smaller than the experimental value, and the deviation between the experimental and the calculated value is in the [-3.79% ; -0.59%] range.

Similarly, the calculated angle is calculated in the [104.5°; 105.9°] range, which is systematically larger than the corresponding experimental value ( ), and the deviation between the experimental and the theoretical value is in the [-2.1%; -1.2%] range. For a whole comparison of the functionals, a new parameter was introduced the total deviation :

δ δ δΔ δ δ
is a measure of the normalized total deviation between the calculated and the experimental values, based on two distances ( and ), one reduced distance ( and two angles ( and ). In the Table 1, the different functionals are classified from the one leading to the smaller to the one leading to the larger . The smallest value is obtained with the BP86 functional (with or without the use of the D3 empirical correction of Grimme with Becke and Johnson damping), whereas the use of the M06-L functional leads to the larger value. Moreover, the values are below 25 % for all the considered functionals. Thus, from a geometric point of view, the 12 chosen functionals correctly describe the Ti +III (1) compound. Two major points will be further discussed: the influence of the use of the D3 empirical correction of Grimme with Becke and Johnson damping, and the effect of the class (local, GGA, hybrid GGA, meta-GGA and rangeseparated hybrids) of the functionals on the geometric description of this system.

Influence of the D3 empirical correction of Grimme with Becke and Johnson damping

The agostic interaction has often been compared to the hydrogen bond. In the latter case, the dispersion forces may notably contribute to the stabilization of the system. A substantial effort was recently made to take into account these forces in DFT calculations. In the present study, the effect of the D3 empirical correction of Grimme with Becke and Johnson damping was evaluated in the case of one GGA functional (BP86), two hybrid functionals (PBE1PBE and B3LYP), and one meta-GGA (TPSSTPSS) and one range-separated hybrid functional (LC-ωPBE). To evaluate the effect of the D3 empirical correction of Grimme with Becke and Johnson damping, the difference between the selected distances and angles calculated with and without this correction are presented in the Table S1.

Four tendencies can be drawn from these comparisons:

 For a given functional, the use of the D3 empirical correction of Grimme with Becke and Johnson damping systematically leads to smaller Ti-N and Ti-B distances. However, this shortening remains smaller than 0.02 Å with the five considered functionals,  The D3 empirical correction of Grimme with Becke and Johnson damping has almost no effect on the B-H and B-H agost. distances,  The Ti-H agost. distance is systematically predicted to be smaller when the D3 empirical correction of Grimme with Becke and Johnson damping is used. As in the case of the Ti-N and Ti-B distances, this shortening remains smaller than 0.02 Å with the five considered functionals,  The angles and are predicted to be 0.1 to 0.5° larger when geometry optimizations are carried out without the D3 correction. Conversely, the is predicted to be 0.1 to 0.2° larger when the D3 correction is used.

Thus, the use of the D3 empirical correction of Grimme with Becke and Johnson damping do not dramatically affect the geometry of the Ti +III (1) compound, and "classical" DFT calculations can suitably describe this 3C/2e interaction. This suggests that the dispersive forces do not play a prominent role in the agostic interaction herein investigated.

Comparison of the calculated geometric parameters depending on the class of the functional used

In our prototype system, the electrostatic nature of the interaction may play a predominant role. Thus, in the context of DFT studies, the treatment of the electron correlation may affect the description of the interaction. More precisely, we were interested in the influence of the class of the functional on the accuracy of the geometric description of the interaction. The deviation of distances and angles with the experimental data are shown in the Table S2 for each class of functionals:  in the case of the local M06-L functional, the deviation on the Ti-N distance is five times higher than that on the Ti-B distance. This functional leads to the highest value of parameter among the considered functionals,  the GGA BP86 functional leads to an accurate geometric description of the system, and the parameter is particularly low, suggesting that this functional is particularly adapted for the description of the system,  the GGA BP86-D3 functional also leads to an accurate description of the interaction. On the other hand, the Ti-N distance is slightly overestimated, whereas the Ti-B distance is slightly underestimated,  the hybrid B3LYP functional leads to a similar overestimation of the Ti-N and Ti-B distances. In the same time, the lengthening of the B-H bond due to the agostic interaction is correctly reproduced,  the combination of this functional with the D3 empirical correction of Grimme with Becke and Johnson damping does not lead to a decrease of the parameter,  the hybrid PBE1PBE functional, taken alone or with the D3 empirical correction of Grimme with Becke and Johnson damping, leads to an overestimation of the Ti-N distance, whereas the Ti-B distance is underestimated,  in the case of the TPSSh hybrid functional, the deviation on the Ti-N distance is almost three times larger than the deviation on the Ti-B distance,  the meta-GGA TPSSTPPS and TPSSTPSS-D3 functionals also lead to an overestimation of the Ti-N distance that is much higher than the deviation on the Ti-B distance,  the range-separated LC-ωPBE and LC-ωPBE-D3 hybrid functionals lead to an underestimation of the Ti-B distance that is much more important than the deviation on the Ti-N distance.

Finally, only two functionals lead to a similar deviation of the Ti-N and Ti-B distances, in combination with an accurate lengthening of the B-H bond involved in the agostic interaction, and with a low value of the parameter: the GGA BP86 functional, and the hybrid B3LYP functional. If we consider a 0.03 Å uncertainty for distances and a 3° uncertainty for angles as acceptable, then the popular B3LYP hybrid functional satisfactorily describe the Ti +III (1) compound. This functional was selected for the further evaluation of the effects of basis sets.

In the Table 1 are also reported the calculated geometric parameters for the (1) compound. As in the case of the Ti +III (1) compound, no dramatic change is observed with respect to the choice of the functional. No experimental data are available for this system, but these results show that all the considered functionals lead to a similar geometric description of the Ti +III (1) and

(1) compounds. More precisely, the variations of the calculated distances and angles for the Ti +III (1) and

(1) compounds are presented in the Table S3 The calculated Ti-N and Ti-B distances are systematically longer in the diamagnetic

(1) compound with respect to the paramagnetic Ti +III Thus, the theoretical calculations suggest that, in the case of the E isomer, the magnetic properties of the titanium atom do not affect the agostic and non-agostic B-H distances in the titanocene (1) complexes.

b. Influence of the basis set and of the relativistic treatment in combination with the B3LYP functional

The Ti +III (1) and

(1) compounds were chosen as case studies, because in addition with experimental data available for the Ti +III (1) compound, a large choice of basis sets is available, including triple-and quadruple-ζ all electron basis sets and relativistic pseudopotentials. However, for organometallic complexes containing heavier group 4 metallic centres, pseudopotentials should be used for calculations in currently available calculators, in order to reduce the number of primitive Gaussians. Moreover, relativistic effects are obviously non-negligible in the case of organometallic complexes. In addition with relativistic pseudopotentials (SDD and LanL2DZ), the scalar relativistic Douglas-Kroll-Hess Hamiltonian was used.

As in the case of the study of the effect of the functional on the geometric description of the agostic interaction, the choice of the basis set does not affect the angles between the B, N, H and Ti atoms. The effects of the basis set on the previously chosen distances are presented in the Table 2. At a first glance, it appears that the Ti-N and Ti-B experimental distances are correctly reproduced, with a deviation of ± 0.03 Å. The lengthening of the B-H bond involved in the agostic interaction is also correctly reproduced, with a deviation of ± 0.01 Å. Thus, from a geometrical point of view, all the considered basis sets combined with the B3LYP functional lead to a suitable description of the Ti +III (1) system. Four Alrichs' basis sets were chosen for this study: Def2-TZVP, Def2-TZVPP, Def2-TZVPPD and Def2-QZVP. Very similar geometries were obtained will these four basis sets: the Ti-N distance is calculated to be 2.182 with all these basis sets, whereas the Ti-B distance is calculated to be in the [2.539 Å; 2.541 Å]. This corresponds to a slight overestimation of the experimental distances, and the deviations between the theoretical and experimental values is +1.4% for the Ti-N distance, and in the [0.8% ; 0.9%] range for the Ti-B distance.

These Alrichs' basis sets also lead to an underestimation of the lengthening of the B-H bond due to the agostic interaction, and the δ ΔdB-H is in the [-7.0% ; -5.0%] range, with: Two Pople's basis sets were considered in our study: the 6-311++G(2d,2p) basis set, alone and in combination with the D3 empirical correction of Grimme with Becke and Johnson damping, and the 6-311++G(3df,3pd) basis set. As in the case of the Alrichs' basis sets, the Ti-N and Ti-B distances are slightly overestimated, and the and are in the [+0.9% ; +1.4%] and [+0.8% ; +1.0%] ranges, respectively. The lengthening of the B-H bond due to the agostic interaction is underestimated with these basis sets, and the is in the [-9.0% ; -7.0%] range.

The use of either LanL2DZ or SDD pseudopotential for the metallic centre, in combination with the 6-311++G(2d,2p) basis set for the other atoms, leads to an improvement of the accuracy of the calculated distances. The use of the LanL2DZ pseudopotential for the metallic centre in combination with the D3 empirical correction of Grimme with Becke and Johnson damping, leads to an overestimation of the Ti-N distance, and an underestimation of the Ti-B distance.

The influence of the use of a scalar relativistic Hamiltonian was also studied. To this end, the two-component relativistic Douglas-Kroll-Hess Hamiltonian was considered, in combination with the DZP-DKH and TZP-DKH basis sets. For a sake of comparison, non-relativistic calculations were also carried out with the DZP and TZP basis sets. Due to a limitation of variables for such calculations with the Guassian09 software, only partial optimizations were carried out. All but nine parameters were frozen, and the positions of the Ti, N, B and H atoms were optimized. The same variables were frozen for the relativistic DKH2/DZP-DKH, DKH2/TZP-DKH and for the non-relativistic DZP and TZP calculations. In the case of the non-relativistic DZP calculations, the Ti-N and Ti-B distances are 2.176 and 2.540 Å, respectively. Compared with the experimental data, this corresponds to a slight overestimation. ( = 1.1 % and = 0.8 %). The lengthening of the B-H bond involved in the agostic interaction is also slightly overestimated (δ ΔdB-H = 8.0 %). Similar distances are calculated when this basis set is used in the context of relativistic DKH calculations ( = 1.0 %, = 0.63 % and = 10.0 %). No dramatic change is observed between the structures partially optimized at the B3LYP/DZP (respectively B3LYP/TZP) level of theory and the structures partially optimized at the relativistic DKH2 B3LYP/DZP-DKH (respectively B3LYP/TZP-DKH) level of theory. This suggests that the relativistic effect has a limited impact on the agostic interaction in the Ti +III (1) compound.

Thus, among the functionals and the basis sets selected for our study, all the levels of theory tested lead to an accurate description of the agostic interaction in the Ti +III (1) compound, from a geometric point of view. The Ti-N, Ti-B and the lengthening of the B-H bond due to the agostic interaction are reproduced with an accuracy better than ± 0.04 Å with all the considered DFT methods. The effect of the method on the topological characterization of the interactions will be further investigated. (a) : . All the calculations were carried out with the B3LYP functional.

Topological characterization of the 3C/2e interaction

For the characterization of a 3C/2e and an estimation of the strength of an agostic interaction, the previously proposed methodology consists in studying the protonated basin. [START_REF] Zins | Activation of C-H and B-H bonds through agostic bonding: an ELF/QTAIM insight[END_REF] We shall briefly remind the reader of the main steps of this topological methodology.

First of all, the 3C/2e nature of the interactions should be ascertained. To this end, the protonated basin within the ELF topological framework should be a trisynaptic basin with a total population close to 2 electrons. Then, a QTAIM/ELF study is carried out to determine the atomic contributions of the metallic centre, the hydrogen atom and the boron atom in the present case, to the protonated basin. The strength of the agostic interaction is evaluated on the basis of the atomic contribution of the metallic centre. Thus, besides an accurate geometric characterization of the organometallic complex, the electronic description of the system obtained from DFT calculations should be precise enough prior to the topological investigation. To evaluate the accuracy of this description, the topological characterization of the protonated basin involved in the agostic interaction was studied for the Ti +III (1) and

(1) compounds:

 with all the above considered functionals in combination with the 6-311++G(2d,2p) basis set,  with all the above considered basis sets, in combination with the B3LYP functional.

a. Influence of the functional in combination with the 6-311++G(2d,2p) basis set

The influence of the choice of the functional on the topological description of the protonated basins, for the hydrogen atoms involved in the agostic interaction, was studied in the context of DFT calculations with the 6-311++G(2d,2p) basis set. The characteristic results obtained for the Ti +III (1) and

(1) compounds are summarized in the Table 3.

First of all, we would like to underline the fact that, whatever the functional used with the 6-311++G(2d,2p) basis set, the hydrogen atom involved in the agostic interaction is characterized, within the ELF topological framework, by a trisynaptic protonated basin with a total population in the 1.92 ≤ V(Ha) ≤ 1.94 electron range, for the Ti +III (1) compound. This is fully consistent with a 3C/2e interaction. Thus, the analysis of the topology of the electron density and the electron density ELF function indeed demonstrates the presence of an agostic interaction between a σ(B-H) bond and the titanium atom. The contribution of the titanium atom in this basin, noted V(Ha)-M, varies from 0.11 to 0.14 electron. The lowest value is calculated with the M-06L/6-311++G(2d,2p) level of theory, whereas the highest value is obtained with the BP86/6-311++G(2d,2p) and BP86-D3/6-311++G(2d,2p) levels of theory. With all the other functionals, the calculated contribution of the metallic centre in the protonated basin (V(Ha)-M) is calculated in the [0.12 e; 0.13 e] range for the Ti +III (1) compound, depending on the functional. With all the levels of theory presented in the Table 3, the total population of the hydrogen atom is smaller in the case of the (1) compound, and V(Ha) is calculated in the [1.88 e ; 1.90 e] range. However, as in the previous case, a trisynaptic basin is obtained, in full agreement with a 3C/2e agostic interaction. The atomic contribution of the metallic centre within this protonated basin is calculated to be in the [0.10 e; 0.12 e] range for the compound, with all the considered functionals. Thus, consistent descriptions of the agostic interactions are obtained from the topological QTAIM/ELF approach. The M06-L/6-311++G(2d,2p) level of theory is likely to slightly underestimate the strength of the agostic interaction in the Ti +III (1) compound. On the contrary, the BP86/6-311++G(2d,2p) and BP86-D3/6-311++G(2d,2p) levels of theory probably slightly overestimate the strength of the agostic interaction in the Ti +III (1) compound. Globally, the strengths of the agostic interactions in the Ti +III (1) and

(1) compounds are calculated to be very similar, in full agreement with the geometric data. It seems however that the agostic interaction may be slightly higher in the case of the Ti +III (1) compound. Indeed, if we exclude the LC-ω-PBE-D3/6-311++G(2d,2p) level of theory, the contribution of the metallic centre on the protonated basin is systematically slightly higher, by 0.01 to 0.03 electron, in the case of the Ti +III (1) compound.

V The B3LYP functional, previously selected on the basis of geometric criteria, seems to satisfactorily describe the agostic interaction of the Ti +III (1) and

(1) compounds , from an electronic, topological point of view.

b. Influence of the basis set in combination with the B3LYP functional

As in the case of the geometric study, the B3LYP functional was further selected to evaluate the influence of the basis set on the electronic characterization of the agostic interaction. The main results are presented in the Table 4.

With all the basis sets selected for this study, calculations carried out with the B3LYP functional leads to the characterization of a trisynaptic protonated basin with a total population in the [1.92 e; 1.94 e] range, for the paramagnetic Ti +III (1) compound. Thus, whatever the level of theory chosen, the topological analysis leads to the identification of a 3C/2e agostic interaction. However, the calculated contribution of the metallic centre on this protonated basin very much depends on the basis set. At the B3LYP/6-311++G(2d,2p)+LanL2DZ, B3LYP-D3/6-311++G(2d,2p)+LanL2DZ and B3LYP/6-311++G(2d,2p)+SDD levels of theory, the atomic contribution of the Ti atom in the hydrogen atom, is calculated to be in the [0.06 e; 0.07 e]. With all the other selected basis sets, i.e. without the use of a pseudopotential for the metallic atom, the atomic contribution of the Ti atom in the hydrogen atom, is calculated to be 0.12 electron. We thus conclude that the use of a relativistic pseudopotential for the metallic centre leads to an accurate description of the agostic interaction from a geometric point of view, but not from a topological point of view. The use of a basis set that takes into account the core electrons is necessary prior to any topological investigation of an agostic interaction. For a sake of computational time saving, we carried out a single point calculation with the relativistic DKH2/DZP-DKH basis set, at the geometry optimized with the B3LYP functional, in combination with the LanL2DZ basis set for the metallic atom and the 6-311++G(2d,2p) basis set for all the other atoms. When the topological study is carried out from this single point calculation, the atomic contribution of the metallic centre in the trisynaptic agostic basin is calculated to be 0.11 electron. Thus, a single point calculation using the DKH2 Hamiltonian in combination with the B3LYP/DZP-DKH level of theory from a correct geometry obtained with a pseudopotential for the metallic centre is sufficient to accurately describe the agostic interaction from a topological point of view. Since the DKH2/DZP-DKH basis set is available for all the atoms involved in the compounds selected for the present study, the following procedure will be further considered:

1. a geometry optimization with the B3LYP functional, in combination with the LanL2DZ basis set for the metallic atom and the 6-311++G(2d,2p) basis set for all the other atoms, 2. a single point calculation using the DKH2 Hamiltonian in combination with the B3LYP/DZP-DKH level of theory, 3. the generation of a .wfn file that will be further used for the topological study of the agostic interaction, 4. the ELF identification of a trisynaptic protonated basin.

We would like to point out the fact that the use of the DKH2 and RESC Hamiltonians lead to similar description of the agostic interaction in terms of geometric as well as topological descriptors, and the use of the RESC Hamiltonian is much less time consuming than the DKH2 one. Furthermore, for the present study, we can anticipate that the QTAIM topological approach is a straightforward tool for a comparison of the agostic strength, since only β-agostic interactions are considered in the present study. Such interactions are generally characterized by a bond critical point (BCP) between the hydrogen involved in the interaction and the metallic centre. For the sake of clarity, this BCP will be further denoted as BCP M-H agostic . The electron density ρ, the Laplacian of the electron density 2 ρ, the ellipticity ε, and the total energy density H, at this BCP M-H agostic can be used for an accurate description of the interaction.

V(Ha)

Atomic contribution V(Ha)-Ti V(Ha)-B V(Ha)-Ha

Characterization of the strength of the interaction a. Influence of the functional in combination with the 6-311++G(2d,2p) basis set

Before using the QTAIM approach to estimate the strength of agostic interactions, one initial point to consider is the impact of the level of theory on the accuracy of the QTAIM topological description of (i) the free B-H bonds, (ii) the agostic B-H bond and (iii) the M-H interaction. The results obtained for the Ti +III (1) compound with the selected functionals in combination with the 6-311++G(2d,2p) basis set are presented in the Table 5.

i. Influence of the functional on the QTAIM topological description of the free B-H bonds

A high electron density at the BCP, in combination with a negative 2 ρ, is a signature of a covalent bond. Thus, a correct level of theory should lead to high ρ and negative 2 ρ at the BCP of the free B-H bonds (BCP B-H free ). The results reported in the Table 5 show that the electron density and the 2 ρ at the BCP B-H free are in the [0.169; 0.174], and [-0.159; -0.270] ranges, respectively. The covalent character of the free B-H bonds is thus well reproduced with all the selected functionals for the Ti +III (1) compound, even if the Laplacian of the electron density is more sensitive to the choice of the functional than the electron density itself. More specifically, the M06-L local functional, the TPSSh hybrid functional, and the TPSSTPSS, TPSSTPSS-D3 meta-GGA functionals lead to smaller Laplacian of the electron density at the BCP B-H free , relatively to the other functional.

ii. Influence of the functional on the QTAIM topological description of the agostic B-H bond

Compared with the free B-H bond, the agostic B-H bond is characterized by lower values of ρ and 2 ρ at the BCP. The difference between the mean values obtained for the free B-H bond and the agostic B-H bond will be denoted as Δ ρ= ρ(BCP B-H free ) -ρ(BCP B-H agostic ) and Δ 2 ρ = 2 ρ(BCP B-H free ) -2 ρ(BCP B-H agostic ), respectively. Using these definitions, ρ(BCP B-H agostic ) is calculated in the [0.122 ; 0.130] range, and Δ ρ is in the [0.047 ; 0.049] range, except in the case of calculations with the M06-L functional, for which Δ ρ is calculated to be Δ ρ= 0.039 a.u. As in the case of the free B-H bond, the value of the Laplacian of the electron density at the BCP B-H agostic is more sensitive to the choice of the functional than the value of the electron density itself. Furthermore, the Laplacian 2 ρ(BCP B-H agostic ) is closed to zero, whatever the choice of the functional is. Using the M06-L local functional, the TPSSh hybrid functional, and the TPSSTPSS, TPSSTPSS-D3 meta-GGA functionals, weakly positive values are obtained, whereas the Laplacian is weakly negative where calculations are carried out with the other functionals. Furthermore, the total energy density is calculated to be in the [-0.117 ; -0.108] range. This result demonstrates that the interaction of the B-H bond with the metallic centre leads to a weakening of the B-H bond. iii.

Influence of the functional on the QTAIM topological description of the Ti-H interaction

First of all, it is worth noticing that the agostic interaction is characterized by a BCP between the metallic centre and the agostic hydrogen atom, with all the selected functionals. The weak value of the electron density at the BCP M-H agostic (ρ(BCP Ti-H agostic ) is calculated in the [0.055 ; 0.061] range), together with the positive value of the Laplacian of the electron density ( 2 ρ(BCP Ti-H agostic ) is calculated in the [0.143 ; 0.162] range) and the very small value of the energy density (H is calculated in the [-0.011 ; -0.006] range), suggest that the agostic interaction is a closed-shell (non-covalent) interaction.

As expected for an agostic interaction, the ellipticity associated with the BCP M-H agostic is substantially larger than in the case of the B-H or M-N bonds. [START_REF] Jablonski | QTAIM-Based Comparison of Agostic Bonds and Intramolecular Charge-Inverted Hydrogen Bonds[END_REF] The results summed up in the Table 6 also demonstrate that all the selected functionals lead to consistent descriptions of the interaction. The B3LYP hybrid functional was selected for a further evaluation of the effect of the basis set on the QTAIM description of the Ti +III (1) compound.

b. Influence of the basis set in combination with the B3LYP functional

The QTAIM topological characteristics of the M-H interaction, as well as the agostic B-H, free B-H and M-N bonds, obtained from calculations using each of the selected basis sets, are summed up in the Table 6.

i. Influence of the basis set on the QTAIM topological description of the free B-H bonds

When the B3LYP functional is used, the electron density and its Laplacian at the BCP B-H free , are calculated to be in the range [0.160; 0.177] and [-0.051; -0.325], respectively. These values are consistent with the description of a covalent bond. Thus, as with the selected functionals combined with the 6-311++G(2d,2p) basis set, all the selected basis sets combined with the B3LYP hybrid functional lead to a consistent description of the free B-H bond.

ii. iii.

Influence of the basis set on the QTAIM topological description of the agostic B-H bond

Influence of the basis set on the QTAIM topological description of the Ti-H interaction

A BCP was found between the metallic centre and the agostic hydrogen, with all the basis sets selected for the resent study. The electron density at the BCP is calculated to be weak, whatever the basis set used (ρ(BCP Ti-H agostic ) is calculated in the [0.056 ; 0.060] range), and the Laplacian of the electron density is calculated to be positive ( 2 ρ(BCP Ti-H agostic ) is calculated to be in the [0.135 ; 0.159] range). Furthermore, the energy density is calculated to be in the [-0.007; -0.013] range, which suggests that the agostic interaction corresponds to a weak polar shared interaction. The results reported in the Tables 5 and6 suggest that a single point calculation with the relativistic DKH2/DZP-DKH basis set, from a geometry optimized with the B3LYP functional, in combination with the LanL2DZ basis set for the metallic atom and the 6-311++G(2d,2p) basis set for all the other atoms, leads to a consistent QTAIM topological description of β B-H agostic interactions, in the case of the family of compounds herein considered. This is the reason why the following methodology was selected:

1. First of all, the geometry of the compounds were built up from the information available from the literature, and fully optimized with the B3LYP functional, in combination with the LanL2DZ basis set for the metallic atom and the 6-311++G(2d,2p) basis set for all the other, 2. Then, a single point calculation was carried out using the DKH2 Hamiltonian in combination with the B3LYP/DZP-DKH level of theory, 3. The ELF method was further used to ascertain the presence of a trisynaptic, agostic, protonated basin, 4. Finally, the M-H interaction and the agostic B-H bond were further probed and characterized thanks to their signatures within the QTAIM framework, in terms of the electron density ρ, the Laplacian of the electron density 2 ρ, the ellipticity ε, and the total energy density H, at the BCP's.

c. Influence of the nature of the substituents and the metallic centre

To further understand the impact of the nature of the metal and of the ligand on the agostic character of a σ(B-H) bond in β position, additional compounds were considered and finally 54 complexes were selected for the present study (Figure 1). We would like to underline the fact that, beyond the nine experimentally identified complexes, the existence of 45 the other structures involving a β agostic interaction has yet to be demonstrated.

To begin with, it is interesting to compare the geometries calculated at the B3LYP/6-311++G(2d,2p)+ LanL2DZ level of theory for the 54 considered species. The Cartesian coordinates are given in Supporting Information. A general feature concerns the relative positions of the M,X,B and agostic H atoms : in the case of the compounds M(1), M(2), M(4), M(5),and M(6) species, these four atoms are coplanar. The M(3) species appear as an exception to this geometrical property. Indeed, for the (3), Zr +III (3), Hf +III (3), (3) ,

(3) and (3) species, a geometric distortion occurs and the four atoms are no more coplanar.

The mono-agostic compounds were further characterized by means of the ELF and QTAIM topological approaches. The ELF analysis systematically confirms the presence of a trisynaptic basin for the agostic hydrogen atom, with a population close to 2 electrons, and with a non-negligible contribution of the metallic centre (Table S4). Finally, the strength of the β B-H agostic interaction in each species was quantitatively characterized by means of the QTAIM topological approach (Table S5). For all the considered compounds, a BCP between the M and H atoms (BCP M-H agostic ) further confirms the existence of a β B-H agostic interaction. Within a same family of compounds, a stronger interaction will be characterized by a higher electron density at the BCP M- H agostic , and a shorter M-H agostic distance. Thus, for a better comparison of the strength of the interaction, in each compound, the M-H agostic distance was plotted as a function of the ρ(BCP M-H agostic ) (Figure 2). In order to evaluate both the influence of the borane ligand and the metallic centre, two graphs were built up with the same data. The variation of the M-H agostic distance as a function of the ρ(BCP M-H agostic ) is shown for each borane ligands in the Figure 2A, and for each of the metallic centre in the Figure 2B.

M N H 2 BH 2 H M N H 2 BH 2 H H M N H 2 BH 2 H H M +III (1) (1) (1)
The figure 2A clearly suggests that the β B-H agostic interaction is systematically stronger with amine-boranes compared to phosphine-boranes. The comparison of the agostic character of the species depending on the nature of the metallic centre (Figure 2B) clearly suggests that the agostic interaction is stronger in the titanocene compounds compared with the zirconocene and hafnocene species. [START_REF] Pun | Amineborane dehydrogenation promoted by isolable zirconium sandwich, titanium sandwich and N 2 complexes[END_REF] Furthermore, the agostic interaction in the paramagnetic titanium species may be as strong as in diamagnetic species.

It is interesting to compare these theoretical results with experimental observations, based on the study of the whole dehydrogenation process (global turn-over of the reaction, identification of the products, and structural characterization of intermediates). The experimental results have unveiled several features of the amine-borane dehydrogenation:

 In a study centered on Rh-containing organometallic catalysts, Manners et al. have demonstrated that dehydrogenation reaction proceeds more efficiently amine-boranes compared to phosphine-borane, 69  The turnovers of the dehydrogenation reactions with titanocene catalysts are often reported to be bigger compared to the ones obtained with zirconocenes and hafnocenes, 68, 70  The study of the dehydrogenation reaction using a metallocene catalyst involving a paramagnetic metallic centre has proven that such centres can be as efficient as diamagnetic species. [START_REF] Helten | Paramagnetic titanium(III) and zirconium(III) metallocene complexes as precatalysts for the dehydrocoupling/dehydrogenation of amine-borane[END_REF] The close agreement between the calculated strength of the agostic bond and the experimental efficiency of the dehydrogenation processes may suggest that the agostic intermediate might have a prominent role in the reaction mechanism.

Conclusion

The aim of the present work was to evaluate the influence of different parameters on the strength of an agostic B-H interaction. A calibration study based on the Cp 2 TiNH 2 BH 3 titanocene compound was carried out to estimate the ability of several DFT levels of theories to reproduce geometric and topological properties of the agostic interaction involved in this compound. We were particularly interested in identifying the smallest level of theory leading to a consistent description of the system. A comparison with experimental (crystallographic) data has shown that the B3LYP/6-311++G(2d,2p)+LanL2DZ level of theory leads to an accurate description of the system from a geometric point of view. Further topological characterizations using the electron localization function at the same level of theory leads to an underestimation of the contribution of the metallic centre in the agostic protonated basin. Thus, a whole-electron basis set should be used for an accurate ELF characterization of the agostic interaction. For a comparative study of titanocenes, zirconocenes and hafnocenes, we thus recommend the use of the DKH relativistic Hamiltonian, in combination with the DZP-DKH basis set. In order to save computational time, a single point relativistic calculation can be carried out from a geometry optimized at the B3LYP/6-311++G(2d,2p)+LanL2DZ level of theory.

Using the B3LYP/DZP-DKH//(B3LYP/6-311++G(2d,2p)+LanL2DZ level of theory, 54 compounds of the group 4 metallocenes, involving differently substituted amine-and phosphine-boranes, were studied. In the case of the (3) compound, the optimized structure does not correspond to a mono-agostic, but to a bi-agostic species. For all the other species considered, a mono-agostic interaction was confirmed by the topological analysis. Indeed, a trisynaptic protonated basin involving ~ 2 electrons with a non-negligible contribution of the metallic centre was identified thanks to the ELF analysis. This is the signature of a 3C/2e interaction. Furthermore, using the QTAIM approach, a BCP was identified between the agostic hydrogen atom and the metallic centre in each of the compounds. The properties of this BCP M-H agostic (the electron density ρ, the Laplacian of the electron density2 ρ, the ellipticity ε, and the total energy density H) were then used to further characterize the interaction.

General trends emerge from this study of the influence of the nature of the metallic centre and the substitution of amine-and phosphine-boranes on the strength of the agostic interaction. Stronger interactions were identified in the case of amine-boranes compared with phosphine-boranes. As far as the metallic centre is concerned, stronger interactions were identified in the case of titanocenes compared with zirconocenes and hafnocenes. The agostic interactions in paramagnetic compounds can be as strong as in the case of the diamagnetic compounds.

The present study demonstrates the feasibility of a comparative study of B-H agostic interactions in a wide set of compounds, including first-, second-, and third-row metallic centres. The use of DFT calculations in combination with the ELF and QTAIM approaches allows the characterization of the 3C/2e interaction. This work constitutes a first step toward a comprehensive study of the whole reaction mechanisms involved in the dehydrogenation processes of boranes.

  2 ρ(BCP B-H agostic ) and Δ 2 ρ are calculated in the [ -0.051 ; 0.027] and [-0.249 ; -0.184] ranges, respectively.

Table 5 :

 5 Effect of the functional on the QTAIM topological description of the protonated basin for the hydrogen atom involved in the agostic interaction with the metallic centre, in the Ti +III (1) compound. All the calculations were carried out with the 6-311++G(2d,2p) basis set. All the values are given in atomic units.

  Regardless of the basis set used, the B-H bond involved in the agostic interaction is characterized by ρ(BCP B-H agostic ) is calculated in the [0.118 ; 0.128] range, and 2 ρ(BCP B-H agostic ) in the [-0.068 ; 0.097] range. The difference between the mean values obtained for the free B-H bond and the agostic B-H bond (Δ ρ= ρ(BCP B-H free ) -ρ(BCP B-H agostic ) and Δ 2 ρ = 2 ρ(BCP B-H free ) -2 ρ(BCP B-H agostic )) are in the ranges [0.042 ; 0.050] for Δ ρ and [-0.270 ; -0.148] for Δ 2 ρ. These values clearly underline the weakening of the B-H bond due to the agostic interaction. Furthermore, the B-H bond involved in the agostic interaction is also characterized by a total energy density of H(BCP B-H agostic ) in the [-0.125 ; -0.096] range. The weakly negative value of the energy density suggests that the agostic B-H bond is characterized by a weak covalent character.

Figure 1 :

 1 Figure 1: Group 4 metallocene compounds involving an agostic interaction with a σ(B-H) bond in β position that were considered for the present study. The compounds for which experimental data are available are presented in a frame.

Figure 2 :

 2 Figure 2: Variation of the M-H agostic distance as a function of the ρ(BCP M-H agostic ): A : for each borane ligands presented in the Figure 1; B: for each of the metallic centre (i.e. paramagnetic Ti, Zr and Hf, or diamagnetic (Z or E) Ti, Zr and Hf).

173/2.203 2.514/2.552 1.211/1.212 1.311/1.309 0.100/0.097 1.872/1.878 102.9/105.0 83.1/84.4 105.6/104.5 4.3 BP86 2.184/2.203 2.530/2.552 1.212/1.212 1.312/1.309 0.100/0.097 1.882/1.878 103.4/105.3 83.4/84.6 105.4/104.3 4.6 TPSSTPSS-D3 2.172/2.198 2.519/2.544 1.205/1.205 1.304/1.304 0.099/0.099 1.870/1.873 103.6/105.6 83.2/84.5 105.0/103.8 5.4 TPSSTPSS 2.181/2.207 2.532/2.567 1.205/1.206 1.305/1.306 0.100/0.100 1.878/1.882 103.9/105.9 83.4/84.7 104.9/103.7 5.1 B3PW91 2.170/2.193 2.519/2.552 1.206/1.206 1.303/1.303 0.097/0.097 1.880/1.881 103.1/105.1 83.6/84.8 105.5/104.3 6.4 PBE1PBE 2.163/2.184 2.510/2.542 1.207/1.208 1.303/1.303 0.096/0.095 1.879/1.878 102.7/104.6 83.5/84.7 105.7/104.5 7.4 TPSSh 2.174/2.197 2.526/2.559 1.204/1.205 1.302/1.304 0.098/0.099 1.877/1.878 103.7/105.6 83.5/84.8 104.9/103.7 6.5 PBE1PBE-D3 2.157/2.179 2.502/2.534 1.207/1.207 1.302/1.302 0.095/0.095 1.874/1.873 102.5/104.3 83.3/84.4 105.8/104.6 8.5 LC-ωPBE-D3 2.150/2.166 2.492/2.522 1.206/1.206 1.297/1.294 0.091/0.088 1.871/1.869 102.2/104.3 83.1/84.4 106.0/104.6 12.9 LC-ωPBE 2.155/2.171 2.501/2.531 1.206/1.206 1.297/1.295 0.091/0.089 1.877/1.875 102.4/104.4 83.2/84.6 105.9/104.5 12.5 M06 2.163/2.182 2.517/2.554 1.204/1.205 1.294/1.290 0.090/0.085 1.892/1.901 102.8/104.7 83.6/85.1 105.7/104.4 13.0 B3LYP-D3 2.171/2.195 2.524/2.558 1.202/1.202 1.295/1.294 0.093/0.092 1.883/1.885 103.6/105.7 83.6/84.9 105.1/103.9 10.9 B3LYP 2.182/2.207 2.542/2.576 1.203/1.203 1.297/1.297 0.094/0.094 1.895/1.898 104.0/105.9 84.0/85.2 104.9/103.7 10.9 M06-L 2.178/2.197 2.525/2.561 1.200/1.201 1.282/1.280 0.082/0.079 1.899/1.909 103.4/105.1 83.4/84.9 105.8/104.7 21.9 Exp 2.153 2.52 1.13 & 1.09 1.21 0.1 84.5 107.1 Table 1: Influence of the functional on the geometric description of the Ti +III (1) (in bold) and (1) (in italic) compounds. The distances are in Å, and the angles are in Degrees. (a) : , (b) : , with All the calculations were carried out with the triple-ζ 6-311++ G(2d,2p) basis set. The different functionals are classified from smaller to larger values. All the calculations were carried out with the 6-311++G(2d,2p) basis set.

	BP86-D3	2.

180/2.209 2.535/2.576 1.204/1.204 1.300/1.299 0.096 1.885/1.892

	Type of basis set	Basis set						a
		Def2-TZVP	2.182/2.207	2.540/2.574	1.204/1.205	1.299/1.299	0.095	1.894/1.895
	Alrichs' basis sets	Def2-TZVPP	2.182/2.207	2.541/2.575	1.203/1.204	1.298/1.297	0.095	1.895/1.898
		Def2-QZVPP	2.182/2.201	2.540/2.575	1.202/1.203	1.297/1.297	0.095	1.894/1.897
		6-311++G(2d,2p)	2.182/2.207	2.542/2.576	1.202/1.203	1.297/1.297	0.095	1.895/1.898
	Pople's basis sets	6-311++G(3df,3pd)	2.182/2.206	2.540/2.574	1.203/1.203	1.297/1.297	0.094	1.894/1.897
	Pople basis sets	6-311++G(2d,2p)+LanL2DZ	2.177/2.204	2.529/2.556	1.202/1.203	1.298/1.299	0.096	1.879/1.885
	with a relativistic
	pseudopotential for	6-311++G(2d,2p)+SDD	2.174/2.199	2.535/2.571	1.202/1.203	1.297/1.297	0.095	1.889/1.893
	the metallic centre
		DZP(9v)	2.176/2.203	2.540/2.581	1.219/1.219	1.312/1.310	0.093	1.894/1.899
		DKH2/DZP-DKH (9v)	2.173/2.201	2.536/2.582	1.219/1.219	1.314/1.311	0.095	1.890/1.899
	Double and triple zeta basis sets	RESC/DZP-DKH(9v)	2.174/2.201	2.537/2.582	1.219/1.219	1.314/1.311	0.095	1.891/1.899
		TZP(9v)	2.183/2.211	2.539/2.275	1.204/1.204	1.301/1.300	0.097	1.887/1.891
		DKH2/TZP-DKH (9v) 2.Exp	2.153	2.52	1.13 & 1.09	1.21	0.1

Table 2 : Effect of the basis set on the geometric description of the Ti +III (1) and (1) compounds. The distances are in Å, and the angles are in Degrees.

 2

Table 3 : Effect of the functional on the ELF topological description of the protonated basin for the hydrogen atom involved in the agostic interaction with the metallic centre, in the Ti +III (1), in bold, and (1), in italic, compounds. All the calculations were carried out with the 6-311++G(2d,2p) basis set.

 3

		(Ha)	V(Ha)-Ti	Atomic contribution V(Ha)-B	V(Ha)-Ha
	B3LYP	1.93/1.89	0.12/0.10	0.24/0.23	1.56/1.53
	B3LYP-D3	1.93/1.90	0.12/0.11	0.24/0.24	1.55/1.54
	B3PW91	1.93/1.89	0.13/0.11	0.25/0.26	1.54/1.52
	TPSSh	1.93/1.89	0.12/0.11	0.22/0.22	1.55/1.55
	TPSSTPSS	1.93/1.89	0.12/0.10	0.23/0.23	1.55/1.54
	TPSSTPSS-D3	1.93/1.89	0.13/0.11	0.23/0.22	1.55/1.54
	M06-L	1.94/1.90	0.11/0.10	0.24/0.22	1.57/1.56
	M06	1.93/1.89	0.12/0.11	0.25/0.24	1.55/1.54
	BP86	1.92/1.88	0.14/0.12	0.26/0.25	1.51/1.50
	BP86-D3	1.92/1.88	0.14/0.11	0.25/0.25	1.51/1.49
	LC-ωPBE	1.94/1.90	0.12/0.11	0.24/0.23	1.56/1.53
	LC-ωPBE-D3	1.94/1.89	0.12/0.12	0.23/0.22	1.56/1.53
	PBE1PBE	1.93/1.89	0.13/0.11	0.24/0.23	1.55/1.52
	PBE1PBE-D3	1.94/1.89	0.13/0.11	0.24/0.23	1.55/1.52

93/1.88 0.12/0.10 0.23/0.23 1.56/1.54 Def2-TZVP 1.92/1.88 0.12/0.10 0.23/0.22 1.56/1.54 Def2-TZVPP 1.92/1.89 0.12/0.10 0.23/0.22 1.56/1.54

	6-311++G(2d,2p)+LanL2DZ	1.92/1.89	0.06/0.06	0.23/0.23	1.61/1.58
	6-311++G(2d,2p)+SDD	1.93/1.89	0.06/0.07	0.24/0.23	1.61/1.58
	6-311++G(2d,2p)	1.93/1.89	0.12/0.10	0.24/0.23	1.561.53
	6-311++G(3df,3pd) 1.Def2-TZVPPD	/1.89	/0.11	/0.22	/1.54
	Def2-QZVPP	1.92/1.89	0.12/0.10	0.23/0.23	1.57/1.57
	DKH2/DZP-DKH (9v)	1.94/1.89	0.12/0.10	0.21/0.19	1.58/1.57
	RESC/DZP-DKH(9v)	1.94/1.89	0.12/0.10	0.24/0.22	1.58/1.57
	DKH2/TZP-DKH (9v)	1.94/1.89	0.12/0.10	0.23/0.21	1.56/1.55
	DZP(9v)	1.94/1.89	0.12/0.10	0.22/0.20	1.58/156
	TZP(9v)	1.94/1.89	0.12/0.10	0.23/0.21	1.58/1.55
	6-311++G(2d,2p)+LanL2DZ
	+ single point with	1.93/1.89	0.11/0.10	0.21/0.21	1.59/1.57
	DKH2/DZP-DKH

Table 4 : Effect of the basis set on the ELF topological description of the protonated basin for the hydrogen atom involved in the agostic interaction with the metallic centre, in the Ti

 4

+III

(1), in bold, and

(1), in italic, compounds.

All the calculations were carried out with the B3LYP functional. All the values are given in e.

Table 6 : Effect of the basis set on the QTAIM topological description of the protonated basin for the hydrogen atom involved in the agostic interaction with the metallic centre, in the Ti +III (1) compound. All the calculations were carried out with the B3LYP functional. All the values are given in atomic units.

 6

		BCP Ti-H agostic				BCP B-H agostic			BCP B-H free			BCP Ti-N
		ρ	2 ρ	ε	H	ρ	2 ρ	ε	H	ρ	2 ρ	ε	ρ	2 ρ	ε
	6-311++G(2d,2p)+LanL2DZ 0.059 0.154 0.482 -0.013	0.125 -0.006	0.162 -0.112 0.174	-0.259	0.044	0.068	0.240 0.301
	6-311++ G(2d,2p)+SDD	0.058 0.135 0.457 -0.012	0.125 -0.007	0.158 -0.112 0.174	-0.260	0.045	0.070	0.217 0.302
	6-311++G(2d,2p)	0.057 0.150 0.436 -0.009	0.125 -0.014	0.149 -0.112 0.174	-0.259	0.044	0.068	0.229 0.293
	6-311++G(3df,3pd)	0.058 0.146 0.452 -0.010	0.127 -0.045	0.127 -0.117 0.176	-0.315	0.040	0.069	0.219 0.279
	Def2-TZVP	0.056 0.157 0.426 -0.007	0.126 -0.011	0.158 -0.118 0.174	-0.257	0.049	0.068	0.230 0.279
	Def2-TZVPP	0.057 0.157 0.440 -0.007	0.126 -0.014	0.147 -0.119 0.175	-0.253	0.045	0.068	0.229 0.278
	Def2-QZVPP	0.058 0.138 0.435 -0.012	0.128 -0.068	0.123 -0.125 0.177	-0.325	0.041	0.069	0.220 0.285
	DKH2/DZP-DKH (9v)	0.059 0.140 0.429 -0.013	0.118	0.074	0.149 -0.096 0.160	-0.080	0.055	0.070	0.224 0.303
	RESC/DZP-DKH (9v)	0.059 0.140 0.429 -0.013	0.118	0.074	0.150 -0.096 0.160	-0.079	0.052	0.070	0.223 0.302
	DKH2/TZP-DKH (9v)	0.058 0.156 0.436 -0.009	0.126 -0.045	0.138 -0.125 0.176	-0.287	0.046	0.069	0.225 0.280
	DZP(9v)	0.058 0.138 0.433 -0.012	0.118	0.068	0.151 -0.096 0.160	-0.086	0.054	0.070	0.222 0.299
	TZP(9v)	0.058 0.156 0.420 -0.009	0.126 -0.038	0.140 -0.125 0.176	-0.280	0.045	0.068	0.224 0.277
	6-311++G(2d,2p)+LanL2DZ
	+ single point with	0.060 0.145 0.420 -0.013	0.121	0.097	0.157 -0.098 0.164	-0.051	0.056	0.069	0.222 0.307
	DKH2/DZP-DKH

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Int Ed 49: 7170-7179


Sophie Tarbouriech

Alexandre Seuret
email: aseuret@laas.fr

Luciano G Moreira
email: luciano.moreira@ufrgs.br

João-Manoel Gomes Da Silva

João-Manoel Gomes
email: jmgomes@ufrgs.br

Observer-based event-triggered control for linear systems subject to cone-bounded nonlinearities

Keywords: Event-triggered control, cone-bounded nonlinearities, observer, stability, LMI

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

Observer-based event-triggered control for linear systems subject to cone-bounded nonlinearities S. Tarbouriech * A. Seuret * L. G. Moreira * * J. M. Gomes da Silva Jr * *

INTRODUCTION

In the context of the implementation of modern control systems through digital communication networks, event-triggering strategies have been proposed mainly to deal with communication, energy consumption and computational constraints (see, for example, [START_REF] Postoyan | Triggering mechanism using freely selected sensors for linear time-invariant systems[END_REF]; Postoyan et al. (2015); [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]; [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF] and references therein.) In the event-triggered control framework, one can basically consider two approaches. The first one, which corresponds to the so-called emulation problem, considers that the controller is given a priori (see, for example, [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF], [START_REF] Wang | Event design in eventtriggered feedback control systems[END_REF], Postoyan et al. (2015), [START_REF] Seuret | Event-triggered sampling algorithms based on a Lyapunov function[END_REF] [START_REF] Abdelrahim | Stabilization of nonlinear systems using eventtriggered output feedback controllers[END_REF] and references therein). In this case, the focus is dedicated to the design of the event-triggering rule or function that leads to a stable closed-loop behavior while avoiding the occurrence of Zeno phenomenon. The second approach, which is referred as a codesign problem, performs the design of both the controller and the event-triggering rule, simultaneously, and is addressed in a few papers [START_REF] Aström | Event based control[END_REF]; [START_REF] Seuret | Event-triggered sampling for linear closed-loop systems with plant input saturation[END_REF][START_REF] Seuret | LQ-based event-triggered controller co-design for saturated linear systems[END_REF], [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF]; [START_REF] Abdelrahim | Co-design of output feedback laws and eventtriggering conditions for linear systems[END_REF]; [START_REF] Antunes | Dynamic programming formulation of periodic event-triggered control: Performance guarantees and co-design[END_REF]).

A large amount of papers addressing asymptotic stability of event-trigger control schemes consider state-feedback laws (see for instance [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]; [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF]; [START_REF] Lehmann | Extension and experimental evaluation of an event-based state-feedback approach[END_REF]; [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF]; [START_REF] Aranda-Escolastico | A novel approach for periodic event-triggering based on general quadratic functions[END_REF]; [START_REF] Wang | Event design in eventtriggered feedback control systems[END_REF]; [START_REF] Moreira | Event-triggered state-feedback control for continuous-time plants subject to input saturation[END_REF]). In this case, it is in general easier to prove stability and that Zeno behavior does not happen. However, in most practical applications the full state measurement is not available or even possible. The fact of using available information (measured or local signals) in order to build the event-triggered control law, from an emulation or a co-design point of view is a challenging problem: see, for example, Lehmann and Lunze (2011a), [START_REF] Almeida | Observer based self-triggered control of an acyclic interconection linear plants[END_REF], [START_REF] Abdelrahim | Stabilization of nonlinear systems using eventtriggered output feedback controllers[END_REF], [START_REF] Donkers | Outputbased event-triggered control with guaranteed L ∞ -gain and improved and decentralised event-triggering[END_REF], [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF], Postoyan et al. (2015), [START_REF] Wang | Event-triggering in distributed networked control systems[END_REF], [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF], [START_REF] Xia | Observer based event-triggered control for certain and uncertain linear systems[END_REF] and references therein. On the other hand, we can also notice that most of the works in the literature consider linear plants. Some papers dealing with generic results for nonlinear systems are [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], Postoyan et al. (2015), [START_REF] Abdelrahim | Stabilization of nonlinear systems using eventtriggered output feedback controllers[END_REF].

In the current paper we focus on the emulation design of eventtriggered control for a particular class of nonlinear systems, which regards linear plants subject to input cone-bounded nonlinearities. This type of input nonlinearities can model, for instance, practical actuators nonlinearities or imperfections (e.g. saturation, deadzone, hysteresis, etc), as well as implementation control constraints such as quantization effects. Moreover, we are concerned by the use of only local information or measurable signals. To do so, a state observer-based approach is considered. In this context, we follow the same vein as in [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF] to tackle the design of the eventtriggering strategy for observer-based state feedback based on the decrease of a Lyapunov function. Recall that in the eventtriggered control context, the plant evolves in continuous time, whereas the control signal is updated depending on discretetime events. Then, the resulting closed-loop system can be cast as a hybrid or impulsive system. Instead of considering the classical hybrid framework to study mixed continuous and discrete dynamics as defined in [START_REF] Goebel | Hybrid Dynamical Systems. Modeling, Stability and Robustness[END_REF], we use an alternative direction as proposed in [START_REF] Tarbouriech | Observer-based event-triggered control codesign for linear systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. In this case, sufficient conditions based on linear matrix inequalities (LMI) are proposed to ensure the global asymptotic stability of the origin of the nonlinear closedloop system under the event-trigger control strategy. Moreover, following the idea presented in [START_REF] Mazo | An ISS selftriggered implementation of linear controllers[END_REF], Zeno behaviors are avoided thanks to the introduction of a minimum dwell-time, which is explicitly forced as a design parameter of the LMI conditions. These conditions are then cast in convex optimization problems to compute the parameters of the eventtrigger rule aiming at reducing the number of control updates. The paper can then be considered as a comprehensive version of [START_REF] Tarbouriech | Observer-based event-triggered control codesign for linear systems[END_REF], since we extend the class of systems under consideration to some particular class of nonlinear systems. The originality of the paper relies on the fact that the approach is based on the satisfaction of a continuoustime stability criterion as well as a discrete-time one, in which a minimum dwell-time is a direct tunable parameter.

The paper is organized as follows. In Section 2, the system under consideration is described, together with the problem we intend to solve. Conditions to solve the event-triggered control problem in an emulation context are proposed in Section 3 first in a generic format and second through linear matrix inequalities, in which the dwell-time appears as a tuning parameter. Section 4 provides a simple optimization criterion, in order to cope with the implicit objective of reducing the number of control updates by playing on the event-trigger rule parameters. In Section 5, an example illustrates the efficiency of the proposed approach. Finally, Section 6 presents some concluding remarks and directions for potential future works.

Notation. Z, R n , R n×n denote respectively the sets of integers, n-dimensional vectors and n × m matrices. For any matrix A, A denotes its transpose and He{A} = A + A . For any square matrix A, trace(A) denotes its trace. For two symmetric matrices of the same dimensions, A and B, A > B means that A -B is symmetric positive definite. I and 0 stand respectively for the identity and the null matrix of appropriate dimensions. For a partitioned matrix, the symbol * stands for symmetric blocks. . stands for the Euclidean norm.

PROBLEM STATEMENT

System data

Consider the following continuous-time plant:

ẋp (t) = A p x p (t) + B p u(t) + B p f f (u(t)), y p (t) = C p x p (t) + D p f f (u(t)), (1)
where x p (t) ∈ R n , u(t) ∈ R m , y p (t) ∈ R p are the state, the input and the output of the plant, respectively. Matrices A p , B p , B p f , C p and D p f are constant and of appropriate dimensions. Pairs (A p , B p ) and (C p , A p ) are supposed to be controllable and observable, respectively. Function f (u) : R m → R l is a nonlinearity affecting the input u. It can represent a neglected nonlinearity, uncertainty or imperfection affecting the input [START_REF] Johansson | Observer-based strict positive real (SPR) feedback control system design[END_REF]). In other words, f (u) allows to represent the error with respect to a linear input. For example, in the case of magnitude saturation of the input, f (u) corresponds to the deadzone sat(u)u (see, for example [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], [START_REF] Zaccarian | Modern anti-windup synthesis[END_REF], [START_REF] Turner | A non-square sector condition and its application in deferred-action anti-windup compensator design[END_REF]). Another interesting case, which can be described through f (u), consists in capturing the fact that only a part of the input is saturated whereas the other part remains linear (see, for example, Tarbouriech et al. (2016a)). One can also cite the case of linear systems in the presence of some other isolated nonlinearities, such as quantization or backlash, which can also be represented by equation (1) (see, for example, [START_REF] Liberzon | Switching in Systems and Control[END_REF], [START_REF] Tarbouriech | Stability analysis and stabilization of systems with input backlash[END_REF], [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF] and references therein). This nonlinearity is therefore supposed to be continuous and to satisfy a cone-bounded property (see, for example, [START_REF] Khalil | Nonlinear systems[END_REF], [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF]). That means that this nonlinearity satisfies the following property f (u) S( f (u) + Ru) ≤ 0 (2) where S ∈ R l×l is any diagonal positive definite matrix. Matrix R ∈ R l×m is supposed to be a diagonal positive matrix that is fixed by the designer and depends on the nonlinearity characteristics. Moreover, it is assumed that Property (2) is globally satisfied, i.e. it is valid for all u ∈ R m .

In this paper, we consider an observer-based state feedback controller to stabilize the plant (1) and therefore to drive the output to zero. This controller is defined by:

   ẋ(t) = A p x(t) + B p u(t) + B p f f (u(t)) -L(y p (t) -ŷ(t)), ŷ(t) = C p x(t) + D p f f (u(t)), u(t) = K x(t),
(3) where x(t) ∈ R n and ŷ(t) ∈ R p are the state and the output of the observer, respectively. Furthermore, L ∈ R n×p and K ∈ R m×n are the observer and controller gains, respectively. By considering the continuous-time system described by (3), the control design is carried out according to the separation principle. The observer gain L is then designed to make A p + LC p Hurwitz. In this case, the state estimation error dynamics is given by ė(t) = (A p + LC p )e(t), (4) where e(t) = x p (t)x(t) ∈ R n , is globally asymptotically stable, i.e. lim t→∞ e(t) = 0. Consequently, the estimation output error e y (t) = C p e(t) also asymptotically converges to zero, i.e. lim t→∞ e y (t) = 0. On the other hand, the dynamics of the observer with the state feedback controller u(t) = K x(t) is given by: ẋ

(t) = (A p + B p K) x(t) + B p f f (K x(t)) -Le y (t). (5)
In this paper, we want to address the problem of an aperiodic event-triggered-based sampled-data implementation of such an observer-based controller. This paper can be viewed as an extension of the technique developed in [START_REF] Tarbouriech | Observer-based event-triggered control codesign for linear systems[END_REF] to the particular class of nonlinear systems represented by equation (1).

Sampled-data control implementation

We are interested in studying the way to implement the control input u. We consider that u is not continuously implemented but is updated at certain instants {t k } k∈N , which form a sequence of strictly increasing positive scalars. The control action is supposed to be held constant between two successive sampling instants (t k and t k+1 ) through a zero order holder. Nevertheless, differently from classical digital control approaches, the sampling interval t k+1t k is not assumed to be constant but can be seen as an additional control action.

Thus, the closed-loop system can be represented by

   ẋp (t) = A p x p (t) + B p u(t k ) + B p f f (u(t k )), ẋ(t) = A p x(t) + B p u(t k ) + B p f f (u(t k )) -Le y (t), u(t k ) = K x(t k ), ∀t ∈ [t k , t k+1 ). (6)
Taking into account (4), performing a change of coordinates, the closed-loop dynamics (6) can be expressed in terms of x and e, as follows:

   ẋ(t) = (A p + B p K) x(t) + B p δ(t) +B p f f (K x(t) + δ(t)) -Le y (t), ė(t) = (A p + LC p )e(t), (7)
where we use the same formulation as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] to define δ(t):

δ(t) = u(t k ) -u(t) = K( x(t k ) -x(t)). ( 9
)
This function δ(t) can be seen as a measure of the difference between the fictive continuous-time control value and its sampled and held version, which is currently implemented. Note that δ(t) depends only on the observer variables and is therefore available at the controller node.

Problem formulation

This work focuses on the event-triggered implementation of the controller represented by (3). Then, an event generator algorithm has to be included in the controller to decide whether or not the control input has to be updated, based on the available information. Adapting the event-triggered control strategy proposed in Tallapragada andChopra (2012, 2013), the sampling instants are determined from the following logic:

t k+1 = min{t ≥ t k + T, s.t. g(δ(t), y a (t)) ≥ 0}.
(10) where y a represents the vector of available information to the controller (which corresponds in our case to y a (t) = [ x(t) , e y (t) ] ) and the function g : R m × R n+p → R has to be efficiently defined such that the asymptotic stability of the closed-loop system (7) under the event-triggered rule described in (10) is ensured. Moreover, note that the trigger criterion given by ( 10) ensures that the next sampling time will occur at least T time units ahead the last one. Thus, T can be seen as a minimal dwell-time, which allows to prevent Zeno solutions. For t ≥ t k + T the control is not be updated until g(δ(t), y a (t)) ≥ 0.

EVENT-TRIGGERED STRATEGY DESIGN

Our aim is to provide an event-triggered strategy to update the control signal applied to the plant based solely on available signals, that is, using only u(t), f (u(t)), x(t) and y p (t). According to (10), this corresponds to design T and g in order to ensure the asymptotic stability of the sampled-data system (6). To do so, we first provide a general formulation, inspired from Tallapragada and Chopra (2012), [START_REF] Tallapragada | On event triggered tracking for nonlinear systems[END_REF], which corresponds to extend Theorem 1 of [START_REF] Tarbouriech | Observer-based event-triggered control codesign for linear systems[END_REF]. With this aim, consider the augmented state x = x e ∈ R 2n . Theorem 1. Consider a positive scalar T , a function g : R m × R n+p → R and the triggering rule

t k+1 = min{t ≥ t k + T, s.t. g(δ(t), y a (t)) ≥ 0} (11) with y a (t) = x(t) e y (t) = I 0 0 C p x(t).
Consider a positive definite function V , for which there exist two positive scalars ε 1 and ε 2 such that, for any x ∈ R 2n , we have

ε 1 x 2 ≤ V (x) ≤ ε 2 x 2 , ( 12
) and assume, in addition, that the function V (x) satisfies, for all t ∈ [t k + T , t k+1 ) and for all k

∈ N V (x(t)) -g(δ(t), y a (t)) -2 f (u(t k )) S( f (u(t k )) + Ru(t k )) < 0, (13)
and, for all k ∈ N

∆V T (x) -2 f (u(t k )) S( f (u(t k )) + Ru(t k )) < 0, (14) with ∆V T (x) = V (x(t k + T )) -V (x(t k )).
Then, the origin of system (7) with the triggering rule ( 11) is globally asymptotically stable and the inter-sampling intervals are lower bounded by T .

Proof. The proof is omitted due to space limitations but can be obtained from the authors.

2

From Theorem 1, a way to design the event-triggered rule using only the available signals is stated below. Theorem 2. Given controller and observer gains K and L and a positive scalar T > 0, suppose that there exist symmetric positive definite matrices P 1 , P 3 , Q ε , Q δ and a matrix P 2 of appropriate dimensions such that the matrix inequalities

Φ 1 =      M    I 0 0 C p 0 0 0 0    * -Q ε      < 0, ( 15
)
Φ 2 =      - P 1 P 2 P 2 P 3 -K R S 0
Λ 1 (T )

P 1 P 2 P 2 P 3 * -2S Λ 2 (T ) P 1 P 2 P 2 P 3 * * - P 1 P 2 P 2 P 3      < 0, ( 16
)
are verified with matrix M is given in ( 8) and

Λ 1 (T ) = e A p -LC p 0 A p +LC p T + T 0 e A p -LC p 0 A p +LC p s ds B p K 0 0 0 Λ 2 (T ) = T 0 e A p -LC p 0 A p +LC p s ds B p f 0 .
(17) Then, the event-triggered sampling rule defined by (11) with

g δ(t), x(t) e y (t) = δ(t) Q δ δ(t) - x(t) e y (t) Q -1 ε x(t) e y (t) (18
) is such that the origin of system ( 7) is globally asymptotically stable. Furthermore, the inter-sampling times are lower bounded by T .

Proof. As for Theorem 1, the proof was removed from reason of place but can be obtained from the authors.

2 Remark 3. Note that Zeno behaviors are avoided from the definition of the event-triggered rule (11) because the intersampling times are lower bounded by the positive scalar T . Actually, T represents a dwell-time implying that there is a minimum time T between two updates.

OPTIMIZATION ISSUES

Let us point out that, differently from [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF], in Theorem 2 the inter-sampling times are directly obtained via the satisfaction of ( 16) without the need of additional a posteriori calculations. Furthermore, if the condition (15) holds then it is always possible to find a small enough T such that conditions ( 16) is verified. The parameter T appears then as a tuning parameter of the event-trigger problem. If T is too large, the conditions may be not feasible. Since a large T can lead to a performance degradation with respect to the

M =    He{P 1 (A p + B p K)} (A p + B p K) P 2 -P 1 LC p + P 2 (A p + LC p ) P 1 B p P 1 B p f -K R S * He{P 3 (A p + LC p ) -P 2 LC p } P 2 B p P 2 B p f * * -Q δ -R S * * * -2S    . (8)
continuous-time implementation, a classical trade-off has to be considered when choosing T . Actually, it is important to note that provided that matrices A p +B p K and A p +LC p are Hurwitz, the existence of a small enough positive scalar T such that the conditions of Theorem 2 are feasible is ensured.

Conditions in Theorem 2 are LMIs provided that K, L and T are fixed, as typically considered in an emulation context. In order to select gains K and L, classical design techniques may be used, possibly considering some performance criteria.

In particular, the observer gain L is chosen to get A p + LC p Hurwitz with fast enough eigenvalues.

Hence, the selection of the parameters Q ε and Q δ can be systematically performed through the following convex optimization problem (given K, L and T > 0): min

P 1 ,P 2 ,P 3 ,Q ε ,Q δ ,S trace(Q δ ) + trace(Q ε ).
s.t. (15), ( 16).

(

) 19
The idea behind the optimization criterion in ( 19) is to get Q δ and Q ε as "small" as possible. With the view of the event triggering rule (18), and condition (15), this also means that the matrix Q δ > 0 is minimized as well as the matrix Q -1 ε > 0 is maximized. Since the control input is updated whenever the function g in ( 18) is positive, this optimization procedure aims at reducing the impact of the first positive contribution in g (Q δ > 0) over the second negative contribution (-Q ε < 0).

Let us emphasize that the dwell-time T is also a design parameter and its role is connected to the expected average sampling rate of the event triggered implementation. It can then be interesting to seek for maximizing the value of T through problem (19) by iteratively increasing T and testing LMI conditions. The implicit goal of such a convex optimization problem proposed is to reduce as much as possible the occurrences of sampling. This aspect is illustrated in Section 5.

NUMERICAL EXAMPLE

System description

Let us consider the following plant:

   ẋp (t) = 0 1 4 0 x p (t) + 0 1 q(u(t)), y p (t) = [1 0] x p (t), (20)
where q(u) is a logarithmic quantization function defined as follows [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF]):

q(u) =        1 ρ j q if 1 ρ j q (1 + η q ) ≤ u < 1 ρ j q (1 -η q ) , j ∈ Z -q(-u) if u < 0 with η q = 1 -ρ q 1 + ρ q , 0 < ρ q < 1
It should be noticed that the quantization error q(u) = q(u)u is restricted to the cone defined by ±η q u (i.e., it satisfies the relation ( q(u) + η q u)( q(u)η q u) < 0) (de Souza et al. ( 2010)).

In order to cast the system in the form (1), with a function f (u) satisfying ( 2), it suffices to consider:

f (u) = q(u) -η q u; A p = 0 1 4 0 ; B p f = 0 1 ; B p = (1 + η q )B p f ; C p = [1 0] ; D p f = 0.
Note that in this case the relation ( q(u) + η q u)( q(u)η q u) < 0 becomes f (u)( f (u) + 2η q u) < 0, i.e., ( 2) is verified with R = 2η q .

Optimization results

We consider the quantization with ρ q = 0.9 (which leads to R = 2η q = 0.105263), the state feedback gain matrix K = [-6 -3], the observer gain matrix L = [-3.5 -7] and the dwell-time T = 0.2. Then, solving the optimization problem (19) with this data, we obtain:

Q ε =   0.452633 1.10763 • 10 -5 -2.19703 • 10 -9 1.10763 • 10 -5 0.275705 1.82799 • 10 -10 -2.19703 • 10 -9 1.82799 • 10 -10 1.51132 • 10 -5   , Q δ = 0.728379.

Influence of T

Now, to show the influence of the parameter T , we solve optimization problem (19) for some values of T in the interval [0.1, 0.7]. Table 1 shows the average number of updates of the control signal obtained for each T chosen, considering 100 different initial conditions and t ∈ [0, 10]. The variation on the number of updates is approximately 12%. A value of T ≥ 0.8 renders optimization problem (19) unfeasible.

Some simulations

We consider several values of T to depict the evolution of the plant and observer states, of the control input and of the sampling instants with the event-triggering rule (10) obtained by Theorem 2. To do this we consider the following initial conditions for the plant and the observer:

x p (0) = -5 0 ; x(0) = 0 0 (21)
Fig. 1a) shows three simulations considering these values for the trigger function and initial conditions defined as in ( 21). The top plots depicts the plant and observer states. It can be noticed that the observer states quickly converge to the plant states, as expected. The control signal is shown in the middle plot. Note that the control signal is effectively held constant between two events. Moreover, the value of the control is subject to a logarithmic quantization. The bottom plot depicts the sampling instants, with the sizes of the bars representing the inter-event times, i.e. the difference between the time of that event and of the previous one. The dashed line marks the dwell-time.

One can see that the trigger strategy is effectively delaying the sampling events, yielding inter-sampling times larger than the dwell-time while ensuring the asymptotic stability of the closed-loop system.

Figs. 1b) and 1c) show simulations for T = 0.4 and T = 0.7, respectively, both considering initial conditions defined as in (21). One can notice that, as T increases, more the inter-sample times are equal to the value of T , i.e. the event-triggered strategy cannot delay the sampling events so effectively anymore.

For T = 0.7 and the initial condition considered, all the intersample times are equal to T . As mentioned before, this value of T is close to the limit of feasibility of optimization problem (19).

CONCLUSION

The paper handled an observer-based event-triggered control strategy for linear systems subject to cone-bounded nonlinearities affecting the input. The developed technique used only available variables. Sufficient conditions based on linear matrix inequalities (LMI) associated to convex optimization problem have been proposed to ensure the asymptotic stability of the closed loop in an emulation context. Moreover, the proposed approach allowed to design the event-triggering rule with a parameter T imposing the inter-sampling times.

These preliminary results let some open issues. In particular, it could be interesting to study the co-design problem, that is the simultaneous design of the control gains (K and L) and the triggering rule (i.e. matrices Q δ and Q ε ). Furthermore, it could be pertinent to study the presence of cone-bounded nonlinearities depending on the state, imposing to change the route used to derive the conditions.

  Fig. 1. Figure representing the state of the plant and the observer, the control input u, and the sampling instants issued from Theorem 2 with T = 0.2 (left), T = 0.4 (middle) and T = 0.7 (right).

  This work was supported by ANR project LimICoS contract number 12 BS03 005 01. L. Moreira is supported by CAPES (PhD scholarship) and J.M. Gomes da Silva Jr. is partially supported by CNPQ (grant PQ 305979/2015-9), Brazil.