Norio Saito

Stéphane Cordier

Pierric Lemoine

Takeo Ohsawa

Yoshiki Wada

Fabien Grasset

Jeffrey S Cross

Naoki Ohashi

Lattice and Valence Electronic Structures of Crystalline Octahedral Molybdenum Halide Clusters-based Compounds, Cs2[Mo6X14] (X = Cl, Br, I), Studied by Density Functional Theory Calculations

Keywords: octahedral cluster, molybdenum halide, DFT calculation, XPS, electronic structure

The electronic and crystal structures of Cs2[Mo6X14] (X = Cl, Br, I) cluster-based compounds were investigated by density functional theory (DFT) simulations and experimental methods such as powder X-ray diffraction (XRD), ultraviolet-visible (UV-Vis) spectroscopy, and X-ray photoemission spectroscopy (XPS). The experimentally determined lattice parameters were in good agreement with theoretically optimized ones, indicating the usefulness of DFT calculations for the structural investigation of these clusters. The calculated band gaps of these compounds reproduced those experimentally determined by UV-Vis reflectance within an error of a few tenths of an eV. Core-level XPS and effective charge analyses indicated bonding states of the halogens changed according to their sites. The XPS valence spectra were fairly well reproduced by simulations based on the projected electron density of states (PDOS) weighted with crosssections of Al Kα, suggesting that DFT calculations can predict the electronic properties of metalcluster-based crystals with good accuracy.

Introduction

Octahedral metal cluster complexes, [M6L

i 8L a 6]
n-(M = Mo, W, or Re; L i = inner ligand; L a = apical ligand), have attracted much attention, being regarded as building blocks for the construction of unique solid state inorganic and hybrid organic/inorganic compounds 1,2 and nanostructures [START_REF] Perrin | The Molybdenum and Rhenium Octahedral Cluster Chalcohalides in Solid State Chemistry: From Condensed to Discrete Cluster Units[END_REF]4 exhibiting interesting chemical and physical properties. [START_REF] Cordier | Inorganic Molybdenum Octahedral Nanosized Cluster Units, Versatile Functional Building Block for Nanoarchitectonics[END_REF] As the size of these complexes is much larger than that of mononuclear ones, they can crystallize with various large inorganic (e.g., Cs + ) and molecular organic counter cations. [START_REF] Cordier | Inorganic Molybdenum Octahedral Nanosized Cluster Units, Versatile Functional Building Block for Nanoarchitectonics[END_REF] Indeed, the combination of these metal cluster complexes and organic molecules enables the design of numerous functional structures, such as meso-scale structured materials 6 and highly dimensional supramolecular assemblies. [START_REF] Kim | Novel Compounds Based on [Re6Q8(L)6] 4-(Q = S, Se, Te; L = CN, OH) and Their Applications[END_REF] An interesting feature of such metal-cluster-based compounds is the tunability of their properties by choosing an appropriate combination of M, L i , and L a [START_REF] Gray | Spectroscopic and Photophysical Properties of Hexanuclear Rhenium(III) Chalcogenide Clusters[END_REF][START_REF] Kirakci | A Highly Luminescent Hexanuclear Molybdenum Cluster -A Promising Candidate toward Photoactive Materials[END_REF][START_REF] Kirakci | A Comparative Study of the Redox and Excited State Properties of (nBu4N)2[Mo6X14[END_REF][START_REF] Kirakci | Hexamolybdenum Cluster Complexes with Pyrene and Anthracene Carboxylates: Ultrabright Red Emitters with the Antenna Effect[END_REF] or by crystallization with various counter cations. [START_REF] Saito | Synthesis and Properties of Charge-Transfer Solids with Cluster Units [Mo6X14] 2-(X = Br, I)[END_REF] For example, some of these compounds show electric conductivity or superconductivity, [START_REF] Chevrel | Superconducting Chevrel Phases: Prospects and Perspectives[END_REF][START_REF] Fischer | Superconductivity in the RexMo6S8[END_REF] while those with a relatively large band gap energy (Eg) show useful luminescence, [START_REF] Maverick | Luminescence and Redox Photochemistry of the Molybdenum(II) Cluster Mo6Cl14 2[END_REF][START_REF] Gray | Highly Emissive Hexanuclear Rhenium(III) Clusters Containing the Cubic Cores [Re6S8] 2+ and [Re6Se8] 2+[END_REF][START_REF] Zietlow | Photophysics and Electrochemistry of Hexanuclear Tungsten Halide Clusters[END_REF] photocatalytic, [START_REF] Barras | Fast Photocatalytic Degradation of Rhodamine B over [Mo6Br8(N3)6] 2-Cluster Units under Sun Light Irradiation[END_REF][START_REF] Barras | One-pot Synthesis of Gold Nanoparticle/Molybdenum Cluster/Graphene Oxide Nanocomposite and its Photocatalytic Activity[END_REF][START_REF] Kumar | Photoreduction of CO2 to Methanol with Hexanuclear Molybdenum [Mo6Br14] 2-Cluster Units under Visible Light Irradiation[END_REF] and redox properties. [START_REF] Gray | Solar Chemistry of Metal Complexes[END_REF] Numerous studies on nanostructure synthesis have been conducted to enhance the useful functions of these cluster compounds. [START_REF] Grasset | Water-in-Oil Microemulsion Preparation and[END_REF][START_REF] Grasset | Metal Atom Clusters" Meet ZnO Nanocrystals: A ((n-C4H9)4N)2Mo6Br14@ZnO Hybrid[END_REF][START_REF] Grasset | One-pot Synthesis and Characterizations of Bi-functional Phosphor-Magnetic @SiO2nanoparticles: Controlled and Structured Association of Mo6 Cluster Units and γ-Fe2O3nanocrystals[END_REF][START_REF] Cordier | Novel Nanomaterials Based on Inorganic Molybdenum Octahedral Clusters[END_REF][START_REF] Aubert | Extended Investigations on Luminescent Cs2[Mo6Br14]@SiO2 Nanoparticles: Physico-Structural Characterizations and Toxicity Studies[END_REF][START_REF] Truong | Visible Tunable Lighting System Based on Polymer Composites Embedding ZnO and Metallic Clusters: From Colloids to Thin Films[END_REF] In particular, [Mo6X i 8X a 6]

2-(X = Cl, Br, I) complexes have been widely studied because of their broadband red luminescence under UV or blue light illumination. [START_REF] Maverick | Spectroscopic, Electrochemical, and Photochemical Properties of Molybdenum(II) and Tungsten(II) Halide Clusters[END_REF] Highly efficient phosphors emitting red light under excitation with blue light-emitting diodes (LEDs) are very important in solid-state lighting devices. Hence, the optimization of chemical compositions and structures of these compounds is promising for their applications as phosphors. In fact Kitamura and his related group have developed novel metal-cluster-based compounds with high luminescence efficiency through modification of cluster's chemical composition. [START_REF] Yoshimura | III) Clusters [Re6(μ3-S)8X6] 4-(X -= Cl -, Br -, I -) are Luminescent at Room Temperature[END_REF][START_REF] Mikhailov | Synthetic Tuning of Redox, Spectroscopic, and Photophysical Properties of {Mo6I8} 4+ Core Cluster Complexes by Terminal Carboxylate Ligands[END_REF] An effective exploration strategy of these complex systems involves the utilization of theoretically predicted composition-structure-property relationships. One of the earlier studies on the molecular orbital (MO) calculation of [Mo6Cl14] 2-cluster was reported by Cotton in 1978. [START_REF] Cotton | Ground State Electronic Structures of Some Metal Atom Cluster Compounds[END_REF] After that electronic properties of the metal cluster complexes such as chemical state and absorption energy have been calculated everywhere since 1990. [START_REF] Honda | Theoretical Study on Electronic Excitation Spectra of Mo and Re Cluster Complexes[END_REF][START_REF] Imoto | Molecular Orbital Calculations of Octahedral Molybdenum Cluster Complexes with the DV-X.alpha[END_REF][START_REF] Robinson | Effect of Coordination Environment on the Electronic Structure and Properties of Mo6-Based Systems: A Density Functional Treatment[END_REF][START_REF] Lin | Structure and Bonding in Face-and Edgebridged Octahedral Transition Metal Clusters[END_REF][START_REF] Costuas | Combined Theoretical and Time-Resolved Photoluminescence Investigations of[END_REF] However, most of those studies were based on MO calculations, where neither the presence of counter cations nor lattice periodicity were considered and electronic structures of spatially isolated and electrically charged metal cluster complexes were calculated, although it is reasonable to expect that exchange of counter cations should cause changes in atomic arrangements and effective charge of the complexes. Hence, in order to investigate electronic structure and structural stability of the metal-cluster-based compounds in crystalline form, it is necessary to perform electronic structure calculations based on crystalline periodicity with charge neutrality. Furthermore, most of those calculations were performed using classical functionals, such as the Hartree-Fock approximation.

Thus, the utilization of updated density functional theory (DFT), e.g., the generalized gradient approximation (GGA), [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] and consideration of lattice periodicity including counter cations is needed for more accurate predictions of composition-structure-property relationships in metal cluster complexes.

In this context, we have recently studied the crystal structure of Cs2[Mo6Cl14] 38 using planewave-based DFT calculations to examine the applicability of DFT for reproducing the structural and electronic features of metal-cluster-based crystalline compounds. As a result, it was shown that the accuracy of conventional DFT is sufficient to predict formation energy differences between polymorphs of crystalline Cs2[Mo6Cl14], implying that DFT calculations can be a useful tool for exploring metal-cluster-complex-based solid systems. Hence, we have attempted to predict the optical properties of metal-cluster-based compounds using updated DFT calculations.

This study describes DFT calculations on Cs2[Mo6X14] (X = Cl, Br, I), aiming to accurately reproduce the effects of halogen substitution using the GGA functional. To examine the consistency between experimental observations and theoretical predictions, the observed lattice parameters were compared with the theoretically optimized ones. Furthermore, the reproducibility of theoretically predicted electronic structures was examined by comparison with experimentally observed X-ray photoemission (XPS) spectra and Eg values.

Materials and methods

Sample preparation

The metal cluster-based solid state ternary halides Cs2[Mo6X14] (X = Cl, Br, I) were synthesized and purified using procedures reported elsewhere. [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF][START_REF] Saito | From Cs2Mo6Cl14 to Cs2Mo6Cl14•H2O and Vice Versa: Crystal Chemistry Investigations[END_REF][START_REF] Kirackci | Synthesis and Characterization of Cs2Mo6X14 (X = Br or I) Hexamolybdenum Cluster Halides: Efficient Mo6 Cluster Precursors for Solution Chemistry Syntheses[END_REF] After purification, all samples were heated at 150 °C for 1 h in a glove box filled with dry N2 gas (< 0.1 ppm H2O). In our previous study, [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF][START_REF] Saito | From Cs2Mo6Cl14 to Cs2Mo6Cl14•H2O and Vice Versa: Crystal Chemistry Investigations[END_REF]

Characterization and analyses

The crystal structures of the obtained powders were characterized by XRD using a RINT-Ultima III diffractometer (Rigaku Co. Ltd., Tokyo, Japan) with monochromatic Cu Kα radiation (λ = 0.15418 nm). The observed diffraction patterns were indexed assuming that the samples crystallized in the trigonal centrosymmetric 31 P c space group. [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF][START_REF] Saito | From Cs2Mo6Cl14 to Cs2Mo6Cl14•H2O and Vice Versa: Crystal Chemistry Investigations[END_REF][START_REF] Kirackci | Synthesis and Characterization of Cs2Mo6X14 (X = Br or I) Hexamolybdenum Cluster Halides: Efficient Mo6 Cluster Precursors for Solution Chemistry Syntheses[END_REF] Finally, the lattice parameters of Cs2[Mo6X14] were refined using the method of least squares.

The diffuse reflectance spectra of Cs2[Mo6X14] in the near-UV to near-infrared (near-IR) range

were measured using a SolidSpec-3700 spectrophotometer (Shimadzu Corp., Tokyo, Japan) equipped with a BaSO4 integrating sphere. Both core-level and valence band range XPS spectra of samples were obtained using a Σ-probe spectrometer (Thermo Fisher Scientific K.K, Yokohama, Japan) equipped with a monochromated Al Kα X-ray source (hν = 1486.6 eV). The pass energy was set at 20.0 eV for all spectra, with the instrumental resolution of electron energy being ~0.2 eV. Samples were attached on an adhesive tape containing carbon particles. When obvious charging under X-ray irradiation occurred, a charge neutralizer employing both Ar + ion and electron emitters was used to eliminate charging. All binding energies were calibrated with respect to the C 1s peak of adventitious carbon at 284.6 eV. After subtraction of the spectral background using Shirley's method, [START_REF] Shirley | High-Resolution X-Ray Photoemission Spectrum of the Valence Bands of Gold[END_REF] the observed spectra were fitted by mixed Gaussian-Lorentzian functions using the method of least squares to calculate the integrated peak intensities and relative shifts of peak positions.

Computational methods

All DFT calculations were performed for the trigonal form of Cs2[Mo6X14] (X = Cl, Br, I), which belongs to the 31 P c centrosymmetric space group, [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF][START_REF] Saito | From Cs2Mo6Cl14 to Cs2Mo6Cl14•H2O and Vice Versa: Crystal Chemistry Investigations[END_REF][START_REF] Kirackci | Synthesis and Characterization of Cs2Mo6X14 (X = Br or I) Hexamolybdenum Cluster Halides: Efficient Mo6 Cluster Precursors for Solution Chemistry Syntheses[END_REF] as shown in Fig. 1 S1.

All DFT calculations were performed with the CASTEP code [START_REF] Clark | First Principles Methods Using CASTEP[END_REF] utilizing the plane wave pseudo-potential method. [START_REF] Milman | Electronic Structure, Properties, and Phase Stability of Inorganic Crystals: A Pseudopotential Plane-Wave Study[END_REF] The norm-conserving pseudo-potentials (NCPPs) [START_REF] Hamann | Conserving Pseudopotentials[END_REF] were generated using the OPIUM code. [START_REF] Hamann | Generalized Norm-Conserving Pseudopotentials[END_REF][START_REF] Rappe | Optimized Pseudopotentials[END_REF] The revised Perdew-Burke-Ernzerhof GGA functional for solid-state compounds (PBEsol) [START_REF] Perdew | Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces[END_REF] was employed for energy minimization using the Pulay density mixing scheme. [START_REF] Pulay | Convergence Acceleration of Iterative Sequences. The Case of SCF Iteration[END_REF] The cell parameters and atomic coordinates were relaxed using a quasi-Newton method and the Broyden-Fletcher-Goldfarb-Shanno Hessian update scheme. [START_REF] Pfrommer | Relaxation of Crystals with the Quasi-Newton Method[END_REF] Electron spin polarization was considered for all DFT calculations. The plane wave cut-off energy was set to 830 eV, a Monkhorst-Pack grid [START_REF] Monkhorst | Special Points for Brillouin-Zone Integrations[END_REF][START_REF] Pack | Special Points for Brillouin-Zone Integrations"-A Reply[END_REF] (4 × 4 × 3 mesh) was employed for Brillouin zone sampling, and a 96 × 96 × 144 mesh was adopted for the fast Fourier transformation. The convergence tolerances were set to 5 × 10 -5 nm for atomic displacement, 5 × 10 -6 eV/atom for total energy, 0.1 eV/nm for maximum interatomic force, and 0.02 GPa for pressure. The effective charges of constituent elements were estimated using two methods, i.e., Mulliken [START_REF] Segall | Population Analysis of Plane-Wave Electronic Structure Calculations of Bulk Materials[END_REF][START_REF] Segall | First-Principles Simulation: Ideas, Illustrations and the CASTEP Code[END_REF] and Hirshfeld population analyses. [START_REF] Hirshfeld | Bonded-Atom Fragments for Describing Molecular Charge Densities[END_REF][START_REF] Wiberg | Comparison of Atomic Charges Derived via Different Procedures[END_REF] To compare the observed spectra with calculated results, additional simulations of optical and electronic spectra were performed. Optical absorption spectra were simulated by computing inter-band transition probabilities using the results of DFT calculations. The number of k-points involved in the simulation of optical properties was set to 21 per irreducible Brillouin zone.

Valence band XPS spectra were also simulated by assuming a previously reported photoionization probability. 56

Results and discussion

Structural analyses

The observed XRD patterns shown in Fig. 2 were identified to be the trigonal phase as previously reported in references. [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF][START_REF] Saito | From Cs2Mo6Cl14 to Cs2Mo6Cl14•H2O and Vice Versa: Crystal Chemistry Investigations[END_REF][START_REF] Kirackci | Synthesis and Characterization of Cs2Mo6X14 (X = Br or I) Hexamolybdenum Cluster Halides: Efficient Mo6 Cluster Precursors for Solution Chemistry Syntheses[END_REF] However, for the sample with X = Cl, some peaks (denoted by arrows in Fig. 2), not assignable to Cs2[Mo6Cl14], were found. As previously reported, [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF] these extra peaks were from the secondary phase(s) crystallizing during synthesis and/or purification. The secondary phase was assigned to be Cs2[Mo6Cl14].H2O as reported in a separate report. [START_REF] Saito | From Cs2Mo6Cl14 to Cs2Mo6Cl14•H2O and Vice Versa: Crystal Chemistry Investigations[END_REF] As the fraction of this secondary phase in the currently prepared sample should be negligibly small, we assume that the obtained results represent the characteristics of the trigonal Cs2[Mo6Cl14] phase. The refined lattice parameters of each sample, listed in Table 1,

were almost identical to those reported in literature, [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF][START_REF] Saito | From Cs2Mo6Cl14 to Cs2Mo6Cl14•H2O and Vice Versa: Crystal Chemistry Investigations[END_REF][START_REF] Kirackci | Synthesis and Characterization of Cs2Mo6X14 (X = Br or I) Hexamolybdenum Cluster Halides: Efficient Mo6 Cluster Precursors for Solution Chemistry Syntheses[END_REF] being only slightly smaller. This deviation was attributed to the removal of volatile impurities, such as water related molecules.

We previously reported that insertion of water molecules from ambient condition into the trigonal lattice of Cs2[Mo6Cl14] caused slight expansion of the lattice volume, and the water impurities could be removed by heating in an atmosphere of dry N2, along with contraction of lattice volume. [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF] Hence, the samples prepared in this study are assumed to be very close to their intrinsic states.

The results of structural optimization and energy minimization converged well for all calculations, with the optimized lattice parameters being close to experimental values, as listed in Table 1. The calculated lattice parameters were well-matched with the experimental ones along the a-axis, being somewhat longer along the c-axis. We note that the overestimation of lattice parameters is due to a general tendency of the GGA functional. [START_REF] Haas | Calculation of the Lattice Constant of Solids with Semilocal Functionals[END_REF] The detailed optimized lattice structures and fractional positions of all atoms in the lattice are summarized in Table S2 in the Supporting Information. These parameters are also consistent with the values reported in literature, [START_REF] Saito | Theoretical and Experimental Determination of the Crystal Structures of Cesium-Molybdenum Chloride[END_REF][START_REF] Kirackci | Synthesis and Characterization of Cs2Mo6X14 (X = Br or I) Hexamolybdenum Cluster Halides: Efficient Mo6 Cluster Precursors for Solution Chemistry Syntheses[END_REF] indicating that the DFT structural optimization was successful, and the lattice structures were consistently reproduced by DFT calculations employing the PBEsol functional.

Electronic structures

The valence band structure of Cs2[Mo6X14] varied with X, as seen from the PDOS of Cs2[Mo6X14] (Fig. 3) obtained using the PBEsol functional after structural optimization. As upspin and down-spin orbitals were identical, this figure shows the total PDOS profiles, featuring a summation of up-and down-spin orbitals. As can be seen, the Mo 4d states were dominantly found around the valence band maximum (VBM), while the valence p-orbitals of X were distributed in the relatively deeper parts of the valence band. In particular, the valence p-states of X i s were located in deeper parts of the valence band compared to those of X a s. Moreover, the sorbitals of X in the semi-core region (> 9 eV) showed similar behavior: the s states of X i s were deeper than those of X a s. These results indicate that the chemical states of X a s and X i s are obviously different. In addition, the energy dispersion of valence electrons also varied with X.

For instance, the energy dispersion of the valence p-orbitals of X a s obviously changed with X (Fig. 3). Indeed, the population of X p-orbitals within VBM increased with the atomic number of X. Moreover, the dispersion of valence X p-orbitals seemed being broader with the increase of the atomic number of X. These valence band structure changes are reflected in the results of charge analyses shown in Table S3. The charge of X became more positive with increasing atomic number, mirrored by the charge of Mo becoming more negative. These results indicate that Mo-X bonds are highly ionic for X = Cl and less ionic for X = Br and I. Although the results of charge analyses suggest changes in chemical bonds between Mo and X, the calculated charge values seem to be inconsistent from the viewpoint of experimental chemistry. Namely, the positively charged X and negatively charged Mo atoms do not correspond to the intuitively expected results, since Mo and X are usually considered as a cation and anions, respectively.

Such inconsistencies between intuitive and calculated results might be due to the charge analysis methodology adopted in this study, implying that the PDOS obtained by projection of the total electron distribution to the assumed atomic orbitals should feature a similar problem.

Considering the difficulties in evaluating reasonable effective charges, [START_REF] Sanchez-Portal | Projection of Plane-Wave Calculations into Atomic Orbitals[END_REF] the characteristics of Mo-X bonds are discussed below in terms of real-space orbital population distribution.

Figure 4 shows the highest occupied molecular orbital (HOMO) of Cs2[Mo6X14]. For X = I, the HOMO was dominated by Mo 4d and I 5p states at X a sites (I a -5p). On the other hand, for X = Cl and Br, the HOMO featured contributions from the p-states of X i . In particular, the HOMO of highly localized on all atoms, and the Mo-Cl bonds seem to be highly ionic. In support of this, the valence orbital overlap of Cl and Mo is negligibly small, as shown by the iso-surfaces of electron orbitals. In contrast, the electron orbitals in Cs2[Mo6I14] show covalent bond character.

For instance, the electron orbital involved in ΦO is located at the center of the Mo-I bond, and all orbital distributions are not homogeneous, but rather anisotropic, as seen in ΦJ, ΦK, and ΦM.

Another characteristic feature found in Cs2[Mo6I14] is the hybridization of I i and I a , as exemplified by ΦK. The difference between Φ14 and ΦO is also interesting. As mentioned above, Φ14 showed a very narrow energy dispersion compared to that of ΦO. Typical orbital distributions of bands involved in Φ14 and ΦO are compared in Figs. 6(o) and (p). All Cl i atoms contribute nearly equally to every electron orbital in Φ14, while the contribution of I i atoms to electron orbitals of ΦO differ from each other. This is an indication that the electron orbitals of Cs2[Mo6I14] possess a strong directionality, which is characteristic of covalent bonds. Indeed, the calculated total electron density in Cs2[Mo6Cl14] showed a spherical distribution centered at every Cl, while that in Cs2[Mo6I14] was distorted from spherical shape (Fig. S1). Hence, it was concluded that Mo-X bonds become more covalent with increasing halogen atomic number.

Reproducibility of typical electronic spectra

Figure 7 compares calculated and observed optical absorption spectra. Herein, experimental spectra were obtained from the observed diffuse reflectance shown in Fig. S2 using the KM function. [START_REF] Wendlandt | Reflectance Spectroscopy[END_REF][START_REF] Tandon | Measurement of Forbidden Energy Gap of Semiconductors by Diffuse Reflectance Technique[END_REF] The calculated spectra were obtained by applying Gaussian broadening with 0.05 eV in width. The observed and calculated absorption edges agreed within an error of less than 0.2 eV, indicating that the calculated Eg well reproduced the experimental values. It has been suggested that Eg calculated by conventional DFT methods tends to be smaller than the experimentally obtained values. [START_REF] Perdew | Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems[END_REF][START_REF] Heyd | Energy Band Gaps and Lattice Parameters Evaluated with the Heyd-Scuseria-Ernzerhof Screened Hybrid Functional[END_REF] However, for this system, the calculated Eg showed good agreement with experimental observations. It should be also noted that the HOMO-LUMO gaps evaluated in the other study employing a molecular orbital method were much wider than the experimental values of Eg. For instance, the HOMO-LUMO gap for X= Cl evaluated by the molecular orbital calculation was reported to be 3.3 eV in the reference. [START_REF] Ramirez-Tagle | Electronic Structure and Molecular Properties of the [Mo6X8L6[END_REF] The comparison of present results with experimental results and the other reported calculations indicates that the corresponding energy band structure calculations based on DFT are appropriate for predicting band gap energies in Cs2[Mo6X14] systems.

It has to be reminded that we have employed energy band structure calculation involving lattice periodicity. The most definite difference between the previous studies [START_REF] Cotton | Ground State Electronic Structures of Some Metal Atom Cluster Compounds[END_REF][START_REF] Honda | Theoretical Study on Electronic Excitation Spectra of Mo and Re Cluster Complexes[END_REF][START_REF] Imoto | Molecular Orbital Calculations of Octahedral Molybdenum Cluster Complexes with the DV-X.alpha[END_REF][START_REF] Robinson | Effect of Coordination Environment on the Electronic Structure and Properties of Mo6-Based Systems: A Density Functional Treatment[END_REF][START_REF] Lin | Structure and Bonding in Face-and Edgebridged Octahedral Transition Metal Clusters[END_REF][START_REF] Costuas | Combined Theoretical and Time-Resolved Photoluminescence Investigations of[END_REF] using the cluster calculation based on molecular orbital concept and present study is that present one dealt with neutral lattice with crystalline periodicity. As this study is based on crystalline lattice, effective charge of the metal cluster complex and counter cation, i.e., Cs, varied with chemical compositions as described above. Hence the effective charge of the metal cluster complexes could be converged into reasonable values. We suppose that this is the most significant merit of this study where the lattice periodicity has been considered. Indeed the accordance between experimental and calculated absorption spectra is good news for researchers investigating the electric and optical properties of metal-cluster-based compounds, since DFT simulation based on lattice periodicity is useful for material design, e.g., obtaining compounds with the desired Eg by substitution of X by another halogen (X') to form Cs2[Mo6(X, X')14]. Namely, we can expect that prediction of the stable structure and physical properties of metal complex compound with various counter cation is possible with plane wave DFT calculations based on lattice periodicity.

The observed XPS core-level spectra of Cs2[Mo6X14] are shown in Fig. 8. The core-level spectra of halogens featured two doublets, denoted P1 and P2. Since these doublets originated from spin-orbit coupling, profile fitting was performed assuming the presence of two different halogen species. Indeed, the presence of two halogen ion components in XPS spectra has been reported, [START_REF] Hamer | X-Ray Photoelectron Spectra of Inorganic Molecules. IX. Distinction Between Bridging and Terminal Metal-Chlorine Bonds in Metal Halide Clusters of Rhenium(III) and Molybdenum(II)[END_REF][START_REF] Walton | Some Recent Aspects of the X-Ray Photoelectron Spectra and Chemistry of Low Oxidation State Molybdenum Halides and Their Complexes[END_REF][START_REF] Best | X-Ray Photoelectron Spectra of Inorganic Molecules. 22. Halogen Core Electron Binding Energies of Low Oxidation State Molybdenum Bromide and Molybdenum Iodide Clusters and Niobium and Tantalum Chlorides Containing the [M6Cl12] n+ Cores[END_REF] with these two peaks attributed to X a and X i . The results of peak fitting are summarized in Table 2. The integrated intensity ratio of P1 and P2 (S1/S2) ranged from 0.74 to 0.77, consistently corresponding to 0.75 (= 6/8), which is reasonably attributed to X a and X i in

the [Mo6X i 8X a 6]
2-complex. The energy difference between P1 and P2 (Eb2 -Eb1) equaled 2.1 eV for X = Cl and Br, being equal to 1.6 eV for X = I. As the current DFT study used pseudopotential and not all-electron calculations, it was not possible to determine the orbital energies of the observed core levels. On the other hand, the above experimentally observed energy difference is comparable to the energy difference of semi-core states calculated by DFT. Based on the calculated DOS shown in Fig. 3, the values of Eb2 -Eb1 estimated by DFT simulations equaled 2.1 eV for X = Cl, 2.0 eV for X = Br, and 1.7 eV for X = I. These estimated values were in agreement with the experimental ones within an error of 0.1 eV, providing further evidence that the results of DFT calculations were consistent with experimental observations. The results of the corresponding XPS studies also indicate that the variation of the electronic structure of Cs2[Mo6X14] with X discussed above was a reasonable consideration. As displayed in Fig. 6(l), electron orbitals with contributions from p-orbitals of both I i and I a were found in the results of the DFT study on Cs2[Mo6I14]. This indicates that the relatively small energy separation between the levels originating from I i and I a resulted in hybridization between I i and I a .

Here, it should be noted that the variation of ionicity of halogen at X a and X i site indicated by DFT calculation enables us to explain observed reactivity of the metal cluster complexes. From the viewpoint of chemical reaction for elimination and exchange of halogens in the metalcluster-based compounds, halogen at X a site in the complex with X i = Cl or Br can be replaced by other halogens, [START_REF] Preetz | Structure and Properties of the Cluster Anions[END_REF][START_REF] Preetz | KriDarstellung, Kristallstruktur und spektroskopische Eigenschaften der Clusteranionen[END_REF] but halogen substitution in [Mo6I14]-complex seems to be difficult. As the complex with X = Cl and Br possesses relatively strong ionicity as indicated by DFT calculation, substitution of X a site in [Mo6Cl14]-and [Mo6Br14]-complex should be allowed when the electric charge of X a site is maintained. However, replacement of I at X a site by other halogen is expected to be limited because relatively high electronegativity of Cl and Br is not compatible to covalent bond formed in [Mo6I14]-complex. Hence, we can conclude that DFT calculation which visualize characteristics of chemical bonds is very useful tool for understanding of the chemical reactivity of the metal cluster complex compounds.

In addition to core-level XPS, the observed valence band XPS spectra and calculated PDOS were also compared, as shown in Fig. 9. Herein, the calculated XPS spectra were obtained by multiplying the reported photoionization probability listed in Table S4 with the PDOS shown in well reproduced in the calculated ones, although the valence band widths of the former were much wider than those of the latter. Note that the presence of two peaks in the Cs 5p region of the observed spectra resulted from spin-orbital coupling, not being an indication of the presence of two different electronic states of Cs. In the observed spectra, the semi-core levels of X i and X a were not well resolved for X = Br and I due to the overlap of doublets resulting from spin-orbit coupling. However, it is evident that the energy separation between the semi-core states of Cl observed in the XPS spectrum was well reproduced in the calculated spectrum. Hence, we can safely conclude that the band structure of Cs2[Mo6X14] was well reproduced by DFT.

Accordingly, the above discussion on valence band electron orbitals obtained by DFT calculations was meaningful, since the calculation results well reproduced the XPS and optical absorption spectra.

Conclusions

In this study, both experimental and theoretical characterizations were carried out to reveal the electronic structure of cluster-based Cs2[Mo6X14] ternary solid state compounds. The calculated structural features (lattice parameters) and optical properties (Eg), as well as the calculated DOS of these compounds were consistent with the experimental results, indicating that DFT simulations are appropriate for reproducing and predicting the above properties. The results of DFT calculations indicated that the electronic structure of Cs2[Mo6X14] solids depend on the nature of X, although their crystal structures were fairly independent of the nature of X. The most significant electronic structure change due to the variation of X was found to be the difference in the degree of hybridization of Mo-X bonds.

As the calculated results were in good agreement with experimental observations, the DFT technique is considered appropriate for the prediction of electronic and optical properties of -5

-4 -3 -2 -1 0 1 Γ Energy (eV) H L M K H A Γ -6 -6 -5 -4 -3 -2 -1 0 1 2 Γ Energy (eV) H L M K H A Γ Φ1(4)
Φ2( 8) Φ5( 2) Φ7( 18) Φ9( 6) Φ11( 8) Φ12( 12) Φ13( 6) Φ14( 12) Φ15( 6) Φ3( 4) Φ6( 6) Φ8( 4) Φ10( 4) Φ4( 6) Φ16 (2) (a) X=Cl (b) X=I ΦA( 4) ΦC( 6) ΦE( 2) ΦG( 6) ΦI( 4) ΦB( 6) ΦD( 12) ΦF( 12) ΦH( 6) ΦJ( 12) ΦK( 4) ΦL( 6) ΦM( 2)ΦN( 6) ΦP( 6) ΦO (12) ΦQ(2)

  (a), with structural optimizations performed by applying structural constraints to maintain 31 P c symmetry. It should be noted that this simplified model was not perfectly compatible with the experimentally refined crystal structure of Cs2[Mo6Br14]. 40 Indeed, as shown in Fig. 1(b), the XRD characterization of Cs2[Mo6Br14] indicated splitting of Cs sites along the c-axis and the presence of interstitial Cs sites at (0, 0, 0.25) with a 12% occupancy. 40 However, the simplified model shown in Fig. 1(a) was adopted for DFT calculations of all Cs2[Mo6X14] complexes (including Cs2[Mo6Br14]) due to limited computational resources. The structural parameters used to construct the initial structural model are listed in Table

Cs2[

  Mo6Cl14] was dominated by Mo 4d and Cl 3p states at X i (Cl i -3p). Note that the HOMO of Cs2[Mo6Br14] was intermediate between that of Cs2[Mo6Cl14] and Cs2[Mo6I14]. These results indicate that the electronic structure of the valence band (VB) of Cs2[Mo6X14] obviously changes with X despite the structural similarity.The valence band structures of Cs2[Mo6Cl14] and Cs2[Mo6I14] are shown in Fig.5to investigate the effect of X in Cs2[Mo6X14]. As seen in this figure, the electronic bands within the VB are classified into bundles. As the electronic states of every cluster are highly localized, every band possesses a small dispersion in k-space, which is the reason behind the formation of bundles. Thus, the observed electronic structure resembles the energy levels of isolated molecules. As seen from Fig.5, the energetic order of the bundles was fairly independent of X; for instance, ΦD and ΦE of Cs2[Mo6I14] resembled Φ4 and Φ5 of Cs2[Mo6Cl14], respectively. However, the number of bands in the ΦA-ΦE bundles in Cs2[Mo6I14] and that of Φ1-Φ5 in Cs2[Mo6Cl14] equaled 28 and 22, respectively, implying that several bands located above ΦE in Cs2[Mo6I14] had energies below that of Φ5 in Cs2[Mo6Cl14]. Indeed, ΦF and Φ6 show different numbers of bands and dissimilar energy dispersions. Another typical difference between Cs2[Mo6Cl14] and Cs2[Mo6I14] can be found in bundles located relatively deep in the valence band. Both Φ14 and ΦO bundles are close to the bottom of the valence band and consist of 12 electron bands. However, the energy dispersion of Φ14 is obvioulsly narrower than that of ΦO. Differences in the electronic band structure of Cs2[Mo6Cl14] and Cs2[Mo6I14] are likely due to the variable degree of Mo-X hybridization. Since Fig. 5 indicates that the energy dispersion of all bundles was wider for Cs2[Mo6I14] than for Cs2[Mo6Cl14], the electrons in Cs2[Mo6Cl14] are presumably more strongly localized than those in Cs2[Mo6I14]. This understanding is consistent with the variation of orbital shapes shown in Fig. 6. The wave functions in Cs2[Mo6Cl14] are

Fig. 3 .

 3 Fig.3. Details of the simulation method are described in the Supporting Information. The shape

FIGURESFigure 1 .

 1 FIGURES

Figure 2 .

 2 Figure 2. XRD patterns of Cs2[Mo6X14]: (a) X = Cl, (b) X = Br, and (c) X = I. Arrows denote

Figure 3 .

 3 Figure 3. PDOS of Cs2[Mo6X14]: (a) X = Cl, (b) X = Br, and (c) X = I.

Figure 4 .

 4 Figure 4. Electron orbital distribution of the highest occupied orbital in Cs2[Mo6X14] for (a) X = Cl, (b) X = Br, and (c) X = I. The yellow clouds indicate iso-surfaces normalized at 0.08 e/Å 3 , and the red and blue arrows indicate the clouds representing the contributions of X i and X a ,

Figure 5 .

 5 Figure 5. Electron energy band structure of Cs2[Mo6X14] in the valence band range for (a) X =

Figure 6 .

 6 Figure 6. Iso-surface plot of typical electron orbitals in Cs2[Mo6X14] for X = Cl and I. The

Figure 7 .

 7 Figure 7. Calculated and observed absorption spectra of Cs2[Mo6X14] (X = Cl, Br, I).

Figure 8 .

 8 Figure 8. Core-level XPS spectra of Cs2[Mo6X14]: (a) Cl 2p region for X = Cl , (b) Br 3p region

Figure 9 .

 9 Figure 9. Comparison between the observed and calculated XPS valence spectra of Cs2[Mo6X14]

Table 1 .

 1 Refined lattice parameters of trigonal Cs2[Mo6X14] (X = Cl, Br, I) in hexagonal expression.

		Lattice parameter
		a / nm	c / nm	V / nm 3
	Experimentally determined in this study
	Cs2[Mo6Cl14]	0.9789(1) 1.4195(3) 1.1779(1)
	Cs2[Mo6Br14]	1.0156(2) 1.5073(5) 1.3465(2)
	Cs2[Mo6I14]	1.0794(2) 1.6249(5) 1.6394(3)
	Calculated in this study
	Cs2[Mo6Cl14]	0.980	1.475	1.227
	Cs2[Mo6Br14]	1.014	1.550	1.380
	Cs2[Mo6I14]	1.084	1.636	1.665
	Indicated in literature [38,40]
	Cs2[Mo6Cl14]	0.9789(1) 1.4214(2) 1.1796(2)
	Cs2[Mo6Br14]	1.01925(1) 1.50690(3) 1.35574(3)
	Cs2[Mo6I14]	1.0804(5) 1.6258(5) 1.6435(1)

Table 2 .

 2 Spectral parameters (Eb for peak position and S for relative integrated peak intensity) of halogens in Cs2[Mo6X14] (X = Cl, Br, I) obtained by profile fitting under the assumption that all core level profiles comprised two components (P1 and P2).

			P1		P2
	X	Core level				(Eb2 -Eb1) / eV S1/S2
			Eb1/eV S1	Eb2/eV S2
	Cl 2p 3/2	197.8	1090 199.9	1460 2.1	0.75
	Br 3p 3/2	180.8	1730 182.9	2240 2.1	0.77
	I	3d 5/2	619.0	7790 620.6	10520 1.6	0.74

particularly Dr. David Lechevalier, Dr. Mari Kono, and Dr. Benjamin Dierre of Saint-Gobain KK (Tokyo, Japan), Noée. Dumait, Serge Paofai of ISCR for their contributions to contributions to the synthesis of metal clusters precursors. N.S. would also like to thank Prof. Junzo Tanaka of Tokyo Tech for support and encouragement. The studywas financially supported by Saint-Gobain Company(France), CNRS, Université de Rennes 1 and NIMS through the Laboratory for Innovative Key Materials and Structures (LINK UMI 3629), with additional contributions from Region Bretagne (France), and Fondation Langlois.

Funding Sources

Saint-Gobain company, Centre National de la Recherche Scientifique (CNRS), Université de Rennes 1 and NIMS through the Laboratory for Innovative Key Materials and Structures (LINK UMI 3629) with additional contributions from Region Bretagne (France), and Fondation Langlois.

Notes

Any additional relevant notes should be placed here.

AUTHOR INFORMATION

Corresponding Author

* E-mail: ohashi.naoki@nims.go.jp

Author Contributions

All authors have given their approval to the final version of the manuscript.

Supporting Information Available: Listings of crystal graphic parameters for theoretical calculation studies, the results for the effective charge analyses, contour map of electron density, experimental UV-Vis diffuse reflection data, and photoionization probability data.

ACKNOWLEDGMENT

This work was carried out as part of the France-Japan international collaboration framework (UMI 3629-LINK Center). The authors thank people involved in LINK and related activities,

For Table of Contents Only

The electronic and crystal structures of Cs2[Mo6X14] (X = Cl, Br, I) ternary cluster-based compounds were investigated by density functional theory (DFT) simulations and experimental methods. The experimentally refined lattice parameters were in good agreement with theoretical optimized ones. The calculated band gaps also reproduced those photo-spectroscopically determined within an error of a few eV. The X-ray photoemission valence spectra were fairly reproduced from projected density of state (PDOS) weighted with cross-sections of Al Kα.


M Sigalotti
email: mario.sigalotti@inria.fr

Keywords: Optimal control, geometric control, second-order optimality conditions, time-optimal, bang arc, singular arc

Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems

Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems

INTRODUCTION

The regularity of optimal trajectories is a much studied problem in control theory, because the presence of discontinuities has obvious implications on the implementability and robustness of optimal feedbacks. The regularity of the value function is also strongly related with the regularity of the optimal trajectories (see, for instance, [START_REF] Schättler | Geometric optimal control[END_REF]). When the value function encodes a distance in a length-space, the regularity of geodesics carries information on the properties of balls and other relevant geometric objects (see [START_REF] Agrachev | A Comprehensive Introduction to Sub-Riemannian Geometry[END_REF]).

We study here a class of control systems which can be written in the form q(t) = u 1 (t)X 1 (q(t)) + u 2 (t)X 2 (q(t)), (1) where the state q evolves on a smooth manifold M , X 1 , X 2 are smooth vector fields, and the control u = (u 1 , u 2 ) takes values in the square [-1, 1] 2 . This setup corresponds to driftless two-input systems for which the modulus of each control parameter has an independent bound. Under some natural Lie algebra rank condition, the time-optimal value function for (1) can also be seen as the length distance associated with a sub-Finsler structure (see [START_REF] Barilari | Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions[END_REF]).

Little is known on the general structure of time-optimal trajectories for systems of the type (1) apart from some homogeneous cases studied in [START_REF] Breuillard | On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry[END_REF] (where M is the Heisenberg group and X 1 , X 2 are two left-invariant generators of the horizontal distribution), [START_REF] Barilari | Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions[END_REF] (where sub-Finsler versions of the Grushin plane and the Martinet distribution have been considered), and [START_REF] Ardentov | Sub-Finsler geodesics on the Cartan group[END_REF] (where the Cartan group is considered). The inhomogeneous 2D case has been studied in Abdul-Latif [START_REF] Ali | Local (sub)-Finslerian geometry for the maximum norms in dimension 2[END_REF].

More is known in the case where the control u = (u 1 , u 2 ) takes values in a ball, since this leads to the sub-

The project leading to this publication has received funding from the ANR project SRGI ANR-15-CE40-0018.

Riemannian framework (see [START_REF] Agrachev | A Comprehensive Introduction to Sub-Riemannian Geometry[END_REF]; [START_REF] Jean | Control of nonholonomic systems: from sub-Riemannian geometry to motion planning[END_REF]; [START_REF] Rifford | Sub-Riemannian geometry and optimal transport[END_REF]). It should however be mentioned that even in this case the minimal regularity of timeoptimal trajectories is still an open problem (see [START_REF] Barilari | On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures[END_REF]; [START_REF] Hakavuori | Non-minimality of corners in subriemannian geometry[END_REF]; [START_REF] Monti | On tangent cones to length minimizers in Carnot-Carathéodory spaces[END_REF] for recent results and a state of the art about this longstanding question). When the control takes values in a ball and a drift is added to the dynamics fewer results are known (see Agrachev andBiolo (2017, 2018); [START_REF] Caillau | Minimum time control of the restricted three-body problem[END_REF]).

Another case where more results are available is when one of the two control inputs u 1 or u 2 is constant. In this case we recover the case of a single-input controlaffine system with control in a compact interval. For such kind of systems on a two-dimensional manifold, the situation has been deeply analyzed and general results covering the generic and the analytic case can be found in Sussmann (1987b,a). A monograph where 2D optimal syntheses are studied in details is [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF]. The results in the 3D case are less complete and do not cover the generic case, but only the local behavior of time-optimal trajectories near points where the brackets of the vector fields X 1 and X 2 have degeneracies of corank 0 (i.e., near generic points) or 1 (see [START_REF] Agrachev | Nonlinear controllability and optimal control[END_REF]; [START_REF] Agrachev | On the local structure of optimal trajectories in R 3[END_REF]; [START_REF] Krener | The structure of small-time reachable sets in low dimensions[END_REF]; Schättler (1988); Schättler (1988); [START_REF] Sussmann | Envelopes, conjugate points, and optimal bang-bang extremals[END_REF]) and some but not all cases of corank larger than 1 [START_REF] Bressan | The generic local time-optimal stabilizing controls in dimension 3[END_REF]; [START_REF] Sigalotti | Local regularity of optimal trajectories for control problems with general boundary conditions[END_REF]). In [START_REF] Sigalotti | Local regularity of optimal trajectories for control problems with general boundary conditions[END_REF], the regularity of time-optimal trajectories near generic points in dimension 4 is also characterized.

The main type of result contained in the papers mentioned above is a guarantee that small-time optimal trajectories are the concatenation of at most a given number of bang and singular arcs (with limitations on the possible concatenation). One of the relevant consequences of this kind of result is that it allows to rule out the appearance of the Fuller phenomenon (also called chattering). This kind of rather radical singularity of optimal trajectory is known to be typical (i.e., not removable by small perturbations) in large enough dimension (see [START_REF] Kupka | The ubiquity of Fuller's phenomenon[END_REF]; [START_REF] Zelikin | Theory of chattering control. Systems & Control: Foundations & Applications[END_REF]; [START_REF] Zelikin | Geometry of neighborhoods of singular trajectories in problems with multidimensional control[END_REF]). Some restriction on the wildness of the Fuller behavior (in particular, on the iteration order of accumulations of switching times) can be found in [START_REF] Boarotto | Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities[END_REF]; [START_REF] Boarotto | Fuller singularities for generic control-affine systems with an even number of controls[END_REF].

In this paper we provide a bound on the worst-regularity behavior of time-optimal trajectories of (1) near generic points in the case where M has dimension 3. Our main result, Theorem 3, states that such trajectories are necessarily concatenations of bang and singular arcs, and that the number of such arcs is not larger than 5. It is interesting to observe that in the corresponding homogenous 3D case, the sub-Finsler structure on the Heisenberg group, smalltime optimal trajectories with 5 arcs do exist [START_REF] Breuillard | On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry[END_REF]; [START_REF] Barilari | Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions[END_REF]). This means that the bound that we provide in this paper is sharp. The main technical step in the proof of our main result used the second-order necessary conditions for optimality proposed in [START_REF] Agrachev | Nonlinear controllability and optimal control[END_REF].

STATEMENT OF THE TIME-OPTIMAL PROBLEM AND NECESSARY CONDITIONS FOR OPTIMALITY

Throughout the paper M denotes a smooth (i.e., C ∞ ) complete manifold and X 1 , X 2 are two smooth vector fields on M . We associate with M , X 1 , and

X 2 the dynamics q = u 1 X 1 (q) + u 2 X 2 (q), q ∈ M, u 1 , u 2 ∈ [-1, 1].
(2) We also introduce the notations u = (u 1 , u 2 ) and

X(u) = u 1 X 1 + u 2 X 2 , X + = X 1 + X 2 , X -= X 1 -X 2 .
It is well known that the time-optimal trajectories of a control system of the form (2) heavily depend on the commutativity properties of the vector fields X 1 , X 2 , which are infinitesimally characterized by the iterated Lie brackets between them.

Given two smooth vector fields X and Y we write [X, Y ] to denote the Lie bracket between X and Y , and ad X for the operator [X, •] acting on the space of smooth vector fields by ad

X (Y ) = [X, Y ].
To reduce the notational burden we also set

X 12 = [X 1 , X 2 ], X +12 = [X + , X 12 ], X -12 = [X -, X 12 ].
The Lie algebra generated by {X 1 , X 2 }, denoted by Lie{X 1 , X 2 }, is the minimal linear subspace of the space of all smooth vector fields on M containing {X 1 , X 2 } and invariant both for ad X1 and ad X2 .

The time-optimal control problem for (2) consists in finding the trajectories of (2) connecting two given points q 0 and q 1 of M in minimal time. The existence of at least one trajectory connecting any pair of points and minimizing the time is a consequence of Chow's theorem and Filippov's theorem, under the assumption that {V (q) | V ∈ Lie{X 1 , X 2 }} = T q M for every q ∈ M (see, for instance, [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]).

First-order necessary conditions for optimality: the Pontryagin maximum principle

A well-known optimality condition satisfied by all timeoptimal trajectories is the Pontryagin maximum principle (PMP, for short). In order to fix some notations, let us recall here its statement.

Let π : T * M → M be the cotangent bundle, and s ∈ Λ 1 (T * M ) be the tautological Liouville one-form on T * M . The non-degenerate skew-symmetric form σ = ds ∈ Λ 2 (T * M ) endows T * M with a canonical symplectic structure. With any smooth function p : T * M → R let us associate its smooth Hamiltonian lift

→ p ∈ C(T * M, T T * M ) by the condition σ λ (•, → p ) = d λ p.
(3)

Introducing the control-dependent Hamiltonian function

H : T * M × R 2 → R by H(λ, v) = λ, v 1 X 1 (q) + v 2 X 2 (q) , q = π(λ), (4
) the statement of the PMP for the time-optimal problem associated with system (2) is the following (see, for instance, [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]). Theorem 1. (PMP). Let q : [0, T ] → M be a time-optimal trajectory of (2), associated with a control u(•). Then there exists an absolutely continuous curve λ :

[0, T ] → T * M such that (q(•), λ(•), u(•)) is an extremal triple, i.e., in terms of the control-dependent Hamiltonian H introduced in (4), one has λ(t) ∈ T * q(t) M \ {0}, ∀t ∈ [0, T ], (5)
H(λ(t), u(t)) = max{H(λ(t), v) | v ∈ [-1, 1] 2 } (6) for a.e. t ∈ [0, T ], λ(t) = → H (λ(t), u(t)), for a.e. t ∈ [0, T ]. ( 7
)
For any extremal triple (q(•), λ(•), u(•)), we call the corresponding trajectory t → q(t) an extremal trajectory and the curve t → λ(t) its We associate with an extremal triple the absolutely continuous functions

t → φ i (t) = λ(t), X i (q(t)) , i = 1, 2. A maximal open interval of [0, T ] on which φ 1 and φ 2 are both different from zero is said to be a bang arc. A u 1 - singular (respectively, u 2 -singular ) arc is a maximal open interval of [0, T ] on which φ 1 ≡ 0 while φ 2 is different from zero (respectively, φ 2 ≡ 0 while φ 1 is different from zero).
An arc which is either u 1 -singular or u 2 -singular is said to be singular. A point separating two bang arcs is said to be a switching time. We say that the control u i switches at the switching time τ if u i (which is constant and of modulus 1 on sufficiently small left-and right-neighborhoods of τ ) changes sign at τ . In particular, if u i switches at τ then it follows from the maximality condition of the P M P that φ i (τ ) = 0. A trajectory is said to be bang-bang if it is the concatenation of finitely many bang arcs.

A useful consequence of the PMP is that for every smooth vector field Y on M , for every extremal triple associated with (2) the identity

d dt λ(t), Y (q(t)) = λ(t), [X(u(t)), Y ](q(t)) (8)
holds true for a.e. t.

Second-order optimality conditions

We recall in this section a useful second-order necessary condition for a trajectory with piecewise constant control to be time-optimal, obtained in Agrachev and Gamkrelidze (1990) using time-reparameterizations as variations of the reference control signal.

In order to state the theorem, let us introduce the following notation. Given a control u : [0, T ] → [-1, 1] 2 and two times s, t ∈ [0, T ], denote by P u (s, t) the flow of (2) from time s to time t. Since we are interested in local properties, we can always assume that solutions of (2) exist globally, which ensures that P u (s, t) is defined on the entire manifold M . As a consequence, by standard properties of solutions of ODEs, P u (s, t) : M → M is a diffeomorphism.

For every diffeomorphism P : M → M , every point q ∈ M , and every v ∈ T q M , P * ,q v ∈ T P (q) M denotes the pushforward of v obtained by applying the differential of P at q. For every diffeomorphism P : M → M and every vector field Y , the push-forward of Y by P is the vector field P * Y : q → P * ,P -1 (q) Y (P -1 (q)).

Using these notations, we can deduce from the PMP that, if (q(•), λ(•), u(•)) is an extremal triple on [0, T ], then, for i = 1, 2 and for t, τ ∈ [0, T ],

φ i (t) = λ(τ ), (P u (τ , t) * X i )(q(τ )) . (9
) Theorem 2. Let q : [0, T ] → M be a time-optimal trajectory for (2) and let u(

•) = (u 1 (•), u 2 (•)) be the corre- sponding control function. Assume that u(•) is piecewise constant on [0, T ], with K non-removable discontinuities τ 1 < τ 2 < • • • < τ K in (0, T ).
Denote by u 0 , . . . , u k the successive values of u(•) on the K + 1 bang arcs. Assume that q(•) admits a unique extremal lift λ(•) up to multiplication by a positive scalar. Fix τ in [0, T ] and let h i = P u (τ , τ i ) * X(u i ), (10) for i = 0, . . . , K. Let Q be the quadratic form

Q(α) = 0≤i<j≤K α i α j λ(τ ), [h i , h j ](q(τ )) , (11)
defined on the space

H = α = (α 0 , . . . , α K ) ∈ R K+1 | K i=0 α i = 0, K i=0 α i h i (q(τ )) = 0 . ( 12
)
Then Q ≤ 0. Remark 1. The theorem above can be extended from timeminimal to (locally) time-maximal trajectories. This can be done following the lines of Theorem 2 in [START_REF] Agrachev | On the local structure of optimal trajectories in R 3[END_REF] where the notion of quasi-optimal trajectory is introduced to cover both kind of properties.

BOUND ON THE NUMBER OF ARCS

Let us assume from now on that M has dimension 3. This section contain the main results on the admissible concatenation of arcs, locally near a point where suitable bracket independence conditions are satisfied. We collect the main results in a single statement, Theorem 3 below.

Theorem 3. Let q 0 ∈ M and Ω be a neighborhood of q 0 , compactly contained in M , such that (X 1 , X 2 , X 12 ), (X 1 , X 12 , X +12 ), (X 1 , X 12 , X -12 ), (X 2 , X 12 , X +12 ), and (X 2 , X 12 , X -12 ) are moving bases on the closure Ω of Ω.

Then there exists T > 0 such that every time-optimal trajectory q : [0, T ] → M of (2) contained in Ω and such that T ≤ T is the concatenation of at most 5 bang or singular arcs. Moreover, if q(•) contains a singular arc, then it is the concatenation of at most a bang, a singular, and a bang arc.

The main technical step in the proof of the theorem is contained in the next lemma, which focuses on the situation in which both u 1 and u 2 switch along the timeoptimal trajectory. Notice that in this case we can relax the assumptions on the triples of vector fields which should be linearly independent on the considered neighborhood Ω.

Actually, we just need to assume that X 1 , X 2 , and X 12 are linearly independent on Ω. Lemma 1. Let q 0 ∈ M and Ω be a neighborhood of q 0 , compactly contained in M , such that X 1 , X 2 , and X 12 are linearly independent on the closure Ω of Ω. Then there exists T > 0 such that every bang-bang timeoptimal trajectory q : [0, T ] → M of (2) contained in Ω, undergoing switchings both in u 1 and in u 2 , and such that T ≤ T is the concatenation of at most 5 arcs.

Proof. Consider a bang-bang extremal trajectory q : [0, T ] → M of (2) contained in Ω. Assume that q(•) is the concatenation of 6 bang arcs and that both u 1 and u 2 switch. We are going to show that, for T small enough, q(•) is not optimal.

Let λ(•) be an extremal lift of q(•) and define φ (t) = λ(t), X (q(t)) , t ∈ [0, T ], ∈ {1, 2, 12}.

(13) It follows from (8) that φ (t) = λ(t), [u 1 (t)X 1 + u 2 (t)X 2 , X ](q(t)) (14) for ∈ {1, 2, 12} and for almost every t ∈ [0, T ]. In particular, φ1 (t) = -u 2 (t)φ 12 (t),

φ2 (t) = u 1 (t)φ 12 (t), (15)
for almost every t ∈ [0, T ].

Assume for now that λ(•) is normalized in such a way that max{|φ

1 (0)|, |φ 2 (0)|, |φ 12 (0)|} = 1. ( 16
)
We first claim that, up to taking T small enough, φ 12 does not change sign on [0, T ]. In order to prove the claim, notice that, since λ(•) solves the time-dependent Hamiltonian system (7) described by the Pontryagin maximum principle, its growth admits a uniform bound among extremal trajectories in T * Ω. It then follows from ( 14) that there exist T, C > 0 (independent of q(•)) such that | φ (t)| ≤ C for ∈ {1, 2, 12} and t ∈ [0, T ], provided that T < T . Because of the assumption that both u 1 and u 2 switch along the trajectory q(•), we know that both φ 1 and φ 2 have a zero on [0, T ]. This implies, in particular, that

|φ 1 |, |φ 2 | ≤ CT on [0, T ]. Let T be smaller that 1/C. It follows that |φ 1 |, |φ 2 | < 1 on [0, T ],
and hence, from ( 16), one has |φ 12 (0)| = 1. As a consequence, up to modifying C uniformly with respect to q(•),

||φ 12 | -1| ≤ CT on [0, T ],
and, in particular, φ 12 has constant sign. This concludes the proof of the claim that φ 12 can be assumed not to change sign on [0, T ].

Let us focus now on the admissible concatenations of switches. Assume for a moment that u 2 switches at the same time τ ∈ (0, T ) at which u 1 does. Then φ1 changes sign at τ . This means that φ 1 does not change sign at τ , contradicting the assumption that u 1 switches at τ . We can then assume that the switches of u 1 and u 2 are distinct.

Denote by τ 0 ∈ (0, T ) the first switching time for q(•). Up to exchanging the roles of X 1 and X 2 , we can assume that τ 0 is a switching time for u 1 . Up to exchanging X 2 and -X 2 , we can assume that u 2 = -1 in a neighborhood of τ 0 . Denote by τ 1 the smaller switching time for q(•) with τ 1 > τ 0 . Assume for now that the (constant) sign of φ 12 on [0, T ] is -1. Hence, φ 2 is increasing and φ 1 decreasing on (τ 0 , τ 1 ). This implies that τ 1 is a switching time for u 2 .

The same reasoning allows to continue the argument and deduce that the sequence of constant controls (u 1 , u 2 ) for q(•) on its 6 bang arcs follows the (periodic) pattern 1 -1

→ τ 0 -1 -1 → τ 1 -1 1 → τ 2 1 1 → τ 3 1 -1 → τ 4 -1 -1 , (17)
where we have denoted by τ 0 , . . . , τ 4 the 5 switching times of q(•). The 6 bang arcs correspond then to the intervals I 1 = (0, τ 0 ), I 2 = (τ 0 , τ 1 ), and so on, up to I 6 = (τ 4 , T ).

In the case where φ 12 > 0 on [0, T ], one analogously shows that the pattern is

-1 -1 → τ 0 1 -1 → τ 1 1 1 → τ 2 -1 1 → τ 3 -1 -1 → τ 4 1
-1 and the argument below can be adapted easily.

Let us denote by t i = τ i+1 -τ i , i = 0, 1, 2, 3, the length of the four bang arcs connecting two switching times.

We are going to apply Theorem 2 to the trajectory q(•) taking τ = τ 2 . Notice that φ 1 (τ ) = 0, since u 1 is switching at τ .

It is convenient to consider a new normalization of the covector λ(•) by imposing that

φ 12 (τ ) = -1. ( 18
)
One of the assumptions of Theorem 2 is the uniqueness of the extremal lift for q(•). In order to prove uniqueness, notice that λ(•) depends only on the value λ(τ ) and that λ(τ ), X 1 (q(τ )) = 0, because τ is a switching time for u 1 , and λ(τ ), X 12 (q(τ )) = -1, because of the normalization of λ(•). We are left to prove the uniqueness of the component φ 2 (τ ) of λ(τ ) along X 2 (q(τ )).

Using (9), let us consider, for t ∈ (0, t 2 ), the development

φ 2 (τ + t) = λ(τ ), (e t(X1+X2) * X 2 )(q(τ )) = φ 2 (τ )a(q(•), t) + b(q(•), t), (19)
where the functions a and b depend on t and on the bangbang trajectory q(•), but not on the extremal lift λ(•).

We write below O(t) to denote a quantity which can be bounded from above by a term of the form C|t|, for t small enough, with C uniform with respect to q(•).

With this notation,

a(q(•), t) = 1 + O(t), b(q(•), t) = O(t).
Since φ 2 (τ + t 2 ) = φ 2 (τ 3 ) = 0, we deduce from (19) that φ 2 (τ ) is a function of q(•) only (including the dependence on t 2 ), i.e., that φ 2 (τ ) is uniquely identified by q(•). This completes the proof of the uniqueness of the extremal lift.

Pushing the computations a step further and noticing that

e t2(X1+X2) * X 2 = X 2 + t 2 [X 1 , X 2 ] + O(t 2 2 ) on Ω, we have that a(q(•), t 2 ) = 1 + O(t 2 2 ), b(q(•), t 2 ) = -t 2 + O(t 2 2
). Together with (19) evaluated at t = t 2 , this yields

φ 2 (τ ) = t 2 + O(t 2 2 ).
Repeating the argument on the interval [τ 1 , τ ], we have that

φ 2 (τ + t) = λ(τ ), (e -t(-X1+X2) * X 2 )(q(τ )) = φ 2 (τ )(1 + O(t 2 )) -t + O(t 2 ), for t ∈ [0, t 1 ], yielding t 2 + O(t 2 2 ) = φ 2 (τ ) = t 1 + O(t 2 1 ). Henceforth, t 2 = t 1 + O(t 2 1 ).
Similarly, t 0 , t 3 = t 1 + O(t 2 1 ), and we then deduce that T = O(t 1 ). In particular, max

t∈[0,T ], i=1,2 |φ i (t)| = O(t 1 ).
According to the relation T = O(t 1 ), we are left to prove that q(•) is not optimal if t 1 is small enough.

Using the notation from Theorem 2, we have

h 0 = e -t1(-X-) * e -t0(-X+) * (X -) = X -+ O(t 1 ), h 1 = e -t1(-X-) * (-X + ) = -X + + O(t 1 ), h 2 = -X -, h 3 = X + , h 4 = e t2(X+) * (X -) = X -+ O(t 1 ), h 5 = e t2(X+) * e t3(X-) * (-X + ) = -X + + O(t 1
). We can then evaluate

σ ij = λ(τ ), [h i , h j ](q(τ ))
for 0 ≤ i < j ≤ 5, obtaining

σ 01 = 2 + O(t 1 ), σ 02 = O(t 1 ), σ 03 = -2 + O(t 1 ), σ 04 = O(t 1 ), σ 05 = 2 + O(t 1 ), σ 12 = 2 + O(t 1 ), σ 13 = O(t 1 ), σ 14 = -2 + O(t 1 ), σ 15 = O(t 1 ), σ 23 = 2, σ 24 = O(t 1 ), σ 25 = -2 + O(t 1 ), σ 34 = 2 + O(t 1 ), σ 35 = O(t 1 ), σ 45 = 2 + O(t 1 ).
Decomposing the relation 5 i=0 α i h i (q(τ )) = 0 on the basis X + (q(τ )), X -(q(τ )), X 12 (q(τ )) and collecting the components along X + (q(τ )) and X -(q(τ )), we get

0 =α 0 O(t 1 ) -α 1 (1 + O(t 1 )) + α 3 + α 4 O(t 1 ) -α 5 (1 + O(t 1 )), 0 =α 0 (1 + O(t 1 )) + α 1 O(t 1 ) -α 2 + α 4 (1 + O(t 1 )) + α 5 O(t 1 ).
Considering, in addition, the relation 5 i=0 α i = 0 and solving with respect to α 0 , α 1 , α 2 , we get

α 0 = -α 3 -α 4 + O(t 1 ; α 3 , α 4 , α 5 ), α 1 = α 3 -α 5 + O(t 1 ; α 3 , α 4 , α 5 ), α 2 = -α 3 + O(t 1 ; α 3 , α 4 , α 5
), where (α 3 , α 4 , α 5 ) → O(t 1 ; α 3 , α 4 , α 5 ) denotes a linear function whose coefficients are O(t 1 ) in the sense introduced above.

We claim now that the space H defined in ( 12), taken here with K = 5, is of dimension 3 in R 6 . Indeed, if a vector p is orthogonal to

W = 5 i=0 α i h i (q(τ )) | 5 i=0 α i = 0 ,
then p annihilates, in particular, h i+1 (q(τ )) -h i (q(τ )) for i = 0, . . . , 4. The same computations as those used to prove the uniqueness of the extremal lift of q(•) show that the orthogonal to W is one-dimensional, that is, W is 2dimensional. Hence, the kernel H of the map (α 0 , . . . , α 5 ) → Since M has one positive and two negative eigenvalues, we conclude from Theorem 2 that the trajectory q(•) is not optimal for t 1 small enough. This concludes the proof of Lemma 1.

We are now ready to conclude the proof of Theorem 3.

Proof of Theorem 3. The proof works by considering separately several types of time-optimal trajectories q : [0, T ] → Ω of (2).

First consider the case where either φ 1 or φ 2 never vanish on [0, T ]. Up to exchanging the roles of X 1 and X 2 , let us assume that φ 2 does not change sign and, consequently, u 2 is constantly equal to +1 or -1 on [0, T ]. This means that the trajectory q(•) is time-optimal also for the single-input control-affine system q = f (q) + ug(q), u ∈ [-1, 1], (20) with g = X 1 and f equal either to X 2 or to -X 2 . In particular g, [f, g], and [f + g, [f, g]] are linearly independent on Ω and the same is true for g, [f, g], and [f -g, [f, g]].

We can deduce from Theorem 3 in [START_REF] Agrachev | On the local structure of optimal trajectories in R 3[END_REF] that q(•) is the concatenation of at most 3 bang arcs or a bang, a singular, and a bang arc.

Consider now the case where both φ 1 and φ 12 have at least one zero on [0, T ]. We claim that in this case φ 2 is never zero on [0, T ], and the conclusion then follows by the case just considered. In order to prove the claim, normalize λ(•) in such a way that max(|φ 1 (0)|, |φ 2 (0)|, |φ 12 (0)|) = 1. Since the growth of λ can be uniformly bounded on Ω, we have that |φ 1 | and |φ 12 | can be bounded by CT for some positive constant C independent on the trajectory q(•). By taking T small enough, it follows that |φ 2 (0)| = 1. Since, moreover, according to (15),

| φ2 | ≤ |φ 12 | ≤ CT ,
we can conclude that φ 2 never vanishes, as claimed.

The case where both φ 2 and φ 12 vanish on [0, T ] being completely analogous, we can assume from now on that φ 12 never vanishes on [0, T ].

We claim that in this case q(•) is bang-bang. Indeed, consider first the case where q(•) has a u 1 -singular arc (τ 0 , τ 1 ). We deduce from the expression of φ1 (cf. ( 15)) and the nonvanishing of φ 12 that u 2 must vanish on (τ 0 , τ 1 ). This, in turn, implies that also φ 2 vanishes on (τ 0 , τ 1 ). Notice now that, by ( 15), φ2 = u 1 φ 12 which would also imply that u 1 ≡ 0 on (τ 0 , τ 1 ). But, clearly, the control corresponding to a time-optimal trajectory cannot vanish on a nontrivial interval, invalidating the assumption that q(•) has a u 1 -singular arc. The case of a u 2 -singular arc is completely symmetric.

In order to prove that q(•) is bang-bang we can then assume that there exists an open interval (τ 0 , τ 1 ) contained in [0, T ] on which φ 1 and φ 2 are different from zero. Assume that (τ 0 , τ 1 ) is maximal with this property and that τ 1 < T . Then either φ 1 or φ 2 vanish at τ 1 . If they both vanish at τ 1 , then we deduce from (15) that if one of the two functions φ i changes sign, then also the derivative of the other function φ 3-i changes sign, which means that sign(φ 3-i ) and hence u 3-i stay constant. Moreover, since the sign of φ1 and φ2 do not change on a bang arc, the interval (τ 1 , T ) will be a bang arc. If they do not both vanish at the same point, then an analogous reasoning shows that q| (τ1,T ) is either a single bang arc or that q| (τ0,T ) is the concatenation of bang arcs undergoing switches both in u 1 and u 2 . Reasoning similarly for the interval [0, τ 0 ], we conclude the proof of the claim that q(•) is bang-bang.

Moreover, the argument above shows that, under the condition that φ 12 never vanishes, the bang-bang trajectory q(•) is either the concatenation of at most 2 bang arcs, or it undergoes switches both in u 1 and u 2 . The conclusion of the proof of Theorem 3 then follows from Lemma 1.

CONCLUSION

We considered in this paper the class of time-optimal driftless two-input control problems in which each scalar control has an independent bound on its modulus. When the system is defined on a three-dimensional manifold and the controlled vector fields satisfy some generic Lie bracket independence condition at a given point, we prove that all small-time optimal trajectories near such a point are the concatenation of at most five bang and singular arcs. The result ensures, in particular, that the time-optimal synthesis does not exhibit the Fuller phenomenon. The proposed bound is sharp, as it has been illustrated by previous works in the literature considering the homogeneous case of the Heisenberg group, also known as Brockett integrator in the control literature.

The proof of this fact extensively uses the Hamiltonian formalism provided by the Pontryagin maximum principle and its related second-order sufficient conditions for optimality. Extensions of the technique to more degenerate bracket conditions or to four-dimensional manifolds could be possible, but would involve more intricate computations.

  5 -dim W = 3.The quadratic form Q from Theorem 2 is then Q(α 3 , α 4 , α 5 ) = (α 3 , α 4 , α 5 )(M + O(t 1 ))(α 3 , α 4 , α 5 )