Hanna Oher

Thomas Vercouter

Florent Réal

Chengming Shang

Pascal E Reiller

Valérie Vallet

Influence of alkaline earth metal ions on

come

Influence of Alkaline Earth Metal Ions on

INTRODUCTION

After light excitation in the UV or visible range, most uranium(VI) compounds produce luminescence in the energy range 15 000 -22 000 cm -1 (650-450 nm), which has been extensively studied since the spectra are fingerprints of the coordination of the uranyl ion UO2 2+ . The positions of the luminescence bands, band spacings, bandwidths, and intensities are features that can be interpreted to determine the nature and number of ligands in the first coordination layer of UO2 2+ , and the local symmetry of the complex. This information is looked for by chemists in order to give the best description of uranium speciation both in solutions and in solids. Time-resolved laserinduced fluorescence spectroscopy (TRLFS) has been widely used for the identification and characterization of U(VI) compounds since it is a non-intrusive, sensitive, and selective technique to detect the luminescence of U(VI) species, even with traces of uranium. [START_REF] Moulin | Uranium Speciation in Solution by Time-Resolved Laser-Induced Fluorescence[END_REF][START_REF] Moulin | Time-Resolved Laser-Induced Fluorescence as a Unique Tool for Low-Level Uranium Speciation[END_REF] The main spectroscopic features are usually reported in the literature and spectral attributions have been proposed for many aqueous inorganic complexes [START_REF] Eliet | Characterization of Hydroxide Complexes of Uranium(VI) by Time-Resolved Fluorescence Spectroscopy[END_REF][START_REF] Meinrath | Lifetime and Fluorescence Quantum Yield of Uranium(VI) Species in Hydrolyzed Solutions[END_REF][START_REF] Moll | Interaction of Uranium(VI) with Silicic Acid in Aqueous Solutions Studied by Time-Resolved Laser-Induced Fluorescence Spectroscopy (TRLFS)[END_REF][START_REF] Couston | Speciation of Uranyl Species in Nitric-Acid Medium by Time-Resolved Laser-Induced Fluorescence[END_REF][START_REF] Geipel | Uranium(VI) Sulfate Complexation Studied by Time-Resolved Laser-Induced Fluorescence Spectroscopy (TRLFS)[END_REF][START_REF] Lee | Formation of Ternary CaUO2(CO3)3 2and Ca2UO2(CO3)3(Aq) Complexes under Neutral to Weakly Alkaline Conditions[END_REF][START_REF] Vercouter | Stoichiometries and Thermodynamic Stabilities for Aqueous Sulfate Complexes of U(VI)[END_REF] and minerals. [START_REF] Geipel | Spectroscopic Properties of Uranium(VI) Minerals Studied by Time-Resolved Laser-Induced Fluorescence Spectroscopy (TRLFS)[END_REF][START_REF] Lehmann | Syntheses and Spectroscopic Characterization of Uranium(VI) Silicate Minerals[END_REF][START_REF] Massuyeau | Spectroscopic Markers for Uranium(VI) Phosphates. Part II: The Use of Time-Resolved Photoluminescence[END_REF][START_REF] Wang | A Cryogenic Fluorescence Spectroscopic Study of Uranyl Carbonate, Phosphate and Oxyhydroxide Minerals[END_REF][START_REF] Amayri | Spectroscopic Characterization of the Uranium Carbonate Andersonite Na2Ca[UO2(CO3)3]•6H2O[END_REF][START_REF] Amayri | Spectroscopic Characterization of Alkaline Earth Uranyl Carbonates[END_REF][START_REF] Amayri | Spectroscopic Characterization of Synthetic Becquerelite, Ca[(UO2)6O4(OH)6]•8H2O, and Swartzite, CaMg[UO2(CO3)3]•12H2O[END_REF][START_REF] Arnold | K(UO2)(SiO3OH)(H2O)1.5] and Compreignacite K2[(UO2)3O2(OH)3]2•7H2O Characterized by Laser Fluorescence Spectroscopy[END_REF][START_REF] Deneufville | Selective Detection of Uranium by Laser-Induced Fluorescence: A Potential Remote-Sensing Technique. 1: Optical Characteristics of Uranyl Geologic Targets[END_REF] The case of the carbonate species is particularly interesting. [START_REF] Lee | Formation of Ternary CaUO2(CO3)3 2and Ca2UO2(CO3)3(Aq) Complexes under Neutral to Weakly Alkaline Conditions[END_REF][START_REF] Wang | A Cryogenic Fluorescence Spectroscopic Study of Uranyl Carbonate, Phosphate and Oxyhydroxide Minerals[END_REF][START_REF] Amayri | Spectroscopic Characterization of Alkaline Earth Uranyl Carbonates[END_REF][START_REF] Prat | Uranium Speciation in Drinking Water from Drilled Wells in Southern Finland and Its Potential Links to Health Effects[END_REF][START_REF] Bernhard | Uranyl(VI) Carbonate Complex Formation: Validation of the Ca2UO2(CO3)3(Aq.) Species. Radiochim[END_REF][START_REF] Shang | Determination of Formation Constants and Specific Ion Interaction Coefficients for Can:NUO2(CO3)n (4-2n)-Complexes in NaCl Solution by Time-Resolved Laser-Induced Luminescence Spectroscopy[END_REF] Both anionic complexes UO2(CO3)2 2-and UO2(CO3)3 [START_REF] Meinrath | Lifetime and Fluorescence Quantum Yield of Uranium(VI) Species in Hydrolyzed Solutions[END_REF]-are characterized by an absorption spectrum shifted to longer wavelengths as compared with UO2 2+ aqua ion and its hydrolyzed species. [START_REF] Scanlan | Equilibria in Uranyl Carbonate Systems---II: The Overall Stability Constant of UO2(CO3)2 2and the Third Formation Constant of and the Third Formation Constant of UO2(CO3)3<sup>4[END_REF][START_REF] Meinrath | Direct Spectroscopic Speciation of Uranium(VI) in Carbonate Solutions[END_REF] Meinrath et al.

reported 20 years ago that the luminescence of UO2(CO3)2 2-seemed to be too weak to be easily observed. [START_REF] Meinrath | Lifetime and Fluorescence Quantum Yield of Uranium(VI) Species in Hydrolyzed Solutions[END_REF] Conversely, recent works have shown that the use of modern TRLFS set-up allows measuring UO2(CO3)3 [START_REF] Meinrath | Lifetime and Fluorescence Quantum Yield of Uranium(VI) Species in Hydrolyzed Solutions[END_REF]-at ambient temperature. [START_REF] Shang | Determination of Formation Constants and Specific Ion Interaction Coefficients for Can:NUO2(CO3)n (4-2n)-Complexes in NaCl Solution by Time-Resolved Laser-Induced Luminescence Spectroscopy[END_REF][START_REF] Jung | Time-Resolved Laser Fluorescence Spectroscopy of UO2(CO3)3 4[END_REF] It has been shown that the U(VI) luminescence is increasing when alkaline earth metal ions are added into a solution that contains UO2(CO3)3 4-, i.e., in the presence of dissolved bicarbonate. The repartition of these species (and solids) can be described in e.g. a log10(a(HCO3 -)/a(H + )) vs. log10(a(Ca 2+ )/a(H + )²) plan for Ca. There was a direct evidence of the formation of ternary complexes CaUO2(CO3)3 2-and Ca2UO2(CO3)3 where the calcium ions enter the second coordination layer of the uranyl and interact with the carbonate ligands, as first reported by Bernhard et al., [START_REF] Bernhard | Speciation of Uranium in Seepage Waters of a Mine Tailing Pile Studied by Time-Resolved Laser-Induced Fluorescence Spectroscopy (TRLFS)[END_REF] and investigated later on by several groups. 8,19,21,26- 31 The luminescence of such complexes exhibits a hypsochromic shift of the whole spectrum by about 300 cm -1 compared with the free ion, when most other U(VI) compounds, like phosphates, silicates, hydroxides, or organic complexes, are showing a bathochromic shift. The blue-shifted luminescence of the triscarbonatouranyl(VI) entities results from the strong interaction of UO2 2+

with the carbonate ligands that bind the uranium atom with two out of their three oxygens in the equatorial plane of the uranyl ion, leading to a six-fold-coordinated uranyl structure. Thanks to this unusual feature, TRLFS has enabled the direct identification of alkaline earth triscarbonatouranyl carbonate complexes in many different types of waters like e.g., seepage water of mine tailing pile, [START_REF] Bernhard | Speciation of Uranium in Seepage Waters of a Mine Tailing Pile Studied by Time-Resolved Laser-Induced Fluorescence Spectroscopy (TRLFS)[END_REF] granitic groundwater, [START_REF] Baik | Determination of Uranium Concentration and Speciation in Natural Granitic Groundwater Using TRLFS[END_REF] seawater, [START_REF] Maloubier | XAS and TRLIF Spectroscopy of Uranium and Neptunium in Seawater[END_REF] natural drinking water from wells, [START_REF] Prat | Uranium Speciation in Drinking Water from Drilled Wells in Southern Finland and Its Potential Links to Health Effects[END_REF] and in sediment pore waters of the Hanford contaminated sites. [START_REF] Wang | Cryogenic Laser Induced Fluorescence Characterization of U(VI) in Hanford Vadose Zone Pore Waters[END_REF] The reason of the increase of the triscarbonatouranyl species luminescence intensity in presence of calcium is still unclear and may originate from either favorable vibronic coupling or diminution of quenching effects from the carbonate ion. Similar observations have been made with the less stable ternary MgUO2(CO3)3 2-and SrUO2(CO3)3 2-complexes, [START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF][START_REF] Beccia | New Insight into the Ternary Complexes of Uranyl Carbonate in Seawater[END_REF] while the confirmation of the existence of the ternary Mg2UO2(CO3)3 complex is still discussed. [START_REF] Jo | Formation of CaUO2(CO3)3 2and Ca2UO2(CO3)3(Aq) Complexes at Variable Temperatures (10-70 °C)[END_REF][START_REF] Geipel | Mixed Complexes of Alkaline Earth Uranyl Carbonates: A Laser-Induced Time-Resolved Fluorescence Spectroscopic Study[END_REF][START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF] Quantum chemical modeling is an appropriate theoretical support for a better understanding of the luminescence of these ternary triscarbonatouranyl complexes with alkaline earth metal cations.

The luminescence of solid uranyl carbonates has been successfully described and analyzed using periodic density functional theory (DFT), particularly in the case of liebigite Ca2UO2(CO3)3•11H2O and the dehydrated Ca2UO2(CO3)3 crystal. [START_REF] Kalashnyk | Combined Experimental and First-Principles Studies of a Hydrated Uranyl Carbonate: Insight into Phonon Spectra for a Core Environmental Class of Uranium Materials[END_REF] While it is established that the luminescence of uranyl compounds is related to transitions from 5f and 5f excited levels, the nature of the ground state orbitals still needs to be more precisely defined for the aqueous uranyl carbonate complexes. This constitutes a first objective of the present work with the support of quantum chemistry modeling. For this aim, triscarbonatouranyl model structures were built with calcium, magnesium and sodium counterions (from 1 to 3 Na + depending on the amount of alkaline earth metal present) to maintain the highest symmetry and reduce the total molecular charge.

Investigating luminescence properties from a uranyl complex requires a step-by-step methodology. The geometry of the complexes first needs to be correctly built both in the ground and excited states, which can be checked by comparison with structural studies on triscarbonatouranyl complexes, with and without calcium. On the one hand, structures and atomic distances were determined by X-Ray Diffraction (XRD), [START_REF] Allen | NMR, Raman, EXAFS, and X-Ray Diffraction Studies of Uranyl Carbonate Complexes in Near-Neutral Aqueous Solution. X-Ray Structure of[END_REF] and Extended X-ray Absorption Fine Structure (EXAFS) [START_REF] Amayri | Spectroscopic Characterization of Alkaline Earth Uranyl Carbonates[END_REF][START_REF] Bernhard | Uranyl(VI) Carbonate Complex Formation: Validation of the Ca2UO2(CO3)3(Aq.) Species. Radiochim[END_REF][START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF][START_REF] Kelly | X-Ray Absorption Spectroscopy Identifies Calcium-Uranyl-Carbonate Complexes at Environmental Concentrations[END_REF][START_REF] Ikeda | Comparative Study of Uranyl (VI) and-(V) Carbonato Complexes in an Aqueous Solution[END_REF] both in crystal form and in solution, respectively. Vibration frequencies are also available from measurements by Raman and Infrared spectroscopy. [START_REF] De | Complexation of the Carbonate, Nitrate, and Acetate Anions with the Uranyl Dication: Density Functional Studies with Relativistic Effective Core Potentials[END_REF][START_REF] Kalashnyk | Exploring Optical and Vibrational Properties of the Uranium Carbonate Andersonite with Spectroscopy and First-Principles Calculations[END_REF][START_REF] Madic | Raman Spectrometric Studies of Actinide (V) and-(VI) Complexes in Aqueous Sodium Carbonate Solution and of Solid Sodium Actinide (V) Carbonate Compounds[END_REF][START_REF] Maya | A Raman Spectroscopy Study of Hydroxo and Carbonato Species of the Uranyl (VI) Ion[END_REF] On the other hand, theoretical methods were applied to triscarbonatouranyl complex by DFT [START_REF] Doudou | Investigation of Ligand Exchange Reactions in Aqueous Uranyl Carbonate Complexes Using Computational Approaches[END_REF] or hybrid DFT, [START_REF] Kubicki | Quantum Mechanical Calculation of Aqueuous Uranium Complexes: Carbonate, Phosphate, Organic and Biomolecular Species[END_REF] Quantum Mechanical Charge Field Molecular Dynamics (QMCF-MD), [START_REF] Tirler | Structure and Dynamics of the Uranyl Tricarbonate Complex in Aqueous Solution: Insights from Quantum Mechanical Charge Field Molecular Dynamics[END_REF] or classical molecular dynamics (MD) simulations. [START_REF] Doudou | Investigation of Ligand Exchange Reactions in Aqueous Uranyl Carbonate Complexes Using Computational Approaches[END_REF][START_REF] Li | Effect of Salt on the Uranyl Binding with Carbonate and Calcium Ions in Aqueous Solutions[END_REF] Then one needs to calculate and identify the excited states that may be involved in the electronic transitions. TD-DFT has demonstrated its ability to reproduce electronic transitions in uranyl compounds from low-lying excited states, including relativistic effects, electron correlation, and spin-orbit coupling. [START_REF] Tecmer | Electronic Spectroscopy of UO2 2+ , NUO + and NUN: An Evaluation of Time-Dependent Density Functional Theory for Actinides[END_REF][START_REF] Oher | Investigation of the Luminescence of [UO2X4] 2-(X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations[END_REF] The calculation of the coupling factors between the different vibrational states of the ground and excited states can be achieved using the Franck-Condon approach for dipole-allowed transitions, to finally describe luminescence spectra.

Quantum chemical methods have already been applied to simulate the luminescence properties of uranyl chloride crystals [START_REF] Watkin | Structure of Dicaesium Tetrachlorodioxouranium (VI)[END_REF] and complexes, [START_REF] Su | Theoretical Study of the Luminescent States and Electronic Spectra of UO2Cl2 in an Argon Matrix[END_REF][START_REF] Görller-Walrand | Spectroscopic Properties of Uranyl Chloride Complexes in Non-Aqueous Solvents[END_REF] uranyl fluoride, [START_REF] Su | Excited States and Luminescent Properties of UO2F2 and Its Solvated Complexes in Aqueous Solution[END_REF] and uranyl glycine complex. [START_REF] Su | Uranyl-Glycine-Water Complexes in Solution: Comprehensive Computational Modeling of Coordination Geometries, Stabilization Energies, and Luminescence Properties[END_REF] We recently applied a methodology developed by Tecmer et al., [START_REF] Tecmer | Electronic Spectroscopy of UO2 2+ , NUO + and NUN: An Evaluation of Time-Dependent Density Functional Theory for Actinides[END_REF] and tested on tetra-halogen compounds UO2Cl4 2-that resulted in simulated luminescence spectra in good accordance with experimental data. [START_REF] Oher | Investigation of the Luminescence of [UO2X4] 2-(X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations[END_REF] Our approach based on DFT and time-dependent DFT, offers a good compromise between the accuracy of the calculations and a rather low computational cost, which enables the application of the method to larger molecular assemblies, especially with actinides.

In the present work, we optimized the geometries of the NamMnUO2(CO3)3 (m+2n-4) (M=Mg, Ca; Luminescence measurements. A 2 mL U(VI) solution aliquot was taken from the prepared test sample and placed in a 1 cm quartz fluorescence cuvette (QS101, Suprasil, Hella Analytics). A preliminary equilibration, period of 10 min at 22℃ for the used cuvette has been reached prior to data acquisition using a thermostated sample holder connected to a thermostat with circulating water (Peter Huber Kältemaschinenbau AG, Germany). As described fully elsewhere, [START_REF] Shang | Determination of Formation Constants and Specific Ion Interaction Coefficients for Can:NUO2(CO3)n (4-2n)-Complexes in NaCl Solution by Time-Resolved Laser-Induced Luminescence Spectroscopy[END_REF] the excitation laser beam was generated by a 355 nm tripled Nd:YAG laser (Surelite, Continuum, USA) coupled to an optical parametric oscillator system (Horizon, Continuum, USA), delivering 5 ns pulses at 10 Hz. The excitation wavelength was set at 450 nm, allowing an optimal luminescence signal -proper balance between quantum yield of U(VI) species and laser energyafter a preliminary test. The excitation energy at 450 nm was monitored using a RJP-734 Joulemeter (Laser Probe, Inc, and was about 23 mJ.

A spectrometer (Acton SP-2358i, Roper Scientific) equipped with an intensified CCD camera (iStar DH734-25F-03, Andor Technology) was used to collect the luminescence signal with either a 300 or a 1800 lines mm -1 grating . The 300 lines mm -1 grating capable of recording the convolution of various fluorescing components was first selected to measure the full uranyl emission spectra covering the wavelength range of 425 nm to 615 nm with a resolution estimated at 0.2 nm for each sample solution. The 1800 lines mm -1 grating providing a resolution estimated at 0.025 nm was then used to focus on specific characteristic peaks with the aim of detecting their possible modifications as a function of Mg/(Ca + Mg) ratios, as well as aqueous compositions in terms of MnUO2(CO3)3 (4-2n)-4 species. The time-resolved measurements were carried out at a delay time of D = 25 ns after the laser pulse and during a measurement gate width of W = 1 µs. Each spectrum was the result of signal accumulated over more than 4000 laser shots depending on the grating used.

COMPUTATIONAL DETAILS

The simulation of the luminescence spectra of uranyl complexes were done using the ezSpectrum 3.0 [START_REF] Mozhayskiy | Iopenshell.Usc.Edu/Downloads[END_REF] program to compute the Franck-Condon factors, corresponding to the overlaps between the luminescent and the ground vibrational wavefunctions. As in our previous work, Duschinksy rotations of the normal modes were included. [START_REF] Oher | Investigation of the Luminescence of [UO2X4] 2-(X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations[END_REF] The numbers of vibrational quanta in excited and ground state were selected to be one and five, respectively to match the number of transitions on experimental spectra. All the stick spectra were computed at 300 K. In order to be able to make a comparison of theoretical results with experimental curves, the calculated peaks were broadened homogeneously using Lorentzian functions with a width corresponding to the experimental values of the full width at half-maximum. For the spectra simulations all data were generated by the ab initio packages as described below.

The interactions between uranyl and its first and second coordination spheres might affect the electronic structure of the uranyl unit. In order to keep reasonable computational costs, the ab initio calculations were performed on model molecular systems built to represent the triscarbonatouranyl negatively charged complex charge-compensated by counter cations in its close environment (Fig. 1). In the present study, the triscarbonatouranyl complex was built with three bidentate carbonate ligands in the equatorial plane of the uranyl ion, matching a D3h symmetry (Fig. 1a), as evidenced by EXAFS solution data. [START_REF] Ikeda | Comparative Study of Uranyl (VI) and-(V) Carbonato Complexes in an Aqueous Solution[END_REF][START_REF] Docrat | X-Ray Absorption Spectroscopy of Tricarbonatodioxouranate(V), [UO2(CO3)3] 5-, in Aqueous Solution[END_REF] Then Na + counter ions were positioned in between the carbonate ligands. The Na3UO2(CO3)3 -model complex was first created to reduce the complex total charge to minus one while keeping the D3h symmetry, and taken as a reference structure (Fig. 1b), instead of the bare UO2(CO3)3 4-complex, as such a highly charged anion is necessarily stabilized in solution with charge-balancing cations. Then structures in which one or two Na + cations are substituted by Ca 2+ or Mg 2+ were investigated (Fig. 1c andd, respectively), leading to neutral and mono-charged cationic complex, respectively. The geometries of these model systems were computed with inclusion of solvent effect to imitate experimental conditions. The QM methods were used as described and justified in our previous study. [START_REF] Oher | Investigation of the Luminescence of [UO2X4] 2-(X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations[END_REF] All the groundstate and excited-state molecular geometries were optimized including the relativistic effects at the spin-free level using the PBE0 functional of the density, [START_REF] Ernzerhof | Assessment of the Perdew--Burke--Ernzerhof Exchange-Correlation Functional[END_REF] and accounting for water solvent effects via the CPCM 59 (the permittivity of water εr = 78.35). Note that spin-orbit coupling effects (SOC)

are not expected to affect uranyl ground and excited state geometries as discussed by Pierloot et al. [START_REF] Pierloot | Electronic Structure and Spectrum of UO2 2+ and UO2Cl4 2[END_REF] The structure of the first low-lying excited state was optimized using the time-dependent (TD)-DFT/PBE0 method as implemented in Gaussian 16 code, 59 with equilibrium CPCM solvation. The vibrational harmonic frequencies were computed using an analytic Hessian matrix, and all relaxed geometries considered for the vibronic spectra calculations represent real minima as they have no imaginary frequencies. All optimized structures are available in electronic supplementary information (ESI). In these calculations, the def2-TZVP (second generation of triple-zeta polarization quality) basis sets [START_REF] Weigend | RI-MP2: Optimized Auxiliary Basis Sets and Demonstration of Efficiency[END_REF][START_REF] Weigend | Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy[END_REF] have been used for all light elements i.e., C, O, Na, Mg, and Ca. The uranium atom is described by a small-core Relativistic Effective Core Pseudopotentials (RECP), which accounts for scalar relativistic of the uranium 60 coreelectrons, [START_REF] Küchle | Energy-Adjusted Pseudopotentials for the Actinides. Parameter Sets and Test Calculations for Thorium and Thorium Monoxide[END_REF][START_REF] Cao | Valence Basis Sets for Relativistic Energy-Consistent Small-Core Actinide Pseudopotentials[END_REF] and the valence electrons are described by the def-TZVP (first generation of triplezeta polarization quality) basis set. [START_REF] Eichkorn | Auxiliary Basis Sets for Main Row Atoms and Transition Metals and Their Use to Approximate Coulomb Potentials[END_REF] In the TD-DFT calculations, the Gaussian program defaults for frozen core orbitals were overridden to freeze the U (5s, 5p, 5d), C (1s), O (1s), Na (1s), Mg (1s), and Ca (1s, 2s, and 2p) orbitals.

According to benchmark calculations carried out by Tecmer et al. [START_REF] Tecmer | Electronic Spectroscopy of UO2 2+ , NUO + and NUN: An Evaluation of Time-Dependent Density Functional Theory for Actinides[END_REF][START_REF] Tecmer | Charge-Transfer Excitations in Uranyl Tetrachloride [UO2Cl4] 2-: How Reliable Are Electronic Spectra from Relativistic Time-Dependent Density Functional Theory?[END_REF][START_REF] Tecmer | Reliable Modeling of the Electronic Spectra of Realistic Uranium Complexes[END_REF] on a series of uranium(VI)-based compounds, the CAM-B3LYP 68 exchange-correlation functional was found to be more accurate than PBE0 for uranyl valence transition energies, for similar optimized structures. Thus, to accurately position the "hot bands", CAM-B3LYP TD-DFT single-point calculations including spin-orbit coupling (SOC) effects with the ZORA Hamiltonian, [START_REF] Van Lenthe | Relativistic Regular Two-Component Hamiltonians[END_REF] and modeling the water solvent with the conductor-like screening model (COSMO) model [START_REF] Klamt | A New Approach to Dielectric Screening in Solvents with Explicit Expressions for the Screening Energy and Its Gradient[END_REF][START_REF] Klamt | Conductor-like Screening Model for Real Solvents: A New Approach to the Quantitative Calculation of Solvation Phenomena[END_REF][START_REF] Klamt | Treatment of the Outlying Charge in Continuum Solvation Models[END_REF] were performed at the spin-free ground and excited-state structures, with the Amsterdam Density Functional package (ADF2018.01). 73 All atoms were described by TZ2P Slater-type basis sets (triple zeta with two polarization functions quality), [START_REF] Van Lenthe | Optimized Slater-Type Basis Sets for the Elements 1-118[END_REF] without the frozen core approximation.

There are controversies in the literature regarding the nature of the "hot band" in the luminescence spectra of uranyl-based complexes. [START_REF] Su | Uranyl-Glycine-Water Complexes in Solution: Comprehensive Computational Modeling of Coordination Geometries, Stabilization Energies, and Luminescence Properties[END_REF] We considered that it corresponds to an average of the lowest-lying electronic transitions, which is a hypothesis difficult to confirm experimentally, but appears reasonable in the case of uranyl compounds. Indeed, at the SOC level, four excited states in the luminescent energy range are very close in energy (within a 900 cm -1 range). They correspond to excitations of an electron out from the  highest occupied molecular orbital (HOMO) into the four non-bonding orbitals noted f-,+, f-,+, -we refer to orbitals, to ease the chemical discussion, while we should formally refer to spinors -that are nearly degenerate. In the present work the "hot band" is defined after averaging the two pairs among the first four excited states including the Zero-Point-Energy (ZPE) corrections estimated for the ground state and the lowest spin-free triplet excited state, assuming the ZPE being almost identical for all the excited states considered here. The latter assumption is reasonable since the excited states are of similar nature and the previously published theoretical results proved, at least for the symmetric stretches, [START_REF] Pierloot | Electronic Structure and Spectrum of UO2 2+ and UO2Cl4 2[END_REF][START_REF] Réal | Theoretical Investigation of the Energies and Geometries of Photoexcited Uranyl(VI) Ion: A Comparison between Wave-Function Theory and Density Functional Theory[END_REF] that the excited-state frequencies only vary by few wave numbers (< 10 cm -1 ).

RESULTS AND DISCUSSIONS

Experimental luminescence spectra. Figure 2 shows the fluorescence spectra acquired with respectively 300 lines mm -1 grating for the global emission spectra (a), and 1800 lines mm -1 grating for the five specific characteristic peaks (b)-(f) at D = 25 ns. Table 1 lists the aqueous concentrations of alkaline earth metal ions and the speciation calculations results in terms of ternary complexes for each sample solution with published thermodynamic data using PhreeqC, [START_REF] Parkhurst | User's Guide to PHREEQC (Version 2) -A Computer Program for Speciation, Batch-Reaction, One-Dimensional Transport, and Inverse Geochemical Calculations[END_REF] with which the use of SIT (Specific ion Interaction Theory) [START_REF] Guillaumont | Update on the Chemical Thermodynamics of Uranium, Neptunium, Plutonium, Americium and Technetium[END_REF] for modelling the influence of ionic As shown in Figure 2a, the M-U(VI)-CO3 (M = Mg, Ca) spectra confirm the existence of desirable species. Successive substitution of calcium for magnesium in the ternary triscarbonatouranyl moiety results in markedly enhanced luminescence intensity as the test solutions evolve from the condition where MgnUO2(CO3)3 (4-2n)-predominate to where CanUO2(CO3)3 (4-2n)-are in high proportion (Table 1). The formation constants of Ca complexes are slightly larger than those of Mg complexes. Using reported data, [START_REF] Dong | Determination of the Formation Constants of Ternary Complexes of Uranyl and Carbonate with Alkaline Earth Metals (Mg 2+ , Ca 2+ , Sr 2+ , and Ba 2+ ) Using Anion Exchange Method[END_REF] one can express the following equilibrium and complexation constant.

MgUO2(CO3)3 2-+ Ca 2+ = CaUO2(CO3)3 2-+ Mg 2+ logK = 1.07
This explains that a low quantity of calcium compared to that of magnesium in sample No.2 leads steadily to a relatively large fraction of Ca-UO2-CO3 species (49%). On addition of Ca, the increasing luminescence intensity suggests that CanUO2(CO3)3 (4-2n)-exhibits higher capacity to fluoresce than MgnUO2(CO3)3 (4-2n)-. Nevertheless, the use of the 300 lines mm -1 grating does not allow discriminating the spectral differences since the characteristic peak positions, determined by a Gaussian-based decomposition, differ within the uncertainties among samples. Therefore, the use of the 1800 lines mm -1 grating was required to probe the possible differences recorded at ambient temperature.

Table 1 The aqueous compositions of prepared sample solutions calculated with published thermodynamic data from Shang et al [START_REF] Shang | Determination of Formation Constants and Specific Ion Interaction Coefficients for Can:NUO2(CO3)n (4-2n)-Complexes in NaCl Solution by Time-Resolved Laser-Induced Luminescence Spectroscopy[END_REF] ., Lee et al. [START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF] and Dong et al. [START_REF] Dong | Determination of the Formation Constants of Ternary Complexes of Uranyl and Carbonate with Alkaline Earth Metals (Mg 2+ , Ca 2+ , Sr 2+ , and Ba 2+ ) Using Anion Exchange Method[END_REF] No.

[ A closer inspection of Figure 2 b-d shows very small hypsochromic shifts of the first three peaks in the whole spectra when the observed signal evolves from the spectral components from MgnUO2(CO3)3 (4-2n)-to those of CanUO2(CO3)3 (4-2n)-. This could only be detected with the 1800 lines/mm gratings. The observed emission bands at longer wavelengths respond even less sensitively to this evolution of signal. More precisely, the center wavelength of the first peak, i.e., the "hot band", representing the electronic transitions 11→00 undergoes a small shift from 466 nm (𝜈̅ = 21 459 cm -1 ) down to 464.7 nm (𝜈̅ = 21 519 cm -1 ). The second and the third peaks, corresponding to 10→00 and 10→01, are respectively shifted from 484.7 nm (𝜈̅ = 20 631 cm -1 ) down to 483.7 nm (𝜈̅ = 20 674 cm -1 ), and from 505.2 nm (𝜈̅ = 19 794 cm -1 ) to 504.6 nm (𝜈̅ = 19 818 cm -1 ). For the last two peaks -10→02 and 10→03 (Figure 2 e,f) -the center wavelengths remain nearly invariable. What stands out also in the emission spectra is that the influence of MgnUO2(CO3)3 (4-2n)-complexes on the center wavelength shift is rather limited once the fraction of Ca complexes is over 50%. As observed in experiments, the hypsochromic shift wavelengths is no longer significant when the fraction of Ca complexes increases from 49% (No.2) to 98% (No.5).

Ca 2+ ] (m) CaUO2(CO3)3 2-(m) Ca2UO2(CO3)3(aq) (m) Ca-UO2-CO3 1 0 0 0 0% 2
These experimental results first reveal that the interaction of alkaline earth metal ions with triscarbonatouranyl group does not seem to alter the inner-sphere coordination mode of the central uranyl ion in terms of conformational changes, supported by the fact that the emission spectra of M-U(VI)-CO3 complexes generally display the typical spectral profiles of UO2(CO3)3 4-species.

Second, only minor modifications in characteristic emission bands have been detected using the 1800 lines mm -1 grating when Mg 2+ is replaced by Ca 2+ in the ternary species. The most intense peaks of vibronic transitions 10→0ν (ν = 0, 1), as well as the "hot band" are more sensitive to the 2 along with the experimental data available in literature. To discuss the effect of the second-sphere counter cations on the first coordination shell bond lengths, we take the Na3UO2(CO3)3 -complex as reference. In its groundstate structure, the following atomic distances were computed: the axial U-Oax bond length at 1.791 Å, the equatorial U-Oeq at 2.433 Å, the U-C at 2.902 Å, and the U-Odist (Odist being the most distant oxygen) at 4.141 Å. The U-Na bond distance was equal to 3.823 Å. Our calculated values are in good agreement with EXAFS data obtained in solution. [START_REF] Amayri | Spectroscopic Characterization of Alkaline Earth Uranyl Carbonates[END_REF][START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF] One can point out that the use of another functional of the density, namely B3LYP, and slightly different basis sets, [START_REF] Ikeda | Comparative Study of Uranyl (VI) and-(V) Carbonato Complexes in an Aqueous Solution[END_REF] has a small impact on the computed distances, as we observe variations not larger than 0.03 Å for the distances within the uranyl first-coordination shell, and 0.14 Å within the second-coordination sphere.

Let us start with general observations on the structural changes induced by the replacement of one Na + ion by magnesium or calcium in the second coordination sphere, using the notations of Figure 2. The uranyl subunit, initially linear, was found to be slightly bent, with an Oax-U-Oax angle of about 176°, and all the U-C bond lengths are no longer equivalent because of the different electrostatic interactions (Figure 3). In general, the U-Oeq bond lengths with the oxygen atoms closer to the sodium ion (r1) shortened, whereas the ones with the oxygen atoms closer to the alkaline earth metal (r3) lengthened; the U-Oeq bond lengths (r2) located between a sodium ion and an alkaline earth ion remained almost unchanged. A similar behavior was obtained for U-C and U-Odist bond lengths: they got shorter close to sodium environment (rA) and longer if magnesium or calcium is near (rB).

The substitution of one sodium in the reference Na3UO2(CO3)3 -by one magnesium decreased the U-Oax bond length by 0.009 Å. The U-C bond lengths vary depending on the counterion present in the vicinity as described previously. In the Na2MgUO2(CO3)3 complex, the computed U-Mg distance is underestimated by our method (about 0.31 Å) as compared to experimental data. [START_REF] Guillaumont | Update on the Chemical Thermodynamics of Uranium, Neptunium, Plutonium, Americium and Technetium[END_REF] The replacement of a second Na + by magnesium-induced identical shortening of the U-Oax bond length as compared to the complex with one magnesium. The U-Mg bond remained the same for both Na2MgUO2(CO3)3 and NaMg2UO2(CO3)3 + complexes. To the best of our knowledge, the structure of the Mg2UO2(CO3)3 in solution has never been determined experimentally.

By looking at the series of calcium complexes, a similar tendency in the ground state geometries was observed. Like in case of the complex with one calcium, a slightly shorter decrease of axial U-Oax bond length about 0.006 Å was observed, while the U-C bonds remained essentially unchanged. Unlike the Na2MgUO2(CO3)3 complex, for which U-Na was longer than U-Mg, the U-Ca, and U-Na bonds are found to be almost identical i.e. ~ 3.8 Å in the Na2CaUO2(CO3)3 complex, which is in a good agreement with EXAFS analysis. [START_REF] Amayri | Spectroscopic Characterization of Alkaline Earth Uranyl Carbonates[END_REF] This elongation of 0.286 Å of the distance to the alkaline earth metal dication is in line with the fact that Mg 2+ has a smaller ionic radius (0.72 Å) than Ca 2+ (1.00 Å). [START_REF] Shannon | Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chaleogenides[END_REF] The NaCa2UO2(CO3)3 + complex formed by the addition of a second Ca 2+ in the outer shell showed that the uranyl unit and first coordination sphere took a similar geometry to that of the NaMg2UO2(CO3)3 + complex. The difference in U-Mg bond lengths in the complexes with one and two Ca 2+ ions was found to be 0.061 Å, while the difference in the U-Ca bond lengths was 0.022 Å between the complexes with one and two Ca 2+ ions.

Looking at the available experimental structures including either Na + , Ca 2+ or Mg 2+ , one can conclude that the U-Oax, U-Ocarbonate and U-C are rather insensitive with respect to the nature of the cation, and the observed variations when changing the counter cations are of similar magnitude as the ones reported theoretically. Our calculations sometimes underestimate by approximately 0.1 Å at most the distances between the uranium atom and the carbonates ligands and the counter cations. We suspect that this minor structural discrepancy might be imputable to the choice of the functional or to the fact that our chemical model is too limited, as explicit solvent (water molecules) are not included. However, we may assume that our calculated geometries are reliable enough to reproduce trends for the luminescence properties. The bond lengths of the excited-state structures are reported in Table 2, and are compared to the bond lengths found in the ground-state geometries. For the reference Na3UO2(CO3)3 -complex and both series of magnesium and calcium complexes, the U-Oax bond elongates by 0.035-0.036 Å upon excitation. One can see that for the Mg/Ca systems, the bond lengths of the U-Oeq are not degenerate any longer. An increase of the U-Oeq, U-C and U-Odist bonds between the ground and excited states was found to be in the range from 0 to 0.01 Å. The positions of Na + and alkaline earth cations in the excited state were found to be rather similar after excitation of an electron. The U-Na bond lengths shortened by up to 0.023 Å along the series, while the excitation process resulted in a small elongation of the U-M bond by at most 0.01 Å. Additional analyses of the structural changes are presented in a table form in the ESI. As there is no literature data available for the excited state structures of triscarbonatouranyl complexes, it is difficult to quantitatively estimate the accuracy of the TD-DFT relaxed calculations. Nevertheless, small elongations of atomic bonds were observed as could be expected as a result of the transition from bonding to nonbonding orbital of the uranium, as observed for the uranyl tetrahalide complexes. [START_REF] Oher | Investigation of the Luminescence of [UO2X4] 2-(X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations[END_REF] As the main objective here is to describe the evolution of the luminescence properties in the series, the methodology appears to be trustworthy.

The intensity distributions observed in the luminescence spectra of U(VI)-based complexes are closely related to the uranyl vibrational modes, mostly the uranyl symmetrical stretching 𝜈 𝑠 . Thus, we have reported in Table 3 the computed frequency values of the ground and excited state uranyl bending 𝜈 𝑏 , symmetrical 𝜈 𝑠 and anti-symmetrical 𝜈 𝑎 stretching modes.

The values for 𝜈 𝑠 and 𝜈 𝑎 in the K4UO2(CO3)3 crystal were measured by Raman and IR spectroscopies at 300 K. 8080 The experimental solid-state symmetrical and anti-symmetrical stretching frequencies amounted to 812 and 889 cm -1 , and agreed well with ours, 837 and 877 cm - 1 , computed for the Na3UO2(CO3)3 -complex with inclusion of long-range hydration effects. The 𝜈 𝑏 for this complex was computed at 286 cm -1 but it could not be compared to any experimental data. It is noteworthy that with the addition of counterions, the uranyl bending motion is coupled with the libration and deformation motions of the carbonate ligands. for which they reported 𝜈 𝑏 in the 225-285 cm -1 range. We calculated 255-282 cm -1 for the Na2CaUO2(CO3)3 complex, which appears to be consistent. Moreover, as the charge of the complex increases with the addition of Mg 2+ or Ca 2+ in the second coordination sphere, gradual shifts to larger vibrational energies of the uranyl stretching modes 𝜈 𝑠 and 𝜈 𝑎 were observed when sodium was replaced by alkaline earth metal, with a shift of about +17-21 cm -1 for the magnesium complexes, and +13-14 cm -1 for the calcium complexes. In the first excited state, the uranyl vibrational stretching frequencies are smaller than in the ground state whereas the bending ones are nearly identical. For the reference Na3UO2(CO3)3 - complex, symmetrical and anti-symmetrical stretching modes decreased by 125 and 100 cm -1 , respectively. A similar tendency, within 10 cm -1 , was observed for the other NamMnUO2(CO3)3 (4- Experimental and theoretical emission energies. In experimental luminescence spectra the first observed electronic transition, usually called the "hot band" is located near 465 nm, and can be detected at room temperature. [START_REF] Wang | A Cryogenic Fluorescence Spectroscopic Study of Uranyl Carbonate, Phosphate and Oxyhydroxide Minerals[END_REF] The study of the electronic structure of uranyl-based complexes shows that the luminescent state usually corresponds to the first low-lying triplet states that are nearly degenerate. Even though the triplet-singlet transitions are forbidden by the spin selection rule in the spin-orbit free picture, this rule is relaxed in the intermediate coupling scheme when spin-orbit coupling is accounted for and can be observed experimentally. The analysis of the molecular orbitals involved in excitations to the first four triplet states (1a-4a) confirmed that uranyl luminescence originates from transitions involving an electron placed in one of the four non-bonding 5f orbitals of uranium(VI) to an orbital that has a uranyl σ (5f, 6d) character mixed with the 2p atomic orbitals of the carbonate oxygens (Figure 4). Thus, the emission process in the triscarbonatouranyl bidentate complexes corresponds to a local transition from one of the nonbonding to bonding uranyl orbitals, with a moderate metal-to-ligand charge transfer character. The origin of the spectral envelope usually shifts along the energy scale as a function of the nature of the equatorial ligands, the stoichiometry and the complex structure. Several spectroscopic studies evidenced a luminescence blue shift of the carbonate complexes as compared to many other uranyl species such as phosphates, halides or hydrates. [START_REF] Bernhard | Uranyl(VI) Carbonate Complex Formation: Validation of the Ca2UO2(CO3)3(Aq.) Species. Radiochim[END_REF][START_REF] Geipel | Mixed Complexes of Alkaline Earth Uranyl Carbonates: A Laser-Induced Time-Resolved Fluorescence Spectroscopic Study[END_REF][START_REF] Wang | Cryogenic Laser Induced Fluorescence Characterization of U(VI) in Hanford Vadose Zone Pore Waters[END_REF][START_REF] Wang | A Cryogenic Fluorescence Spectroscopic Study of Uranyl Carbonate, Phosphate and Oxyhydroxide Minerals[END_REF] From previous studies on triscarbonatouranyl complexes [START_REF] Lee | Formation of Ternary CaUO2(CO3)3 2and Ca2UO2(CO3)3(Aq) Complexes under Neutral to Weakly Alkaline Conditions[END_REF][START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF] , the blue shift was estimated to 7, 15, and 25 nm (in approximately 300 -1100 cm -1 ) with respect to UO2(H2O)5 2+ , UO2Cl4 2-and UO2(NO3)2 complexes, respectively. Wang et al. [START_REF] Wang | A Cryogenic Fluorescence Spectroscopic Study of Uranyl Carbonate, Phosphate and Oxyhydroxide Minerals[END_REF] inferred that ionic interactions between the uranyl ion and anions with stronger basicity lead to larger decrease of the symmetric O=U=O stretching frequency, along with a larger redshift of the luminescence spectra. This explanation holds for carbonate ligands that are softer bases than chlorides, for instance. [START_REF] Parr | Absolute Hardness: Companion Parameter to Absolute Electronegativity[END_REF] Theoretically, the vertical emission energy, averaged by their nature E1a,2a and E3a,4a should explain the position of the "hot band." The theoretical E1a,2a and E3a,4a values together with the experimental emission maxima are shown in Table 4. The experimental "hot band" values for the series of magnesium and calcium triscarbonatouranyl complexes agreed well with the previous spectroscopic studies [START_REF] Lee | Formation of Ternary CaUO2(CO3)3 2and Ca2UO2(CO3)3(Aq) Complexes under Neutral to Weakly Alkaline Conditions[END_REF][START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF] and usually remain constant within 2 nm. In aqueous solution, Mg 2+ exhibits a stronger polarizing effect than Ca 2+ due to its higher charge density. When M 2+ counterions interact with UO2(CO3)3 4-, the global electron density of the negative ion group, especially of the equatorial oxygen atoms in carbonates, would be more distorted towards the Mg 2+ than Ca 2+ . The effect of distortion could change the bonding character from ionic to covalent if enough valence electrons are attracted to the positive ions. [START_REF] Clugston | Advanced Chemistry[END_REF] The electronic redistribution occurred in Mg-UO2-CO3 complexes results in a less symmetric structure, which probably reasons out the weak shift of emission spectra to longer wavelengths. In a previous study, the effect of molecular symmetry on fluorescence results has been specially studied by comparing the experimental parameters of a set of organic molecules. [START_REF] Nizhegorodov | Effect of Molecule Symmetry on Fluorescence Parameters and on the Intercombination Conversion Constant[END_REF] It was shown that the more symmetric molecules often give rise to larger quantum yield and longer decay time because of several transitions forbidden by symmetry. For the more symmetric Ca-UO2-CO3 complexes, the decrease in the number of non-radiative pathways theoretically contributes to less energy losses between excitation and emission, thus should cause the hypsochromic shift in emission spectra. [START_REF] Lakowicz | Principles of Fluorescence Spectroscopy[END_REF] Table 4 Experimental and computed "hot band" energies (in nm and cm -1 ) of NamMnUO2(CO3)3 (4- m-2n)-(M=Mg, Ca; m,n = 0-2) complexes in water. Theoretical values were obtained at all-electron SOC CAM-B3LYP level of theory, modeling the water solvent with the COSMO model.

Exp.

E1a,2a a E3a,4a a nm cm -1 nm cm -1 nm cm a the reported E1a,2a and E3a,4a values correspond to the average of the two first pairs of vertical emission energies (See Figure 4, see ESI) computed at the lowest triplet state geometries, with addition of the zero-point energy corrections of the ground and first excited states; b this work; c in n-dodecane; d E1a,2a and E3a,4a values were obtained at TD-DFT/PBE0 level of theory in the corresponding media

The comparison of the E1a,2a and E3a,4a theoretical energies showed that the E3a,4a values agree better with experimental "hot band" values than the E1a,2a. However, the E1a,2a energies are in the range of the triscarbonatouranyl spectral envelope, thus they can contribute to the experimental spectra. For the Mg 2+ and Ca 2+ triscarbonatouranyl complexes our calculations showed a redshift from Na3UO2(CO3)3 -to Na2MUO2(CO3)3 and NaM2UO2(CO3)3 + as a result of negative and positive charges in our model structures used for calculations, while the complexes in solution are stabilized by counter-ions present in the working solution. It is worth noting that our calculations did not show any significant impact of the alkaline earth metal on the position of the emission maxima. It is still difficult to predict emission energies of actinides with a high precision, but our method allowed to reproduce accurately the blue shift of a luminescence profile of triscarbonatouranyl complexes. As compared to similar calculations of the E3a,4a performed in our group at the same level of theory for the [A336]2[UO2Cl4] in n-dodecane and UO2(NO3)2 . 2H2O in water media, our method is qualified to reproduce experimental "hot band" position within ±10 nm (~500 cm -1 ) accuracy.

Vibronic progressions. The luminescence spectrum of U(VI) complexes typically shows wellspaced vibronic progressions that overlap with the pure electronic transition originates from the luminescent state to the ground state. The vibronic progression means that there is a coupling to symmetrical stretching mode of uranyl unit during the emission process. The theoretical and experimental luminescence spectra of Mg 2+ and Ca 2+ triscarbonatouranyl complexes are shown in Fig. 5. The theoretical spectra were all shifted to the position of the experimental "hot band" for the sake of spectra comparison. The calculated spectrum of Na3UO2(CO3)3 -was not reported because it was considered not reliable due to the geometrical distortion of some electron configurations leading to a deficiency of degeneracy.

The intensity of luminescence depends on the quantum yield, which is basically a ratio of the rate of de-excitation by photons emission to the rate of de-excitation by both radiative and nonradiative processes. It is usually much lower than unity for uranyl species since non-radiative transitions are the major part of relaxation of the excited molecules. [START_REF] Su | Uranyl-Glycine-Water Complexes in Solution: Comprehensive Computational Modeling of Coordination Geometries, Stabilization Energies, and Luminescence Properties[END_REF] This is why the prediction of real experimental intensities from quantum chemical model is necessarily incomplete, and only relative peak intensities and trends are relevant.

For all the triscarbonatouranyl complexes the vibronic part starts at the 10→00 transition with the band spacing equal to the ground state uranyl stretching frequency 𝜈 𝑠 𝐺𝑆 , while the 11→00 "hot band" appears at higher energy from the 10→00 transition. The analysis of E1a,2a and E3a,4a

theoretical emission energies has showed that the E1a,2a agreed well with the experimental 10→00 transition, while E3a,4a is much closer to the 11→00 value and spaced from E1a,2a by a 𝜈 value ranging from 729 to 798 cm -1 (See Table 2). Such observation is supporting the possibility of the simultaneous contributions of two excited states to the spectrum. Thus, it has been assumed that there are two vibronic progressions starting at E1a,2a and E3a,4a energies and contributing to the 10→00 and 11→00 transitions, respectively. To reconstruct theoretical spectra, two vibronic progressions, computed out of the first low-lying excited state, were superimposed on the same energy scale.

The analysis of computed spectra showed no other significant contributions than uranyl symmetrical stretching mode, which means that the coupling scheme remains the same for Mg 2+ and Ca 2+ triscarbonatouranyl complexes. The detailed assignment of the peaks is provided in the ESI. The experimental and theoretical spectral characteristics such as positions of the peaks and the band-spacing values are reported in Table 5. with respect to experimental data, what is deduced from the spacing between the 11→00 and 10 →00 transitions, and they are very similar to the excited state symmetrical stretching mode. The higher values attributed to 𝜈 from experimental spectra could originate from small additional contributions to the vibronic coupling that are not accounted for. Indeed, our theoretical model does not account for explicit water molecules in the structure's calculations, and anharmonicity effects were not accounted for in the spectra computations. Although the model could be improved, but with additional calculation cost, we conclude that the agreement between the peak positions in the experimental and computed spectra is satisfactory for such modelling of luminescence peaks from the uranyl complexes. a,b The 𝜈 𝑠 𝐺𝑆 corresponds to the averaged band-spacing values between vibronic transitions, the 𝜈 corresponds to spacing between the 11→00 and 10→00 transitions. The spectral characteristics in nm is available in the Tables S8 and S9 of ESI; c,d the 11→00 was taken from experimental Mg-UO2-CO3 and Ca-UO2-CO3 emission spectra to match position of theoretical peaks with experimental ones It is worth analyzing the change of the theoretical and experimental luminescence intensities of the complexes not in absolute terms but relative ones for the two alkaline earth metal ions. First, the comparison of theoretical intensities of Mg 2+ and Ca 2+ complexes between each other is scientifically wrong because the used theoretical models do not account for many other experimental factors that have an influence on luminescence intensities (pH, ionic strength etc.). Moreover, experimental luminescence intensities of the whole spectral envelope depend on additional factors like the effective laser excitation of each species, influence of experimental apparatus such as slot width, or quenching effects, which are not included in the theoretical calculations.

The spectrum of the Mg2UO2(CO3)3 complex was fully calculated whereas we were unsuccessful in calculating the vibronic progressions in Na2MgUO2(CO3)3 in the framework of the Franck-Condon harmonic approximation. Indeed, we observed a distortion of the UO2(CO3)3 4- moiety upon Mg 2+ complexation in Na2MgUO2(CO3)3, that was reflected in the out-of-plane molecule distortion caused by increasing of a polarizing effect of Mg 2+ in water. The full theoretical spectrum of Na2MgUO2(CO3)3 could be obtained using gas-phase geometries of S3 in ESI we observed that U-Oax bond displacement from the ground to the luminescent state is identical and equal to 0.036 Å. Thus, our theoretical model showed that even if the Mg2UO2(CO3)3 could form in experimental conditions, its luminescence spectrum will be hardly distinguished by TRLFS from the one by MgUO2(CO3)3, except if they have significantly different luminescence lifetimes. On the contrary, the addition of a second Ca 2+ ion to the triscarbonatouranyl complexes induced an enhancement of the experimental luminescence intensities (Fig. 5b), as shown otherwise. [START_REF] Lee | Formation of Ternary CaUO2(CO3)3 2and Ca2UO2(CO3)3(Aq) Complexes under Neutral to Weakly Alkaline Conditions[END_REF][START_REF] Shang | Determination of Formation Constants and Specific Ion Interaction Coefficients for Can:NUO2(CO3)n (4-2n)-Complexes in NaCl Solution by Time-Resolved Laser-Induced Luminescence Spectroscopy[END_REF][START_REF] Lee | Formation, Stability and Structural Characterization of Ternary MgUO2(CO3)3 2and Mg2UO2(CO3)3(Aq) Complexes[END_REF] This enhancement was well reproduced in the computed spectra comparing the Na2CaUO2(CO3)3 and NaCa2UO2(CO3)3 + model systems (Fig. 5d). This result is consistent with the fact that the U-Oax bond changes are slightly different and equal to 0.035 and 0.036 Å for complexes with one and two Ca 2+ , respectively.

The theoretical results confirmed that the luminescence intensities of the triscarbonatouranyl complexes depend on the structure of uranyl unit with a major contribution from the U-Oax bond displacement, and only a minor one from equatorial ligands. The intensity trends of the 10→00 transition were correctly reproduced by the quantum chemical simulations without any fitting.

CONCLUSIONS

The interaction of the triscarbonatouranyl complex UO2(CO3)3 4-with alkaline earth metal ions M 2+ (Mg and Ca) in solution is known to induce stable complex of stoichiometry MnUO2(CO3)3 (4- 2n)-(n=1, 2), which has important implication on uranyl speciation in different environments. These species have specific luminescence features, as a result of the electronic configurations of the ground and excited states, the vibronic states and vibrational coupling. The luminescence of the complexes with Mg 2+ (n=1) and Ca 2+ (n=1, 2) were measured by TRLFS and compared for the first time with simulated spectra obtained using quantum mechanics applied to model structures NamMnUO2(CO3)3 (4-m-2n)-(M = Mg, Ca; m, n = 0-2). The ground and excited state geometries of each structure were optimized by DFT/PBE0 and TD-DFT/PBE0 at the spin-free level including relativistic effects and water solvent effects (CPCM). The first low-lying excited states were successfully computed together with the vibrational harmonic frequencies for Na2MgUO2(CO3)3, NaMg2UO2(CO3)3 + , Na2CaUO2(CO3)3, and NaCa2UO2(CO3)3 + . The 11→00 "hot band" positions were determined by the more accurate TD-DFT/CAM-B3LYP with single-point calculations. This theoretical approach enabled to calculate the main luminescence emissions of the complexes with the corresponding assignment of the electronic transitions and vibronic modes involved. The resulting calculated spectra showed a very good agreement with experimental band positions and band spacing attributed to MgUO2(CO3)3 2-, CaUO2(CO3)3 2-, and Ca2UO2(CO3)3. Particularly, the calculations reproduce fairly well the blue shift of the triscarbonatouranyl complexes luminescence as compared with other uranyl complexes. The calculations also confirmed the similarities of the spectra for Mg 2+ and Ca 2+ complexes, and correctly reproduced the increase of the luminescence intensity upon addition of a second Ca 2+ as a result of a slightly large increase of the U-Oax distance upon excitation and change of polarizing effect. In the case of Mg 2+ complexes, the U-Oax bond change was found to be the same for both Na2MgUO2(CO3)3 and Mg2UO2(CO3)3, the polarizing effect of magnesium is found to have more impact on the U-Oyl unit, suggesting that the luminescence of these complexes would hardly be distinguished experimentally, unless they are showing very different decay times.

The QM method applied in this work offered an appropriate compromise between the calculation cost and the accuracy of the results in comparison with experimental data. Further works could improve the representativity of the model structures by adding explicit water molecules in the structures for instance that will affect the positions of the alkaline earth metal ions and vibronic coupling.

ASSOCIATED CONTENT

Luminescence spectra of Na2MgUO2(CO3)3, NaMg2UO2(CO3)3 + , Na2CaUO2(CO3)3, and NaCa2UO2(CO3)3 + were computed with ab initio methods including long-range water solvation effect. The evolution of theoretical intensities is in a good agreement with the recorded experimental luminescence spectra obtained by time-resolved laser-induced fluorescence spectroscopy (TRLFS) in aqueous phase. The analysis of the theoretical results reveals the importance of alkali earth-counter cations' polarizing effect as well as their influence on the structures and luminescence intensities.

Figure TOC. For Table of Contents Only

  Structures and Luminescent Properties of NamMnUO2(CO3)3 (4-m-2n)-(M = Mg, Ca; m , n = 0-2): Time-Resolved Fluorescence Spectroscopy and Ab Initio Studies Hanna Oher, Thomas Vercouter, Florent Réal, Chengming Shang, Pascal E Reiller, Valérie Vallet To cite this version: Hanna Oher, Thomas Vercouter, Florent Réal, Chengming Shang, Pascal E Reiller, et al.. Influence of Alkaline Earth Metal Ions on Structures and Luminescent Properties of NamMnUO2(CO3)3 (4-m-2n)-(M = Mg, Ca; m , n = 0-2): Time-Resolved Fluorescence Spectroscopy and Ab Initio Studies. Inorganic Chemistry, 2020, 59, pp.15036-15049. 10.1021/acs.inorgchem.0c01986. hal-02955418

m,n = 0- 2 )

 2 complexes by DFT, and compared with literature data to evaluate the reliability of the uranyl distances and harmonic frequencies and their change between the ground and luminescent states to capture the evolution of intensities. The luminescence properties calculated by TD-DFT methods were compared to experimental data acquired with our TRLFS set-up, and from the literature. Finally, the present results from quantum chemical modeling are discussed and provide a better understanding of the luminescence bands and intensities of the ternary alkaline earth metal triscarbonatouranyl complexes. EXPERIMENTAL DETAILS Reagents and solution. Five samples of varying [Mg]/([Ca] + [Mg]) ratios from 100% to about 50% were prepared for further acquisition of resulting luminescence signals of MenUO2(CO3)3 (4- 2n)-(Me = Mg 2+ , Ca 2+ ; n = {1,2}). The ionic strength was fixed at 0.

Figure 1 .

 1 Figure 1. The structures of uranyl bidentate tri-carbonate complexes NamMnUO2(CO3)3 (4-m-2n)-

  strength is possible. The sum of [Mg] and [Ca] is maintained at around 0.04 mol kgw -1 for this series with varying ratios of [Mg 2+ ]/([Ca 2+ ] + [Mg 2+ ]) from 1 to 0.5.

Figure 2 .

 2 Figure 2. Measured luminescence spectra of U(VI) at various concentration ratios [Mg 2+ ]/([Mg 2+ ]

  change of counter ions in the second coordination sphere. The slight bathochromic shifts in the case of Mg (compared to Ca) indirectly give indications that M 2+ /UO2(CO3)3 4-interactions can generate very subtle differences in U(VI) fluorescence behavior where a detailed quantitative description of the M 2+ /UO2 2+ /CO3 2-system with quantum chemical modelling provides a useful account of the experimental results. Theoretical ground and excited-state structures. All the key distances for the M(II)n,Na(I)m[UO2(CO3)3] (4-m-2n)-are reported in Table

Figure 3 .

 3 Figure 3. Schematic notation of U-CO3 bonds in a) Na2MUO2(CO3)3 and b) NaM2UO2(CO3)3 + (M

  It is thus inappropriate to discuss this vibrational frequency as pure bending mode of the uranyl unit. The same observation was made by Kalashnyk et al. for the andersonite mineral Na2CaUO2(CO3)3•xH2O (x=5.3 -5.6) 42

Figure 4 .

 4 Figure 4. The highest occupied and lowest unoccupied molecular orbitals of Na3UO2(CO3)3 -

Figure 5 .

 5 Figure 5. Deconvoluted experimental spectra of (a) MgUO2(CO3)3 2-, and (b) CaUO2(CO3)3 2-and

  Na2MgUO2(CO3)3 including water structures corrections, and then plotted in Fig 5c. The detailed procedure of a spectrum prediction is available in the ESI. On Fig 5c the calculated intensity of the 10→01 peak in the magnesium complexes, decreases more than the experimental Mg-UO2-CO3 intensity because the calculations might underestimate the theoretical U-Oax bond length changes upon excitation. From Table

Table 2

 2

		U-Oax	U-Oeq	U-C	U-Odist	U-M	U-Na	Method
			(r1/ r2/ r3)	rA/ rB	rA/ rB
	Ground state
	Na3UO2(CO3)3 -	1.791	2.433	2.902	4.141		3.823	PBE0/CPCM
		1.806	2.457	2.930	4.188		3.961	B3LYP/PCM 40
	Na2MgUO2(CO3)3 -	1.782	2.387/2.439/	2.857/2.9	4.089/4.	3.515	3.801	PBE0/CPCM
			2.477	48	172
	NaMg2UO2(CO3)3 + 1.774	2.390/2.429/	2.900/2.9	4.119/4.	3.511	3.740	PBE0/CPCM
			2.470	80	193
	Na2CaUO2(CO3)3	1.785	2.404/2.433/	2.874/2.9	4.108/4.	3.865	3.818	PBE0/CPCM
			2.469	31	162
	NaCa2UO2(CO3)3 +	1.779	2.406/2.437/	2.902/2.9	4.129/4.	3.864	3.796	PBE0/CPCM
			2.462	53	176
	UO2(CO3)3 4-	1.81(1) 2.44(1)	2.92(1)	4.17(1)			EXAFS 40
		1.80(2) 2.44(2)	2.89(2)	4.20(2)			EXAFS 20

. Ground and excited-state geometries (in Å) of NamMnUO2(CO3)3 (4-m-2n)-(M=Mg, Ca; m,n = 0-2) complexes computed with the PBE0 functional in the CPCM water solvent, and compared to selected literature data. The experimental uncertainties are given in parentheses.

Table 3

 3

	Ground and excited-state vibrational frequencies (in cm -1 ) of NamMnUO2(CO3)3 (4-m-2n)-
	(M=Mg, Ca; m,n = 0-2) complexes computed with the PBE0 functional in the CPCM water
	solvent.
		𝜈 𝑏	𝜈 𝑠	𝜈 𝑎 Method
	Ground state
	Na3UO2(CO3)3 -	286	837 877 PBE0/CPCM
	Na2MgUO2(CO3)3	230-285 857 898 PBE0/CPCM
	NaMg2UO2(CO3)3 +	250-305 874 918 PBE0/CPCM
	Na2CaUO2(CO3)3	255-282 850 891 PBE0/CPCM
	NaCa2UO2(CO3)3 +	242-288 862 905 PBE0/CPCM
	K4UO2(CO3)3		806 882 IR/Raman 80
	Na2CaUO2(CO3)3,xH2O (x=5.3 -5.6) 225-285 833 931 Raman 42
	Excited state
	Na3UO2(CO3)3 -	276	712 777 TD-PBE0/CPCM
	Na2MgUO2(CO3)3 -	238-278 722 796 TD-PBE0/CPCM
	NaMg2UO2(CO3)3 +	224-307 739 812 TD-PBE0/CPCM
	Na2CaUO2(CO3)3	228-275 720 791 TD-PBE0/CPCM
	NaCa 2 UO 2 (CO 3 ) 3 +	233-278 728 801 TD-PBE0/CPCM

Table 2 .

 2 Experimental and theoretical spectral characteristics of triscarbonatouranyl complexes.Experimental 𝜈 𝑠𝐺𝑆 and 𝜈 values were obtained by Lorentzian fitting procedure.

	Complex	Spectral maxima (cm -1 )				𝜈 𝑠 𝐺𝑆 a (cm -1 )	𝜈 b (cm -1 )
		11→00 10→00	10→01	10→02	10→03
	Experimental
	Mg-UO2-CO3	21459	20631	19794	18986	18202	826	828
	Ca-UO2-CO3	21519	20674	19818	18986	18202	832	845
	Theoretical
	Na2MgUO2(CO3)3	21459 c	20737	19822	18904	17978	857	792
	NaMg2UO2(CO3)3 + 21459 c	20720	19847	18972	18099	874	739
	Na2CaUO2(CO3)3	21519 d	20799	19949	19099	18249	850	798
	NaCa2UO2(CO3)3 +	21519 d	20790	19928	19066	18205	862	729

m-2n)-(M = Mg, Ca; m,n = 0-2) compounds. A complete analysis of vibrational frequencies behavior is available in the ESI.

Extraction of the pure spectra of the (Mg/Ca-) UO2-CO3) species from the raw TRLFS data; procedure to estimate the spectrum of the Na2MgUO2(CO3)3 complex in water; assignments for theoretical spectra. An independent ZIP file contains the Cartesian coordinates of all calculated structures.

Supporting Information

The following file supplemental-uo2co3.pdf is available free of charge. It contains:

AUTHOR INFORMATION

Corresponding Authors


Keywords:

For joint estimation of states and parameters in time varying time-delay systems (TDS) involving both distributed and lumped time-delays, a general approach is proposed in this paper to transforming existing (non adaptive) observers to adaptive observers. In addition to the convergence conditions of the considered existing observers, a persistent excitation condition is introduced in order to ensure the convergence of parameter estimation. In contrast to implicitly formulated convergence conditions, which are usually assumed jointly for both state and parameter estimations in most TDS adaptive observers, the persistent excitation condition in the proposed approach is explicitly formulated and decoupled from the conditions initially assumed for state estimation.

INTRODUCTION

Due to the increasing importance of time-delays in control systems equipped with more and more digital components, a great deal of interest has been given to state observer design for time-delay systems (TDS). In this respect, systems with time-delays involved in state equations have been studied in e.g. [START_REF] Germani | An asymptotic state observer for a class of nonlinear delay systems[END_REF], [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF], [START_REF] Yeganefar | Input-to-state stability of timedelay systems: a link with exponential stability[END_REF]. Similar interests have been given to systems with delayed output measurements [START_REF] Kazantzis | Nonlinear observer design in the presence of delayed output measurments[END_REF], [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF], [START_REF] Cacace | An observer for a class of nonlinear systems with time varying observation delay[END_REF], [START_REF] Cacace | A chain observer for nonlinear systems with multiple time-varying measurement delays[END_REF], [START_REF] Kahelras | Sampled-data chain-observer design for a class of delayed nonlinear systems[END_REF].

For TDS involving unknown parameters, the problem of adaptive observer design is still not sufficiently investigated. In [START_REF] Mondal | Adaptive observer for a class of nonlinear systems with time-varying delays[END_REF], an adaptive observer design, following an approach similar to nonlinear adaptive observer design [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF], has been proposed for a class of time-invariant TDS. In [START_REF] Sassi | Design of a functional adaptive observer for bilinear delayed systems[END_REF], a functional adaptive observer has been proposed for a bilinear TDS. In each of these recalled results, the asymptotical convergence of the designed adaptive observer is ensured by some implicitly formulated condition: the existence of a solution to some inequalities, which are specific to each design method. This paper proposes a general approach to transforming (non adaptive) observers to adaptive observers for joint state-parameter estimation in linear time varying (LTV) TDS. In the case of classical (delay-free) LTV state-space systems, a general method has been proposed in [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] for such a transformation. The results of this paper are essentially an extension of this method to TDS, but some non trivial difficulties have to be overcome to achieve this extension, as highlighted in the following sections.

Basically, it is assumed that • if the parameters involved in the considered TDS were all known, then an exponentially convergent (non adaptive) observer would be available for state estimation; • an explicitly formulated persistent excitation (PE) condition is satisfied.

It is well known in classical adaptive estimation problems that PE is essential for parameter estimation Narendra and Annaswamy [1987], Shi, Nar, [START_REF] Karl | Adaptive control[END_REF]. This paper proposes a natural extension of the classical PE condition to the design of TDS adaptive observers. The considered existing observers may have been designed with different methods, yet their transformations to adaptive observers follow the same general approach. This generality covering existing observers designed with different methods, and the decoupling of the parameter estimation condition (the PE condition) from the state estimation condition, are important particularities which distinguish the presented results from existing results.

The results of this paper have been inspired by the preliminary idea summarized in the working note [START_REF] Zhang | State and parameter estimation in linear systems with delays[END_REF].

The paper is organized as follows: the problem under study is formulated in Section II; the observer design and analysis are respectively dealt with in Sections III and IV. The conclusion in section VI.

PROBLEM STATEMENT

Consider the class of time varying linear differential time-delay systems (TDS) in the form of ẋ

(t) = A 0 (t)x(t) + A 1 (t)x(t -h) + Φ(t)θ (1a) y = C 0 (t)x(t) + C 1 (t)x(t -h) (1b
) where x(t) ∈ R n denotes the states, y ∈ R m the outputs, θ ∈ R p the unknown parameters, h a positive constant specifying the time delay, A 0 (t), A 1 (t), C 0 (t), C 1 (t), Φ(t) are matrices of appropriate sizes filled with real-valued functions which are bounded and piecewise continuous in t.

All quantities in this model are known or accessible from sensor measurements, except the state vector x(t) and the parameter vector θ.

An additional term B(t)u(t) could be inserted into the TDS (1) in order to represent (control) inputs. Because such a term would imply trivial modifications of the results of this paper, it is omitted for lighter presentations.

As mentioned in the introduction, such a TDS is an infinite dimensional system. By abuse of terminology, the term "state" is still used to refer to the finite dimensional vector x(t), by analogy to classical state-space systems, in order to ease the presentation. The "state equation" (1a) is sometimes called a delay differential equation (DDE).

The problem considered in this paper is to design an adaptive observer for joint estimation of x(t) and θ under two assumptions:

• the existence of an exponentially convergent (non adaptive) state observer for the TDS (1) in the case where θ is known, and • a persistent excitation condition, as typically required in parameter estimation problems.

The designed adaptive observer will be in the form of a DDE, so that it can be numerically solved with efficient DDE solvers.

The first assumption is more accurately formulated below, whereas the second one will be completed later. Assumption 1. There exists a (time varying) bounded matrix gain L(t) ∈ R n×m such that the following DDE :

ξ(t) = (A 0 (t) -L(t)C 0 (t))ξ(t) +(A 1 (t) -L(t)C 1 (t))ξ(t -h) (2)
is exponentially stable in the sense that [START_REF] Yegafner | Input-to state stability of time-delay systems: A link with exponential stability[END_REF]) :

|ξ(t)| ≤ sup t 0 ≤s≤t 0 +h |ξ(s)|M 0 e -σt , ∀t > t 0 + h (3)
for some positive scalars M 0 and σ. 2

Remark 1. This assumption simply means that, if the true parameter vector θ was known, then an exponentially convergent (non adaptive) observer of system (1) (for state estimation only) would be available in the form of

ẋ(t) = A 0 (t)x(t) + A 1 (t)x(t -h) + Φ(t)θ -L(t)ỹ(t) (4a) ỹ(t) = C 0 (t)x(t) + C 1 (t)x(t -h) -y(t), (4b)
where L(t) is the same gain matrix as in Assumption 1. 2

Lemma 1. Add an additive input term d(t) into the homogeneous system (2), then Assumption 1 implies that the resulting TDS

ξ(t) = (A 0 (t) -L(t)C 0 (t))ξ(t) +(A 1 (t) -L(t)C 1 (t))ξ(t -h) + d(t) (5)
is input-to-state stable, i.e., ∀t > t 0 + h,

|ξ(t)| ≤ β( sup t 0 ≤s≤t 0 +h |ξ(s)|, t) + γ(ess. sup t 0 +h≤s<t |d(s)|) (6)
where s → γ(s) is a K function (γ is strictly increasing and γ(0) = 0) and (t, s) → β(t, s) is a KL function (for each fixed t, the function s → β(s, t) is a K function, and for each fixed s the function t → β(s, t) is non increasing and goes to zero as t → ∞) . 2

This result is a simple consequence of the Theorem 3.2 in [START_REF] Yegafner | Input-to state stability of time-delay systems: A link with exponential stability[END_REF]).

ADAPTIVE OBSERVER DESIGN

The proposed method for adaptive observer design will be based on the state-parameter decoupling transformation approach, initially introduced in the case of classical (delay-free) LTV systems [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF].

Adaptive observer structure

In order to transform the (non adaptive) state observer (4) to an adaptive observer for joint estimation of x(t) and θ, a natural idea is to replace in (4) the unknown parameter vector θ by its estimate θ(t), which is computed by a somehow designed adaptive law. The following design will essentially follow this idea, except an extra term v 0 (t) in the state estimation equation, yielding an estimator in the form of ẋ(t) = A 0 (t)x(t) + A 1 (t)x(t -h)

+ Φ(t) θ(t) -L(t)ỹ(t) + v 0 (t) (7a) ỹ(t) = C 0 (t)x(t) + C 1 (t)x(t -h) -y(t).
(7b) The extra term v 0 (t) is added in order to compensate the parameter estimation error introduced in the state estimation equation (7a), due to the fact that the true θ has been replaced by its estimate θ(t).

Before designing an adaptive law computing the parameter estimate θ(t), it will be first shown that an appropriately chosen v 0 (t) will greatly help the convergence analysis of the designed algorithm.

Decoupling transformation

Define the state and parameter estimation errors

x(t) x(t) -x(t) (8) θ(t) θ(t) -θ,
(9) then it is straightforward to derive the state estimation error dynamics equation

ẋ(t) = (A 0 (t) -L(t)C 0 (t))x(t) + (A 1 (t) -L(t)C 1 (t))x(t -h) + Φ(t) θ(t) + v 0 (t). ( 10
)
The key step of the error analysis is the decoupling transformation, which has been proposed originally for the design of adaptive observers of classical (delay-free) systems [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], in the form of η(t) = x(t) -Υ(t) θ(t), (11) where Υ(t) ∈ R n×p is a time varying matrix to be specified so that the transformed error variable η(t) is decoupled from θ. Some simple computations then lead to

η(t) = [A 0 (t) -L(t)C 0 (t)][η(t) + Υ(t) θ(t)] + [A 1 (t) -L(t)C 1 (t)][η(t -h) + Υ(t -h) θ(t -h)] +Φ(t) θ(t) -Υ(t) θ(t) -Υ(t) θ(t) + v 0 (t) = ω 1 (t) + ω 2 (t) + ω 3 (t), ( 12
)
with

ω 1 (t) = [A 0 (t) -L(t)C 0 (t)]η(t) + [A 1 (t) -L(t)C 1 (t)]η(t -h) ω 2 (t) = (A 0 (t) -L(t)C 0 (t))Υ(t) + Φ(t) -Υ(t) θ(t) ω 3 (t) = (A 1 (t) -L(t)C 1 (t))Υ(t -h) θ(t -h) -Υ(t) θ(t) + v 0 (t).
At this point, if the decoupling technique proposed in [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] was naively followed, then one would choose Υ(t) ensuring ω 2 (t) = 0 for all t ≥ t 0 , as a solution of the differential equation

Υ(t) = (A 0 (t) -L(t)C 0 (t))Υ(t) + Φ(t); ( 13
)
and then choose v 0 (t) so that ω 3 (t) = 0 for all t, yielding

v 0 (t) = Υ(t) θ(t) -(A 1 (t) -L(t)C 1 (t))Υ(t -h) θ(t -h). ( 14
)
With ω 2 (t) and ω 3 (t) annihilated, equation ( 12) would become simply

η(t) = [A 0 (t) -L(t)C 0 (t)]η(t) + [A 1 (t) -L(t)C 1 (t)]η(t -h), (15
) and η(t) governed by this equation would converge exponentially to zero, according to Assumption 1. This decoupling technique worked well for classical (delay-free) systems, as in [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], [START_REF] Li | An adaptive observer for joint estimation of states and parameters in both state and output equations[END_REF], where the variable v 0 (t) as chosen in ( 14) was simply v 0 (t) = Υ(t) θ(t), ( 16) because the terms in (14) involving delays did not exist. Moreover, θ(t) = θ(t) -θ and θ = 0, therefore v 0 (t) was simply written, in the classical case, as

v 0 (t) = Υ(t) θ(t), (17)
which could be readily computed with an appropriately designed parameter adaptation law.

However, for the systems considered in this paper, the situation is more complicated due to the presence of the terms involving time-delays. The trouble is that the variable v 0 (t) chosen in ( 14) cannot be computed in practice, since θ(t -h) = θ(t -h) -θ would require the true value of θ, which is of course unknown in the considered estimation problem! Apparently, this is a non trivial difficulty that did not exist in the classical case as considered in [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] and [START_REF] Li | An adaptive observer for joint estimation of states and parameters in both state and output equations[END_REF].

Fortunately, this difficulty can be addressed as follows.

Add the term (A 1 (t) -L(t)C 1 (t))Υ(t -h) θ(t) to ω 2 (t) and subtract the same term from ω 3 (t), so that the sum in (12) remains unchanged. Accordingly, the second term of v 0 (t) in ( 14)

-(A 1 (t) -L(t)C 1 (t))Υ(t -h) θ(t -h),
(18) which could not be computed due to the dependence of θ(t -h) on the unknown true parameter vector θ, then becomes

(A 1 (t) -L(t)C 1 (t))Υ(t -h)[ θ(t) -θ(t -h)] = (A 1 (t) -L(t)C 1 (t))Υ(t -h)[ θ(t) -θ -θ(t -h) + θ] (19) = (A 1 (t) -L(t)C 1 (t))Υ(t -h)[ θ(t) -θ(t -h)], (20)
which no longer involves the unknown θ, and therefore can be computed with a parameter adaptation law to be designed.

Then ω 2 (t) and ω 3 (t) become

ω 2 (t) = (A 0 (t) -L(t)C 0 (t))Υ(t) + Φ(t) -Υ(t) + (A 1 (t) -L(t)C 1 (t))Υ(t -h) θ(t) (21) ω 3 (t) = [A 1 (t) -L(t)C 1 (t)]Υ(t -h)[ θ(t -h) -θ(t)] -Υ(t) θ(t) + v 0 (t). ( 22
)
Accordingly, let Υ(t) be generated through the TDS Υ(t) = (A 0 (t) -L(t)C 0 (t))Υ(t) 23) with an arbitrary initial condition, so that ω 2 (t) = 0 for all t > t 0 + h. Moreover, choose v 0 (t) as

+ (A 1 (t) -L(t)C 1 (t))Υ(t -h) + Φ(t) (
v 0 (t) = Υ(t) θ(t) -[A 1 (t) -L(t)C 1 (t)]Υ(t -h)[ θ(t -h) -θ(t)] = Υ(t) θ(t) -[A 1 (t) -L(t)C 1 (t)]Υ(t -h)[ θ(t -h) -θ(t)]
so that ω 3 (t) = 0 for all t > t 0 + h, by noticing that θ(t) = θ(t) -θ = θ(t) and θ(t -h) -θ(t) = θ(t -h) -θ(t). Now in (12), ω 2 (t) = ω 3 (t) = 0, therefore η(t) is indeed governed by the TDS (15), which is exponentially stable according to Assumption 1.

At this point, it is only shown that the decoupled estimation error η(t) as defined in (11) converges to zero exponentially. Based on this result, the convergences of the state estimation error x(t) and of the parameter estimation error θ(t) will be analyzed. For this purpose, the adaptive law for parameter estimation should designed appropriately.

Parameter estimation law design

If the output error ỹ(t) was available, then a linear regression parameter estimation algorithm inspired by the classical least squares estimator would be in the form of

θ(t) = -R(t)Λ(t) T • ỹ(t) -C 1 (t)Υ(t -h)[ θ(t -h) -θ(t)] (24a) Ṙ(t) = λR(t) -R(t)Λ(t) T Λ(t)R(t) (24b) Λ(t) = C 0 (t)Υ(t) + C 1 (t)Υ(t -h) (24c)
where the scalar λ > 0 is an exponential forgetting factor.

Of course, the output error ỹ(t) as defined in (7b) cannot be obtained without computing x(t) and θ(t). Therefore, parameter estimation should be jointly performed with state estimation, by combining (24a) with ( 7).

The whole adaptive observer for joint state-parameter estimation is then expressed in the following algorithm, where x 0 (t) : [t 0 , t 0 + h] → R n is the initial state estimate, θ 0 ∈ R p the initial parameter estimate, 0 n×p the n × p zero matrix, I p the p × p identity matrix, r 0 > 0 a scalar value, and λ > 0 a forgetting factor.

This result then implies that the homogeneous part of the LTV system (31), as expressed in (32), is exponentially stable. Therefore, the state θ(t) of the LTV system (31) driven by exponentially vanishing η(t) and η(t -h) converges exponentially to zero.

Finally, it follows from (11), the exponential convergences to zero of η(t) and of θ(t), and the boundedness of Υ(t) that

x(t) = η(t) + Υ(t) θ(t) (39)

converges exponentially to zero. 2

CONCLUSION

A general approach has been proposed in this paper to transforming existing (non adaptive) state observers to adaptive observers, for linear time varying TDS. This approach greatly simplifies the design of adaptive observers for joint state-parameter estimation, under an explicitly formulated persistent excitation condition for parameter estimation, which is decoupled from the conditions initially assumed by existing state observers.

Adaptive observer algorithm

For t ∈ [t 0 , t 0 + h]:

x(t) = x 0 (t), θ(t) = θ 0 , Υ(t) = 0 n×p .

(25)

For t = t 0 + h: R(t) = r 0 Ip (26)

For t > t 0 + h:

ADAPTIVE OBSERVER CONVERGENCE ANALYSIS

Before analyzing the convergence of the state and parameter estimation errors, it is important to ensure that the auxiliary variables computed in the adaptive observer ( 27) are all bounded.

Boundedness of auxiliary variables

Let us start with the recursively computed Υ(t). Proposition 1. Under Assumptions 1, the recursively generated matrix Υ(t) through the DDE (27d) from bounded matrices

Further boundedness results will require a persistent excitation assumption, as typically required in parameter estimation problems. Assumption 2. There exist positive scalar constants T and δ, such that for all t > t 0 + h,

where the inequality X ≥ Y means that X -Y is a positive semidefinite matrix. 2 Remark 2. The symmetric integral in ( 28) is by construction a positive semidefinite matrix. Assumption 2 states that it is positive definite with a strictly positive lower bound. This integral is also upper bounded due to the already established boundedness of Λ(t).

2

Remark 3. The matrix Λ(t) is generated from Φ(t) through (27d) and (27b). For more clarity, let us group these two equations as

(29b) It is then clear that Λ(t) is the output of a linear TDS driven by the input Φ(t). In other words, Λ(t) is obtained by linearly filtering Φ(t) through a TDS. Therefore, the persistent excitation condition stated in Assumption 2 is indeed an assumed property of Φ(t). In practice, inequality (28) can be monitored by numerically solving the TDS (29) with a DDE solver.

2 Proposition 2. Under Assumptions 1 and 2, the recursively generated matrix R(t) through the ODE (27e) from Λ(t) is bounded for all t > t 0 + h. Moreover, R(t) is an invertible matrix and its inverse is also bounded for all t > t 0 + h. 2

Convergence of estimation errors

The convergence of θ(t) and x(t) will be analyzed, based on the already established convergence of η(t). Proposition 3. Under Assumptions 1 and 2, both the state estimation error x(t) = x -x(t) and the parameter estimation error θ(t) = θ-θ(t) of the adaptive observer ( 27) converge to zero exponentially when t → +∞.

2

Proof. Let us first analyze θ(t). Notice that θ = θθ = θ, then

where Λ(t) is as defined in (27b).

By viewing R(t)Λ(t) T Λ(t) as a (bounded) time varying matrix, θ(t) is the state of a linear time varying (LTV) system, driven by the inputs η(t) and η(t -h), which tend exponentially to zero. Before studying the behavior of this LTV system, let us first consider its homogeneous part, namely θ = -R(t)Λ(t) T Λ(t)ϑ(t).

(32) In order to study the stability of this homogeneous LTV system, define a Lyapunov function candidate