Ahmad Naïm

Yacine Bouhadja

Miguel Cortijo

Elen Duverger-Nédellec

Howard D Flack

Eric Freysz

Philippe Guionneau

Antonio Iazzolino

Amine Ould Hamouda

Patrick Rosa

Ahmad Naim

Olaf Stef- Ańczyk

Ángela Valentín-Pérez

Mehdi Zeggar

Design and study of structural linear and nonlinear optical properties of Chiral [Fe(phen)3] 2+ complexes

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

Introduction

Photophysical properties of Fe(II) complexes with polyimine ligands, such as [Fe(bipy)3] 2+ , [Fe(phen)3] 2+ , or [Fe(terpy)2] 2+ , where bipy = 2,2'-bipyridine, phen = 9,10-phenantroline and terpy = 2,2':6',2"-terpyridine, have been the subject of intense scrutiny recently. [START_REF] Zhang | Tracking Excited-State Charge and Spin Dynamics in Iron Coordination Complexes[END_REF][START_REF] Sousa | Ultrafast Deactivation Mechanism of the Excited Singlet in the Light-Induced Spin Crossover of [Fe(2,2-Bipyridine)3]2+[END_REF][START_REF] Papai | Theoretical Investigation of the Electronic Structure of Fe ( II ) Complexes at Spin-State Transitions[END_REF][START_REF] Vankó | Detailed Characterization of a Nanosecond-Lived Excited State: X-Ray and Theoretical Investigation of the Quintet State in Photoexcited [Fe(Terpy)2]2+[END_REF][START_REF] Tribollet | Transient Absorption Spectroscopy of the Iron(II) [Fe(Phen)3]2+complex: Study of the Non-Radiative Relaxation of an Isolated Iron(II) Complex[END_REF][START_REF] Gallé | Transient Absorption Spectroscopy of the [Fe(2 CH3-Phen)3]2+complex: Study of the High Spin ↔ Low Spin Relaxation of an Isolated Iron(II) Complex[END_REF] These studies are of course of fundamental interest for the understanding of photophysics in 3d transition metal complexes, but have also application in the understanding of optical writing/magnetic reading in spin crossover (SCO) compounds. [START_REF] Létard | Towards Spin Crossover Applications[END_REF] Moreover such complexes were thought as potential substitutes for Grätzel-type dye-sensitized photovoltaic cells. [START_REF] Monat | Femtosecond Excited-State Dynamics of an Iron(II) Polypyridyl Solar Cell Sensitizer Model[END_REF] Indeed these diamagnetic low-spin d [START_REF] Gallé | Transient Absorption Spectroscopy of the [Fe(2 CH3-Phen)3]2+complex: Study of the High Spin ↔ Low Spin Relaxation of an Isolated Iron(II) Complex[END_REF] diimine complexes are considered as models for Fe(II) SCO complexes which quite commonly present N6 coordination spheres based on heteroaromatic ligands. Comparatively, SCO complexes usually show much smaller zero-point energy difference ∆HHL (≈1000 cm -1 ) between the high-spin (HS) configuration, for low ligand-field (LF) strength, and the low-spin (LS) configuration, for high LF strength. [START_REF] Hauser | Ligand Field Theoretical Considerations[END_REF][START_REF] Gütlich | Spin Crossover-An Overall Perspective[END_REF] It is known for SCO complexes that in the solid state the low temperature LS ground state may be photoconverted to a long-lived metastable HS state, the so-called Light-Induced Excited Spin State Trapping (LIESST) effect. [START_REF] Decurtins | Light-Induced Excited Spin State Trapping in a Transition-Metal Complex: The Hexa-1-Propyltetrazole-Iron (II) Tetrafluoroborate Spin-Crossover System[END_REF][START_REF] Hauser | Intersystem Crossing in the[END_REF] It was shown even earlier that such photoconversion could take place in solution at about room temperature using pulsed irradiation with corresponding fast relaxations. 13 Fe(II) LS polyimine complexes, while having much higher ∆HHL (≈6000 cm -1 ), show nevertheless the same conversion to a metastable HS state by ultrafast optical pumping, with even faster corresponding relaxations. [START_REF] Zhang | Tracking Excited-State Charge and Spin Dynamics in Iron Coordination Complexes[END_REF][START_REF] Vankó | Detailed Characterization of a Nanosecond-Lived Excited State: X-Ray and Theoretical Investigation of the Quintet State in Photoexcited [Fe(Terpy)2]2+[END_REF][START_REF] Gallé | Transient Absorption Spectroscopy of the [Fe(2 CH3-Phen)3]2+complex: Study of the High Spin ↔ Low Spin Relaxation of an Isolated Iron(II) Complex[END_REF][START_REF] Lemke | Coherent Structural Trapping through Wave Packet Dispersion during Photoinduced Spin State Switching[END_REF][START_REF] Zerdane | Comparison of Structural Dynamics and Coherence of D-d and MLCT Light-Induced Spin State Trapping[END_REF][START_REF] Sousa | Theoretical Evidence for the Direct 3 MLCT-HS Deactivation in the Light-Induced Spin Crossover of Fe( Ii )-polypyridyl Complexes[END_REF][START_REF] Field | Spectral Signatures of Ultrafast Spin Crossover in Single Crystal [FeII(Bpy)3](PF6)2[END_REF][START_REF] Auböck | Sub-50-Fs Photoinduced Spin Crossover in [Fe(Bpy)3] 2+[END_REF][START_REF] Brady | Time-Resolved Relaxation Studies of Spin Crossover Systems in Solution[END_REF][START_REF] Juban | Ultrafast Dynamics of Ligand-Field Excited States[END_REF][START_REF] Gawelda | Structural Analysis of Ultrafast Extended X-Ray Absorption Fine Structure with Subpicometer Spatial Resolution: Application to Spin Crossover Complexes[END_REF][START_REF] Gawelda | Structural Determination of a Short-Lived Excited Iron(II) Complex by Picosecond X-Ray Absorption Spectroscopy[END_REF][START_REF] Gawelda | Ultrafast Nonadiabatic Dynamics of [Fe II (Bpy) 3 ] 2+ in Solution[END_REF][START_REF] Consani | Vibrational Coherences and Relaxation in the High-Spin State of Aqueous [FeIIbpy3]2[END_REF][START_REF] Bressler | Femtosecond XANES Study of the Light-Induced Spin Crossover Dynamics in an Iron(II) Complex[END_REF] While these compounds exhibit many interesting features for optical applications, their use in optics is still limited due to the centrosymmetry that most of Fe(II) complexes with polyimine ligands usually adopt in solid state. This centrosymmetry prevents these compounds from exhibiting interesting properties such as piezoelectric and electro-optic effects, as well as second order non-linear optical phenomena giving rise to second harmonic and sum frequency generation. To further extend the application of Fe(II) complexes, it is necessary to break their natural centrosymmetry. One strong point of molecular materials is the endless chemical playground they represent for grafting new properties. Notably, we have been investigating the combination of SCO and chirality for some years now, [START_REF] Naim | Rational Design, Synthesis and Characterization of Chiral Spin Crossover Compounds[END_REF] a combination sometimes obtained by serendipity [START_REF] Kumar | Emerging Trends in Spin Crossover (SCO) Based Functional Materials and Devices[END_REF] and obtained rationally only using optically active ligands. 27- 34 Due to their extensively studied photophysics, the model complexes [Fe(bipy)3] 2+ , [Fe(phen)3] 2+ , or [Fe(terpy)2] 2+ were evident targets for attractive low-cost non-centrosymmetric materials showing non-linear optical properties, and for the study of their eventual dependence with the spin state of the complex. Indeed some data were recently published on a SCO compound, but puzzlingly the reported crystalline structure is centrosymmetric, and the magnitude of the effect reported was quite weak. [START_REF] Bonhommeau | Magnetism and Molecular Nonlinear Optical Second-Order Response Meet in a Spin Crossover Complex[END_REF] In view of those results, we decided to pursue the study of the chiral induction of the model complex [Fe(phen)3] 2+ by a series of chiral anions. We used two families of chiral anions (Figure 1): D2-symmetric anions based on cheap and easily available tartaric acid such as arsenyl/stibyl tartrate adducts, and D3-symmetric anions, tris(catechol)phosphate(V) (TRISCAT), tris(catechol)arsenate(V) (TRISCAS) and 3,4,5,6-tetrachlorocatechol phosphate(V) (TRISPHAT). TRISCAT is easily accessible, but it epimerizes quickly in solution and loses its optical activity. [START_REF] Cavezzan | ACTIFS : MISE EN EVIDENCE D'UN PROCESSUS IRREGULIER[END_REF] However, TRISCAS was reported to be stable in solution and was resolved to ∆ and Λ isomers using cinchonine and cinchonidine alkaloids respectively. [START_REF] Rosenheim | Optically Active Tribenzene Catechol-Arsenic Acid S[END_REF][START_REF] Ryschkewitsch | Synthesis of Asymmetric Boron Cations and Resolution with As(C6H4O2)3-Anion[END_REF] Likewise, TRISPHAT was resolved with high diastereomeric excess using chiral alkaloids, [START_REF] Lacour | Asymmetric Recognition of TRISPHAT Anion. Unusually High Difference in Reactivity of the Pseudoenantiomers of Cinchona Alkaloids[END_REF][START_REF] Favarger | Large-Scale Synthesis and Resolution of [ Tris ( Tetrachlorobenzenediolato ) Phosphate ( V )] Anion Ure 1 ), First Identified by Allcock in the Crude Reaction Mixture of Tricyclic Hexachlorocyclotriphosphazene and Pyrocatechol , 10 Was Later Shown to Be S[END_REF] and has been used in asymmetric chemistry as resolving agent for high levels of induction or recognition in molecular [START_REF] Jodry | Supramolecular Stereocontrol of Octahedral Metal-Centered Chirality. Ligand Modulation[END_REF][START_REF] Reddy | Determination of Labile Chiral Supramolecular Ion Pairs by Chromatographic NMR Spectroscopy[END_REF] or supramolecular processes. [START_REF] Jodry | Efficient Resolution of a Dinuclear Triple Helicate by Asymmetric Extraction/Precipitation with TRISPHAT Anions as Resolving Agents[END_REF] Surprisingly, they have been used so far only sparsely for the synthesis of materials. [START_REF] Sénéchal-David | Supramolecular Solid-State Interactions between Helicene-like Terpyridinium Cation and Tris(Tetrachloro-Benzenediolato)Phosphate(V) (TRISPHAT) Anion[END_REF] There are only few reports of diastereoselective ion pairing of the TRISPHAT anion with diimine low-spin Fe(II) complexes, either in solution, [START_REF] Jodry | Supramolecular Stereocontrol of Octahedral Metal-Centered Chirality. Ligand Modulation[END_REF][START_REF] Reddy | Determination of Labile Chiral Supramolecular Ion Pairs by Chromatographic NMR Spectroscopy[END_REF][START_REF] Lacour | Diastereoselective Ion Pairing of TRISPHAT Anions and Tris(4,4'-Dimethyl-2,2'-Bipyridine)Iron(II)[END_REF][START_REF] Bark | Synthesis and Stereochemical Properties of Chiral Square Complexes of Iron(II)[END_REF] or in the solid state. [START_REF] Jodry | Efficient Resolution of a Dinuclear Triple Helicate by Asymmetric Extraction/Precipitation with TRISPHAT Anions as Resolving Agents[END_REF][START_REF] Pérollier | Sugar Derived Hexacoordinated Phosphates: Chiral Anionic Auxiliaries with General Asymmetric Efficienc[END_REF][START_REF] Monchaud | Ion-Pair-Mediated Asymmetric Synthesis of a Configurationally Stable Mononuclear Tris(Diimine)-Iron(II) Complex[END_REF] On the other hand, the D2-symmetric As(III) [START_REF] Bott | Group 15 Complexes with α -Hydroxy Carboxylic Acids : 7 * The Preparation and Structure Determination of Sodium ( + ) -Tartrato Arsenate ( III ), [ Na 8 As 10 ( C 4 H 2 O 6 ) 8 ( C 4 H 3 O 6 ) 2 ( H 2 O ) 19 ] N ; Silver ( I ) ( + ) -Tartrato Arsenate[END_REF] or Sb(III) [START_REF] Palenik | Bond Valence Sums and Structural Studies of Antimony Complexes Containing Sb Bonded Only to O Ligands[END_REF] tartrate adducts were used in the past for the resolution of metal complexes, [START_REF] Lacour | Diastereoselective Ion Pairing of TRISPHAT Anions and Tris(4,4'-Dimethyl-2,2'-Bipyridine)Iron(II)[END_REF][START_REF] Dwyer | The Resolution of the Tris o,Phenantroline Ferrous Ion and the Oxidation of the Enantiomorphous Forms[END_REF][START_REF] Zalkin | Crystal Structure of L-Tris(1,10-Phenathroline) Iron(II) Bis(Antomony(III)d-Tartrate) Octahydrate[END_REF][START_REF] Watson | Cis-[Ru(Phen)2(CH3CN)2] 2 +[END_REF][START_REF] Koda | No Title[END_REF][START_REF] Matsumoto | The Crystal Structure of (-)589-Acetylacetonatobis(Trimethylenediamine)Cobalt(III) Arsenic(V) (+)-Tartrate Monohydrate[END_REF] mainly through chiral HPLC, [START_REF] Watson | Cis-[Ru(Phen)2(CH3CN)2] 2 +[END_REF][START_REF] Sarma | The Stereochemistry of Complex Inorganic Compounds. XVIII. A New Method for the Preparation of Inorganic Complexes in Their Optically Active Forms[END_REF][START_REF] Sakaguchi | Structure , Thermal Racemization , and Ion Association of the [1,1,1-Tris(5a-Mino-2-Azapentyl)Ethane]Cobalt (III) Complex[END_REF][START_REF] Yamazaki | IV. COMPLETE RESOLUTION OF THE NEUTRAL COMPLEX, MERIDIONAL ISOMER OF TRIS(B-ALANINATO)COBALT(III)[END_REF] with C2 and C3 association models proposed for the chiral recognition of octahedral metal complexes by [Sb2(tartrate)2] 2-. [START_REF] Miyoshi | Mechanism of Chiral Recognition of Octahedral Metal Complexes Effected by Bis(Mu-d-Tartrato)Diantimonate(III) Anion in Solution. 2. Cage Complexes of the Type [Co(N)6]3+[END_REF][START_REF] Miyoshi | Mechanism of Chiral Recognition of Octahedral Metal Complexes Effected by Bis(Mu-d-Tartrato)-Diantimonate(III) Anion in Solution. I. Association Models and Their Application to Some [Co(N)6]3+-Type Complexes[END_REF] Their synthesis is based on easily available and cheap tartaric acid. Potassium (+)antimonyl tartrate is even commercially available, but its insolubility in organic solvents limits severely its usefulness. The chirality of these anions can be simply labelled by extending the IUPAC skew line system (Scheme 1), [START_REF] Iupac | Tentative Proposals for Nomenclature1 of Absolute Configurations Concerned with Six-Coordinated Complexes Based on the Octahedron[END_REF] with one line joining the carboxylate coordinated oxygen atoms of one tartrate ligand and the second line the twofold axis intersecting the As atoms. # This is indeed similar to the accepted convention for tris(bidentate) octahedral complexes, 62 when considering one ligand and the threefold axis of the complex (Scheme 1). [START_REF] Iupac | Tentative Proposals for Nomenclature1 of Absolute Configurations Concerned with Six-Coordinated Complexes Based on the Octahedron[END_REF] Chiral [Fe(phen)3] 2+ complexes show strong non-linear optical properties as shown in our recently published study on second-harmonic generation, its correlation with the electronic absorption spectrum, and sum-frequency generation for drop-cast films of the complex resolved with the arsenyl tartrate anion that is published elsewhere. [START_REF] Iazzolino | Nonlinear Optical Properties and Application of a Chiral Spin Crossover Compound[END_REF] Herein we present a full report on the synthesis and characterization in solution and in the solid state of such materials resolved with two families of anions (see table of compounds in Supplementary Information). Dichroic optical studies allowed us to compare the chiral induction in the solid state and in solution. While optically pure complexes with TRISPHAT could not be isolated, complexes containing the TRISCAT and TRISCAS anions were shown to crystallize as inversion twins in the Sohncke R32 space group, the Flack/batch parameter being close to the optical purity of the starting anion. The arsenyl tartrate anions yielded instead structures in the enantiomorphic P3121/P3221 space groups. Those molecular materials, soluble in some common organic solvents, can be easily drop-cast as films. Non-linear optical studies on drop-cast films of the complex resolved with the TRISCAS anion revealed very strong second-harmonic generation. The intensity of the signal monitored in function of the excitation wavelength showed a strong correlation with the electronic spectrum and its dichroic counterpart. This work opens the way to novel studies at room temperature on non-linear optical properties dependence upon modulation of the spin state 64 by ultrafast pumping.

O O E O O E O O O O O O O O a a b b a a b b Λ ∆ O O E O O E O O O O O O O O E = A s , Sb
Scheme 1. Definitions of the ∆ and Λ configurations in arsenyl/stibyl tartrate anions and octahedral complexes.

Results and discussion

Synthesis and characterization of chiral anions.

Syntheses of TRISCAT, TRISCAS and TRISPHAT were reproduced or adapted from previous literature and are reported in Supplementary Information. While synthesis of the arsenyl tartrate adducts readily yields ∆-1, Λ-1, or rac-1 as confirmed by elemental analysis (Scheme 2), instead reactions with antimony trioxide did not give satisfactory analyses. In agreement with previously published structures for other derivatives of this dianion, [START_REF] Bott | Group 15 Complexes with α -Hydroxy Carboxylic Acids : 7 * The Preparation and Structure Determination of Sodium ( + ) -Tartrato Arsenate ( III ), [ Na 8 As 10 ( C 4 H 2 O 6 ) 8 ( C 4 H 3 O 6 ) 2 ( H 2 O ) 19 ] N ; Silver ( I ) ( + ) -Tartrato Arsenate[END_REF][START_REF] Palenik | Bond Valence Sums and Structural Studies of Antimony Complexes Containing Sb Bonded Only to O Ligands[END_REF] they demonstrate D2 symmetry: (-)-(S,S)-tartaric acid gives the ∆ isomer while the adduct with (+)-(R,R)-tartaric acid is the Λ isomer, # as shown by crystal structures solved in the Sohncke & P21212 space group (see details in Table SI2 and Figure SI3 in Supporting Information). Polarimetry and FTIR show that anion 1 in solution is likely to partially decoordinate, but regains the D2 structure upon solvent removal (see details in Supporting Information). Accordingly, a further study of the arsenic and antimonyl tartrate anions in solution by vibrational dichroic studies will be reported elsewhere. [Fe(phen)3](X)2 (X = TRISCAT (2); TRISCAS : rac (3), ∆ (4), Λ (5)) No Fe(II) complexes based on TRISCAT or TRISCAS have been reported in literature before, the only examples with TRISCAS being Cu(II) complexes. [START_REF] Lemus | Resolution and Characterization of Helicate Dimer and Trimer Complexes of 1,3-Bis(9-Methyl-1,10-Phenanthrolin-2-Yl)Propane with Copper(I)[END_REF][START_REF] Habermehl | Asymmetric Transformation of a Double-Stranded , Dicopper ( I ) Helicate Containing Achiral Bis ( Bidentate ) Schiff Bases[END_REF] The synthetic procedure consisted in all cases in layering a methanolic iron sulphate solution over solutions of TRISCAT.HNBu3 or TRISCAS racemic/∆/Λ salts and 1,10-phenanthroline (phen) in a mixture of methanol and dichloromethane. Red single crystals were formed in a few weeks. Diffraction data for [Fe(phen)3](rac-TRISCAT)2 (2), [Fe(phen)3](X-TRISCAS)2 (X = rac (3), ∆ (4), Λ (5)) were collected. As shown by almost identical cell parameters between complexes 2-5, they are isomorphous. The data could be satisfactorily integrated in the trigonal system, but scaling and merging could be performed either in the -3 or -3m Laue class. In order to select the correct space group within the corresponding point groups (-3m, 3m, 32, -3 and 3) belonging to those two Laue classes, a dataset collected on complex 4 integrated in trigonal settings but scaled and merged in point group P1 was further analyzed (see details and Table SI6 in Supplementary Information). [START_REF] Flack | Absolute-Structure Determination: Past, Present and Future[END_REF] The intensity data were then sorted into sets of 12 reflections symmetry-equivalent under point group -3m, of which 45 complete sets were obtained. We calculated then for each point group Rmerge for |Fobs| [START_REF] Sousa | Ultrafast Deactivation Mechanism of the Excited Singlet in the Light-Induced Spin Crossover of [Fe(2,2-Bipyridine)3]2+[END_REF] , A and D (A and D being the average and the difference of Friedel opposites respectively). [START_REF] Flack | The Mean-Square Friedel Intensity Difference in P1 with a Centrosymmetric Substructure[END_REF][START_REF] Flack | Applications and Properties of the Bijvoet Intensity Ratio[END_REF][START_REF] Flack | The Use of X-Ray Crystallography to Determine Absolute Configuratio[END_REF][START_REF] Shmueli | Concise Intensity Statistics of Friedel Opposites and Classification of the Reflections[END_REF][START_REF] Flack | Practical Applications of Averages and Differences of Friedel Opposites[END_REF][START_REF] Parsons | Analysing Friedel Averages and Differences[END_REF][START_REF] Parsons | Use of Intensity Quotients and Differences in Absolute Structure Refinement[END_REF] There was a clear preference for non-centrosymmetric point groups 3 and 32. Systematic extinctions led to potential R3 and R32 space groups respectively. The structures could be solved in either space group. Full convergence required the inclusion of an inversion twin law for all complexes 2-5, and final refinement statistics were systematically slightly better for the R32 space group. The X-ray crystallographic refinement data for complexes 2-5 are shown in Table SI9 in Supporting Information. Flack parameters were obtained after full convergence of the structures: for complexes 2-5, it deviates significantly from zero at 0.45(2), 0.473 (15), 0.163(13) and 0.277(2) respectively. Those values were apparently close to the optical purity of the starting anion: TRISCAS diastereomeric excesses were 0% for 2-3, 88% for 4, while it was 66% for 5. Nevertheless, when we performed a second synthesis of complex 4 with an 86% d.e. batch of ∆-TRISCAS, the resulting crystal analyzed yielded a Flack parameter statistically equal to zero (complex 4a at -0.006(3)), evidencing that the link between crystals and starting anion optical purity is not completely straightforward. The fact that complexes 2-3 did not crystallize in a centrosymmetric space group as expected led us to check the veracity of the existence of these inversion twins. Indeed, it has been found true that most non-centrosymmetric crystal structures with Flack parameter values between 0.25 and 0.75 published in Acta Cryst. C in 2011 and 2012 were in fact twinned by inversion. [START_REF] Flack | Centrosymmetric and Pseudo-Centrosymmetric Structures Refined as Non-Centrosymmetric[END_REF][START_REF] Flack | Centrosymmetric Crystal Structures Described as Non-Centrosymmetric: An Analysis of Reports in Inorganica Chimica[END_REF][START_REF] Flack | Absolute-Structure Reports[END_REF] We thus analysed the data further, with refinement completed without the twin law for complexes 3 and 4 (see more details and Figures SI7 and SI8 in Supporting Information). Observed and modelled data were then separated in centric and acentric reflexions. [START_REF] Flack | The Mean-Square Friedel Intensity Difference in P1 with a Centrosymmetric Substructure[END_REF][START_REF] Flack | Applications and Properties of the Bijvoet Intensity Ratio[END_REF][START_REF] Flack | The Use of X-Ray Crystallography to Determine Absolute Configuratio[END_REF][START_REF] Shmueli | Concise Intensity Statistics of Friedel Opposites and Classification of the Reflections[END_REF][START_REF] Flack | Practical Applications of Averages and Differences of Friedel Opposites[END_REF][START_REF] Parsons | Analysing Friedel Averages and Differences[END_REF][START_REF] Parsons | Use of Intensity Quotients and Differences in Absolute Structure Refinement[END_REF][START_REF] Flack | Centrosymmetric and Pseudo-Centrosymmetric Structures Refined as Non-Centrosymmetric[END_REF][START_REF] Flack | Centrosymmetric Crystal Structures Described as Non-Centrosymmetric: An Analysis of Reports in Inorganica Chimica[END_REF][START_REF] Flack | Absolute-Structure Reports[END_REF] The slopes given by the plots of Dobs (difference for paired observed Friedel reflections) vs. Dsingle (difference for paired Friedel reflections calculated from a single domain model) were in perfect agreement with the Flack parameter issued from the least-square refinement including the racemic twin law, evidencing that those compounds are indeed racemic twins. One must note that crystals observed under polarized light did not evidence macroscopic domains of opposite handedness. The inversion twin implies then the presence of microscopic domains of opposite handedness that are intimately mixed. The global description of all crystal structures is similar. As expected, the interaction of the bulky TRISCAT or TRISCAS counter ions with the [Fe(phen)3] 2+ system does not modify the spin state of the tris-phenanthroline iron(II) system. The average iron-nitrogen bond distance is Fe-Navg ≈ 1.97 Å, corresponding to the LS state due to the strong σ-donating and π-back bonding capabilities of the ligand molecule. [START_REF] Goodwin | Spin Crossover in Iron(II) Tris(Diimine) and Bis(Terimine) Systems[END_REF] The [Fe(phen)3] 2+ complex and two TRISCAT/TRISCAS anions form triads by interacting strongly along the complex 3-fold axis (Figure 2), the distances between iron and anion being dFe-P = 5.6889(8) Å and dFe-As = 5.7162(16) Å for complexes 2 and 3 respectively. Each complex is surrounded by six anions, forming a shell along the crystallographic c axis (see more details in Supporting Information, Figure SI11), with each anion of this shell itself part of another triad. Those triads are staggered alternatively above and below (Figure 3). The dichloromethane molecules fill the void spaces between the triads. The average contacts and distances between cations and anions for complexes 2-5 are very similar (Supporting Information, Table SI10). We observe heterochiral interactions within the triads, mediated mainly through hydrogen bonding between the α-acidic hydrogens of the phenanthroline ligand and the oxygen atoms of the anion (C-H•••O ≈ 3.3 Å) and some aromatic carbons (C-H•••C(π) ≈ 3.5-3.7 Å), completed by a strong oxygen-carbon π-π interaction (O(π)•••C(π) ≈ 3.1-3.2 Å). X-ray powder diffractograms for complexes 3-5 confirmed that the crystalline phases we evidenced are the only ones present: experimental and calculated diffraction patterns from the corresponding X-ray structures agree perfectly (Supporting Information, Figure SI12). While trying to obtain drop-cast films for the NLO measurements described below, solutions of twinned crystals of 3 and 4 in acetonitrile yielded isostructural solvatomorphs, as acetonitrile solvated crystal structures (4b-5b and 4c-5c, Table SI13 in Supporting Information). Spontaneous resolution occurs, yielding conglomerates of enantiopure crystals. This is in contrast with the recently reported copper complexes resolved using the TRISCAS anion which evidenced enhanced racemization of the anion in a mixture of dichloromethane and acetonitrile. [START_REF] Lemus | Resolution and Characterization of Helicate Dimer and Trimer Complexes of 1,3-Bis(9-Methyl-1,10-Phenanthrolin-2-Yl)Propane with Copper(I)[END_REF] Nevertheless, R32 being an achiral Sohncke space group, crystals of opposite handedness show achiral morphologies and cannot be sorted mechanically. [Fe(phen)3](rac-TRISPHAT)2 (6) As we mentioned, there have been a few reports studying the chiral induction of the TRISPHAT anion on some diimine Fe(II) complexes, but those were limited to solution studies (NMR and CD measurements). [START_REF] Jodry | Supramolecular Stereocontrol of Octahedral Metal-Centered Chirality. Ligand Modulation[END_REF][START_REF] Reddy | Determination of Labile Chiral Supramolecular Ion Pairs by Chromatographic NMR Spectroscopy[END_REF][START_REF] Lacour | Diastereoselective Ion Pairing of TRISPHAT Anions and Tris(4,4'-Dimethyl-2,2'-Bipyridine)Iron(II)[END_REF][START_REF] Bark | Synthesis and Stereochemical Properties of Chiral Square Complexes of Iron(II)[END_REF] In order to obtain structural data, we obtained crystals in a few days by layering a methanol solution of iron(II) sulphate over a solution of the phen ligand and the corresponding salt of TRISPHAT anion. When using rac-TRISPHAT, we observed two distinct crystals morphologies: on one hand hard deep-red well-defined prisms, and on the other hand crumbly redorange flat crystals. With enantiopure ∆-TRISPHAT, only the flat orange crystals were obtained, while a Λ-TRISPHAT batch with a low d.e. excess, and thus containing a significant amount of the racemate, yielded again a mixture of phases. Despite many trials, only the hard-red prisms allowed us to solve and refine the corresponding structure in the centrosymmetric monoclinic C2/c space group. The reaction with ∆-TRISPHAT would have suggested that the red-orange phase may be the looked-for non-centrosymmetric one, and that some spontaneous resolution may occur even with the racemic anion. Unfortunately this phase is very fragile and likely to be very heavily solvated, and crystals do not survive either manipulation or being outside the mother liquor. Dichroic spectra measured in a 99:1 CHCl3/DMSO mixture did not show any signal, though this solvent mixture was reported to yield a d.e.>95% for TRISPHAT. [START_REF] Jodry | Supramolecular Stereocontrol of Octahedral Metal-Centered Chirality. Ligand Modulation[END_REF][START_REF] Reddy | Determination of Labile Chiral Supramolecular Ion Pairs by Chromatographic NMR Spectroscopy[END_REF] The centrosymmetric structure corresponds to the expected Fe(II) complex (see Figure 4 and Table SI14 in Supporting Information for further details), with half of it and a TRISPHAT anion making up the asymmetric unit. This phase is also heavily solvated, with huge solvent-accessible spaces accounting for almost 1/3 of the unit cell volume, and the SQUEEZE procedure [START_REF] Spek | SQUEEZE: A Tool for the Calculation of the Disordered Solvent Contribution to the Calculated Structure Factors[END_REF] had to be applied to allow a satisfactory refinement, yielding [Fe(phen)3](rac-TRISPHAT)2.(CH2Cl2)n (6) (4 ≤ n ≤ 16). The low-spin Fe(II) complex has an average Fe-N bond length of 1.982(1)Å. When looking at the crystal packing, with the provision that interactions with the solvent molecules cannot be analyzed due to disorder, no particularly relevant interaction between iron complex and phosphate anions can be observed. Overall those results are really surprising considering previous solution NMR and CD studies which stated that "diastereoselective induction from TRISPHAT anions onto tris(diimine)iron(II) complexes occurs with better selectivity with phen rather than bipy ligands". [START_REF] Jodry | Supramolecular Stereocontrol of Octahedral Metal-Centered Chirality. Ligand Modulation[END_REF][START_REF] Reddy | Determination of Labile Chiral Supramolecular Ion Pairs by Chromatographic NMR Spectroscopy[END_REF] [Fe(phen)3](X) (X = As2(tartrate)2 : ∆ (7), Λ(8), racemic mixture (9)

A solution of the corresponding (NBu4)2[∆/Λ-As2(tartrate)2] anion 1 was layered with a solution of [Fe(phen)3] 2+ to form crystals of complexes [Fe(phen)3](X), X= ∆-As2(tartrate)2 (7) and Λ- (8). The crystal structures could be solved and refined to convergence in the enantiomorphic P3121/P3221 space groups respectively. Crystallographic data are reported in Table SI15 in Supporting Information. Unit cells of 7 and 8 are very similar to the analogous complex of [Fe(phen)3] 2+ with [Λ-Sb2(tartrate)2] 2-that crystallizes in the same trigonal chiral space group P3221. [START_REF] Zalkin | Crystal Structure of L-Tris(1,10-Phenathroline) Iron(II) Bis(Antomony(III)d-Tartrate) Octahydrate[END_REF] These phases showed again to be heavily solvated, with huge solvent-accessible spaces accounting for almost 1/3 of the unit cell volume. No satisfactory discrete disorder model was achievable, so the SQUEEZE procedure had also to be applied to allow a satisfactory refinement, yielding a likely composition [Fe(phen)3][As2(tartrate)2](MeOH)(CH2Cl2)4. Crystal structures of 7 and 8 are enantiomorphous with the same unit cell, each asymmetric unit consists of half a cation and half an anion (Figure 5) lying over crystallographic twofold axes. The cation molecular symmetry is nevertheless very nearly threefold. The iron atom is clearly low-spin in both case with an average Fe-N bond length of 1.968(3) Å and 1.970(6) Å for 7 and 8 respectively. Flack parameters for 7 and 8 are statistically different from zero but quite close to it, showing that the crystals are almost enantiopure. Synthesis with an equimolar racemic mixture of (NBu4)2[∆/Λ-As2(tartrate)2] yielded similar looking crystals of complex 9 (Figure 5). The crystal structure could be solved and refined as a racemic twin in the P3121 space group, with a batch factor of 0.492(13), as already observed for [Fe(phen)3](rac-TRISCAS)2 (3). Cation and anion interact homochirally in complexes 7 and 8 (Figure SI16 in Supporting Information): the helicity of [Fe(phen)3] 2+ is induced by the configuration of the D2 arsenyl tartrate. This is again similar to the previously reported structure with antimonyl tartrate. [START_REF] Zalkin | Crystal Structure of L-Tris(1,10-Phenathroline) Iron(II) Bis(Antomony(III)d-Tartrate) Octahydrate[END_REF] The crystal packing is identical for 7-9. The cations form layers between which [As2(tartrate)2] 2-anions are embedded in order to compensate the charge of the layers (Figure SI17 in Supporting Information). Six anions are interacting with each [Fe(phen)3] 2+ and have the same configuration as the cationic complex. Four of them are almost equidistant (dFe-As ≈ 6.1 Å) and two are farther away (dFe-As ≈ 9.5 Å). This is opposite to the isostructural [Λ-Fe(phen)3](Λ-Sb2(tartrate)2) complex where the Fe-Sb distances of the four nearest anions are unequal (5.884 Å and 6.233 Å) and other two anions are at longer distances (9.80 Å). [START_REF] Zalkin | Crystal Structure of L-Tris(1,10-Phenathroline) Iron(II) Bis(Antomony(III)d-Tartrate) Octahydrate[END_REF] The four closest anions are interacting with the cationic complex along two of the twofold and threefold symmetry axes of the complex (Figure 6). The homochiral interaction is mediated through hydrogen bonds between the oxygen atoms of the tartrate and hydrogen atoms of the aromatic rings of the complex (Oanion•••C-Hcomplex ≈ 3.146(4) Å, 3.337(3) Å). Theses distances are similar to those found for antimonyl tartrate (Oanion•••C-Hcomplex ≈ 3.198(4) Å, 3.317(3) Å). Powder X-ray diffractograms measured for complexes 7-9 show an excellent match between experimental and calculated patterns from the single crystal structures (Supporting Information, Figure SI18), confirming the presence of one single phase in the bulk product.

Electronic absorption and circular dichroism of complexes 2-5 and 7-9

The tris-chelation of a metal by conjugated aromatic ligands is well understood, leading to exciton splitting [START_REF] Harada | Electronic CD Exciton Chirality Method: Principles and Applications[END_REF] of the three degenerate ligand π -π * transitions in the UV region in a so-called CD couplet with opposite rotatory strengths, with a pattern correlated with the helicity of the ligands disposition. [START_REF] Kaizaki | Applications of Electronic Circular Dichroism to Inorganic Stereochemistry[END_REF] Indeed the intensity of the two dichroic bands is induced by the helical disposition of the π -π * transitions and the resulting charge displacement by coulombic coupling. Reference spectra are available for TRISCAS (water solution): [START_REF] Mason | Electronic Absorption and Circular Dichroism Spectra and Absolute Stereochemistry of Tris-Catechyl-Arsenate(V) Ion[END_REF][START_REF] Ito | The Absolute Configuration of (-)589-Tris(1,2-Benzene-Diolato)Arsenate(V) Ion[END_REF] two π-π* catechol transitions were observed in water solution at 220 and 276 nm, but only the high-energy transition has dipole moments oriented in such a way to interact and give a couplet. Only the couplet low energy component could be measured at 226 nm with positive Cotton effect (∆ε = +68 Lmol -1-1 cm -1 ) for the Λ configuration. [START_REF] Mason | Electronic Absorption and Circular Dichroism Spectra and Absolute Stereochemistry of Tris-Catechyl-Arsenate(V) Ion[END_REF][START_REF] Ito | The Absolute Configuration of (-)589-Tris(1,2-Benzene-Diolato)Arsenate(V) Ion[END_REF] We measured CD spectra for both anion 1 and the easily accessible K2[Sb2(tartrate)2] (Figure SI19 to SI21 in Supporting Information), as compared to published spectra of tartrate. Dianionic tartrates show bands at 194 and 211 nm assigned to n-π* and n-σ* forbidden transitions of the carboxylate groups, with no excitonic coupling seen outside supramolecular packings. [START_REF] Brizard | Molecular and Supramolecular Chirality in Gemini-Tartrate Amphiphiles Studied by Electronic and Vibrational Circular Dichroisms[END_REF][START_REF] Zhang | Protein-Sized Chiral Fe 168 Cages with NbO-Type Topology H NMR ( Figure S14 ; See Supporting Information )[END_REF][START_REF] Hoffmann | Extending the Applications of Circular Dichroism in Structure Elucidation: Aqueous Environment Breaks the Symmetry of Tartrate Dianion[END_REF] The CD spectra in water for K2[Sb2(tartrate)2] and MeCN for anion 1 show instead two excitonic couplets: one at lower energies, centered at 223 nm for K2[Sb2(tartrate)2] and at 219 nm for anion 1; the second one at 200 and 197 nm respectively. In both cases, excitonic couplets are of positive (respectively negative) chirality for the Λ (respectively ∆) anion configuration, which upon inspection of the crystal structures do not correspond to the coupling of the transitions observed for dianionic tartrate. This combined with the observed bathochromic shift supports that electronic transitions are different for As and Sb tartrate adducts, with the deprotonated hydroxyles of the tetraanionic tartrates likely to play a role. Anion 1 is sparingly soluble in water, and the corresponding spectra are seen to be strikingly similar to those of (NBu4)2(tartrate), supporting that according to the solvent partial or total decoordination of the tartrate may occur. Further dichroic studies in solution will be reported elsewhere. In the solid state (Figure SI22 in Supporting Information), KBr pellets of ∆-1 and Λ-1 show what could be part of excitonic couplets below 230 nm. The difference in the Cotton effect between ∆-1 and Λ-1 is likely due to inevitable experimental errors in preparing the KBr pellets. To the best of our knowledge no further CD spectra were previously reported for tartrate adducts with As(III) or Sb(III). Reference spectra are available for [Fe(phen)3] 2+ in water solution: one π-π* phen transition was observed to give a couplet at 260 nm and 272 nm, which are A2 and E coupled in the D3 symmetry point group. [START_REF] Mccaffery | Optical Rotatory Power of Co-Ordination Compounds. Part XII Spectroscopic and Configurational Assignments for the Tris-Bipyridyl and -Phenanthroline Complexes of the Di-and Tri-Valent Iron-Group Metal Loins[END_REF] The two bands show opposite Cotton effect, negative at 272 nm for the ∆ configuration and inversely positive for the Λ configuration (∆ε = +715 Lmol -1-1 cm -1 ), confirmed from the crystallographic structure of [Λ-Fe(phen)3] 2+ . [START_REF] Zalkin | Crystal Structure of L-Tris(1,10-Phenathroline) Iron(II) Bis(Antomony(III)d-Tartrate) Octahydrate[END_REF] Electronic spectra for complexes 2-5 and 7-9 in acetonitrile solutions are shown in Figure 7 and8, with further details in Figure SI23 to Table SI26 in Supporting Information. The TRISPHAT asymmetric induction towards [Fe(phen)3] 2+ in solution has been extensively studied by circular dichroism and 1H-NMR, [START_REF] Jodry | Supramolecular Stereocontrol of Octahedral Metal-Centered Chirality. Ligand Modulation[END_REF][START_REF] Reddy | Determination of Labile Chiral Supramolecular Ion Pairs by Chromatographic NMR Spectroscopy[END_REF] so we studied the effectiveness of optically active TRISCAS towards the same cation. Efficient asymmetric induction was evidenced towards robust dinuclear helicate copper(II) species, but it was pointed out that asymmetric induction was affected by the possible protonation of the anion and the donating character of the solvent. [START_REF] Lemus | Resolution and Characterization of Helicate Dimer and Trimer Complexes of 1,3-Bis(9-Methyl-1,10-Phenanthrolin-2-Yl)Propane with Copper(I)[END_REF][START_REF] Habermehl | Asymmetric Transformation of a Double-Stranded , Dicopper ( I ) Helicate Containing Achiral Bis ( Bidentate ) Schiff Bases[END_REF] The [FeN6] 2+ core is known for its lability. [START_REF] Dwyer | The Resolution of the Tris o,Phenantroline Ferrous Ion and the Oxidation of the Enantiomorphous Forms[END_REF] Nevertheless we measured the CD spectra in acetonitrile for complexes 4 and 5 (Figure 7, top) and complexes 7 and 8 (Figure 8, top). It is clear from Figure 7 and 8 that in acetonitrile solution no [Fe(phen)3] 2+ MLCT bands are present. [Fe(phen)3](TRISCAS)2 (Figure 7) shows only three Cotton effects at 209 nm, 234 nm and 280 nm (∆ε of +1004, +643 and +50 Lmol -1 cm -1 respectively for 4), inverted for 5 (∆ε of -758, -755 and -48 Lmol -1 cm -1 respectively). Those features correspond only approximately to the TRISCAS catechol π -π * transitions reported previously. [START_REF] Kaizaki | Applications of Electronic Circular Dichroism to Inorganic Stereochemistry[END_REF] Indeed the couplet reported at 220 nm is not apparent here, and transitions appear to be much more intense. Nevertheless Λ/∆-TRISCAS alkaloid salts when measured in acetonitrile instead of water were seen to show transitions with pronounced hypsochromic shifts and enhanced intensities (Figure SI27 in Supporting Information). The low-energy band of the couplet is shifted to 206 nm, the uncoupled transition to 235 nm, and a further transition is seen at 280 nm, with ∆ε = -96, -46 and -3 Lmol -1 cm -1 , +151, +73 and +5 Lmol -1 cm -1 for cinchoninium(∆-TRISCAS) and cinchonidinium(Λ-TRISCAS), respectively. The absence of any Cotton effect corresponding to the MLCT bands in the visible region and the similarity with the anion CD spectra indicates for the complex that dissociation of the ion pair occurs with subsequent racemization of the ∆or [Λ-Fe(phen)3] 2+ complexes through the well-known Bailar or Rây-Dutt twist mechanisms. [START_REF] Rzepa | In Search of the Bailar and Ra ˆ y -Dutt Twist Mechanisms That Racemize Chiral Trischelates : A Computational Study of Sc III , Ti IV , Co III , Zn II , Ga III , and Ge IV Complexes of a Ligand Analogue of Acetylacetonate[END_REF] The link of these UV features with TRISCAS is confirmed by the complete loss of optical activity for the [Fe(phen)3](∆-TRISCAS)2 solution (Figure 7) upon addition of a few drops of trifluoroacetic acid: the acidic racemization of ∆-TRISCAS is well documented. [START_REF] Kolp | Crystal Structure Analysis and Chiral Recognition Study of Δ[END_REF] For [Fe(phen)3](As2(tartrate)2), the CD spectra for 7 and 8 (Figure 8) are clearly very similar to the ones we recorded for anion 1 in MeCN (Figure SI20 in Supporting Information). The ion-pair dissociation and subsequent quick racemization observed here is puzzling since antimonyl(tartrate) was used to prepare optically pure [Fe(phen)3] 2+ samples on which dichroic spectra were originally measured in much more polar water. [START_REF] Mccaffery | Optical Rotatory Power of Co-Ordination Compounds. Part XII Spectroscopic and Configurational Assignments for the Tris-Bipyridyl and -Phenanthroline Complexes of the Di-and Tri-Valent Iron-Group Metal Loins[END_REF] We also checked whether any chiral induction could be seen in solution by recording 1 H NMR spectra for complexes 3 and 4 in CD3CN and (CD3)2CO (Figures SI28 in Supporting Information). In both cases spectra are seen to differ for hydrogen atoms in ortho and para positions from the spectrum recorded for [Fe(phen)3]Cl2 in CD3CN, [START_REF] Pazderski | 1H NMR Assignment Corrections and 1H, 13C, 15N NMR Coordination Shifts Structural Correlations in Fe(II), Ru(II) and Os(II) Cationic Complexes with 2,2-Bipyridine and 1,10-Phenanthroline[END_REF] suggesting close ion pairing in solution. Nevertheless, the absence of [Fe(phen)3] 2+ MLCT bands in the CD measurement in both acetonitrile and acetone would indicate that this interaction is not close enough to prevent racemisation of [Fe(phen)3] 2+ . The low solubility of complex 4 in non-polar solvents prevented studying the effect of the polarity of the solvent. As shown by the crystal structure, there is clearly an efficient asymmetric induction in the solid state. While measurement of dichroic spectra in the solid state is usually very tricky, due to macroscopic anisotropies such as linear dichroism and birefringence, which may be of similar or greater amplitude than the CD signal, and which may couple with non-ideal characteristics of the polarization modulation, [START_REF] Kuroda | Chirality in Crystals[END_REF] dispersion of microcrystalline powder in KBr pellets solves partially this issue. [START_REF] Kuroda | A Solid-State Dedicated Circular Dichroism Spectrophotometer: Development and Application[END_REF] We measured CD spectra of KBr pellets with complexes 4 and 5 (7.2 mmol/dm 3 , thickness = 0.35 mm) at room temperature (Figure 7). The [Fe(phen)3] 2+ MLCT bands are observed this time, as alternating bands at 555, 470 and 405 nm (∆ε of -178, +114 and -61 Lmol -1 cm -1 respectively for 4, and mirroring intensities of +284, -291 and -0.5 Lmol -1 cm -1 for 5).

The intense π-π* absorption bands in the UV-region (200-400 nm) correspond to electronic transitions of TRISCAS and 1,10-phenanthroline that are overlapping. Nevertheless, couplets for both chromophores can be observed. Indeed, for complex 4 the couplet corresponding to the high-energy π-π* phen transitions is observed at 280 and 260 nm (∆ε +220 and -346 Lmol -1 cm -1 respectively), its sign corroborating the Λ configuration observed in the X-ray structure. Surprisingly, the low-energy transition gives significant dichroic signal at 296 nm (∆ε +236 Lmol -1 cm -1 ). The ∆-TRISCAS catechol couplet high-energy component is observed at 212 nm (∆ε -226 Lmol -1 cm -1 ), with the uncoupled transition at higher wavelength at 238 nm (∆ε -287 Lmol -1 cm -1 ). The Λ-TRISCAS complex 5 shows correspondingly opposite ∆ε values: -595 and +611 (phen couplet), +608 (296 nm), +395 (catechol couplet) and +391 Lmol -1 cm -1 (238 nm). When recrystallizing the complex in acetonitrile, the CD spectrum of a KBr pellet prepared from a single crystal showed very similar CD spectra to the parent compound (Figure SI29 in Supporting Information). ∆ε values for the single crystal pellet are puzzlingly very close to the ones obtained from the starting complex prior to recrystallization, which supposedly was less optically pure. Mirror images CD spectra for KBr pellets of 7 and 8 are shown on Figure 8. The [Fe(phen)3] 2+ MLCT bands are again observed at approximately 555, 470 and 405 nm (∆ε of +97, -67 and +24 Lmol - 1 cm -1 respectively for 7, -70, +40 and -6 Lmol -1 cm -1 respectively for 8). In the UV range, five Cotton effects are seen at 350, 298, 285, 260 and 233 nm (∆ε -30, -142, -119, +102 and -38 Lmol -1 cm - 1 respectively for 7, +27, +96, +81, -59 and +19 Lmol -1 cm -1 respectively for 8). The 285 and 260 nm are the high-energy phen couplet and its sign correspond to the crystallographic ∆and [Λ-Fe(phen)3] 2+ for complexes 7 and 8 respectively. The lower energy transition at 298 nm was already observed in complexes 4 and 5, but the well-defined feature at 350 nm was not present and remains unexplained. The higher Cotton effect at 233 nm may be linked to the [As2(tartrate)2] dianion.

Second-harmonic generation experiments

As discussed above, one of the main interests in the use of chiral anions is that they will ensure that a given material will crystallize in a non-centrosymmetric space group that may exhibit second-order nonlinear optical properties. We checked that indeed the chiral complexes we synthesized exhibit such properties by performing second harmonic generation (SHG) experiments. The principle of these experiments is very simple: exciting the sample with intense laser pulses, the second-harmonic signal (SHS) is recorded at the exit of sample or in reflection at its surface. In short, the experimental set-up we used is as follow. Laser pulses provided at 60 kHz repetition rate by an optically parametric amplifier were focused on the sample by mean of a 20 cm focal length lens. The energy of 160 fsec pulses, tunable in the 2.4 µm -0.6 µm spectral range, can be as high as 60 µJ. The SHS generated is collected in transmission at the exit of the sample and it is focused at the entrance slit of a spectrometer connected to a CCD camera. To reject the light at the fundamental laser frequency a filter was inserted in front of the spectrometer. Samples were prepared by drop casting a small volume of solutions of complexes 4 and 5 (see Supplementary Information for further details). We found that acetonitrile solutions resulted in the formation of rather large microcrystallites. Instead with N,N-dimethylformamide we observed rather good quality films on the dried samples, with small crystallites no larger than a few micrometers in size randomly dispersed (see Figure SI30-31 in Supporting Information). Preservation of the molecular structure was evidenced by Raman spectroscopy on those films as compared with bulk single-crystals, and conservation of the structure was checked by Xray diffraction (see Figure SI32 in Supporting Information). The spectra of the normalized SHS we recorded for films prepared from [Fe(phen)3](∆-TRISCAS)2 ( 4) and [Fe(phen)3](Λ-TRISCAS)2 (5) when the excitation wavelength is fixed at λ = 1004 nm are displayed in Figure 9. These spectra are centred at λ/2 = 502 nm (Figure 9a). Moreover, as expected for a non phasematched optical second-order nonlinear phenomena, the amplitude SHS in both samples evolves as the square of the fundamental pulse intensity (Figure 9b). This ensemble of results clearly indicates that the recorded signal results indeed from SHG. We have performed the same experiment on a drop-cast of a racemic solution of [Fe(phen)3](TRISCAS)2 (3). Under the same experimental conditions, we could not detect any SHG signal. This experiment confirms that the racemic sample is not optically active and therefore does not exhibit any second order nonlinear optical properties. Finally, we also measured the evolution of the SHS when we tuned the central wavelength of the excitation pulses. The SHS we recorded at λ = 1050 nm is displayed against the signal we recorded at λ = 1350 nm and λ = 800 nm (Figure 10). These three signals were normalized against the signal we recorded in α-Quartz under the same experimental conditions. This latter crystal belongs to the 32 space group and thus exhibits the same nonlinear coefficients than [Fe(phen)3](∆-TRISCAS)2 (4) and [Fe(phen)3](Λ-TRISCAS)2 (5) micro-crystallites under study. Moreover, the second-order optical susceptibility of α-Quartz is almost flat over the whole visible and near I.R spectral range. At variance with the SHS we recorded in α-Quartz, the SHS we recorded for either [Fe(phen)3](∆-TRISCAS)2 ( 4) or [Fe(phen)3](Λ-TRISCAS)2 ( 5) strongly increases as we tune the central wavelength of the laser pulses towards λ = 1050 nm. As previously mentioned, these two samples present a small and broad absorption peak centered at λ = 520 nm related to the d-d transition of iron(II) that overlaps with the more intense MLCT absorption band due to the phenanthroline ligand. Hence the increase of the SHS when the excitation wavelength is about λ = 1050 nm is related to the resonant enhancement of the second-order susceptibility of this material. We published elsewhere extensive details about the experimental set-up, a more detailed study on films made of 7 and 8 of the SHG dependence on the irradiation wavelength, and the possibility of performing Sum Frequency Generation on two femtosecond laser pulses, a property with applications in nonlinear optics. [START_REF] Iazzolino | Nonlinear Optical Properties and Application of a Chiral Spin Crossover Compound[END_REF] Overall complexes 4,5, 7 and 8 present similar NLO characteristics. Indeed the behavior and the spectral dependence we have observed in the visible range is due to the MLCT transitions, which have similar intensities for all those complexes.

Conclusions

As initially desired, the use of a variety of chiral anions based on hexavalent P/As(V) (TRISCAT, TRISPHAT, TRISCAS) and tetravalent As(III) ([As2(tartrate)2] 2-) have allowed us to perform the chiral resolution of the well-known [Fe(phen)3] 2+ complex. TRISCAS and a new lipophilic derivative of [As2(tartrate)2] 2-we synthesized [START_REF] Srinivasan | Enantiomeric Resolution and X-Ray Optical Activity of a Tricobalt Extended Metal Atom Chain[END_REF] allow easy resolutions by selective crystallization. We found that TRISCAS yields dichloromethane solvates that crystallize in the Sohncke space group R32 as inversion twins, with molar fractions reflecting the initial optical purity of the chiral anion. Recrystallization in acetonitrile yields the corresponding acetonitrile solvate which spontaneously resolve. Instead the [As2(tartrate)2] 2-anion yields methanol/dichloromethane solvates that crystallize in the enantiomorphic P3121/P3221 space groups, even when starting from a racemic mixture of the chiral anion. With the chiral anion TRISPHAT, we observed racemization of the anion in solution which prevented the resolution of the complex. Dichroic spectroscopic studies evidenced the racemization of the [Fe(phen)3] 2+ complex in solution, while chiral induction was preserved in the solid state upon (re)crystallization. In order to study the second-order nonlinear optical properties, we tested also the dropcasting of those complexes. We showed that using N,N-dimethylformamide solutions allow easy growth of good-quality films. Preservation of the chiral character of the complex was evidenced by very strong signals in second-harmonic generation experiments when irradiated with femtosecond laser pulses. Interestingly the second-harmonic signal intensity dependence with the excitation wavelength hinted at a resonant enhancement of the second-order susceptibility of this material. The [Fe(phen)3] 2+ photophysics is presently very much studied, and we are now exploring the possibility of photoconverting the complex to its metastable HS state by ultrafast pumping, and then checking SHG in that transient state. Preliminary results show that it is a good candidate for an ultrafast switch for non-linear optics, and they will be reported elsewhere in due course.

ASSOCIATED CONTENT

Supporting information contains experimental details of anions and complexes synthesis and physical characterization: crystallographic data, electronic absorption, dichroic and NMR spectra, details on second-harmonic generation experiments, imaging, X-ray diffraction and Raman spectra of drop-casts films. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1 .

 1 Figure 1. Schematic representation of D3-symmetric chiral anions (TRISCAT, TRISCAS, TRISPHAT) and D2-symmetric chiral arsenyl tartrate anion.

Scheme 2 .

 2 Scheme 2. Synthetic path to chiral anion 1.

Figure 2 .

 2 Figure 2. Axial interactions of two ∆-TRISCAS with the 1,10-phenanthroline α acidic hydrogens of a [Λ-Fe(phen)3] complex in complex [Fe(phen)3](rac-TRISCAS)2 (3).

Figure 3 .

 3 Figure 3. Staggered interactions between a given triad (van der Waals representation, ∆ TRISCAS in blue and Λ Fe(II) in red) and two triads of its shell, as seen perpendicularly to the c axis.

Figure 4 .

 4 Figure 4. Complex [Fe(phen)3](rac-TRISPHAT)2 (6) measured at 120 K. H atoms have been omitted for clarity, with displacement ellipsoids drawn at the 50% probability level.

Figure 5 .

 5 Figure 5. Asymmetric unit of complex [Fe(phen)3](As2(tartrate)2) (9) at 120K. Displacement ellipsoids are drawn at the 50% probability level.

Figure 6 .

 6 Figure 6. Interaction of the four closest [∆-As2(tartrate)2] 2-anions with [∆-Fe(phen)3] 2+ along 3-and 2-fold symmetry axes of complex 7.Powder X-ray diffractograms measured for complexes 7-9 show an excellent match between experimental and calculated patterns from the single crystal structures (Supporting Information, FigureSI18), confirming the presence of one single phase in the bulk product.

Figure 7 .

 7 Figure 7. (top) CD spectra : 4 (blue shades) and 5 (red shades) in KBr pellets (full lines) and in acetonitrile solution (short dotted line), and 4 in acetonitrile solution + 10 eq TFA (short dotted green line); (bottom) corresponding UV-visible spectra of 4 and 5 in acetonitrile solution.

Figure 8 .

 8 Figure 8. (top) CD spectra : 7 (red shades) and 8 (blue shades) in KBr pellets (full lines) and in acetonitrile solution (short dotted line); (bottom) corresponding UV-visible spectra of 7 and 8 in acetonitrile solution.

Figure 9 .

 9 Figure 9. a) Spectra of the SHS recorded for the two enantiomers (4 and 5) and the racemic mixture (3) excited by femtosecond pulses centered at λ=1050 nm. b) Evolution of the SHS intensity versus the intensity of the fundamental pulse. The blue and red dots are the experimental data for the two enantiomers, the straight line is a fit considering a quadratic dependence.

Figure 10 :

 10 Figure 10: a) Spectra of the SHS recorded in [Fe(phen)3](∆-TRISCAS)2 4 under the same experimental condition when the excitation wavelength is centred at λ = 800 nm, 1050 nm and 1350 nm respectively. These signals are normalized against the signal recorded in α-Quartz under the same experimental conditions.

ACKNOWLEDGMENT

This work was supported by the CNRS, the University of Bordeaux, the Conseil Régional de Nouvelle Aquitaine, the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 706556 CHIMMM (Posdoctoral fellowship for MC), the ANR project CHIROTS ANR-11-JS07-013-01 (PR, PhD fellowship for AN). The authors warmly thank L. Etienne and A. Fargues for technical assistance.

AUTHOR INFORMATION

Author Contributions

The manuscript was written through contributions of all authors.

Notes † Died February 2017. # See e.g structure DKSTBR in the CSD, where R,R-tartrates give a torsion angle between coordinated O atoms and the twofold axis bisecting the tartrates of +81.15° corresponding to the Λ enantiomer, while S,S-tartrates give -82.51°and the ∆ enantiomer. & We follow here the IUCr convention for spacegroups exempt of improper rotations, see e.g. http://reference.iucr.org/dictionary/Sohncke_groups


Simone Baldi
email: s.baldi@tudelft.nl

Muhammad Ridho
email: muhammadridhorosa@tudelft.nl

Paolo Frasca
email: paolo.frasca@gipsa-lab.fr

Elias B Kosmatopoulos

Platooning merging maneuvers in the presence of parametric uncertainty

Keywords: Adaptive synchronization, heterogeneous uncertain agents, adaptive control

Recently, adaptive platooning strategies to cope with uncertain vehicle parameters have been proposed. However, in line with most platooning literature, only acyclic graphs have been considered. This work addresses the merging maneuver in the presence uncertain vehicle parameters: during this maneuver, a cyclic communication graph is instantiated, which must be handled in a suitable way. Ideas used to handle this situation and corresponding results are illustrated using a platoon of three vehicles implementing a merging maneuver.

INTRODUCTION

In automated driving research, a recognized approach for improving road throughput is grouping vehicles into platoons controlled by one leading vehicle [START_REF] Günther | Platooning at traffic lights[END_REF]). In Cooperative Adaptive Cruise Control (CACC) platooning is enabled by inter-vehicle communication in addition to on-board sensors [START_REF] Marsden | Towards an understanding of adaptive cruise control[END_REF]; [START_REF] Li | Dsrc based vehicular platoon control considering the realistic V2V/V2I communications[END_REF]). Several studies have been conducted to develop CACC strategies that guarantee synchronized behavior of vehicle platoons. Under the assumption of vehicle-independent driveline dynamics (homogeneous platoon), [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF] synthesized a one-vehicle only look-ahead CACC; [START_REF] Hafez | Effect of leader information broadcasted throughout vehicle platoon in a constant spacing policy[END_REF] developed a longitudinal controller based on broadcasting the leading vehicle's acceleration and velocity to all vehicles in the platoon; [START_REF] Kianfar | A control matching model predictive control approach to string stable vehicle platooning[END_REF] integrated safety and physical constraints in CACC by a model predictive controller. Recently, communication failures have been addressed in [START_REF] Acciani | Cooperative adaptive cruise control over unreliable networks: an observerbased approach to increase robustness to packet loss[END_REF], while heterogeneity and uncertainty have been handled adaptively in Harfouch et al. (2017a). Reviews on the practical challenges of CACC were conducted by [START_REF] Dey | A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (cacc)[END_REF]; [START_REF] Larsson | The vehicle platooning problem: Computational complexity and heuristics[END_REF]. Among the challenges, a relevant one is how to include merging maneuvers in the synchronization protocol: in fact, most works consider operating a platoon under acyclic graphs [START_REF] Harfouch | An adaptive switched control approach to heterogeneous platooning with inter-vehicle communication losses[END_REF]). However, if a vehicle wants to

The research leading to these results has been partially funded by the European Commission H2020-SEC-2016-2017-1, Border Security: autonomous control systems, under contract #740593 (ROBORDER) and H2020-ICT-2014-1, FIRE+ (Future Internet Research & Experimentation), under contract #645220 (RAWFIE). This work is companion with Baldi et al. (2018).

merge in the middle of a platoon, a gap must be created for it, so that the merging vehicle needs bidirectional (and thus cyclic) communication for improved safety. Handling cycles in platooning protocols is difficult, because the input of a vehicle turns out to depend on the input of the neighbors (Baldi and Frasca (2018)): this creates algebraic loops that can make the input not well posed, and that is usually solved by assigning priorities to remove the cycles [START_REF] Wang | Hierarchical decomposition based consensus tracking for uncertain interconnected systems via distributed adaptive output feedback control[END_REF]). We remark that, differently from consensus/cooperative internal model frameworks [START_REF] Qu | Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles[END_REF]; [START_REF] Azzollini | Adaptive Synchronization over Uncertain Multi-Agent Systems: A distributed homogenization-based approach[END_REF], we are focusing on a distributed input CACC protocol mutuated from [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF], for which all aforementioned issues are open. Works considering merging maneuvers include [START_REF] Amoozadeh | Platoon management with cooperative adaptive cruise control enabled by VANET[END_REF]; [START_REF] Maiti | A conceptualization of vehicle platoons and platoon operations[END_REF] (vehicle entry and leaving via finite state machines); [START_REF] Scarinci | Definition of a merging assistant strategy using intelligent vehicles[END_REF] (creating merging gaps for on-ramp vehicles); [START_REF] Chien | Regulation layer controller design for automated highway system: Platoon merge and split controller design[END_REF] (platoon merge and split); [START_REF] Rai | Real and virtual leader-follower strategies in lane changing, merging and overtaking maneuvers[END_REF]; [START_REF] Bengtsson | Interaction protocol for highway platoon merge[END_REF]; [START_REF] Goli | Evaluation of lateral trajectories with different controllers for multi-vehicle merging in platoon[END_REF] (lane changing, merging and overtaking). Heterogeneity and uncertainty are often overlooked in the aforementioned works, and the merging maneuver is not embedded in any synchronization protocol: in this work we want to tackle these issues by showing that synchronization can be extended to the merging manuevers. In addition, we show how to exploit the graph structure to guarantee well posedness of the actual inputs at every time instant.

CACC SYSTEM STRUCTURE

Consider a heterogeneous platoon with M vehicles (Fig. 1), where v i and d i represent the velocity (m/s) of vehicle i, and the distance (m) between vehicle i and its preceding vehicle i -1, respectively. Let us define S M = {i ∈ N| 1 ≤ i ≤ M } with i = 0 reserved for the platoon's desired behavior (virtual leading vehicle). In line with most CACC literature, we will focus on the longitudinal dynamics only, while for the lateral dynamics a separate steering controller is assumed to be in place. The following longitudinal model, derived by Ploeg et al. (2011), is used

  ḋi vi ȧi   =   0 1 0 0 0 1 0 0 -1 τi   Ai d i v i a i xi +   0 0 1 τi   bi u i , i ∈ S M (1)
where a i and u i are respectively the acceleration (m/s 2 ) and external input (m/s 2 ) of the i th vehicle, τ i (s) represents each vehicle's driveline time constant. Furthermore, the virtual leading vehicle is defined as

  ḋ0 v0 ȧ0   =   0 1 0 0 0 1 0 0 -1 τ0   d 0 v 0 a 0 +   0 0 1 τ0   u 0   ḋ0 v0 ȧ0   = 0 1 0 0 0 1 a 01 a 02 a 03 Am d 0 v 0 a 0 xm + 0 0 b 00 bm r (2)
where the second equation has been obtained assuming that the lead vehicle is controlled by a state-feedback controller u 0 = k * 0 x m + l * 0 r that makes its dynamic stable: therefore a 01 , a 02 , a 03 are design parameters selected such that the matrix A m is Hurwitz. Note that, under the assumption of a homogeneous platoon, we have τ i = τ 0 , ∀i ∈ S M . In this work, we remove the homogeneous assumption by considering that ∀i ∈ S M , τ i is an unknown parameters. The motivation is that, in practice, τ i sensibly changes according to vehicle and road conditions.

The main goal of every vehicle, except the leading vehicle, is to maintain a desired distance between itself and its preceding vehicle. To this purpose, a constant distance headway (CDH) spacing policy defines the desired distance r ij (t) between vehicles i and j (r ij depends on time because it can change during the merging manuever). Then, it is possible to define the state error (spacing distance, the relative velocity, and relative acceleration) between the j th and the i th vehicle as:

e ji (t) = d j (t) v j (t) a j (t) - d i (t) v i (t) a i (t) + r ji (t) 0 0 . ( 3
)
The error (3) includes the spacing distance, the relative velocity, and relative acceleration. Because we consider platoon formations during merging maneuvers, (3) will be defined among two adiacent vehicles for which a communication link is instantiated. The control objective is to regulate e ji to zero for all such adjacent vehicles. In the next section, we present how r ij and the network topology itself change during the merging manuever.

THE SYNCHRONIZATION PROTOCOL

Fig. 2. Communication graph before/during/after merging

To describe the merging manuever, let us consider the networks in Fig. 2. Three vehicles, denoted with indices 1, 2 and 3, have uncertain dynamics

ẋ1 = A 1 x 1 + b 1 u 1 ẋ3 = A 3 x 3 + b 3 u 3 ẋ2 = A 2 x 2 + b 2 u 2 (4) where A 1 , A 2 , A 3 and b 1 , b 2 , b 3 are
unknown matrices in the form of (1). Vehicle 3 is indicated before vehicle 2 because we aim at merging vehicle 3 in between vehicles 1 and 2. Consider the virtual leader ẋm = A m x m + b m r (5) where A m and b m are known matrices in the form of (2). Fig. 2 shows that, before attempting to merge, vehicle 3 aligns to vehicle 2 (network 1). When the merging starts (network 2), a cyclic graph appears (bidirectional link between vehicles 2 and 3) and vehicle 2 increases its distance from vehicle 1. The bidirectional link between vehicle 2 and vehicle 3 is used for safety reasons by vehicle 2 to watch the behavior of vehicle 3 and vice versa (as it happens in merging manuevers operated by humans). Finally, in network 3, the merging is complete and a new acyclic directed network is established between the three vehicles. The following CDH spacing policies are considered:

• Network 1: r 32 = 0 and r 21 = ρ for a certain design parameter ρ; • Network 2: r 21 increases linearly from ρ to 2ρ, r 32 decreases linearly from 0 to -ρ, and r 31 = ρ; • Network 3: r 31 = ρ and r 23 = ρ.

Being the system matrices in (4) unknown, the synchronization task has to be achieved in an adaptive fashion. The following result, mutuated from (Baldi and Frasca (2018)), justifies that the adaptive problem is well posed in the sense of [START_REF] Tao | Adaptive Control Design and Analysis[END_REF]; [START_REF] Ioannou | Robust Adaptive Control[END_REF]). Proposition 1. [Distributed matching conditions] For dynamics in the form (1) and (2), there exist vectors k

* 1 , k * 2 , k * 3 and scalars l * 1 , l * 2 , l * 3 such that A m = A 1 + b 1 k * 1 , b m = b 1 l * 1 A m = A 2 + b 2 k * 2 , b m = b 2 l * 2 A m = A 3 + b 3 k * 3 , b m = b 3 l * 3 . (6)
In addition, the signs of l * 1 , l * 2 , l * 3 are positive, and there exist vectors

k * 21 = k * 2 -k * 1 l * 2 /l * 1 , k * 31 = k * 3 -k * 1 l * 3 /l * 1 , k * 32 = k * 3 -k * 2 l * 3 /l * 2 , k * 23 = k * 2 -k * 3 l * 2 /l * 3 and scalars l * 21 = l * 2 /l * 1 , l * 31 = l * 3 /l * 1 , l * 32 = l * 3 /l * 2 , l * 23 = l * 2 /l * 3 such that A 1 = A 2 + b 2 k * 21 , b 1 = b 2 l * 21 A 1 = A 3 + b 3 k * 31 , b 1 = b 3 l * 31 A 2 = A 3 + b 3 k * 32 , b 2 = b 3 l * 32 A 3 = A 2 + b 2 k * 23 , b 3 = b 2 l * 23 . (7)
The adaptive controller is now presented: because the controller for the acyclic networks 1 and 3 can be easily derived using the approach in (Baldi and Frasca (2018)), in the following we will focus on the cyclic network 2.

The adaptive controller

The synchronization of vehicle 1 to the reference model is the well-known model reference adaptive control (Tao, 2003, Chap. 4): it amounts to the controller u 1 (t) = k 1 (t)x 1 (t) + l 1 (t)r(t) (8) and to the adaptive laws

k 1 (t) = -γ k b m P e 1 (t)x 1 (t) l1 (t) = -γ l b m P e 1 (t)r(t) (9
) where e 1 = e 10 = x 1 -x m , k 1 , l 1 are the estimates of k * 1 , l * 1 , the scalars γ k , γ l > 0 are adaptive gains, and P is a positive definite matrix satisfying 32 respectively. Then, provided that the inputs are well defined at very time instant, vehicle 2 synchronizes to vehicles 1 and 3, while vehicle 3 synchronizes to vehicles 1 and 2 (i.e. merging is achieved in network 2). Proof 1. Proving synchronization exploits the Lyapunov function

P A m + A m P = -Q, Q > 0. ( 10
u 3 (t) = k 31 (t) x 1 (t) 2 + k 32 (t) x 2 (t) 2 + k 3 (t)
V 1 + V 231 + V 321 , where V 1 = e 1 P e 1 + tr k i ki γ k |l * i | + l2 i γ l |l * i | V 231 = e 231 P e 231 + tr k 21 k21 γ k |l * 2 | + tr k 23 k23 γ k |l * 2 | +tr k 2 k2 γ k |l * 2 | + l2 21 γ l |l * 2 | + l2 23 γ l |l * 2 |
V 321 = e 321 P e 321 + tr

k 31 k31 γ k |l * 3 | + tr k 32 k32 γ k |l * 3 | +tr k 3 k3 γ k |l * 3 | + l2 31 γ l |l * 3 | + l2 32 γ l |l * 3 | . ( 15
)
and the error dynamics, as depicted in Fig. 3 are 16) where e 231 = e 21 +e 23 and e 321 = e 31 +e 32 . Using standard Lyapunov arguments and the Barbalat's lemma we can show V1 + V231 + V321 → 0 as t → ∞ and hence all errors go to zero.

ė1 = A m e 21 + b 1 ( k 1 x 1 + l1 r) ė231 = A m e 231 + b 2 ( k 21 x 1 + k 2 e 21 + l21 u 1 ) +b 2 ( k 23 x 3 + k 2 e 23 + l23 u 3 ) ė321 = A m e 321 + b 3 ( k 31 x 1 + k 3 e 31 + l31 u 1 ) +b 3 ( k 32 x 2 + k 3 e 32 + l32 u 2 ) (

Fig. 3. The synchronization errors

Theorem 1 assumes that the inputs are well defined at very time instant. Therefore, the presence of a cycle in network 2 requires us to find some well-posedness conditions on the input, as discussed in next section.

WELL-POSEDNESS OF THE INPUT

By considering network 2 in Fig. 2, the inputs to the three vehicles can be written as or, in a more compact matrix form

u 1 (t) = k 1 (t)x 1 (t) + l 1 (t)r(t) 2u 2 (t) = k 21 (t)x 1 (t) + k 2 (t)(x 2 (t) -x 1 (t)) + l 21 (t)u 1 (t) + k 31 (t)x 3 (t) + k 2 (t)(x 2 (t) -x 3 (t)) + l 23 (t)u 3 (t) 2u 3 (t) = k 31 (t)x 1 (t) + k 3 (t)(x 3 (t) -x 1 (t)) + l 31 (t)u 1 (t) + k 32 (t)x 2 (t) + k 3 (t)(x 3 (t) -x 2 (t)) + l 32 (t)u 2 (t). ( 17
1 0 0 -l 21 2 -l 23 -l 31 -l 32 2 U u 1 u 2 u 3 = k 1 x 1 + l 1 r (k 21 -k 2 )x 1 + 2k 2 x 2 + (k 31 -k 2 )x 3 (k 31 -k 3 )x 1 + (k 32 -k 3 )x 2 + 2k 3 x 3 .
Even though the vehicles do not need to invert U to obtain their inputs, if we want to guarantee that u 1 , u 2 , and u 3 are well posed at all time steps, we need the matrix U to be invertible. To this purpose, let us calculate the determinant of U , so as to obtain det

1 0 0 -l 21 2 -l 23 -l 31 -l 32 2 = 4 -l 23 l 32 . (18)
In the ideal case (with the actual parameters from Proposition 1) l * 23 l * 32 = 1, giving an ideal determinant equal to 3. However, in the actual case with the estimated parameters, the determinant of U can take arbitrary values and even result equal to 0. This would make the inputs u 1 , u 2 , and u 3 not well posed at all time steps. A simple approach to guarantee well posedness of the inputs at all time steps is to allow vehicle 2 and vehicle 3 to exchange their estimates l 23 (t) and l 32 (t). This way it is possible to project the estimates in such way that l 23 (t)l 32 (t) = 4 and the matrix U is always invertible. The following assumption is made. Assumption 1. The actual parameters l * 23 and l * 32 are known to reside in a convex compact set (call it Ω l ) that does not contain the set l * 23 l * 32 = 4.

An example of Ω l (among infinite other choices) is l * 23 ≥ 0, l * 32 ≥ 0, l * 23 ≤ -l * 32 + 3.99 as represented in Fig. 4. In general, the set Ω l can be written as Ω l = {l 23 , l 32 | g(l 23 , l 32 ) ≤ 0} (19) for some appropriate vector function g(l 23 , l 32 ). The following main result follows. Theorem 2. Consider the merging phase described by network 2 in Fig. 2. Under Assumption 1, consider the three vehicles described by (4) and the leading vehicle described by (5), the controllers (8), ( 11), ( 13) and the adaptive laws (9), ( 12), ( 14) with the following modifications

l23 (t) = P Ω l [-γ l b m P (e 21 (t) + e 23 (t))u 3 (t) δ l23 (t) ] (20) = δ l23 (t) if l 23 (t) ∈ Ω l , or l 32 (t) ∈ δ(Ω l ) with δ l23 ∇g ≤ 0 0 otherwise l32 (t) = P Ω l [-γ l b m P (e 31 (t) + e 32 (t))u 2 (t) δ l32 (t) ] = δ l32 (t) if l 32 (t) ∈ Ω l , or
l 32 (t) ∈ δ(Ω l ) with δ l32 ∇g ≤ 0 0 otherwise where P Ω l has been defined as a projection operator in the set Ω l . In particular, δ(Ω l ) is the border of Ω l and ∇g is the derivative of g with respect to l 23 or l 32 . Then, merging is achieved in network 2, i.e. e 1 , e 21 , e 23 , e 31 , e 32 → 0. Proof 2. The proof exploits again the Lyapunov function ( 15), and it follows the same lines as adaptive control designs with parameter projection (Ioannou and Sun, 2012, Sects. 6.6 and 8.5). In fact, we have V1 + V231 + V321 ≤ -e 1 Qe 1 -e 231 Qe 231 -e 321 Qe 321 + V p where

V p (t)      = 0 if l 23 (t), l 32 (t) ∈ Ω l , or
l 23 (t) ∈ δ(Ω l ) with δ l23 ∇g ≤ 0, or l 32 (t) ∈ δ(Ω l ) with δ l32 ∇g ≤ 0 ≤ 0 otherwise i.e. V p is a term that due to the convexity of the projection set Ω l verifies V p ≤ 0. Therefore, V p can only make the derivative of the Lyapunov function more negative (Ioannou and Sun, 2012, Sects. 6.6 and 8.5). Hence, V1 + V231 + V321 ≤ -e 1 Qe 1 -e 231 Qe 231 -e 321 Qe 321 and stability follows from Barbalat's lemma as in Theorem 1. The details are left to the reader for lack of space.

Remark 1. Theorem 2 basically states that the structure of the network can be exploited to implement appropriate projection laws (cf. (21)) that make the input well posed at every time instant, even in the presence of cycles. The parameters of the reference model are taken as: a 01 = -4, a 02 = -6, a 03 = -4, and b 00 = 1, while the dynamics of the vehicles in (1) are unknown. Table 1 shows the parameter used to simulate each vehicle i, together with their initial conditions. The reference signal r is taken to be a ramp. The simulations are carried out at low speed (around 2.5m/s) only to better visualize the gaps between vehicles : we have verified that the proposed approach works also at higher speeds. The design parameter are taken as: Q = diag(1, 1, 5), ρ = 7m, the adaptive gains γ k = 0.005, γ l = 0.001, and all coupling gains, k i , k ij , l i , l ij , are initialized to 0. The maneuver is organized as:

NUMERICAL EXAMPLE

-0-30 s: vehicle 3 aligns with vehicle 2, while vehicles 1 and 2 achieve the initial formation. -30-50s: vehicle 2 creates an increasing gap for vehicle 3, while vehicle 3 starts the merging. -50-60s: the final formation is achieved.

Theorems 1 and 2 do not consider switching topologies (cf. Fig. 2). During the different merging phases changes, vehicles 2 and 3 end up having a different number of neighbors, which require to implement a different controller, one for each different topology. Therefore, a switching controller scheme, shown in Fig. 5, and resembling the multiple model control architecture [START_REF] Hespanha | Hysteresis-based switching algorithms for supervisory control of uncertain systems[END_REF]; [START_REF] Baldi | Multi-model unfalsified adaptive switching supervisory control[END_REF]) will be used in this work. Note that vehicles 2 and 3 have: one neighbor in network 1 (vehicle 1 and vehicle 3, respectively); two neighbors in network 2 (vehicles 1 and 3 and vehicles 1 and 2, respectively); one neighbor in network 3 (vehicle 3 and vehicle 2, respectively). Therefore, three adaptive controllers are possible for vehicle 2 and vehicle 3, whose activation depends on the active communication graph during the merging phase (cf. Fig. 2). It was demonstrated that each standalone networked adaptive controller (the two ones derived from (Baldi and Frasca (2018)) for the acyclic networks 1 and 3, and the one derived from Theorem 2 for the cyclic network 2) is stable. The stability of the resulting controller in the presence of switching will be the subject of further studies, using tools from adaptive switched control [START_REF] Sang | Adaptive control of piecewise linear systems: the state tracking case[END_REF]; [START_REF] Yuan | Adaptive asymptotic tracking control of uncertain time-driven switched linear systems[END_REF]). Such literature has shown that when switching among stable adaptive systems occurs, there exists a dwell time for which stability can be derived. The simulation in the next section are performed to show the stability of such switched architecture: note that the proposed merging maneuver has a dwell time of 20 s. . 6, 7, 8, and 9 show the response of p i , v i , a i , and u i , respectively. In Fig. 6, we can observe, in the interval 0-30 seconds (network 1), that vehicles 2 and 3 are at a distance ρ from vehicle 1, at the same time vehicle 1 synchronize to reference model. Then, in the interval 30-50 seconds (network 2), vehicle 2 makes a gap by increasing the distance with vehicle 1 in order to allow vehicle 3 to merge in between vehicle 1 and vehicle 2. Finally, in the interval 50-60 seconds (network 3), vehicle 3 is located at

CONCLUSIONS

While most platooning literature has focused on acyclic graphs, the merging maneuver requires to handle a cyclic graph. This makes synchronization more difficult, because the input of a vehicle depends on the input of the neighbors which might create not well-defined inputs. In this work we have shown that it is possible to exploit the graph structure to implement appropriate parameter projection and guarantee well posedness of the actual inputs. Future work will include considering unmatched uncertainties [START_REF] Lymperopoulos | Adaptive control of networked distributed systems with unknown interconnections[END_REF]; [START_REF] Romagnuolo | Estimating uncertainties in cooperative networks[END_REF]). Furthermore, to overcome the technical issues that come from switching (Section 5), it might be interesting to consider smoothing/mixing mechanisms [START_REF] Kuipers | Multiple model adaptive control with mixing[END_REF]; [START_REF] Baldi | Stability margins in adaptive mixing control via a Lyapunov-based switching criterion[END_REF]).

Fig. 1 .

 1 Fig. 1. CACC-equipped heterogeneous vehicle platoon (Ploeg et al. (2014))

  k 21 (t) = -γ k b m P (e 21 (t) + e 23 (t))x 1 (t) k 23 (t) = -γ k b m P (e 21 (t) + e 23 (t))x 3 (t) k 2 (t) = -γ k b m P (e 21 (t) + e 23 (t))(e 21 (t) + e 23 (t)) l21 (t) = -γ l b m P (e 21 (t) + e 23 (t))u 1 (t) l23 (t) = -γ l b m P (e 21 (t) + e 23 (t))u 3 (t) (12) where k 21 , k 23 , k 2 , l 21 , l 23 are the estimates of k * 21 , k * 23 , k * 2 , l * 21 , l * 23 respectively. Also, consider the controller

)Fig. 4 .

 4 Fig. 4. Singular set (red curve) and projection set (shaded blue area)

Fig. 5 .

 5 Fig. 5. The switching adaptive control for vehicle k

Fig. 6 .

 6 Fig. 6. The position response

Figs

  Figs. 6, 7, 8, and 9 show the response of p i , v i , a i , and u i , respectively. In Fig.6, we can observe, in the interval 0-30 seconds (network 1), that vehicles 2 and 3 are at a distance ρ from vehicle 1, at the same time vehicle 1 synchronize to reference model. Then, in the interval 30-50 seconds (network 2), vehicle 2 makes a gap by increasing the distance with vehicle 1 in order to allow vehicle 3 to merge in between vehicle 1 and vehicle 2. Finally, in the interval 50-60 seconds (network 3), vehicle 3 is located at

Fig. 7 .

 7 Fig. 7. The velocity response

  31 , k 32 , k 3 , l 31 , l 32 are the estimates of k *

	+l 31 (t) and the adaptive laws u 1 (t) 2 k 31 (t) = -γ k b m P (e 31 (t) + e 32 (t))x 1 (t) + l 32 (t) u 2 (t) 2 k 32 (t) = -γ 31 , k * (13) 32 , k * 3 , l * 31 , l *

e 31 (t) + e 32 (t) 2 k b m P (e 31 (t) + e 32 (t))x 2 (t) k 3 (t) = -γ k b m P (e 31 (t) + e 32 (t))(e 31 (t) + e 32 (t)) l31 (t) = -γ l b m P (e 31 (t) + e 32 (t))u 1 (t) l32 (t) = -γ l b m P (e 31 (t) + e 32 (t))u 2 (t) (14)

where k

Table 1 .

 1 Vehicles parameters and initial conditions

	Vehicle i	τ i	x i (0)
	Vehicle 1	0.5	[-2,1,0]
	Vehicle 2 0.33 [-15,2,1]
	Vehicle 3	0.2	[-20,2,1]