Dr Julien Massue
email: massue@unistra.fr

Dr Gilles Ulrich

Dr Filippo Monti

Dr Andrea Barbieri
email: andrea.barbieri@isof.cnr.it

Phosphorescent Cyclometalated Iridium(III) Complexes Bearing Ethynyl-Extended 2-(2'-Hydroxyphenyl) Benzoxazole Ancillary Ligands

Keywords: Iridium complexes, Hydroxybenzazoles, Luminescence, Phosphorescence, Charge Transfer Excited State

published or not. The documents may come

Introduction

Cyclometalated iridium(III) complexes displaying sizable phosphorescence emission have been the focus of intense research due to their strategic position at the crossroad of many attractive applications. These phosphorescent emitters have been indeed beneficially used in applicative fields including optoelectronics, [1][2][3][4][5] non-linear optics, [6] sensing, [7][8][9][10] or as particularly biocompatible probes for in vitro bioimaging. [11][12][13] This interest is largely due to the possibility to fine-tune their photophysical properties depending on subtle electronic modifications on the π-conjugated backbone of the ligands coordinated to the metal. [14][15][16][17] Indeed, cyclometalated complexes have been shown to feature a strong interplay between the molecular orbitals of the ancillary ligands and those of the metallic center, leading to either Metal-to-Ligand Charge Transfer (MLCT) or Ligand-to-Ligand Charge Transfer (LLCT) intense transitions. Electronic modifications around the trivalent iridium ion can indeed lead to drastic optical parameter changes such as absorption and/or emission maximum wavelength, phosphorescent quantum yields or excited-state lifetimes. Decorated phenylpyridines are typically used as ligands around the trivalent iridium ion to ensure a strong cyclometalation. The coordination sphere can be completed by more exotic heterocyclic structures in order to fine-tune the electronic properties of the corresponding cyclometalated phosphorescent emitters. Among those π-conjugated scaffolds surrounding the iridium(III) center, several examples can be found in the literature including 1,2,3-triazolylidene, [18] diisocyanoterphenyl, [19] N-heterocyclic carbene [20] or 2-(2-hydroxyphenyl) benzazole [21][22][23] (HBX) derivatives. HBX heterocycles have been extensively used as highly emissive fluorophores due to a beneficial excited-state intramolecular proton transfer (ESIPT) photophysical process, paving the way to many innovative applications. [24][25] Alternatively, they are excellent chelates for a wide range of metallic or metalloid ions. [26][27][28] In this article, a series of cyclometalated iridium(III) complexes bearing a π -extended 2-(2-hydroxyphenyl) benzoxazole (HBO) ancillary ligand with terminal groups of different electronic nature (see Scheme 1) has been synthesized and their photophysical properties studied at room and low temperature. The nature of the transitions was probed in each case by TD-DFT calculations. Singlet oxygen sensitization studies were also performed for all complexes.

Results and Discussion

Synthesis. The synthesis of the cyclometalated Iridium(III) complexes 4-6 and 7-9 is presented on Scheme 2. The synthetic targets can be obtained straightforwardly in one step, starting from the previously reported 2-(2'-hydroxyphenyl)benzoxazole (HBO) ligands 1-3, [29] respectively functionalized by p-ethynyltolyl (R = Me), p-ethynyl-phenyl-N-dibutylamino (R = NBu2) or pethynyl-benzonitrile (R = CN) groups. The formation of the targeted cyclometalated Iridium(III) complexes can be readily achieved by reacting the Iridium(III) chloro-bridged dimers precursors A or B with two equivalents of HBO chelates 1-3, in the presence of an excess of potassium carbonate in a mixture of methanol and dichloromethane at 40 °C (Scheme 2). Pure Iridium(III) complexes were obtained in 47-72% yield after purification on a silica gel column chromatography. All dimers were characterized with 1 H-NMR spectroscopy and highresolution mass spectrometry (HR-MS), reflecting their molecular structure in each case. Formation of the targeted complexes 4-6 and 7-9 was notably confirmed by assessing the disappearance of the downfield H-bonded phenolic proton present in the molecular structure of starting HBO ligands 1-3 by 1 H-NMR spectroscopy. All spectra can be found in the Supporting Information (SI). As reported for other similar Ir(III) cyclometalated complexes, it is expected that complexes 4-9 adopt a distorted octahedral geometry with the Nnitrogen atoms of the pyridine moiety in trans position. [START_REF] Flamigni | Photochemistry and Photophysics of Coordination Compounds Ii[END_REF] Scheme 2. Synthesis of cyclometalated Iridium(III) complexes 4-6 and 7-9

Absorption. Absorption spectra of HBO ligands 1-3 and iridium(III) heteroleptic complexes 4-9, recorded in acetonitrile solution at room temperature, are presented on Figure 1, while relevant main absorption parameters are listed in Table 1.

N N Ir R 1 O N O R 2 R 2 R 2 R 2 R 1 = Me, NBu 2 , CN R 2 = H, F
Ligands 1-3 display a strong absorption band centered around 330-340 nm with relatively high extinction coefficients (Figure 1). This band can be assigned to 1 - * transitions and its position and intensity is clearly affected by the nature of the substituent R (Scheme 2). In particular, the electron-withdrawing -CN and -donating -NBu2 groups, both induce a blue-shift in the absorption of about 140 cm -1 , from 346 to 330 nm (Table 1). The HBO ligand 2, functionalized with a NBu2 moiety shows a second intense band at lower energy appearing as a shoulder.

All iridium(III) complexes 4-9 display an intense absorption band at about 250-260 nm (εmax ~50-60,000 M -1 •cm -1 ) assigned by comparison to the archetypal [Ir(ppy)3] complex to spinallowed 1 LC transitions centered on the phenylpyridine ligands. [START_REF] Flamigni | Photochemistry and Photophysics of Coordination Compounds Ii[END_REF] At lower energy, between 300 and 370 nm the complexes 4, 5, 7 and 8 are characterized by the presence of intense bands that, by comparison with the absorption spectra of the relevant HBO ligands 1 and 2, can be attributed to spin allowed ligand centered ( 1 LC) transitions involving the HBO heterocycle. In complexes 6 and 9, bearing the strong electronwithdrawing substituent -CN, these bands appear to be shifted at lower energy (λmax = 371 and 364 nm, εmax = 29,900 and 40,600 M -1 •cm -1 , for 6 and 9, respectively). The broad and moderately intense bands appearing above 380 nm and extending up to 450-500 nm (εmax ~10,000 M -1 •cm -1 ) likely originate from iridium-based spin allowed metal-to-ligand charge transfer ( 1 MLCT) transitions, with possible weak contribution of formally spin forbidden triplet metal-to-ligand charge transfer ( 3 MLCT) transitions, induced by the presence of the heavy iridium center (Ir = 3,909 cm -1 ). [START_REF] Montalti | Handbook of Photochemistry[END_REF] It should be noted that only a limited effect on the absorption profiles and band positions is induced by the introduction of the fluorine atoms on the phenylpyridine rings. A more pronounced effect, especially on the bands observed between 300 and 370 nm, is induced by the substituents at the terminal position of the p-ethynylphenyl group (R = Me, CN or NBu2). 2 and described more in detail in SI in terms of Natural Transition Orbital (NTO) couples. [START_REF] Martin | [END_REF] All the reported vertical excitations are computed at the ground-state minimum-energy geometry in acetonitrile, using the linearresponse formalism. Regardless of the complex, all the four lowest-energy triplets are essentially 3 LC states, with some minor contribution from the iridium d orbitals. For all compounds, the first two triplets (i.e., T1 and T2) are centered on the benzoxazole-based ancillary ligand, while T3 and T4 mainly involve the two cyclometalating phenylpyridine ligands (see SI, Tables S4. 1-6).

The substituents at the para position of the ethynyl-phenyl moiety of the ancillary ligand strongly affect the energy and the mutual position of the T1 and T2 states. In fact, although both triplet states are centered on the same ancillary ligand, only one of them has a pronounced charge-transfer (CT) nature, while the other is predominantly populated by a local excitation (LE). The CT state formally involves the promotion of one electron from the ethynyl-phenyl moiety of the ancillary ligand to the benzoxazole core. The energy of this CT state therefore strongly depends on the electronic nature of the terminal substituents attached to the ancillary ligand. On the other hand, the LE state is mainly a transition involving the π-π * orbitals of the CC extended spacer of the ancillary ligand and the two nearby aromatic rings. LE states are less sensitive to the different electron-donating or -withdrawing substituents that are chemically attached to the ancillary ligand.

In the case of complexes 4 and 7, functionalized with an innocent methyl group at the terminal position of the ancillary ligand, the lowest triplet (T1) is the 3 LC-CT state, located 2.47-2.48 eV above S0 in 4 and 7, respectively. On the other hand, the 3 LC-LE state is 0.12 eV above T1 in both cases. In complexes 5 and 8, the presence of a strong electron donor substituent (NBu2) leads to the stabilization of the 3 LC-CT state by 0.10-0.11 eV, as compared to 4 and 7. The 3 LC-LE state also is stabilized, but only to a minor extent (0.02-0.03 eV). In the case of complexes 6 and 9, bearing a strong cyano acceptor, an increase of the energy of the 3 LC-CT state, in the range 0.10-0.11 eV is observed, with respect to the methyl-substituted complex 4 and 7. Moreover, the cyano substituent is able to lower the energy of the π * orbitals of the CC linker, owing to an increased electron delocalization, which triggers a stabilization of the corresponding 3 LC-LE state. As a consequence, a state flipping is observed and, for complexes 6 and 9, the lowest triplet state is 3 LC-LE in nature, while the 3 LC-CT state is lifted to T2.

As mentioned before, for all the complexes 4-9, T3 and T4 are both mainly centered on the two cyclometalating ligands, which are nearly equivalent within each complex. As a consequence, these two triplets are almost degenerate and their energy is virtually affected only by the type of cyclometalating ligand. The presence or absence of fluorine on these ancillary ligands does not lead to any observable effects. The energy of T3 and T4 is calculated to be 2.67  0.02 eV for the nonfluorinated complexes 4-6 and 2.82  0.03 eV for the fluorinated complexes 7-9. The energy increase observed in such transitions, by going from ppy to dfppy cyclometalating ligands, is a well-known effect, which is mainly ascribable to the stabilization of the highest-occupied π orbitals of the cyclometalating ligands, prompted by the electron-withdrawing fluorine atoms. [33][34] Figure 2. Energy diagram of the lowest-lying triplet states of complexes 4-9, computed in acetonitrile as vertical excitations from the respective groundstate minimum-energy geometries. For further details see Tables S1-S6.

Emission. The luminescence emission properties analysis for the three ligands 1-3 and the six iridium(III) complexes 4-9 has been performed at room temperature in dilute acetonitrile solutions, in air-equilibrated and air-free conditions (Figure 3) and at low temperature in CH2Cl2:CH3OH (1:1) glassy matrix. The relevant photophysical data are summarized in Table 2. All ligands 1-3 and complexes 4-9 display an almost featureless broad spectrum in acetonitrile solution at room temperature. From the emission profiles, it is possible to identify some similarities in the optical behavior of the six investigated complexes. In particular, as previously observed in the absorption spectra (Figure 1), for complexes 5 and 8, bearing a -NBu2 group as substituent on the ancillary ligand, the emission profiles are the most red-shifted of the series. For the other complexes, the emission shapes and relative maxima are in the same range within the series of complexes with the same cyclometalating ligands, as for 4 and 6, and for 7 and 9 (in this case, the second is just slightly blue-shifted than the former of ~7 nm). In all cases, the fluorinated derivatives 7-9 display a slightly blue-shifted emission wavelength with respect to the non-fluorinated congeners 4-6. These results can be rationalized with the help of DFT calculations. In order to assess the nature of the emitting states, spin-unrestricted DFT calculations were carried out to optimize the lowest triplet state of all the complexes. After full geometry optimization, starting from the Franck-Condon region and imposing a spin multiplicity of 3, all complexes relax to a minimum that displays the same nature of the T1 state computed by vertical excitations from the S0 minimum (i.e., no root-flipping is observed upon relaxation of T1). As a consequence, a 3 LC-CT state is expected to be responsible for the emission of all complexes, with the notable exception of 6 and 9 (R = CN) which should emit from a 3 LC-LE state. The spin-density distributions of the lowest-energy triplet state of complexes 4-9 are reported in the SI (Figure S3.1) and their topologies can be compared to the NTO couples associated to the S0  T1 vertical transitions (SI, Tables S4.1-6).

In line with the experimental data, the emission of complexes 5 and 8 (R = NBu2 group as substituent on the ancillary ligand) is estimated to occur at lower energy with respect to the other complexes (2.21 eV for 5 and 8). These values compare well with the experimental data reported in Table 2 for roomtemperature acetonitrile solutions (i.e., 2.18 and 2.20 eV for 5 and 8, respectively). Such a red-shifted emission is due the presence of the electron-donating amino substituent on the donor side of the ancillary ligand, which stabilizes the related 3 LC-CT emitting state.

On the other hand, the emission of 4 and 6 (R = Me and CN) is predicted to be very similar for both complexes and it is computed at (2.30  0.02) eV. This is in line with experimental data, despite the fact that the values are slightly overestimated when compared to emission spectra (i.e., 2.23 and 2.24 eV for complexes 4 and 6, respectively). This can be easily justified considering that the DFT estimated emission is computed as an adiabatic energy difference between S0 and T1. As for the fluorinated analogues 7 and 9, a minor blue shift in the emission energy is also expected from the calculations, despite the theoretical shift being underestimated compared to the experimental data (i.e., 0.03 vs. 0.09 eV).

It is worth stressing that, although complexes 4 and 6 (and the fluorinated analogues 7 and 9) display almost identical luminescence profiles and energies, they actually emit from triplet excited states of different nature (i.e., a 3 LC-CT state for 4 and 7 and a 3 LC-LE state for 6 and 9). The fact that the emission energy of these complexes is very similar is just a coincidence.

All complexes display bright emission both at room and at low temperature. The PLQY values emission quantum yield () of 4-9 ranges from 0.11 to 0.44, in de-aerated acetonitrile solutions. Moreover, the fluorinated derivatives 7-9 display the highest values of lifetimes and kr in the series (Table 2). As denoted by the calculated values of the radiative constants for all the complexes (kr =  / τ, values from 1.2 to 6.9×10 4 s -1 ), are one order of magnitude lower than those of reported iridium(III) complexes, emitting from a state with an important ligandcentered contribution. [START_REF] Flamigni | Photochemistry and Photophysics of Coordination Compounds Ii[END_REF] [a] Quantum yield and lifetime values in de-aerated (and air-equilibrated) acetonitrile solution; λexc = 420 nm and 370 nm for quantum yield and lifetime measurements, respectively. [b] In DCM:MeOH (1:1) glassy solution, λexc = 400 nm and 370 nm for spectra and lifetime measurements.

All these evaluations and comparisons with theoretical data reveal the suitability of this kind of complexes whose photophysical properties can be efficiently modified through mild structural alteration. The color coordinates in the CIE 1931 color space have been calculated for the different HBO ligands 1-3 and Iridium(III) complexes 4-9 from the relevant irradiance spectra. The results are compared in Figure 4. It should be observed that the emission color of the iridium complexes can be finely tuned, by simply changing the nature of substituent on the ppy and HBO ligands, from yellow (5: 0.499, 0.493) to yellow green (9: 0.389, 0.578). The HBO ligands 1 and 3 afford a yellowish green color (ca. 0.29, 0.55). In glassy matrix at 77 K, the iridium(III) complexes 5 and 8 (R = NBu2) display red-shifted emissions, as compared to the other complexes in the series bearing the same cyclometalating ligands (Figure 5). This feature was already observed on the maximum emission wavelength at room-temperature acetonitrile spectra. For all Iridium(III) complexes, a pronounced structuration of the emission bands is observed at low temperature, revealing a fine vibronic progression, further indicating a strongly ligandcentered nature of the emitting states (Figure 5). This experimental observation strongly supports the 3 LC attribution to the nature of the lowest-lying triplet states. In addition, the high blue shift of the emission profiles recorded at 77 K, with respect to those obtained at room temperature, clearly suggests the involvement of transitions with a high charge-transfer character (rigidochromic effect). These findings are in line with the presence of 3 LC-CT emitting states, as indicated by DFT calculations. It is also worth-mentioning that the excited-state lifetimes of 6 and 9 (R = CN) at 77 K are more than ten times longer with respect of those of the other investigated complexes (R = Me, NBu2). This does indicate that, for the cyanosubstituted complexes 6 and 9, the nature of the emitting state is different from that of the other Iridium(III) complexes 4,5 and 7,8. Again, this experimental observation is in line with DFT predictions, which estimate the 3 LC-LE to be lower in energy with respect to the 3 LC-CT state, only for complexes 6 and 9. Singlet Oxygen sensitization. All the complexes show a strong reduction of luminescence quantum yield in air-equilibrated solution (Table 2); a feature commonly observed for iridium(III) complexes. This indicates the occurrence of a triplet-triplet energy transfer to molecular oxygen, generating the 1 O2 ( 1 g) excited species. The process can be schematized as follow:

where P is the photosensitizer (i.e., the Iridium(III) complexes 4-9), 1 P * and 3 P * are the lowest-lying singlet and triplet excited states and ISC stands for intersystem crossing.

The bimolecular rate constant of the sensitization process, kq, can be determined from the excited-state lifetimes in airequilibrated (q) and O2-free (0) solutions by means of the Stern-Volmer equation: [START_REF] Lakowicz | Principles of Fluorescence Spectroscopy[END_REF] in which [O2] is the oxygen concentration.

By substituting the relevant lifetime values reported in Table 2, one obtains kq ~1×10 10 M -1 s -1 for all complexes 4-9 (Table 3). This is close to the diffusional limit of the quenching rate constant in acetonitrile (kdiff = 1.9×10 10 M -1 s -1 ), [START_REF] Montalti | Handbook of Photochemistry[END_REF] as often observed for iridium(III) complexes. [START_REF] Flamigni | Photochemistry and Photophysics of Coordination Compounds Ii[END_REF] The quantum yields of sensitization of singlet oxygen () can be determined by comparison between the typical luminescence centered at 1280 nm of iso-absorbing (exc = 442 nm) methanol solutions of complexes 4-9, and [Ru(bpy)3] 2+ in the same solvent ( = 0.90). The values obtained for  are reported in Table 3 and the luminescence spectra of 1 O2 are shown in Figure 6. The complexes 6 and 9, bearing the -CN terminal substituent and also showing the longest excited state lifetime in the series, display the highest quantum yield of sensitization. They are therefore, capable of combining an extremely fast reaction rate with a still very high yield of production of the reactive singlet oxygen species (70-74%, Table 3). This interesting feature make them very attractive candidates for photodynamic therapy applications. [START_REF] Mckenzie | [END_REF] The other complexes (i.e., 4, 5, 7 and 8,) although maintaining reaction rates close to the diffusional limit, have lower quantum yield, on the order of 50% (Table 3). 2+ in air-equilibrated methanol solution ( = 0.90).

[37]

Conclusion

A series of six cyclometalated iridium(III) complexes containing a functionalized HBO-based N^O chelate have been synthesized and their photophysical properties fully studied at room and low temperature, in air-equilibrated or de-aearated conditions. These complexes exhibit bright phosphorescence and a fine-tuning of their emission color can be achieved by simply modifying the terminal group of the ancillary ligand. DFT calculations indicate that all the complexes emit from a ligand-centered excited-state involving the HBO ancillary ligand. Anyway, the terminal group on this ligand has a huge impact on the nature of the emitting state: an internal charge-transfer transition is responsible for emission in the case of electron-donating groups (such as Me and NBu2), while a locally-excited state becomes active, when the ancillary ligand is equipped with an electron-withdrawing cyano substituent. This difference in the emission nature is further corroborated by the longer excited-state lifetimes observed in glassy matrix at low temperature for the complexes having the electron-withdrawing CN terminal group. Finally, these iridium(III) complexes display competitive singlet oxygen sensitization parameters making them attractive candidates as sensitizers in photodynamic therapy. Work along these lines is currently in progress.

Experimental Section

Materials and Methods. All reactions were performed under a dry atmosphere of argon. Chemicals were purchased from commercial sources and used without further purification. Reaction solvents were distilled according to common procedures. Thin layer chromatography (TLC) was performed on silica gel or aluminum oxide plates coated with fluorescent indicator. Chromatographic purifications were conducted using 40-63 μm silica gel. All mixtures of solvents are given in v/v ratio.

1 H-NMR (400.1 MHz) spectra were recorded on a Bruker Advance 400 MHz spectrometer with perdeuterated solvents containing residual protonated solvent signals as internal references. The cyclometalated Iridium(III) chloro-bridged dimers A [38] and B [38] and the ethynyl-extended 2-(2'-hydroxyphenyl) benzoxazole ligands 1-3 [29] were synthesized according to reported procedures.

General procedure. In a Schlenk tube, under an inert atmosphere, the cyclometalated Iridium(III) chloro-bridged dimer A or B (0.1 mmol) and potassium carbonate (2 mmol) were dissolved in a mixture of CH2Cl2/CH3OH (1:1, v:v). 5-extended-2-2'(hydroxyphenyl)benzoxazole) 1-3 (0.2 mmol) were then added to the resulting suspensions and the mixtures were subsequently stirred for 24 hours at 40 °C. After cooling down, the crude mixtures were extracted with CH2Cl2, washed with water, dried (MgSO4), and the solvents removed in vacuo. The crude residues were purified by column chromatography on SiO2 eluting with CH2Cl2/pet. ether to yield complexes 4-9 as yellow powders in 47-72% yield.

Complex 4: yellow powder. 58%. The correction curve of the wavelength dependent phototube response between 280 and 1600 nm has been obtained by using a calibrated deuterium / halogen lamp source DH-3plus-CAL-EXT (Ocean Optics). Luminescence quantum efficiencies in solution were evaluated with reference to quinine sulfate (ϕ = 0.53, in air-equilibrated 0.1 N H2SO4). [START_REF] Montalti | Handbook of Photochemistry[END_REF] Band maxima and quantum yields were obtained with uncertainty of 2 nm and 20%, respectively. Luminescence lifetimes were measured with an IBH 5000F time-correlated single-photon counting (TCSPC) device, by using pulsed NanoLED and SpectraLED excitation sources at 331 and 370 nm. The analysis of the luminescence decay profiles against time was accomplished with the DAS6 Fluorescence Decay Analysis Software (Horiba). Experimental uncertainties in the lifetime determinations are estimated to be 10%. Singlet oxygen sensitization efficiencies have been measured with reference to [Ru(bpy)3] 2+ in CH3OH ( = 0.90) [37] by comparing the area of singlet oxygen phosphorescence spectra. Excitation at 442 nm has been performed with a HeCd laser (Kimmon).

DFT Calculations. Density functional theory (DFT) calculations were carried out using the D.01 revision of the Gaussian 09 program package [START_REF] Frisch | Gaussian 09, Revision D.01[END_REF] in combination with the M06 global-hybrid meta-GGA exchange-correlation functional. [START_REF] Zhao | [END_REF][41] The fully relativistic Stuttgart/Cologne energy-consistent pseudopotential with multielectron fit was used to replace the first 60 inner-core electrons of the iridium metal center (i.e., ECP60MDF) and was combined with the associated triple-ζ basis set (i.e., cc-pVTZ-PP basis); [42] for all other atoms, the Pople 6-31G(d,p) basis set was adopted. [43] All the reported complexes were fully optimized without symmetry constraints, using a time-independent DFT approach, in their ground state (S0) and triplet emitting state (T1). All the optimization procedures were performed using the polarizable continuum model (PCM) to simulate acetonitrile solvation effects. [START_REF] Cramer | Solvent Effects and Chemical Reactivity[END_REF][START_REF] Tomasi | [END_REF][46] Frequency calculations were always used to confirm that every stationary point found by geometry optimizations was actually a minimum on their corresponding potential-energy surfaces (no imaginary frequencies). To investigate the nature of the emitting states, geometry optimizations and frequency calculations were performed at the spin-unrestricted UM06 level of theory (imposing a spin multiplicity of 3), using the corresponding S0 minimum-energy geometry as starting point. The emission energy from the lowest triplet excited state was estimated by subtracting the SCF energy of the emitting state (T1) from that of the singlet ground state (S0), both in their minimum-energy conformation (adiabatic energy difference). Time-dependent DFT calculations (TD-DFT), [47][48][49] carried out at the same level of theory used for geometry optimizations, were used to locate the lowest-energy 12 vertical excitations of triplet spin multiplicity, starting from the ground-state minimum-energy geometry of each complex. The nature of the transitions was assessed by the help of Natural Transition Orbital (NTO) analysis. [START_REF] Martin | [END_REF] All the pictures showing molecular geometries, orbitals and spin-density surfaces were created using GaussView 6. [START_REF] Dennington | GaussView[END_REF]

Scheme 1 .

 1 Scheme 1. General structure of complexes

Figure 1 .

 1 Figure 1. Absorption spectra of HBO ligands 1-3 and Iridium(III) complexes 4-9 in acetonitrile solution at rt.

Figure 3 .

 3 Figure 3. Emission spectra in de-aerated acetonitrile solution at room temperature of HBO ligands 1-3 and Iridium(III) complexes 4-9 in acetonitrile at room temperature. The area has been rescaled to the relevant quantum yield.

Figure 4 .

 4 Figure 4. CIE 1931 spectral chromaticity coordinates of 1-9 in CH3CN at rt.

Figure 5 .

 5 Figure 5. Normalized emission spectra in DCM:MeOH 1:1 (v/v) mixture at 77 K.

Figure 6 .

 6 Figure 6. Emission spectra of 1 O2 in air-equilibrated methanol solutions for Iridium(III) complexes 4-9 (Ru is [Ru(bpy)3] 2+ ); the area has been rescaled to the relevant quantum yield of sensitization.

Complex 7 :

 7 Yellow powder. 61%. 1 H-NMR (400 MHz, CDCl3) δ (ppm) = 8.77 (d, 1H, CH Ar, J = 6 Hz), 8.28 (d, 1H, CH Ar, J = 8.8 Hz), 8.19-8.22 (m, 2H, CH Ar), 7.08 (d, 1H, CH Ar, J = 6 Hz), 7.71 (q, 2H, CH Ar, J = 7.2 Hz), 7.49 (d, 1H, CH Ar, J = 8 Hz), 7.40 (d, 2H, CH Ar, J = 8 Hz), 7.34 (dd, 1H, CH Ar, J = 9.2 Hz), 7.20 (t, 1H, CH Ar, J = 8 Hz), 7.08-7.15 (m, 3H, CH Ar), 6.91-6.97 (m, 2H, CH Ar), 6.71 (d, 1H, CH Ar, J = 8.8 Hz), 6.38-6.51 (m, 2H, CH Ar), 6.22 (d, 1H, CH Ar, J = 8 Hz), 5.92 (dd, 1H, CH Ar, J = 8.8 Hz), 5.57 (dd, 1H, CH Ar, J = 8.2 Hz), 2.36 (s, 3H, CH3

Table 1 .

 1 Absorption parameters , nm (, 10 3 M -1 •cm -1 )[a]

	1	281 sh (42.2), 289 (53.8), 305 (44.6), 342 (10.7)
	2	258 (19.8), 302 sh (31.9), 339 (46.2)
	3	230 (21.1), 261 sh (23.4), 271 (25.7), 291 (36.9), 302 (43.4), 322 (37.6),
		350 sh (14.2)
	4	258 (56.2), 295 (47.0), 333 (42.1), 420 (9.8)
	5	259 (49.8), 303 (48.8), 332 (54.9), 350 (52.5), 430 (9.6)
	6	260 (54.7), 274 sh (48.5), 371 (32.8), 435 (9.6)
	7	250 (58.8), 292 (51.8), 330 (44.3), 415 (9.9)
	8	249 (51.3), 302 (47.3), 330 (53.7), 350 (50.3), 425 (9.2)
	9	253 (61.3), 270 sh (52.4), 364 (40.6), 430 (11.1)
	[a] In acetonitrile solution at rt; sh is shoulder
	Lowest-lying triplet states. Since iridium(III) complexes
	ultimately emit from excited states of triplet spin multiplicity, a
	TD-DFT investigation was performed to assess the energy and
	the nature of the lowest-lying triplet states of all the investigated
	complexes. The first four triplet excitations of compounds 4-9
	are summarized in Figure

Table 2 .

 2 Emission parameters

		rt	a			77 K	b
		em, nm		, s (kr, s -1 )	em, nm	, s
	1	517	0.11 (4.8110 -2 )
	2	546	0.10 (0.7510 -2 )
	3	514	0.22 (8.2010 -2 )
	4	555	0.22 (0.3310 -2 )	3.2 (42.010 -3 )	513	7.5
	5	568	0.19 (0.1410 -2 )	5.8 (36.410 -3 )	527	13.8
	6	554	0.18 (0.2410 -2 )	2.9 (40.810 -3 )	513	190
	7	537	0.44 (0.1710 -2 )	12.7 (50.910 -3 )	502	14.3
	8	563	0.11 (0.0810 -2 )	9.2 (43.510 -3 )	525	27.6
	9	530	0.35 (0.0910 -2 )	19.1 (56.510 -3 )	509	420

Table 3 .

 3 Singlet oxygen ( 1 g) sensitization parameters

		kq (M -1 s -1 )	a	 b
	4	1.210 10		0.49
	5	1.410 10		0.53
	6	1.310 10		0.74
	7	1.010 10		0.37
	8	1.210 10		0.54
	9	0.910 10		0.70
	[a] In air-equilibrated methanol solution at room temperature. [b] Calculated
	with respect to [Ru(bpy)3]

  1H, CH Ar, J = 7.6 Hz), 7.05 (t, 1H, CH Ar, J = 6.4 Hz), 6.76-6.96 (m, 6H, CH Ar), 6.71 (d, 1H, CH Ar, J = 8.8 Hz), 6.46 (d, 1H, CH Ar, J = 7.6 Hz), 6.18 (d, 1H, CH Ar, J = 7.6 Hz), 6.09 (d, 1H, CH Ar, J = 8.4 Hz). ESI-HRMS: calcd for C44H27IrN4O2 (M+) 836.1760, found (M+) 836.1730.

	1 H-NMR (400 MHz, CDCl3) δ (ppm) =
	8.74 (d, 1H, CH Ar, J = 5.2 Hz), 8.14 (d, 1H, CH Ar, J = 2.4 Hz), 8.02 (d,
	1H, CH Ar, J = 5.6 Hz), 7.79 (d, 1H, CH Ar, J = 8 Hz), 7.72 (d, 1H, CH Ar,
	J = 8 Hz), 7.51-7.60 (m, 4H, CH Ar), 7.37 (d, 1H, CH Ar, J = 8 Hz), 7.32
	(d, 2H, CH Ar, J = 8 Hz), 7.23 (dd, 1H, CH Ar, J = 9 Hz), 7.05-7.08 (m,
	3H, CH Ar), 6.97 (t, 1H, CH Ar, J = 6 Hz), 6.68-6.87 (m, 6H, CH Ar), 6.62
	(d, 1H, CH Ar, J = 9.2 Hz), 6.40 (d, 1H, CH Ar, J = 7.6 Hz), 6.11 (d, 1H,
	CH Ar, J = 7.6 Hz), 6.01 (d, 1H, CH Ar, J = 8.4 Hz), 2.28 (s, 3H, CH3).
	ESI-HRMS: calcd for C44H31IrN3O2 (M+H) 826.2043, found (M+H)
	826.2052.
	Complex 5: Yellow powder. 72%. 1 H-NMR (400 MHz, CDCl3) δ (ppm) =
	8.74 (d, 1H, CH Ar, J = 5.6 Hz), 8.19 (d, 1H, CH Ar, J = 2.4 Hz), 8.10 (d,
	1H, CH Ar, J = 6 Hz), 7.84 (d, 1H, CH Ar, J = 8 Hz), 7.77 (d, 1H, CH Ar, J
	= 8 Hz), 7.56-7.64 (m, 4H, CH Ar), 7.44 (d, 1H, CH Ar, J = 8 Hz), 7.29-
	7.35 (m, 3H, CH Ar), 7.12 (t, 1H, CH Ar, J = 8 Hz), 7.03 (t, 1H, CH Ar, J =
	6.4 Hz), 6.75-6.94 (m, 6H, CH Ar), 6.69 (d, 1H, CH Ar, J = 8.8 Hz), 6.57
	(d, 2H, CH Ar, J = 8.8 Hz), 6.48 (d, 1H, CH Ar, J = 8.4 Hz), 6.19 (d, 1H,
	CH Ar, J = 7.2 Hz), 6.09 (d, 1H, CH Ar, J = 8.4 Hz), 3.28 (t, 4H, CH2, J =
	7.6 Hz), 1.54-1.62 (m, 4H, CH2), 1.32-1.41 (m, 4H, CH2), 0.97 (t, 6H, CH3,
	J = 7.2 Hz). ESI-HRMS: calcd for C51H46IrN4O2 (M+H) 939.3248, found
	(M+H) 939.3230.
	Complex 6: Yellow powder. 64%. 1 H-NMR (400 MHz, CDCl3) δ (ppm) =
	8.78 (d, 1H, CH Ar, J = 5.6 Hz), 8.25 (d, 1H, CH Ar, J = 2 Hz), 8.08 (d,
	1H, CH Ar, J = 5.2 Hz), 7.87 (d, 1H, CH Ar, J = 8 Hz), 7.79 (d, 1H, CH Ar,
	J = 8.4 Hz), 7.59-7.68 (m, 6H, CH Ar), 7.53 (d, 2H, CH Ar, J = 8.4 Hz),

7.46 (d, 1H, CH Ar, J = 8.4 Hz), 7.31 (dd, 1H, CH Ar, J = 8 Hz), 7.15 (t,

  ). ESI-HRMS: calcd for C44H26F4IrN3O2 (M+) 897.1588, found (M+) 897.1611.

	Complex 8: Yellow powder. 65%. 1 H-NMR (400 MHz, CDCl3) δ (ppm) =
	8.70 (d, 1H, CH Ar, J = 6 Hz), 8.21 (d, 1H, CH Ar, J = 8.4 Hz), 8.09-8.14
	(m, 2H, CH Ar), 8.01 (d, 1H, CH Ar, J = 8 Hz), 7.61-7.67 (m, 2H, CH Ar),
	7.41 (d, 1H, CH Ar, J = 8 Hz), 7.23-7.26 (m, 3H, CH Ar), 7.12 (t, 1H, CH
	Ar, J = 8.4 Hz), 7.02 (t, 1H, CH Ar, J = 7.2 Hz), 6.83-6.88 (m, 2H, CH Ar),
	6.61 (d, 1H, CH Ar, J = 8.8 Hz), 6.50 (d, 2H, CH Ar, J = 8 Hz), 6.30-6.43
	(m, 2H, CH Ar), 6.13 (d, 1H, CH Ar, J = 8 Hz), 5.84 (d, 1H, CH Ar, J = 8.8
	Hz), 5.49 (d, 1H, CH Ar, J = 8.4 Hz), 3.20 (t, 4H, CH2, J = 6.4 Hz), 1.46-
	1.54 (m, 4H, CH2), 1.24-1.33 (m, 4H, CH2), 0.89 (t, 6H, CH3, J = 7.6 Hz).
	ESI-HRMS: calcd for C51H42F4IrN4O2 (M+H) 1011.2871, found (M+H)
	1011.2907.
	= 7.2 Hz), 6.72 (d, 1H, CH Ar, J = 8.8 Hz), 6.39-6.52 (m, 2H, CH Ar), 6.21
	(d, 1H, CH Ar, J = 8.4 Hz), 5.90 (d, 1H, CH Ar, J = 8.4 Hz), 5.56 (d, 1H,
	CH Ar, J = 8.4 Hz). ESI-HRMS: calcd for C44H24F4IrN4O2 (M+H) 909.1462,
	found (M+H) 909.1491.
	Photophysics. Absorption spectra of dilute acetonitrile (VWR Uvasol)
	solutions were recorded on a Perkin Elmer Lambda 950 UV/Vis/NIR
	spectrophotometer. The molar absorption coefficients were calculated by
	applying the Lambert-Beer law to low absorbance spectra (A < 1)
	recorded at successive dilutions. Steady-state photoluminescence
	spectra were measured on an Edinburgh FLS920 Vis/NIR fluorimeter,
	equipped with a Hamamatsu R928P Peltier-cooled (Vis) and a super-
	cooled (193 K) Hamamatsu R5509-72 (NIR) PMTs. The concentration of
	sample solutions was adjusted to obtain absorption values A < 0.1 at the
	excitation wavelength and the solutions de-aerated by bubbling Ar for at
	least 20 minutes.

Complex 9: Yellow powder. 47%. 1 H-NMR (400 MHz, CDCl3) δ (ppm) = 8.73 (d, 1H, CH Ar, J = 6 Hz), 8.30 (d, 1H, CH Ar, J = 8 Hz), 8.20-8.24 (m, 2H, CH Ar), 8.06 (d, 1H, CH Ar, J = 5.6 Hz), 7.70-7.76 (m, 2H, CH Ar), 7.49-7.61 (m, 5H, CH Ar), 7.34 (dd, 1H, CH Ar, J = 8.8 Hz), 7.22 (t, 1H, CH Ar, J = 7.6 Hz), 7.10 (t, 1H, CH Ar, J = 6.4 Hz), 6.95 (q, 2H, CH Ar, J

Acknowledgements

J.M and G.U. are grateful to the Centre National de la Recherche Scientifique (CNRS) for providing research facilities and the Agence Nationale de la Recherche (ANR project GeDeMi) for financial support. Funding from the Italian CNR (project PHEEL) is also acknowledged.

Entry for the Table of Contents

The photophysical features of iridium(III) complexes incorporating an ethynyl-extended benzoxazole-based ancillary ligand are effectively modulated by the simple introduction of electron-donating (Me, NBu2) or -withdrawing (CN) moieties on the terminal endgroup, leading to CT or LC excited-states. The complexes containing the cyano-functionalized ligands features singlet oxygen sensitization parameters, making them attractive candidates for PDT.


interpretation: the relative can be interpreted as denoting an amount  relativizer: only that and Ø are allowed  definiteness: the relative gap can appear as an associate of there (which normally does not allow any strong DPs including traces)  no stacking Amount: the identity of substance is not necessary:

(2) a.

It will take us the rest of our lives to drink the champagne that they spilled that evening.

b.

We will never be able to recruit the soldiers that the Chinese paraded last May Day.

The ambiguity of ACD relatives is also analyzed as a case of amount reading:

(3) Martha put everything she {could, should have, wanted to} in the car. "The amount she could until the car was full"

SYNTAX: THE COVERT DEGREE INSIDE THE RELATIVE

Explanation for the contrast: the gap in (1b) is an e-type variable and definite, the gap in (1a) is a d-type variable (a degree) and forms part of an indefinite NP:

(4) a. (February 10, 2020) Then we are given some vague understanding that the construction should be analyzed as ( 6), yet how does this compose? (6) as many books as there were books on the table

3.

G&L SOLUTION Grosu and Landman 1998: degrees To unpack: normally degrees are points on a scale (intervals, for some people). Here a degree (still on a scale) is relativized to units of measure (which seem to be taken as identical to the corresponding items, but this is not made clear, we'll see)

The interpretation of the head noun

Idea: it is interpreted inside the relative, but the outer copy forces the entire NP to denote the kind of entity that its head denotes, i.e., a set of sums of individuals Two additional operations postulated: SUBSTANCE (applying to the degree CP and extracting the substance measured) and X (composing the degree CP with the head NP as three does):

(13) SUBSTANCE (CP) = {x: |x|, P, x  CP} SUBSTANCE applies by default. Sometimes it doesn't and then we get the identity-of-quantity interpretation (the Chinese soldiers, the spilled wine)

How does the cardinal combine with the NP + maximized degree relative combo? It does not restrict the denotation of the plural NP, it must be identical to the first member of the triple. The compositional semantics is not provided (but it's actually pretty easy) This has to be definite, so existential DPs cannot be formed on the basis of a degree relative

Further complications

The determiner does not have to be the definite article:

(15) I took with me every book that there was on the table.

substance, quantity

Despite the presence of there-construction (15) can only be interpreted as identity of substance

Grosu and Landman 1998 come up with a very complex analysis permitting the combination of ( 14) with other determiners and also cases like (15)

The reason why degree relatives cannot stack lies in the head-raising analysis

Problems

The hypothesis that degrees are triples does not help with the champagne example:

(16) It will take us the rest of our lives to drink the champagne that they spilled that evening.

Mass nouns cannot be used as units of measure

In addition, same interval on the same scale can be two different degrees: Taking a step back from (15):  there-construction + that-relativizer entail a degree analysis  yet semantically this doesn't look like degrees are involved!

ALTERNATIVES

Facts to explain:  the identity of substance reading  the restrictions on the determiner, the relativizer and stacking  compatibility with there-construction  the ambiguity of ACD relatives General picture: there can be no unified analysis

von Fintel 1999

Very good summary of G&L, provides a non-modal example of an amount relative:

(18) There isn't the water in the sink that there is in the bathtub.

Multiple copy interpretation (cf. You drank the only beer there was left.

(24) substance/amount reading not available a.

It will take days to drink the only champagne they spilled that evening. b.

Marv put the only thing(s) he could in his pocket. c. ?? The books cost the only amount of money we had.

Why is amount reading incompatible with only? Because maximization (hence, uniqueness), which is absent from existential relatives Why is the relativizer restricted? Because who and which are disallowed with singleton sets (idea originally due to [START_REF] Heim | Where does the definiteness restriction apply? Evidence from the definiteness of variables[END_REF] or, as explained by McNally 1997, the correlate in thereconstruction is kind-denoting and this denotation is incompatible with which/who Summary: nothing on amount relatives, but good reasons for excluding existential relatives (31) a.

It will take us the rest of our lives to find the champagne that they spilled that evening.

b.

We will never be able to recruit the soldiers that the Chinese paraded last May Day.

Mendia 2017: amount interpretations are a special case of kind interpretations

Evidence and cross-linguistic comparison

Context dependency: manipulating the context can bring into focus either the amount or the kind reading:

(32) We lost the battle because we didn't have the soldiers that the Imperial Army had. Relative degrees and degree relatives, Seminar on relative clauses, LLF/SFL, Paris (February 10, 2020) Amount interpretation can be achieved without a relative clause:

(33) with PPs a.

We lost the battle because we didn't have the soldiers of the Imperial Army. b.

We used to organize a soccer team, but we don't have the students in the department anymore.

(34) with bare DPs a.

We lost the battle because we didn't have the soldiers. b.

We used to organize a soccer team, but don't have the students anymore. (February 10, 2020) However, because they can have the null AMOUNT both inside and outside the relative clause, they can deal with there-relatives as well

CONCLUSION

Multiple types of relatives are gathered under the umbrella of degree relatives 6.

(

  17) a. ⟦3cm⟧ = λx . centimeters (x) ∧ DEGREE centimeter (x) = 3, centimeters, x b. ⟦30mm⟧ = λx . millimeters (x) ∧ DEGREE millimeter (x) = 30, millimeters, x Is this counter-intuitive? I'm not sure. Language might not know that 3cm=30mm

( 30 )

 30 Raising of DegP explains no stacking Spanish facts: what raises is a head (Deg 0 ), which requires pied-piping of the AP predicate 4.6. Mendia 2017: English has no degree relatives, Spanish does Mendia 2017: amount relatives are a case of deferred reference: they refer to the amount and not to the entity denoted by the relativized NP  the relativized DP is a definite amount  the relativized NP is actually indefinite (despite the determiner)  there is always a comparison of two amounts of the same stuff NB: like Grosu and Landman 2017, Mendia includes into consideration amount relatives with an overt amount noun as head These properties also characterize kind readings of examples like (2), as in:

  Relative degrees and degree relatives, Seminar on relative clauses, LLF/SFL, Paris

	Ora Matushansky		2
	(5)	DP
	D		NumP
	the Num		NP	λd
		three	NP	CP
			books	NP	C
			d many books C	TP
				(that)	DP	T
				there T	SC
				were NP	PP
				AP	NPon the table
				d many books

λd . there are [d-many books] on the table b. λd . x [books (x) & |x| = d & on-the-table (x)]

Note the tacit assumption that the relative clause attaches below the cardinal

The head noun books is interpreted inside the relative clause and then raises out of it:

  Relative degrees and degree relatives, Seminar on relative clauses, LLF/SFL, Paris(February 10, 2020) the sum of all books on the table if there are three books on the table

	(14) a.	three books that there were __ on the table
	b.	= {⨆{x  BOOK: ON-THE-TABLE (x)}}
		if |⨆{x  BOOK:ON-THE-TABLE (x)}| = 3;
		undefined otherwise
	c.	the set of

McNally 2008: there is some confusion here

  Relative degrees and degree relatives, Seminar on relative clauses, LLF/SFL, Paris(February 10, 2020)

	4.2. Butler 2001: dynamic semantics
	Focus on ACD relatives, which can be interpreted restrictively or exhaustively:
	(20) Peter ate everything that would fit in his pocket.
	a.	Restrictive reading:
		Peter ate everything (relevant) that was of an appropriately small size.
		∀x(P(x) → A(x))
	b.	Exhaustive reading:	amount
		Peter ate a pocketful of something.
		∃x(P(x) ∧ ¬∃y [x ≠ y ∧ P(y) ∧ □(P(y) → P(x))] ∧ A(x))
	Degree relatives only receive an exhaustive interpretation
	Notes a problem with the maximization analysis: sometimes a minimum is required:
	But is this a degree relative?
	(21) The money that Peter can live on is small.
	Butler's alternative: exhaustification instead of maximization
	Dynamic-semantic account assuming that relative clauses introduce discourse referents which
	can then be equated with the head:
	(22) a.	restrictive reading:
		for every thing that Peter ate it would fit in his pocket
	b.	exhaustive reading:
		what Peter ate would fit in his pocket
	Non-compositional, the determiner simply ignored
	4.3. Carlson 1977: two properties shared between certain types of relative clauses: the restriction
	on the relativizer (that/Ø/*which) and the restriction on the determiner (only strong ones)
	McNally 2008: amount relatives and relatives out of existentials are two different things:
		identity of substance: for amount (champagne) relatives only
		the head noun in relatives out of existentials permits further modification by DP-
		internal only, which blocks amount readings:
	(23) existential relative:
	a.	The only reasons there are are reasons for action…
	b.

[START_REF] Hackl | Comparative Quantifiers[END_REF]

:

(19) the books (that there are n many books on the table)-many

The definite article and the choice of the relativizer are constrained in the same way

  The number of friends that came to your party came to my party.

					for
	kind and amount relatives (but also for existential relatives, see above)
	Kind and amount are also connected in nominal exclamatives:
	(35) a.	It's amazing the cars he owns.
	b.	The cars he owns!
	(36) a.	kind: what's remarkable/surprising is the kinds of cars he owns.
	b.	amount: what's remarkable/surprising is how many cars he owns.
	Mendia 2017 argues against a covert degree operator in English amount relatives (e.g., on
	the basis of their non-islandhood) and provides a way of deriving amount readings from kind
	readings
	Novel empirical data from Spanish
	Spanish has amount relatives, English doesn't:
	(37) [Context: The same number of friends attended both our birthday parties, but they were
	different friends.]
	a.	The friends that came to your party came to my party.	FALSE
	b.	A mi fiesta vinieron los amigos que vinieron a la tuya.	TRUE
		to my party came	the friends that came	to the yours
	Spanish can do what English does and more (see also (26)):
	(38) Me	pregunto las	manzanas que trajo	Pedro.
	1SG.DAT ask	the.F.PL apples	that brought Pedro
	I wonder how many apples Pedro brought.
	4.7

. Grosu and Landman 2017: overview and retrofitting

  This, like the previous, is a very good place to look for a literature review in Grosu and Landman 2017 assimilate amount relatives to relatives with an overt head noun amount: the relative modifies the null noun AMOUNT: Relative degrees and degree relatives, Seminar on relative clauses, LLF/SFL, Paris

	Ora Matushansky		8
	(39)	DP
	D	NP
	the	NP [meas]	NP [of]
	NP [meas]	CP	wine

AMOUNT

It remains a mystery to me how the noun wine gets into the relative clause under this view

Relative degrees and degree relatives, Seminar on relative clauses, LLF/SFL, Paris (February 10, 2020)

Herdan 2008: lay off the champagne

Herdan 2008 looks at ACD relatives:

(25) a.

Marv put in his pocket everything (that)/*which he could. b.

Marv put in his pocket everything/#something that he could. (26) Jose no entendió lo hermosa que era la novela. Jose NEG understood the.N.SG beautiful.F.SG that was the novel.F.SG Jose did not undestand how beautiful the novel was.

Proposal: a head-internal analysis for the DegP only and arguments for it From idioms:

(27) At the morgue, it would take us just a day to get the creeps that a cemetery gives in a year.

A. Literal meaning: creeps are items B. # Idiom reading involving the degree of the amount of creeps Relativization of this idiom is possible:

(28) At the morgue I get the creeps that only a cemetery gives.

Same argument from binding:

(29) It would take us all year to paint the portraits of himself i that John i burned in a fit of paranoia.

A. # type of portrait B. painting the actual burned canvas C. # paint the amount of portraits


Zacharie Ales
email: zacharie.ales@ensta-paristech.fr

Sourour Elloumi
email: sourour.elloumi@ensta-paristech.fr

Compact MILP formulations for the p-center problem

Keywords: p-center, discrete location, equivalent formulations, integer programming

The p-center problem consists in selecting p centers among M to cover N clients, such that the maximal distance between a client and its closest selected center is minimized. For this problem we propose two new and compact integer formulations. Our first formulation is an improvement of a previous formulation. It significantly decreases the number of constraints while preserving the optimal value of the linear relaxation. Our second formulation contains less variables and constraints but it has a weaker linear relaxation bound. We besides introduce an algorithm which enables us to compute strong bounds and significantly reduce the size of our formulations. Finally, the efficiency of the algorithm and the proposed formulations are compared in terms of quality of the linear relaxation and computation time over instances from OR-Library.

Introduction

We consider N clients {C1, ..., CN } and M potential facility sites {F1, ..., FM }. Let dij be the distance between Ci and Fj. The objective of the p-center problem is to open up to p facilities such that the maximal distance (called radius) between a client and its closest selected site is minimized. This problem is very popular in combinatorial optimization and has many applications. We refer the reader to the recent survey [START_REF] Calik | p-Center Problems[END_REF]. Very recent publications include [START_REF] Ferone | On the fast solution of the p-center problem[END_REF][START_REF] Ferone | A new local search for the p-center problem based on the critical vertex concept[END_REF] which provide heuristic solutions and [START_REF] Calik | Double bound method for solving the p-center location problem[END_REF] on an exact solution method. In this paper, we will focus on mixed-integer linear programming formulations of the p-center problem. Let M and N respectively be the sets {1, ..., M } and {1, ..., N }. The most classical formulation, denoted by (P1), for the p-center problem (see for example [START_REF] Daskin | Network and discrete location analysis[END_REF]) considers the following variables:

yj is a binary variable equal to 1 if and only if Fj is open; xij is a binary variable equal to 1 if and only if Ci is assigned to Fj; -R is the radius.

(P1)

                                               min R s.t. M j=1 yj ≤ p M j=1 xij = 1 i ∈ N xij ≤ yj i ∈ N , j ∈ M M j=1 dij xij ≤ R i ∈ N xij, yj ∈ {0, 1} i ∈ N , j ∈ M r ∈ R (1a) (1b) (1c) (1d) (1e)
Constraint (1b) ensures that no more than p facilities are opened. Each client is assigned to exactly one facility through Constraints (1c). Constraints (1d) link variables xij and yj while (1e) ensure the coherence of the objective. A more recent formulation, denoted by (P2), was proposed in [START_REF] Sourour Elloumi | A new formulation and resolution method for the p-center problem[END_REF]. Let D 0 < D 1 < ... < D K be the different dij values ∀i ∈ N ∀j ∈ M. Note that, if many distances dij have the same value, K may be significantly lower than M × N . Let K be the set {1, ..., K}. Formulation (P2) is based on the variables yj, previously introduced, and one binary variable z k , for each k ∈ K, equals to 1 if and only if the optimal radius is greater than or equal to D k :

(P2)                                min D 0 + K k=1 (D k -D k-1 ) z k s.t. 1 ≤ M j=1 yj ≤ p z k + j : d ij <D k yj ≥ 1 i ∈ N , k ∈ K yj, z k ∈ {0, 1} j ∈ M, k ∈ K (2a) (2b) (2c)
Constraints (2c) ensure that if no facility located at less than D k of client Ci is selected, then the radius must be greater than or equal to D k . This formulation has been proved to be tighter than (P1) [START_REF] Sourour Elloumi | A new formulation and resolution method for the p-center problem[END_REF]. However, its size strongly depends on the value K (i.e., the number of distinct distances dij). It also has recently been adapted to the p-dispersion problem which consists in selecting p facilities among N such that the minimal distance between two selected facilities is maximized [START_REF] Sayah | A new compact formulation for the discrete p-dispersion problem[END_REF].

A last formulation, that can be deduced from (P2) by a change of variables, has been recently introduced [START_REF] Calik | Double bound method for solving the p-center location problem[END_REF] and named (P4). It contains, for all k ∈ K, a binary variable u k equal to 1 if and only if the optimal radius is D k (i.e., u k = z k -z k+1 and z k = K q=k uq):

(P4)                                      min K k=1 D k u k s.t. (2b) j : d ij ≤D k yj ≥ k q=1 uq i ∈ N , k ∈ K K k=1 u k = 1 yj, u k ∈ {0, 1} j ∈ M, k ∈ K (3a) (3b) (3c)
They also proposed a weaker version of this formulation, called (P3), obtained by replacing the left-hand side of constraints (3b) by u k . They proved that (P4) leads to the same linear relaxation bound and has the same size as (P2).

The rest of the paper is organized as follows. Section 2 presents our two new formulations. In Section 3 we introduce an algorithm. Finally, Section 4 describes numerical results on instances from the OR-Library.

2 Our new formulations

Formulation (CP 1 )

In (P2), for all k ∈ K, variable z k is equal to 1 if and only if the optimal radius is greater than or equal to D k . As a consequence, the following constraints are valid

z k ≥ z k+1 k ∈ {1, ..., K -1}. ( 4
)
We first show that these inequalities are redundant for (P2). Let (P 2 ) be the formulation obtained when contraints (4) are added to (P2) and let v(F ) be the optimal value of the linear relaxation of a given formulation F . We now prove that adding constraints (4) does not improve the quality of the linear relaxation.

Proposition 1. v(P 2 ) = v(P2)
Proof. We show that an optimal solution (ỹ, z) of the relaxation of (P2) satisfies (4). For each distance D k there exists a client i(k) such that zk +

j : d i(k)j <D k ỹj = 1 (5)
otherwise zk can be decreased and (ỹ, z) is not optimal. We now assume that zk-1 < zk for some index k ∈ {2, ..., K}. It follows that zk-1 +

j : d i(k)j <D k-1 ỹj < zk + j : d i(k)j <D k ỹj = 1
The last equality follows from [START_REF] Sourour Elloumi | A new formulation and resolution method for the p-center problem[END_REF]. Therefore, constraints (2c) for i(k) and k -1 is violated.

We now prove that a large part of constraints (2c) are redundant in (P 2 ).

Let N k i be the set of facilities located at less than D k from client Ci. We can observe that

N k i is included in N k+1 i , for all k ∈ K. Moreover, N k i is equal to N k+1 i if and only if there is no facility at distance D k from client Ci. Let Si be the set of indices k ∈ {1, ..., K -1} such that N k i is different from N k+1 i . Observe that |Si| ≤ min(M, K).
We define Formulation (CP1) as Formulation (P 2 ) where only the constraints (2c) such that k ∈ Si or k = K are kept. 4)

(CP1)                        min D 0 + K k=1 (D k -D k-1 ) z k s.t. (2b), (
z k + j : d ij <D k yj ≥ 1 i ∈ N , k ∈ Si ∪ {K} yj, z k ∈ {0, 1} j ∈ M, k ∈ K (6a) (6b)
The number of constraints is dominated by the number of constraints (6b). This number is bounded by both N M and N K.

The following proposition proves that (CP1) is a valid formulation.

Proposition 2. (CP1) is a valid formulation of the p-center problem.

Proof. We show that the constraints removed from (P 2 ) are dominated.

If

N k i = N k+1 i , then j : d ij <D k yj = j : d ij <D k+1 yj. Since z k ≥ z k+1 , we have: z k + j : d ij <D k yj ≥ z k+1 + j : d ij <D k+1 yj ≥ 1.
As a consequence, the constraint (2c) associated with i and k is dominated by the one associated with i and k + 1.

We now prove that Formulations (P2) and (CP1) lead to the same bound by linear relaxation.

Proposition 3. v(CP1) = v(P2).
Proof. The arguments used in the proof of Proposition 2 can be used again to show that the constraints removed from (P 2 ) do not impact the value of the linear relaxation.

To sum up, (CP1) is a valid formulation that has the same LP bound as (P2). However, as detailed in Table 1, Formulation (CP1) is much smaller since it reduces the number of constraints by a factor of up to N .

Formulation (CP 2 )

We now introduce a second formulation, denoted by (CP2), which contains less variables and constraints than (CP1). We replace the K binary variable z k with a unique general integer variable r which represents the index of a radius:

(CP2)                      min r s.t. (2b) r + k j : d ij <D k yj ≥ k i ∈ N , k ∈ Si ∪ {K} yj ∈ {0, 1} j ∈ M r ∈ {0, ..., K} (7a)
Constraints (7a) play a similar role to Constraints (6b). Formulation (CP2) does not directly provide the value of the optimal radius R but its index r such that D r = R. We now prove that Formulation (CP2) is valid.

Proposition 4. (CP2) is a valid formulation of the p-center problem.

Proof. Let (ỹ, z) be an integer solution of (CP1). We first show that there exists an integer solution (y, r) of (CP2) which provides the same radius by setting y = ỹ and r = K k=1 zk . We need to prove that constraints (7a) are satisfied. We know that zk +

j : d ij <D k ỹj ≥ 1
is satisfied for any client Ci and any distance D k . If zk is equal to 0, the corresponding Constraint (7a) is satisfied, as

j : d ij <D k ỹj ≥ 1.
Otherwise, the same result is obtained since the zk variables are ordered in decreasing order which leads to r ≥ k. These two solutions provide the same radius as

D 0 + K k=1 (D k -D k-1 ) zk = D K k=1 zk .
We now prove that for any solution (ỹ, r) of (CP2) there exists an equivalent solution (y, z) of (CP1). We set y = ỹ and

z k = 1 if and only if r ≥ k. Constraint r + k j : d ij <D k ỹj ≥ k (8)
is satisfied for any k ∈ K. If r is lower than k, then at least one variable ỹj from equation ( 8) is equal to 1 and the corresponding constraint (6b) is satisfied. Otherwise, z k is equal to 1 and the same conclusion is reached.

We now prove that the linear relaxation of (CP1) is stronger than the one of (CP2).

Assumption 1 We shall suppose D 0 = 0 and ∀k ∈ K,

D k -D k-1 = 1.
This assumption is not restrictive, one can transform any instance by replacing any distance D k by its rank k. The transformed problem is equivalent as if the optimal radius is D k * , then the optimal solution of the transformed problem is k * . Under this assumption, problems (CP1) and (CP2) have the same optimal values, both of them compute the rank of the optimal radius. Proposition 5. Let CP1 and CP2 respectively be the LP relaxation of (CP1) and (CP2), v(CP1) ≥ v(CP2) under Assumption 1.

Proof. Let (ỹ, z) be a solution of CP1. We build a solution (y, r) of CP2 with the same value. We take y = ỹ and r = K k=1 zk . We need to prove that constraints (7a) are satisfied. Since the z k variables are ordered in decreasing order by Constraints 4, it follows that r ≥ kz k ∀k ∈ K. This and Constraints (2c) imply that Constraints (7a) are satisfied.

Table 1 summarizes the size of the previously mentioned formulations.

Formulation # of variables # of constraints

(P1) O(N M ) O(N M ) (P2), (P3), (P4) O(M + K) O(N K) (CP1) O(M + K) O(min(N M, N K)) (CP2) O(M ) O(min(N M, N K))
Table 1. Size of the four formulations (K ≤ N M ).

A two-step resolution algorithm

We present, in this section, a two-step algorithm to solve more efficiently the p-center problem.

Let lb be a lower bound of the optimal radius. We suppose that lb is one of the distances D k since, otherwise, lb can be set to the next distance. All the distances dij lower than lb can be replaced by lb.

Similarly, all the distances dij greater than an upper bound ub can be replaced by ub + 1 in order to discard solutions of value greater than ub.

The size of Formulations (P2) and (CP1) strongly depends on K. This value can be reduced by identifying lower and upper bounds. Such bounds can easily be obtained, as mentioned in [START_REF] Sourour Elloumi | A new formulation and resolution method for the p-center problem[END_REF].

Our resolution algorithm, depicted in Figure 1, can be applied to any formulation F of the p-center problem including (P1), (P2), (P3), (P4), (CP1) and (CP2). It is mainly based on the idea that whenever the optimal value v of the linear relaxation of F is not equal to an existing distance, then there exists k ∈ K such that D k-1 < v < D k . In that case, D k constitutes a stronger lower bound than v and the linear relaxation can be solved again. This process is repeated until an existing distance is obtained as the optimal value of the linear relaxation. This constitutes

Step 1 of the algorithm.

The bound obtained when applying this algorithm over (P2) or (CP1) corresponds to the one called LB * , computed by a binary search algorithm in [START_REF] Sourour Elloumi | A new formulation and resolution method for the p-center problem[END_REF].

Step 1 can be further improved by introducing the notion of dominated clients and dominated facilities within some reduction rules. A facility Fa is dominated if there exists another facility F b such that dia ≥ d ib for all clients i. Such a facility can be removed as it will always be at least as interesting to assign a client to F b than to Fa. Similarly, a client Ca is said to be dominated if there exists another client C b such that daj ≤ d bj for all facilities j. Dominated clients can also be ignored. Instructions 3 and 4 are repeated since new dominated clients and facilities may be found when a bound is improved, and vice versa.

Step 2 of Algorithm 1 consists in solving Formulation F to optimality with the improved bounds lb and ub computed in Step 1.

7 lb ← min k {D k : v ≤ D k } 8 until v = lb // until v is one of the existing distances //
Step 2 9 r * ← SolveOptimally(F , lb, ub) 10 return r * Fig. 1. Algorithm used to solve the p-center problem through F, a p-center formulation.

We implement Formulations (P1), (P2), (CP1) and (CP2) as well as Algorithm 1 on an Intel XEON E3-1280 with 3,5 GHz and 32Go of RAM with the Java API of CPLEX 12.7. Following several authors, we consider instances from the OR-Library [START_REF] Beasley | Or-library: distributing test problems by electronic mail[END_REF].

Comparing sizes and computation times on 5 instances

Table 2 presents a comparison of the sizes of the four formulations on the five first instances of the OR-Library with N = M = 100. We use the initial lower bound LB0 = maxi∈N minj∈M dij and initial upper bound U B0 = minj∈M maxi∈N dij introduced in [START_REF] Sourour Elloumi | A new formulation and resolution method for the p-center problem[END_REF]. As expected, the number of variables in (CP1) and (P2) are equal and are significantly lower than in (P1). Formulation (P2) has more constraints than Formulation (P1). Formulation (CP1) has by far less constraints than (P2). All this explains why (CP1) has the best performances in every aspect. Formulation (CP2) is the most compact but this does not fully compensate the poor quality of its LP bound. Table 2. Size and resolution times (1 thread) of the formulations for the five first OR-Library instances with lb = LB0 and ub = U B0.

(P1) (P2) (CP1) (CP2)

Relaxation and computation times on the 40 OR-Library instances

In Table 3, we perform a larger comparison with stronger bounds lb and ub equal to the bounds LB1 and U B1 introduced in [START_REF] Sourour Elloumi | A new formulation and resolution method for the p-center problem[END_REF]. The resolution is then performed by CPLEX with its default parameters but with a maximal CPU time of 1 hour. The first column is the instance number. The three following columns provide N , p and the optimal value of the instances (N = M in these instances). Columns 5 and 6 contain the initial bounds LB and U B. For each formulation, column "b" corresponds the optimal value of the linear relaxation and column "t" to the resolution time in seconds.

We can first observe that Formulations (CP1) and (P2) solve all the 40 instances within 1 hour while ten instances are not solved with (P1) and one instance is not solved with (CP2). We can even observe that (CP1) solves the whole set of instances in less than 50 minutes and (P2) in less than 85 minutes. Formulation (P2) outperforms (CP1) mainly on instances 36 and 39. This is possibly due to some difficulty of the solver to find good feasible solutions.

Results of Algorithm 1

Table 4 presents the results of Algorithm 1 with formulations (CP1) and (CP2). Columns "t1" and "t2" respectively correspond to the time of the first phase and the total time. Formulation (CP2) is now able to solve all the instances within 1 hour. We observe that the total time to solve the 40 instances is reduced by approximately 6 times for (CP1) and 14 times for (CP2) if compared to Table 3.

Conclusion

We introduced two new compact formulations of the p-center problem. We theoretically compared the quality of their LP bounds and their sizes to existing formulations. Numerical experiments confirmed these results and highlighted the fact that our new formulation (CP1) outperforms the previously known formulations (P1) and (P2) at all levels. Our more compact formulation (CP2) suffers from the poor quality of its linear relaxation. Another aspect of our work was to embed the formulations within a two-step algorithm in order to obtain better computation times.

Our future work will focus on improving our compact formulation through polyhedral studies.

N

Algorithm 1 : 2 repeat 3 6 v

 1236 F : formulation of the p-center problem p: maximal number of centers d: distances lb, ub: initial bounds Result: The optimal radius // Step 1 1 repeat Remove dominated clients and facilities // Reduction rules 4 (lb, ub) ← Compute bounds 5 until lb and ub are not improved and no more dominated clients or facilities have been found ← SolveLinearRelaxation(F , lb, ub)

Table 3 .

 3 Comparison of the different formulations with lb = LB1 and ub = U B1. For each instance, the smallest time appears in bold. Symbol "-" means that the instance was not solved within 1 hour.

	p opt lb	ub	b	(P1)	t	b	(P2)	t	(CP1) b	t	(CP2) b	t
	1 100 5 127 59 133 98		2,4 107		75,3 107	1,0 85	4,0
	2 100 10 98 56 117 77		2,9 86		7,3 86	0,5 71	5,2
	3 100 10 93 55 116 74		2,9 84		2,5 84	0,2 69	3,1
	4 100 20 74 41 127 55		0,7 65		7,9 65	0,6 53	3,4
	5 100 33 48 23	87 31		0,8 38		1,0 38	0,1 30	1,5
	6 200 5 84 38	94 68		35,9 75	106,7 75	2,7 59	47,1
	7 200 10 64 34	79 51		20,5 58	100,2 58	1,8 46	26,1
	8 200 20 55 30	72 41		20,7 48		87,2 48	1,6 38	19,6
	9 200 40 37 22	73 28		8,9 33		14,9 33	1,4 27	29,8
	10 200 67 20 11	44 15		1,6 18		0,8 18	0,3 14	5,5
	11 300 5 59 34	67 50		99,0 54		30,4 54	6,2 44	68,1
	12 300 10 51 30	72 43	229,7 48		71,0 48	7,2 39	98,7
	13 300 30 36 20	56 28	114,0 33		44,6 33	4,7 26	106,9
	14 300 60 26 14	60 19	157,1 23		33,4 23	12,9 18	151,7
	15 300 100 18 10	42 13		8,6 16		9,4 16	0,9 13	30,2
	16 400 5 47 26	51 41	403,2 45		25,3 45	3,3 36	54,5
	17 400 10 39 21	47 33	737,8 36		35,0 36	24,9 29	149,2
	18 400 40 28 16	50 22	664,7 25		96,4 25	22,1 20	431,4
	19 400 80 18 10	40 14	226,2 16		81,4 16	18,5 13	116,9
	20 400 133 13 7	32 10		9,0 12		3,0 12	0,9 10	22,5
	21 500 5 40 23	48 35 2581,0 37	118,3 37	13,6 31	194,6
	22 500 10 38 21	49 31		-35	924,4 35	24,6 28	507,8
	23 500 50 22 13	38 17 1375,8 20	212,2 20	38,4 16	481,8
	24 500 100 15 9	35 12	573,7 14		51,0 14	29,6 11	209,2
	25 500 167 11 6	27	8		57,2 10		5,1 10	2,0	8	23,1
	26 600 5 38 21	43 32 3093,6 35	106,0 35	13,6 28	152,4
	27 600 10 32 18	39 28 3118,9 30	104,3 30	48,3 25	341,5
	28 600 60 18 10	33 14		-16	176,2 16	103,3 13	-
	29 600 120 13 7	36 10		-12	130,7 12	77,8	9	893,6
	30 600 200	9 5	29	7	106,5	8	12,4	8	15,7	7	89,8
	31 700 5 30 16	34 27 1793,8 28		68,8 28	12,5 24	139,9
	32 700 10 29 16	35 25		-27	718,7 27	127,3 22	944,5
	33 700 70 15 9	26 13		-14	155,1 14	76,0 12	890,1
	34 700 140 11 6	30	9 2617,9 10	168,7 10	32,8	8	464,9
	35 800 5 30 16	32 27		-29		23,0 29	13,0 23	170,6
	36 800 10 27 16	34 24		-26	130,3 26	821,7 21 1056,6
	37 800 80 15 8	26 12		-14	222,5 14	90,9 11 1706,9
	38 900 5 29 15	35 25		-27		68,8 27	19,0 21	300,1
	39 900 10 23 13	28 20		-22	348,4 22 1190,0 18 1786,4
	40 900 90 13 7	22 10		-12	551,0 12	129,5 10 1059,9
			Total		57699		5129		2991	16390

Table 4 .

 4 Results obtained with Algorithm 1 of Figure1with lb = LB1 and ub = U B1.