Stefano Gulli

Jean-Claude Daran

Rinaldo Poli
email: rinaldo.poli@lcc-toulouse.fr

Jean-Claude Daran

Dr Mr S Gulli

Prof J.-C R Daran

Prof R Poli

Synthesis and Structure of Four-Coordinate Copper(II

Keywords: Functionalized -ketiminato ligands, Controlled Radical Polymerization, ATRP, Polystyrene Copper, Functionalized ketimidato ligands, Controlled Radical Polymerization, ATRP, Polystyrene

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Introduction

Copper coordination chemistry with multidentate ligands has recently experienced a revival after the discovery [1] and intensive development [2][3][4][5] of Cu-mediated controlled radical polymerization. This polymerization strategy rests on the reversible trapping of the growing radical chain, which is accomplished by transfer of a halogen atom from a Cu II -X compound, yielding the corresponding Cu I complex and the halogen-terminated radical chain. The technique is therefore named "atom transfer radical polymerization" (ATRP). The process can be started from stable Cu I complexes, usually made in situ from CuX (X = Cl or Br) and a suitable multidentate ligand, plus a halogenated compound R0-X as initiator, in the presence of the polymerizable monomer (socalled ATRP conditions). Alternatively, it can be started from a stable Cu II complex (also usually made in situ from CuX2 and the appropriate ligand) and a standard radical source, in the presence of the monomer (so-called reverse ATRP conditions).

A recent review, focused on the structural aspects of coppercatalyzed ATRP, shows that the complexes used for this purpose are inevitably made by mixing CuX and CuX2 with neutral supporting ligands, therefore always containing a halogen atom in the coordination sphere. [6] Occasionally, the halide is displaced to yield cationic complexes, such as for instance [Cu(bipy)2] + [7] or [Cu(Me6TREN)] + , [8] but the introduction of the anionic functionality into the multidentate ligand itself does not seem to have so far attracted much attention. We wish to explore here the use of anionic multidentate ligands for copper coordination chemistry, starting with the tridentate "XL2" type. This is the minimum denticity likely to provide stable complexes in both oxidation states, since the Cu II Cl(XL2) coordination environment is well established, for instance in Tp*CuCl (Tp* = tris(3,5dimethylpyrazolyl)borate), [9] and the putative 3-coordinate Cu I XL2 system resulting from the halogen atom transfer may be stabilized by dimerization or by monomer coordination, as for instance in [Tp*Cu]2 or Tp*Cu(olefin) complexes. [10,11] One of the simplest ways to introduce an anionic functionality in a chelating ligand is though deprotonation of a -diketone such as acetylacetone. Since Cu has greater affinity to N ligands, modification of acetylacetone by introduction of a diamine was envisaged, to yield amine-functionalized -ketimines, a well known class of ligands that has already been used to generate Cu I derivatives. [START_REF] Zhao | Beijing Huagong Daxue Xuebao[END_REF][13][14][START_REF] Zhang | Nankai Daxue Xuebao[END_REF] We report here the synthesis and structural characterization of new copper(II) complexes incorporating these ligands, as well as a preliminary exploration of their ability to control the radical polymerization of styrene under reverse ATRP conditions.

Results and Discussion

Condensation of acetylacetone with one equivalent of R2NCH2CH2NH2 (R = Me, Et) yielded ligands HL 1 and HL 2 as shown in Scheme 1. [16][17][18][19] Both ligands have recently been used in coordination chemistry with both main group [17,19] and transition metals. [START_REF] Zhao | Beijing Huagong Daxue Xuebao[END_REF][START_REF] Zhang | Nankai Daxue Xuebao[END_REF]20] The reaction of these ligands with anhydrous CuCl2 in THF, in the presence of one equivalent of triethylamine, provides a high-yield entry into complexes CuCl(L 1 ), 1, and CuCl(L 2 ), 2.

Complex 2 has already been recently described. [START_REF] Zhao | Beijing Huagong Daxue Xuebao[END_REF] It was synthesized in unreported yields by direct interaction of CuCl2 and HL 2 in acetone at 60-65°C without use of a base and it was only characterized by IR spectroscopy. Our method uses NEt3, allowing the elimination of the generated HCl as Et3NHCl, which precipitates in THF and can be easily removed by filtration, while the reaction is complete in less than 2 h at room temperature. The two compounds show a sharp EPR spectrum in dichloromethane solution at room temperature (Figure 1a), with the expected 1:1:1:1 hyperfine splitting caused by the I = 3/2 Cu nucleus. The dimethylamino derivative 1 reveals also an additional and barely discernible superhyperfine splitting, which may be caused by coupling to one of the N atoms ( 14 N: I = 1). The g tensor is tetragonal, as revealed by the spectrum taken at 120 K (Figure 1b), with a large a║ and a non discernible a┴. angle is also rather small. The Cl-Cu-O plane makes a dihedral angle of 21.47 (7)° with the N1-Cu-N2 plane. This distortion appears quite natural for this specific coordination environment. A search on the Cambridge Data Base for other 4-coordinate Cu complexes with a ClON2 donor set has revealed 42 mononuclear geometries, for which the average value of this specific dihedral angle is 11±7°. Compound 1, on the other hand, is dinuclear, with bridging Cl atoms and 5-coordinate Cu centers. The bridging Cu2Cl2 moiety is in fact quite asymmetric, with very different Cu-Cl distances. The shorter Cu-Cl distance is only slightly lengthened relative to the Cu-Cl distance in compound 2. Upon closer look, is it clear that the compound can be described as the dimerization of CuCl(L 1 ) moieties that have a geometry very closely relate to that of the mononuclear compound 2, through loose Cu-Cl i and Cu i -Cl interactions. Indeed, the coordination geometries around the Cu center in 1 and 2 are strikingly similar, including the deviation of the Cl-Cu-N1 angle from linearity and the dihedral angle between the N1-Cu-N2 and Cl-Cu-O planes from coplanarity (15.19(6)°). An interesting question therefore arises: why this nuclearity difference? The existence of the Cu-Cl i interactions demonstrates the preference of a dinuclear structure, but their weakness suggests that their establishment may be discouraged by small perturbations. For compound 2, for which the shortest intermolecular contacts of the Cl atom are established with C-H bonds of neighboring diethylaminoethyl moieties, the small perturbation is undoubtedly caused by the steric repulsion that the ethyl substituents would experience in a dinuclear structure. Indeed, the space filling model of compound 1, shown in Figure 4, indicates how one of the NMe2 methyl groups in one half of the molecule (right arrow) finds itselves in close proximity with the methyl group of the keto group in the other half og the molecule (left arrow), suggesting that a similar structure may not be sufficiently favoured with the bulkier NEt2-substituted ligand. It is then also interesting to re-examine the related mononuclear Cu structures containing a ClON2 ligand set already described in the literature, to identify the perturbation impeding dimerization in those cases. For a few compounds, the authors reported a 4-coordinate structure without realizing the presence of a loose axial interaction with an interstitial solvent molecule. This is the case for a few "methanol solvates", with Cu-O distances to the axially bonded MeOH ligand as little as 2.27 Å. [21][22][23] For another three compounds, the structure was described as mononuclear but it is in fact halide-bridged dinuclear, just like 1, with "loose" Cu-X distances of 2.61 [24] and 2.84 [25] Å (X = Cl), or 3.09 [26] (X = Br). In other cases, although axial donor atoms are far away for detection as bonding interactions (distance greater than the sum of van der Waals radii), the donor atom (Cl but sometimes also O) is correctly positioned to provide some axial electron density to the Cu center in 2D or 3D packing arrangements. [27][28][29][30][31][32] Interesting situations are presented by other structures that, like that of compound 2, are genuinely mononuclear, without significant additional contacts to the 4-coordinate Cu II center. A few of theses are polyaromatic compounds with delocalized planar ligands, which always display a -stacking arrangement. [33][34][35][36][37][38][39][40] This indicates that intermolecular -staking interactions provide better stabilization to the crystal structure than the halide-bridge formation. In other cases, the dinuclear structure simply does not form because of the presence of a bulky group near the Cu center, either an NEt2 group like in compound 2, [41,42] or another bulky group. [31,43,44] The structure of compound 1 is identical to that of compound [Cu(L 1 )(O2CPh)]2, [START_REF] Zhang | Nankai Daxue Xuebao[END_REF] where the benzoate ligand occupies the same position as the Cl ligand in 1 by adopting a bridging monodentate (κ 1 :μ) coordination mode, also with very different distances to the two Cu atoms (1.984(2) Å for the strong equatorial bond, 2.788 Å for the loose axial bond). The Cu-N2 bond (2.064(3) Å) is essentially identical to that in 1, whereas the Cu-N1 bond (1.914(3) Å) is significantly shorter and the Cu-O1 (1.944(3) Å) is significantly longer. The structure of 2, on the other hand, may be directly compared with that of compound CuCl[Et2NCH2CH2N-C(Me)CHC(Ph)O], the two compounds differing only by the nature of the substituent at the keto position (Ph vs. Me). [31] In the Ph substituted compound, the Cu-Cl distance is significantly longer (2.300 Å), while the Cu-N1 distance (1.956 Å) is marginally longer and the Cu-O (2.107 Å) and Cu-N2 (1.992 Å) distances are marginally shorter. The trans Cl-Cu-N1 angle (164.8°) is quite similar to that in 1.

Compounds 1 and 2 have been tested as controlling agents for the radical polymerization of styrene under reverse ATRP conditions. The first experiments carried out at 80°C, with primary radicals generated from 2,2'-azobisisobutyronitrile (AIBN, t1/2 = ca. 70 min), are shown in Figure 5a. Several features of the two plots shown in Figure 5 deserve discussion. The first and most important one is that polymerization occurs and is sustained well beyond the time required to completely consume the radical initiator (6-7 h) in the presence of both compounds. Together with the continued increase of the number-average molecular weight and the low polydispersities of the resulting polymers at high conversion, this represents evidence that the radical chains are continuously generated from a dormant species and that the polymerization is controlled. Note that the first order kinetics plot indicates linearity in the high conversion region (after completely consuming the initiator), in agreement with a constant radical concentration in solution and with the occurrence of the ATRP mechanism. At low conversions (short times), on the other hand, the polymerization is faster. In addition, the molecular weights at low conversions are much greater than the expected values, but subsequently continue to grow at approximately the expected growth rate, while the polydispersity remains narrow throughout the polymerization process (< 1.2), see Figure 5b. This phenomenon can be ascribed to a slow trapping rate, generating chains of much higher molecular weight than expected during the initial phase of initiator decomposition. The relatively long initial phase of initiator decomposition (6-7 half-lives, namely 6-7 hours) entails the continuous generation of new radical chains throughout this period, but the molecular weight distribution does not appear to be too negatively affected by this initial phase or poor control.

The polymerization proceeds at approximately the same rate in the presence of compounds 1 and 2, both before and after depletion of the radical initiator (see Figure 5a), indicating that the activation/deactivation equilibrium is very similar for both copper compounds. In relation to the above described structural study of the two Cu II complexes, it is clear that the weak dichlorido-bridged interaction revealed for compound 1 does not play a major role in this process, as also indicated by the identical EPR spectroscopic properties of the two compounds in solution.

The putative Cu I catalyst generated by Cl atom transfer to the radical chain is a three-coordinated complex where the constraints of the ligand bite do not allow the metal center to adopt a preferred trigonal planar configuration. Although, as mentioned in the introduction, the complex may be stabilized by dimerization or by coordination of a monomer molecule, we wondered whether the addition of Lewis bases such as pyridine (py) or bipyridine (bipy) could affect the polymerization rate and controllability. Because of the identical solution properties and polymerization results of compounds 1 and 2, additional experiments in the presence of py or bipy were only carried out with the ethylsubstituted complex 2. The results (Figure 6) indicate a negligible effect on the reaction rate when using only 1 equiv of either additive. The difference between the three experiments may be considered to fall within the margins of the experimental error. The trends of Mn and PDI as a function of conversion were also sensibly identical for the three experiments. When the py/Cu ratio was increased to 10, however, the polymerization was significantly retarded in the second period, after full consumption of the initiator, whereas the behaviour was once again identical before full consumption of the radical initiator. Therefore, it appears that the three-coordinate Cu I complex generated by Cl atom transfer, Cu(L 2 ), can be stabilized by the Lewis base additive as shown in Scheme 2, the overall effect being a shift of the deactivation equilibrium toward the Cu I form. The absence of a notable effect on the rate during the first polymerization period is in agreement with the prediction that the Lewis additive action is restricted to the Cu I stabilization and does not affect the rate of Cl atom transfer from compound 2 to the growing radical chain (deactivation, kd). Figure 6. First-order kinetic plot for the radical polymerization of styrene in the presence of compound 2 alone (diamonds) or together with py (1 equiv, triangles; 10 equiv, spheres) or bipy (1 equiv, squares). Conditions are the same as in Figure 5.

An additional experiment was carried out in the presence of compound 2 and py (1 equiv) at lower temperature (30°C). In this case, 2,2'-azobis(4-methoxy-2,4-dimethylvaleronitrile), V-70, was used as radical initiator (t1/2 = 10 h). The results are compared with those obtained at 80°C in Figure 7. The polymerization rate is slower as expected. Once again, the process is sustained beyond the time needed to fully deplete the radical initiator (ca. 70 h for V-70 at 30°C), with a visible breakpoint in the reaction rate, indicating reversible reactivation of the dormant chains by atom transfer to Cu I . However, the controllability is not improved, as shown by the fact that the polymer Mn is still large relative to the theory while the PDI is greater relative to the experiment at 80°C. The reason for this result is that, while controllability in the second polymerization period is expected to increase, the lower temperature also causes a slower deactivation rate, negatively affecting the molecular weight distribution in the first polymerization period. The slow radical trapping by complexes 1 and 2 may be directly related to the unsuitable geometry of the three-coordinate Cu I species that is formed by Cl atom transfer. As it has been previously pointed out, the best systems for rapid radical deactivation, which is crucial for good controllability, are those where the coordination geometry changes in a minimal way on going from Ln/Cu II -Cl to Ln/Cu I , while the electronic properties and the coordination geometry of the Ln coordination sphere are most suitable to the Cu I species. In our case, the situation is quite the opposite, since the L 1 and L 2 ligands yield a suitable coordination environment for the Cu II system in combination with the Cl atom but are not at all adapted to the Cu I system.

Conclusions

We have prepared and characterized two neutral 4-coordinated Cu II monochloride complexes, disclosing new structural features related to weak intermolecular Cu•••Cl bonding and discussing the criteria allowing this interaction to occur in these and other related complexes previously described in the literature. These complexes have been tested as reversible trapping agents for growing polystyrene radical chains under reverse ATRP conditions. While the systems are negatively affected by a slow trapping rate, the principle of reversible atom transfer leading to a controlled radical polymerization has been demonstrated. The trapping rate is expected to be accelerated by using other tridentate "XL2" ligands with greater bites, thus adapting to a trigonal planar coordination for Cu I , or perhaps even better by moving to tetradentate "XL3" ligand that would stabilize Cu I in a tetrahedral environment. Research in our laboratory is now oriented along these directions.

Experimental Section

General: All solvents used in the reactions were distilled under argon. The NMR spectra were recorded on Bruker ARX 250 and AV300 instruments. Chemical shifts are expressed in ppm downfield from Me4Si. Coupling constants are given in Hertz. The EPR spectra were measured on a Elexsys E500 BRUKER spectrometer (X-band) equipped with both a frequencemeter and gauss-meter. Acetylacetone, N,N-dimethylethylenediamine, N,Ndiethylethylenediamine and CuCl2 were purchased from Sigma-Aldrich and used as received. Styrene (Sigma-Aldrich) was dehydrated by passing through a neutral alumina column and then stirred on CaH2 for 1 day, followed by distillation under argon.

Synthesis of Me2NCH2CH2NHC(Me)=CHC(Me)=O, HL 1

In a 500 ml round-bottomed flask equipped with a condenser, 29.3 g (0.29 mol) of acetyl acetone and 26 g (0.29 mol) of N,N-dimethylethylenediamine in 250 ml of benzene were refluxed for 48 h to yield an intense yellow solution. The conversion was monitored through measurement of the water removed from the reaction through a Dear-Stark apparatus. The reaction starts immediately, as suggested by the color change but two days of reflux were necessary to complete it. The solvent was then removed to yield an intense yellow-colored liquid. Distillation at 115 ° C yielded the product as a yellow oil. Yield: 47.1 g (95.5%). 1 H-NMR (C6D6, 250 MHz): δ 11.1 (s(br), 1H, NH), 4.98 (s, 1H, CH3CCHCCH3), 2.93 (q, 2H, J = 6.2 Hz, NHCH2CH2), 2.15 (t, 2H, J = 6.5 Hz, NHCH2CH2), 2.08 (s, 3H, CH3CO), 2.06 (s, 6H, N(CH3)2), 1.63 (s, 3H, CH3CN). 13 C-NMR (C6D6, 62.90 MHz): δ 193.6, 161.5, 95.0, 58.9, 45.2, 40.9, 28.7, 18.4.

Synthesis of Et2NCH2CH2NHC(Me)=CHC(Me)=O, HL 2

In a 500 ml round-bottomed flask equipped with a condenser, 29.3 g (0.29 mol) of acetylacetone and 34 g (0.29 mol) of N,N-diethylethylenediamine in 250 ml of benzene were refluxed for 48 h to yield an intense yellow solution. The conversion was monitored through measurement of the water removed from the reaction through a Dear-Stark apparatus. The reaction starts immediately, as suggested by the color change but two days of reflux were necessary to complete it. The solvent was then removed to yield an intense yellow-colored liquid. Distillation at 145°C yielded the product as a yellow oil. Yield: 49.7 g (87.6%). 1 H-NMR (C6D6, 300 MHz): δ 11.1 (s, 6H), 4.98 (s, 1H, CH3CCHCCH3), 2.93 (q, 2H, J = 6.2 Hz, NHCH2CH2), 2.37 (q, 4H, J = 7.2 Hz, N(CH2CH3)2), 2.34 (t, 2H, J = 6.6 Hz, NHCH2CH2), 2.08 (s, 3H, CH3CO), 1.64 (s, 3H, CH3CN), 0.98 (t, 6H, J = 7 Hz, N(CH2CH3)2). 13 C{ 1 H}-NMR (C6D6, 75.47 MHz): δ 193.5, 161.4, 95.0, 52.9, 47.3, 41.5, 28.7, 18.5, 12.1.

Synthesis of compound CuCl(L 1 ), 1

To a solution of CuCl2 (5 g, 0.038 mol) in 10 ml of THF, a solution of HL 1 (7.5 g, 0.038 mol in 10 ml of THF) was added, followed by stirring for 1 h at room temperature. The solution became dark green. Then, 5.5 ml (0.04 mol) of Et3N was added, followed by stirring for another 1 h, resulting in the formation of a white precipitate (Et3NH + Cl -), which was then removed by filtration. The solvent was partially removed under reduced pressure to about 5 ml and slow diffusion of hexane gave black crystals suitable for analysis and for the X-ray diffraction analysis. Yield: 9.

Synthesis of compound CuCl(L 2 ), 2

To a solution of CuCl2 (5 g, 0.038 mol) in 10 ml of THF, a solution of HL 2 (6.5 g, 0.038 mol in 10 ml of THF) was added, followed by stirring for 1 h at room temperature. The solution became dark green. Then, 5.5 ml (0.04 mol) of Et3N was added, followed by stirring for another 1 h, resulting in the formation of a white precipitate (Et3NH + Cl -), which was removed by filtration. The solvent was partially removed under reduced pressure to about 5 ml and slow diffusion of hexane gave black crystals suitable for analysis. Yield: 4.8 g (57.9%). Anal. Calcd. for C11H21ClCuN2O: C, 44.59; H, 7.14; N, 9.45 %. Found: C, 44.0; H, 7.7; N, 9.4 %. EPR (CH2CH2, rt): g = 2.098; aCu = 81.4 G. EPR (CH2CH2, 120 K): g║ = 2.20 (a║ = ca. 180 G), g┴ = 2.09.

Standard procedure for the polymerization experiments

The polymerizations were conducted in pre-dried Schlenk flasks. The solids (complexes, initiators and bipyridine) and a stirring bar were added, and then the flask was sealed with a glass septum. Oxygen was removed from the flask by applying vacuum and backfilling with argon (three cycles). The monomer and the solvents (pyridine, toluene and dodecane) were added via syringe. The solution was degassed by three freeze-pump-thaw cycles. A first sample was removed as reference and the flask immersed in an oil bath held by thermostat at the desired temperature. At timed intervals, samples were removed from the flask by syringe and diluted into THF. Every sample was filtered through neutral alumina to remove the catalyst. A small amount of this THF solution was used to determine the monomer conversion by gas chromatography (GS) with dodecane as an internal standard. From the remaining sample, the polymer product was precipitated by adding into excess MeOH. The precipitate was collected and analyzed by GPC-SEC.

X-ray crystallography

A single crystal of each compound was mounted under inert perfluoropolyether on the tip of a glass fibre and cooled in the cryostream of either an Oxford-Diffraction XCALIBUR CCD diffractometer for 1 or an Oxford-Diffraction GEMINI CCD diffractometer for 2. Data were collected using the monochromatic MoK radiation (= 0.71073). The structures were solved by direct methods (SIR97) [45] and refined by least-squares procedures on F 2 using SHELXL-97. [46] All H atoms attached to carbon were introduced in calculation in idealised positions and treated as riding models. The drawing of the molecules was realised with the help of ORTEP32. [47] The crystal data and structure refinement parameters for both compounds are listed in Table 1. Crystallographic data (excluding structure factors) have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC 800731 & 800732. Copies of the data can be obtained free of charge on application to the Director, CCDC, 12 Union Road, Cambridge CB2 1EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk).

Figure 1 .

 1 Scheme 1.

Figure 2 .

 2 Figure 2. An ORTEP view of the molecule of compound 2, also showing the numbering scheme used. Thermal ellipsoids are drawn at the 50 % probability level. Main geometrical parameters: Cu-Cl, 2.2458(4); Cu-O, 1.9126(10); Cu-N1, 1.9404(12); Cu-N2, 2.0563(12) Å; Cl-Cu-O, 91.92(4); Cl-Cu-N1, 163.62(4); Cl-Cu-N2, 93.46(4); O-Cu-N1, 92.92(5); O-Cu-N2, 166.32(5); N1-Cu-N2, 85.38(5)°.

Figure 3 .

 3 Figure 3. An ORTEP view of the molecule of compound 1, also showing the numbering scheme used. Thermal ellipsoids are drawn at the 50 % probability level. Main geometrical parameters: Cu-Cl, 2.3202(4); Cu-Cl i = 2.8442(4); Cu-O, 1.917(1); Cu-N1, 1.9642(13); Cu-N2, 2.0610(13) Å; Cl-Cu-Cl i = 85.77(1); Cl-Cu-O, 88.97(3); Cl-Cu-N1, 170.47(4); Cl-Cu-N2, 92.65(4); Cl i -Cu-O, 97.56(4); Cl i -Cu-N1, 103.40(4); Cl i -Cu-N2, 95.25(4); O-Cu-N1, 92.34(5); O-Cu-N2, 167.17(5); N1-Cu-N2, 84.02(5)°.

  Figure 3. An ORTEP view of the molecule of compound 1, also showing the numbering scheme used. Thermal ellipsoids are drawn at the 50 % probability level. Main geometrical parameters: Cu-Cl, 2.3202(4); Cu-Cl i = 2.8442(4); Cu-O, 1.917(1); Cu-N1, 1.9642(13); Cu-N2, 2.0610(13) Å; Cl-Cu-Cl i = 85.77(1); Cl-Cu-O, 88.97(3); Cl-Cu-N1, 170.47(4); Cl-Cu-N2, 92.65(4); Cl i -Cu-O, 97.56(4); Cl i -Cu-N1, 103.40(4); Cl i -Cu-N2, 95.25(4); O-Cu-N1, 92.34(5); O-Cu-N2, 167.17(5); N1-Cu-N2, 84.02(5)°.

Figure 4 .

 4 Figure 4. A space-filling view of the structure of compound 1.

Figure 5 .

 5 Figure 5. (a) First-order kinetic plot and (b) Mn (filled symbols) and PDI (hollow symbols) vs. conversion for the radical polymerization of styrene in the presence of compounds 1 (triangles) or 2 (diamonds) The straight line in plot (b) indicates the theoretical molecular weight (MR°+ MM•(nM/nCu)•p +MCl; where MR°, MM and MCl are the molar masses of primary radical, monomer and Cl atom, respectively, p is the degree of conversion, and (nM/nCu) is the molar ratio of monomer and Cu). Conditions: 50% (v/v) in toluene, 80°C, [styrene]:[AIBN]:[Cu] = 200:0.8:1.

Figure 7 .

 7 Scheme 2.

New

  Cu II systems for ATRP Neutral monochloride Cu II complexes with dialkylaminoethyl-functionalized -ketiminato ligands have been synthesized and shown to control the radical polymerization of styrene under reverse ATRP conditions. A structural comparison between the dimethyl and diethylamino compounds reveals a delicate balance between mononuclear and dinuclear arrangements in the solid state. Stefano Gulli, Jean-Claude Daran, Rinaldo Poli* ….. Page No. -Page No. Synthesis and structure of 4-coordinate copper(II) complexes stabilized by ketiminato ligands and application in the reverse ATRP of styrene

  1 g (98.1%). Anal. Calcd. for C9H17ClCuN2O: C, 40.30; H, 6.39; N, 10.44 %. Found: C, 39.6; H, 6.8; N, 10.3 %. EPR (CH2CH2, rt): g = 2.097; aCu = 83.4 G. EPR (CH2CH2, 120 K): g║ = 2.20 (a║ = ca . 190 G), g┴ = 2.08.

Table 1 .

 1 Crystal data for compounds 1 and 2. Tables of conversions, molecular weights and polydispersity index for the all polymerization experiments (2 pages).

	Compound	1	2
	Empirical formula	C9H17ClCuN2O	C11H21ClCuN2O
	Formula weight	268.24	296.30
	Temperature, K	180(2)	180(2)
	Wavelength, Å	0.71073	0.71073
	Crystal system	Monoclinic	Monoclinic
	Space group	P 21/n	P 21/n
	a, Å	10.2397(4)	10.2616(3)
	b, Å	9.6133(4)	9.7014(3)
	c, Å	12.3296(5)	14.1431(4)
	, °	110.201(4)	103.053(3)
	Volume, Å 3	1139.03(8)	1371.59(7)
	Z	4	4
	D (calcd), Mg/m 3	1.564	1.435
	Abs. coeff., mm -1	2.122	1.770
	F(000)	556	620
	Crystal size, mm 3	0.12x0.11x0.07	0.44x0.34x0.31
	Theta range, °	2.75 to 28.32°.	3.48 to 29.12.
	Reflts collected	12338	15349
	Unique reflts [R(int)]	2646 (0.0266)	3263 (0.0228)
	Completeness, %,	93.3	94.2
	Abs. correction	Multi-scan	Multi-scan
	Max. / min. transm.	1.00000,	1.00000, 0.86185
		0.74280
	Refinement method	F 2	F 2
	Data / restr. / param.	2416 / 0 / 131	3204 / 0 / 149
	Goodness-of-fit on F 2	1.101	1.103
	R , wR2 [I>2(I)]	0.0224, 0.0601	0.0217, 0.0607
	R , wR2 (all data)	0.0264, 0.0616	0.0289, 0.0619
	Resid. density, e.Å -3	0.287 and -0.611 0.415 and -0.269
	Supporting Information:

Acknowledgments

We are grateful to the Agence National de la Recherche (Programme ANR Blanc "OMRP", 2010-13), the Centre National de la Recherche Scientifique (CNRS) and the Institut Universitaire de France (IUF) for support of this work and to the Politecnico, Milan, Italy for an Erasmus exchange fellowhip to SG. We also thank Dr. Zhigang Xue and Mr Andrés F. Cardozo for technical assistance. ____________


F Bréhard

M Joldes
email: joldes@laas.fr

J B Lasserre

S Laurens
email: sophie.laurens@cnes.fr

A Rondepierre
email: aude.rondepierre@insa-toulouse.fr

Polynomial superlevel set approximation of swept-volume for computing collision probability in space encounters

D. Arzelier, F. Bréhard, M. Joldeş, J.-B. Lasserre, S. Laurens, A. Rondepierre

Abstract-Computing long-term collision probability in space encounters is usually based on integration of a multivariate Gaussian distribution over the volume of initial conditions which generate collisions in the considered time interval. As this collision set is very difficult to determine analytically, for practical computation various simplifications are made in the literature. We present a new method for computing the collision probability based on two steps. Firstly, a higher-order outer-approximation of the swept-volume by a polynomial superlevel set is obtained as an optimal solution of a polynomial optimization problem. This has the advantage of providing approximate closed-form descriptions of the collision-prone states which can then be effectively used for long-term and repeated conjunctions analysis. From a computational viewpoint, one has to solve a hierarchy of linear matrix inequality problems of increasing size, which provide approximations (i) of increasing accuracy and (ii) convergent in volume to the original set. Secondly, once such a polynomial representation is computed, a high-order quadrature scheme for volumes implicitly defined by a polynomial superlevel sets is employed. Finally, the method is illustrated on some numerical examples borrowed from the literature.

I. INTRODUCTION

Since the collision between the Russian satellite COS-MOS 1934 and one debris of COSMOS 926 in December 1991, no less than eight orbital collisions have been reported between operational satellites. Space agencies and operators of the field have established alert procedures to assess the risks of collision for controlled satellites, and to authorize avoidance maneuvers if the predicted risk, measured by a probability of collision, exceeds some tolerance threshold. Risk evaluation is also performed after an avoidance maneuver to assess its benefit. When assessing conjunctions between two objects, the information usually available is: (1) a bound on the radius of the involved objects assumed to be spherical; [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF] normal probability distributions (mean and covariance values) of the state vectors of the objects at the Time of Closest Approach (TCA, when their nominal relative distance is estimated to be minimal).

In this context, two classes of encounters are defined.

For the so-called short-term encounters [START_REF] Foster | A Parametric Analysis of orbital Debris Collision Probability and Maneuver Rate for Space Vehicles[END_REF], [START_REF] Akella | Probability of collision between space objects[END_REF], [START_REF] Chan | Spacecraft Collision Probability[END_REF], [START_REF] Serra | Fast and accurate computation of orbital collision probability for short-term encounters[END_REF], the objects' relative velocity is assumed to be very high (several km/s), so that the relative motion is considered rectilinear on the encounter time interval. When the cross-correlations between the estimated states of the two objects, as well as their velocity uncertainty are neglected, the formulation of the collision probability is greatly simplified. In brief, the relative positions which generate collisions lie in a three-dimensional cylinder (also called collision tube), on which a Gaussian density is to be integrated. Finally, this reduces to computing a two-dimensional Gaussian integral on a disk. In the second class, called long-term encounters [START_REF] Chan | Spacecraft Collision Probability[END_REF], [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF], the relative velocity is of the order of m/s and both objects spend a significant amount of time in proximity to each other. This type of encounter is more common in the context of formation flying or proximity operations. Computing collision probabilities, in this so-called nonlinear framework, is considerably more difficult than for the short-term case. Even though the state distribution at TCA is a Gaussian one, the integration volume is no-longer a cylinder and can be very intricate (see Section III for details). Due to this, generalization attempts were proposed for specific cases of configurations [START_REF] Patera | Satellite collision probability for nonlinear relative motion[END_REF], [START_REF] Mckinley | Development of a nonlinear probability of collision tool for the Earth observing system[END_REF], [START_REF] Chan | Spacecraft Collision Probability[END_REF], but these approaches are relatively limited because of their characterization for particular relative trajectories. In [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF], V. Coppola proposes a different mathematical formalization and generalization. Roughly speaking, using a change of variable, the complicated integration volume at TCA is mapped via the dynamics at each time on the so-called hard-body sphere (which is a three-dimensional sphere of known radius equal to sum of the objects radii). In turn, this implies propagating the TCA distribution (both position and velocity) via the non-rectilinear dynamics. This change of variable is correct when imposing for each relative trajectory at most one entry crossing the hard-body sphere, which means that multiple encounters between the two objects are excluded. To tackle a practical implementation, the propagated distribution is assumed to remain Gaussian during the encounter interval. These two assumptions provide a rather restricted framework, which is however currently one of the most accomplished in the literature. Another approach, sketched in the works of Chan [START_REF] Chan | Hovering collision probability[END_REF], [START_REF] Chan | Spacecraft collision probability for long-term encounters[END_REF], consists in focusing on a different mathematical description of the integration volume at TCA, which is also called the swept-volume. Generated by the propagation of the hard-body during the encounter duration, it is defined as a union of ellipsoids (see Section III-A) and numerically characterized by its envelope in the 3dimensional case, or in an ad-hoc manner, with various trivial simplifications for lower dimensional cases. However, a general method relying on both (i) an effective characterization of the swept-volume (when its shape is not trivially reduced to a cylinder), and (ii) computation of the subsequent integral of the Gaussian density over such a volume, is missing in the literature. Following up this intuition, in this article we invoke polynomial optimization to provide approximate closedform descriptions of the collision-prone states, which can be effectively used for long-term and repeated conjunctions. This is a generalization with respect to Coppola's formulation which cannot handle multiple conjunctions correctly. Actually, even a visual accurate outer-approximation of the swept volume can provide important insights on the practical type of encounter. For instance, a straight cylinder form can confirm some of the encounter assumptions of the short-term framework.

The proposed method is based on two steps: (1) higherorder implicit1 outer-approximation of the swept-volume by a Polynomial Superlevel Set (PSS). From a computational viewpoint, one has to solve a hierarchy of linear matrix inequality problems (each providing an outerapproximation of increasing accuracy) with convergence in volume to the original set. (2) Once such a polynomial representation has been computed, a high-order quadrature scheme for volumes implicitly defined by a PSS is employed. With PSS approximation, highly non-convex shapes can be outer-approximated accurately, which in turn allows for further analysis of so-called longterm encounters. The method is illustrated on numerical examples borrowed from the literature.

II. ENCOUNTER MODELING AND PROBLEM STATEMENT

Consider an operational spacecraft (called primary and denoted by p) in orbit around the Earth and a space debris (called secondary and denoted by s). Their state is described by their position and velocity vectors r ⋆ and v ⋆ , in a reference frame R (⋆ = p or ⋆ = s). Classically the objects are modeled as spheres [START_REF] Akella | Probability of collision between space objects[END_REF], [START_REF] Patera | Satellite collision probability for nonlinear relative motion[END_REF], [START_REF] Chan | Spacecraft Collision Probability[END_REF], [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF], of known radii R ⋆ , since this allows for factoring out their orientation (attitude) and, for a conservative modeling of the secondary object, whose geometry is often poorly known (see Figure 1). With this assumption, a collision occurs when the relative distance between r p and r s is less than the so-called hard-body radius R = R p + R s . Based on this notion, it is natural to focus on the

x x x x R v p v s r p r s
Fig. 1: Encounter between two spherical objects.

relative state vector x T r = (r sr p ) T , (v sv p ) T ∈ R 6 , whose dynamics are given by:

ẋr (t) = f (t, x r (t)), t ∈ [t 0 , t f ], x r (t 0 ) = x 0 r . (1)
where f is a real Lipschitz continuous vector field and T := [t 0 , t f ] is the given time interval of the encounter. In general, these equations include the Newtonian gravitational central field and possible orbital perturbations (non spherical Earth, atmospheric drag, e.g.). It is assumed that, for each given relative initial condition x 0 r ∈ R 6 , the solution x r (t|x 0 r ) (also called trajectory or sample path) of the system (1) exists and is unique for t ∈ T . The initial conditions x 0 r ∈ R 6 are usually subject to uncertainties, and so, they are supposed to be distributed according to a given probability measure µ I , with density ρ I . As mentioned in the introduction, in practice the uncertainty distribution is usually given and estimated to be Gaussian at t T CA ∈ T . In the following, without loss of generality, we suppose that t T CA = t 0 (the proposed algorithms are easily tunable for the case t 0 < t T CA < t f , as illustrated by examples in Sec. VI).

Assumption 1 (Initial Gaussian distribution):

The relative initial conditions are normally distributed:

ρ I (x 0 r ) := e -1 2 (x 0 r -mI ) T P -1 I (x 0 r -mI ) (2π) 3 det(P I ) , (2)
with mean relative vector m I and covariance matrix P I .

In relative dynamics, the notion of collision transcribes to a relative trajectory entering a forbidden region X R :

X R = {x T r = (r T r , v T r ) ∈ R 6 | r r
Definition 1 (Collision): Given a relative initial condition x 0 r ∈ R 6 , a time interval T and a forbidden region X R , a collision occurs if there exists t ∈ T such that x r (t|x 0 r ) ∈ X R . Definition 2 (Collision domain/swept-volume): The domain of collision X 0 T is the set of relative initial conditions leading to collision on the time interval T , namely:

X 0 T = {x 0 r ∈ R 6 | ∃ t ∈ T , x r (t|x 0 r ) ∈ X R }. (4)
The problem of computing the collision probability is formally stated as: Problem 1 (General formulation): Let the dynamics in (1), a time interval T and a forbidden region X R . Provided that the initial conditions x 0 r ∈ R 6 are distributed according to a given probability measure µ I , compute the probability that a collision occurs:

P c (T ) := P(x 0 r ∈ X 0 T ) = µ I (X 0 T ) = X 0 T dµ I . ( 5
)
Note that with Assumption 1, Eq. ( 5) becomes:

P c (T ) = X 0 T ρ I (x r )dx r . ( 6
)
Problem 1 is in general very difficult. A first issue is to determine the domain of integration, which strongly depends on the chosen model for the relative dynamics [START_REF] Chan | Spacecraft Collision Probability[END_REF].

A theoretical solution was proposed for a polynomial vector field f , in the framework of moments-measures, sum-of-squares (SOS) in a series of works [START_REF] Streif | Probabilistic and set-based model invalidation and estimation using lmis[END_REF], [START_REF] Korda | Moment-sum-of-squares hierarchies for set approximation and optimal control[END_REF], [START_REF] Arzelier | Global probability of collision: problem modelling via occupation measure[END_REF]: using Liouville's equation, the nonlinear dynamics is lifted into a linear equation on measures, which is then solved using the Lasserre moment-SOS hierarchy of relaxations. However, in addition to the inherent numerical complexity of the present high-dimensional situation, we have also encountered numerical issues. This led us to consider other numerically-tractable solutions with a more practical utility, as discussed in what follows.

III. SWEPT-VOLUME AS A UNION OF BASIC SEMI-ALGEBRAIC SETS

The main goal is to obtain a more tractable characterization of the integration domain X 0 T . From Equation ( 4), one obtains a description of X 0 T as a union of sets, by retro-propagating with the inverse flow ϕ t0 t of the relative dynamics the set X R at each time t ∈ T :

X 0 T = t∈T ϕ t0 t (x r ) | x r ∈ X R . (7)
However, due to the complicated nature of the inverse flow (no closed-form solution in general), this set can hardly be analytically described, so various additional assumptions have been made in the literature [START_REF] Chan | Spacecraft Collision Probability[END_REF], [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF], [START_REF] Chan | Hovering collision probability[END_REF]. In this work, the assumption of linearized relative dynamics is employed, which is quite commonly used for uncertainty propagation in such applications [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF] (see Sec. VI for a specific example of linearization). Assumption 2 (Linearized relative dynamics): The relative dynamics flow is linear and invertible. The solution of the relative dynamics equation is therefore known via a given state transition matrix Φ(•, t 0 ) : T → R 6 :

x r (t|x 0 r ) = Φ(t, t 0 )x 0 r , for t ∈ T . (8) From Equations ( 8) and ( 4), the swept-volume becomes:

X 0 T = x 0 r ∈ R 6 : ∃ t ∈ T , R 2 -x 0 T r Φ(t, t 0 ) T I 11 Φ(t, t 0 )x 0 r 0 , (9)
where the matrix I 11 ∈ R 6×6 is defined by I 11 := I 3 0 0 0 . This appears in the formula simply because only the positions (first 3 coordinates of x r (t|x 0 r )) are constrained to belong to X R . An observation made in [START_REF] Chan | Hovering collision probability[END_REF] is that when the last 3 coordinates of x 0 r are fixed (no velocity uncertainty), the above inequality describes an ellipsoid for each time t. Since this simplification is important in practice, we detail it in what follows.

A. No velocity uncertainty

Denote the initial state vector

x 0 r ∈ R 6 by x 0 r T := (r 0 T r , v 0 T r )
, where the relative velocity v 0 r ∈ R 3 is exactly known (not a random vector). Firstly, note that the integral in Equation ( 6 Assume that Φ 11 (t, t 0 ) is invertible for each t ∈ T . The swept-volume X 0 rT ∈ R 3 , containing all the relative positions r 0 r ∈ R 3 which lead to collisions during the time interval T , is described by a union of ellipsoids,

X 0 rT = t∈T E t,t0 , with E t,t0 := r 0 r ∈ R 3 : R 2 -(r 0 r -c(t, t 0 )) T Q(t, t 0 ) -1 (r 0 r -c(t, t 0 )) 0 , where c(t, t 0 ) = -Φ 11 (t, t 0 ) -1 Φ 12 (t, t 0 )v 0 r , Q(t, t 0 ) = Φ 11 (t, t 0 ) -1 Φ 11 (t, t 0 ) -T .

Remark 1 (The 3D swept-volume as a compact set):

Provided that the matrix Φ 11 (t, t 0 ) is invertible for each t ∈ T , each ellipsoid E t,t0 is proper i.e., Q(t, t 0 ) has full rank, and thus their union is compact. This is important for practical implementations which are developed in Sec. IV-C. Otherwise, the swept-volume can still be described by the union of non-necessary proper 3D quadratic forms.

B. Gaussian integral of union of semi-algebraic sets

Motivated by the previous description of the sweptvolume as a union of ellipsoids, one observes that in the general case, by taking a sufficiently fine discretization of size N ,

τ N := {t 0 • • • t i • • • t f }, the constraints describing subsets K i ⊆ X 0 T : Ki := {x 0 r ∈ R 6 : R 2 -x 0 T r Φ(ti, t0) T I11Φ(ti, t0)x 0 r 0}, (10)
provide an approximate description of X 0 T as a union of basic semi-algebraic sets (which are neither disjoint, nor compact in general):

K := N i=1 Ki ⊆ X 0 T . (11)
The main advantage of expressing the swept-volume as a union of basic semi-algebraic sets, is that it can be exploited in the framework of Polynomial Optimization [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF]. The simplified formulation of Pb. 1 reads: Problem 2 (Integration on union of semi-algebraic sets): Given a union of basic semi-algebraic sets as in Eq. ( 11), [START_REF] Henrion | Approximate volume and integration for basic semialgebraic sets[END_REF], and a Gaussian probability measure µ I compute the integral:

Pc (T ) := µ I (K) = K dµ I . ( 12
)
This problem has a theoretical interest of its own and has been already addressed in the literature [START_REF] Lasserre | Semidefinite relaxations for lebesgue and gaussian measures of unions of basic semialgebraic sets[END_REF] based on the measure-moments framework. In brief, it can be proven that Problem (2) is equivalent to an infinite-dimensional linear program on positive measures. Firstly, let us fix some necessary notations. Given a Borel set Ω ⊂ R n , let M(Ω) be the Banach space of finite signed Borel measures on Ω, equipped with the total variation norm. For µ, ν ∈ M(Ω), denote by supp(µ) the support of µ (i.e. the smallest closed set Γ of Ω s.t. |µ|(Ω \ Γ) = 0); denote µ ν when µ(A) ν(A) for any measurable set A ⊆ Ω; finally, a positive measure µ is denoted by µ 0. Proposition 2 (Thm. 3 [START_REF] Lasserre | Semidefinite relaxations for lebesgue and gaussian measures of unions of basic semialgebraic sets[END_REF]): The following problem has an optimal solution and κ * K = Pc (T ).

κ * K = sup µ1,...,µN ∈M(R 6 ) N i=1 µ i (K i ) , s.t. N i=1 µ i µ I , supp(µ i ) ⊆ K i , i = 1, . . . , N µ i 0, i = 1, . . . , N. (13)
Roughly speaking, this problem aims at maximizing the sum of masses of N positive measures µ i (with supp(µ i ) ⊆ K i ) whose sum is dominated by the known Gaussian measure µ I . Problem ( 13) is numerically solved via a monotone sequence of upper bounds converging to the optimal value of (13). Each upper bound is obtained by solving a truncated-moment problem similar to the one applied in [START_REF] Henrion | Approximate volume and integration for basic semialgebraic sets[END_REF] and [START_REF] Lasserre | Computing gaussian and exponential measures of semi-algebraic sets[END_REF] for the computation of respectively Lebesgue or Gaussian measures of a basic semi-algebraic set. The monotonic convergence, which is rather slow in general, may be significantly improved using a technique based on Stokes' formula. This requires to use a polynomial which vanishes on the algebraic boundary of the considered set [START_REF] Lasserre | Semidefinite relaxations for lebesgue and gaussian measures of unions of basic semialgebraic sets[END_REF]. Unfortunately this acceleration technique cannot be directly used here, due to the high number of sets K i involved. Indeed a polynomial that vanishes on the boundary of the union can have a potential very high degree, which results in intractable additional Stokes' constraints. Another limitation of directly solving [START_REF] Lasserre | Computing gaussian and exponential measures of semi-algebraic sets[END_REF] is that when its optimal value is very small, very high order relaxations are needed. The latter have a large size and are numerically ill-conditioned because of high values of some moments of the Gaussian to be computed. Hence, motivated also by the practical question of obtaining a closed-form representation of the sweptvolume, we chose instead another similar approach for Problem 2, with two main steps: (1) Find an implicit representation of the integration domain K by a PSS [START_REF] Dabbene | Simple approximations of semialgebraic sets and their applications to control[END_REF], [START_REF] Lasserre | Semidefinite relaxations for lebesgue and gaussian measures of unions of basic semialgebraic sets[END_REF]; (2) Compute the integral ( 12) with a high-order quadrature for volumes implicitly defined by a PSS [START_REF] Saye | High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles[END_REF].

IV. PSS APPROXIMATIONS OF THE SWEPT-VOLUME

It was shown in [START_REF] Dabbene | Simple approximations of semialgebraic sets and their applications to control[END_REF] (see also [START_REF] Henrion | Approximate volume and integration for basic semialgebraic sets[END_REF], [START_REF] Lasserre | Semidefinite relaxations for lebesgue and gaussian measures of unions of basic semialgebraic sets[END_REF] for similar works) that a union K = N i=1 K i ⊆ R n , of compact basic semialgebraic sets K i (given by a conjunction of polynomial inequalities), can be efficiently approximated by a PSS.

Let R[x] d be the vector space of polynomials in the variables x = (x 1 , . . . , x n ) over reals, of total degree at most d. Moreover, suppose that K can be outerbounded by a hyper-rectangle

K ⊆ B := [a, b] = {x ∈ R n , a i x i b i , for i = 1, . . . , n} , a, b ∈ R n . Definition 3 (PSS): A degree d-PSS approximation for K is defined by a polynomial p d ∈ R[x] d , s.t. K ⊆ PSS p d := {x ∈ B : p d (x) 1}. (14)
A first technical issue is that the swept-volume in Eq. ( 11) is not compact in general. An exception is the 3D case (no velocity uncertainty, cf. Remark 1) for which the method of [START_REF] Dabbene | Simple approximations of semialgebraic sets and their applications to control[END_REF] can be directly applied. For the 6D general case, a straightforward solution is to rely on the fact that the approximation computed for K is to be used afterwards for integrating the multivariate Gaussian density [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF]. Hence, one can consider a suitable h-σ ellipsoid corresponding to the given covariance matrix P I (say h = 8.5 in practice, to be tuned depending on numerical requirements) and bound it by a hyper-rectangle B. Then, one uses the sets K i = K i ∩ B, which are compact. With this additional approximation, we next proceed with a description of the method and algorithms adapted from [START_REF] Dabbene | Simple approximations of semialgebraic sets and their applications to control[END_REF], [START_REF] Lasserre | Semidefinite relaxations for lebesgue and gaussian measures of unions of basic semialgebraic sets[END_REF] to our case, to obtain a PSS of fixed degree d via semidefinite optimization.

A. PSS approximations of the 6D swept-volume

The polynomial optimization problem reads: Problem 3 (Approximate PSS for the swept-volume): Let the semi-algebraic set K = i=1,...,N K i be given by the union of N basic compact semi-algebraic sets K i , a given bounding hyper-rectangle B ⊇ K and also a fixed degree d. Solve the optimization problem

w * d,K = inf p∈R[x 0 r ] d p 1 = B p(x 0 r )dx 0 r , s.t. p 0 on B, p 1 on K 1 , . . . , p 1 on K N . (15)
The main result is the following (its proof is very similar to the one given in [8, Thm. 2]).

Theorem 1: The infimum in Problem ( 15) is attained for

a polynomial p * d,K ∈ R[x 0 r ] d . Moreover, PSS p * d,K ⊇ K, w * d+1,K w * d,K and lim d→∞ w * d,K = vol(K).
The polynomial p * d,K can be seen as an approximation of the indicator function 1 K of the set K. Such an approximation can be obtained by solving a convex optimization problem whose constraints are Linear Matrix Inequalities (LMIs). Moreover, as the degree of the approximation d increases, the sequence (p d ) d 1 converges in L 1 -norm, almost uniformly and almost everywhere to the indicator function of the set K of interest. This can be thought as a direct generalization of classical approximation by ellipsoids. Indeed if degree-2 PSS approximations are used, we exactly recover wellknown semi-definite optimization-based approaches. Note that Problem (15) has a dual infinite-dimensional linear problem on measures, namely, denoting the classical Lebesgue measure on B by λ B :

v * K = sup µ1,...,µN ∈M(B) N i=1 µ i K i , s.t. N i=1 µ i λ B , supp(µ i ) ⊆ K i , i = 1, . . . , N µ i 0, i = 1, . . . , N. (16)
In addition, v * K = lim d→∞ w * d,K = vol(K) [START_REF] Dabbene | Simple approximations of semialgebraic sets and their applications to control[END_REF]. Remark 2: It is very interesting to observe that Pb. ( 13) and ( 16) are very similar: the main difference resides only in the initial measure considered. For Pb. [START_REF] Lasserre | Computing gaussian and exponential measures of semi-algebraic sets[END_REF], one aims at computing the integral of a Gaussian measure µ I to a union of semi-algebraic sets K, while for Pb. [START_REF] Mckinley | Development of a nonlinear probability of collision tool for the Earth observing system[END_REF] the Lebesgue measure λ B is used instead, which simply corresponds to computing the integration of a union of semi-algebraic sets. We provide in what follows the basic details of the numerical computation of the solution of Problem [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF], which is more or less standard in the field of Polynomial Optimization [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF].

B. SOS relaxations of Problem (3)

Recall that the constraints of Problem (3) are:

-the polynomial p is positive on B, -p -1 is positive on K i , ∀ i = 1, • • • , N . A common strategy for enforcing positivity is by requiring the polynomial to be sum-of-squares (SOS). Let us denote the convex cone of real polynomials that are SOS by

Σ 2 [x] ⊂ R[x] and respectively, Σ 2 [x] 2k ⊂ R[x] 2k
, its subcone of SOS polynomials of degree at most 2k. Using Putinar's Positivstellensatz [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF], [START_REF] Lasserre | An Introduction to Polynomial and Semi-Algebraic Optimization[END_REF], Problem (15) becomes, when fixing ℓ ∈ N:

w * 2ℓ,d,K = inf p∈R[x 0 r ] d B p(x 0 r )dx 0 r , s.t.      p -σ0,B - 1 j 6 gj,B σj,B = 0 σ0,B ∈ Σ 2 [x 0 r ] 2ℓ , σj,B ∈ Σ 2 [x 0 r ] 2(ℓ-1) , ∀j = 1, • • • , 6,          p -σ0,K 1 -g1σ1,K 1 - 1 j 6 gj,Bσ1,j,B = 1 σ0,K 1 ∈ Σ 2 [x 0 r ] 2ℓ , σ1,K 1 ∈ Σ 2 [x 0 r ] 2(ℓ-1) , σ1,j,B ∈ Σ 2 [x 0 r ] 2(ℓ-1) , ∀j = 1, • • • , 6, . . .          p -σ0,K N -gN σN,K N - 1 j 6 gj,B σN,j,B = 1 σ0,K N ∈ Σ 2 [x 0 r ] 2ℓ , σN,K N ∈ Σ 2 [x 0 r ] 2(ℓ-1) , σN,j,B ∈ Σ 2 [x 0 r ] 2(ℓ-1) , ∀j = 1, • • • , 6 (17)
where the constraints defining the sets K i are:

K i := {x 0 r ∈ R 6 : g i (x 0 r ) 0, g j,B 0, j = 1, . . . , 6}, i = 1, . . . , N, (18)
and the polynomial g i is obtained from Equation [START_REF] Henrion | Approximate volume and integration for basic semialgebraic sets[END_REF],

g i (x 0 r ) := R 2 -x 0 T r Φ(t i , t 0 ) T I 11 Φ(t i , t 0 )x 0 r , i = 1, . . . , N, (19)
The polynomials g j,B form the constraints defining the hyper-rectangle B, Proof: The proof can be found in [START_REF] Lasserre | Semidefinite relaxations for lebesgue and gaussian measures of unions of basic semialgebraic sets[END_REF]. Let us briefly discuss the simplification obtained when the Gaussian uncertainty on the velocity can be neglected in the six dimensional relative dynamics (position, velocity), during the encounter time interval T .

g j,B (x 0 r ) := (x 0 r j -a j )(b j -x 0 r j ), j = 1, . . . , 6. (20

C. No velocity uncertainty

In this simplified setting, the swept-volume is described as a compact union of 3-dimensional ellipsoids according to Proposition 1. Algorithm 1 summarizes the computation of the PSS approximation in this case. Its correctness follows from Propositions 1 and 3. The only technicality (described for completeness in Lines 3 -4), resides in computing a bounding box B.

Algorithm 1 PSSAPPROX3D(τ N , Φ(t, t 0 ), v 0 r , R, d, l)
Input: time grid τN , Φ(t, t0) with invertible upper-left block for t ∈ τN , known initial relative velocities

v 0 r ∈ R 3 , radius R, degrees 2ℓ d, d 1. Output: p d ∈ R[x] d is a PSS approximation of the dis- cretized collision set {r 0 r ∈ R 3 : ∃t ∈ τN s.t. x 0 r = [r 0 T r v 0 T r ] T , x 0 T r Φ(t, t0) T I11Φ(t, t0)x 0 r R 2 } . ⊲ Define ellipsoids Et i ,t 0 := r 0 r ∈ R 3 : R 2 -(r 0 r -c(ti, t0)) T Q(ti, t0) -1 (r 0 r -c(ti, t0)) 0 1: c(ti, t0) ← -Φ11(ti, t0) -1 Φ12(ti, t0)v 0 r , for ti ∈ τN ; 2: Q(ti, t0) ← Φ11(ti, t0) -1 Φ11(ti, t0) -T , for ti ∈ τN ; ⊲ Find a bounding box Br := {x ∈ R 3 : a x b} 3: δ(ti, t0) ← diag 1 R 2 Q(ti, t0) , for ti ∈ τN ; 4: [a, b] ← min t i ∈τ N (c(ti, t0) -δ(ti, t0)), max t i ∈τ N (c(ti, t0) + δ(ti, t0)) ; ⊲ Solve the optimization problem 5: gt i ← R 2 -(x -c(ti, t0)) T Q(ti, t0) -1 (x -c(ti, t0)) for ti ∈ τN ; 6: g j,B 0 r ← (xj -aj) (bj -xj), for j = 1, 2, 3; w * 2ℓ,d = min p∈R[x] d B 0 r p(x)dx, s.t.      p -σ 0,B 0 r - 1 j 3 g j,B 0 r σ j,B 0 r = 0 σ 0,B 0 r ∈ Σ 2 [x 0 r ] 2ℓ , σ j,B 0 r ∈ Σ 2 [x 0 r ] 2(ℓ-1) , ∀j = 1, 2, 3,    p -σ0,t i -gt i σ1,t i = 1 σ0,t i ∈ Σ 2 [x 0 r ] 2ℓ , σ1,t i ∈ Σ 2 [x 0 r ] 2(ℓ-1) , . (21)
7: return p * 2ℓ,d = argmin [START_REF] Serra | Fast and accurate computation of orbital collision probability for short-term encounters[END_REF] For the general 6-dimensional case a similar algorithm can be designed, which solves Problem 17, with the ad-ditional requirement (and approximation) that a compact bounding box needs to be provided as input. Moreover, these algorithms are easily tractable in software.

D. Implementation Details

Firstly, the objective function [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF]. Finally, this boils down to solving only a semi-definite programming problem (whenever the degrees d and l are fixed), which was done with the Mosek SDP solver [START_REF] Mosek Aps | [END_REF].

B p(x)dx = 0 |i| d p i B x i dx

V. GAUSSIAN INTEGRATION ON THE SWEPT-VOLUME

While a compact description of the swept-volume as a PSS is interesting in itself, recall that the complete goal of Problem 2 was the integration of a Gaussian distribution on this swept-volume. To this end, besides the formulation of the optimization Problem (13), we have developed two different strategies that use the obtained PSS directly, and depend upon the assumption on velocity uncertainties: -The 3D case: Algorithm 1 returns a polynomial p * 2ℓ,d , which provides an implicit representation of the approximated volume. This is used as input for [20, Algorithm 3], which automatically determines a highorder accurate numerical quadrature for the evaluation of integrals over volumes, whose geometry is defined implicitly via a fixed level set of a smooth function φ : R 3 → R. Obviously in our case, φ = p * 2ℓ,d . -The 6D case: Similarly, after Problem 17 is solved for an optimal p * 2ℓ,d , the integral of a Gaussian distribution over the volume PSS p * 2ℓ,d has to be evaluated. Since the code of [START_REF] Saye | High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles[END_REF] is currently restricted to 3D, a basic Monte Carlo sampling is done, which consists in simply checking whether p * 2ℓ,d (X s,i ) 1 for each sample X s,i . This procedure is to be replaced by a 6D implementation of the algorithm in [START_REF] Saye | High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles[END_REF]. Moreover, to cross-check the validity and quality of the proposed methods, a brute-force Monte Carlo process was also designed. Given for completeness in Algorithm 2, it uses the very idea of the representation of the swept-volume as a union K of compact sets K i . It is simply based on testing whether an initial condition x 0 r ∈ R 6 leads to a collision in the time interval T , i.e. whether x 0 r belongs to one of the K i (i = 1, . . . , N ).

VI. NUMERICAL EXAMPLES

Two numerical examples borrowed from the paper by S. Alfano [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF] are used to illustrate the proposed approach. The Gaussian distribution of both the primary and secondary is known at TCA. A nonlinear two-body Keplerian analytical propagation is used for both the primary and secondary mean vectors, then the norm of the mean relative position (miss distance) is computed on the time interval of the encounter and plotted. The transition matrix Φ(•, t 0 ) is computed by performing a linearization of the Keplerian dynamics with respect to the primary object trajectory, following the classical algorithm of Shepperd [START_REF] Shepperd | Universal Keplerian State Transition Matrix[END_REF]. For completeness, the Gaussian distribution of the relative state at TCA (with or without velocity uncertainty) is propagated using the transition matrix on the time interval of the encounter. Then, the so-called instantaneous collision probability is plotted for both cases. This indicator is useful in practice since it shows the probability of collision at each given instant. The formula consists in computing a Gaussian integral over a 3D ball and semi-analytical efficient algorithms are available [START_REF] Serra | Fast and accurate computation of orbital collision probability for short-term encounters[END_REF]. Finally, the 3D sweptvolume shape and its PSS approximation are depicted.

A. Example 1

The first example is the case 7 from [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF], involving nonlinear relative motion for two satellites in Low Earth Orbits (LEO) where the mean miss distance at TCA is less than the hard-body radius and the relative velocity at TCA is low (0.19 m/s). This example was intended to test the sampling methods of the literature, since a low probability of collision is expected and a great number of Monte Carlo samples is needed to get an accurate result. Let us first check the impact of velocity samples gives a collision probability of 0.000158, confirming that we get a good approximation. Note that this result agrees with the ones presented in [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF] where the probability is evaluated as 0.000161 for a highly computationally demanding Monte Carlo approach (662 million Monte Carlo runs needed) and 0.000164 for the voxels method. Finally, the results are consistent with Coppola's formulation when assuming no velocity uncertainty (formula (40) of [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF]) which gives 0.000159.

B. Example 2

We consider the encounter [2, Example No. 9], of two objects in highly-eccentric orbits (HEO) where the mean miss distance at TCA is greater than the combined object radius. The computed instantaneous collision probability is given in Figure 3(b) and (c) with/without relative velocity uncertainty considered at TCA. Firstly, our method is applied assuming that there is no velocity uncertainty (the 3D case). In Figure 3 Finally, the code of [START_REF] Saye | High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles[END_REF]Algorithm 3] requires less than 10 seconds to evaluate the Gaussian integral over the PSS computed before, leading to Pc = 0.2825, which is confirmed by brute-force simulations. Formula (40) in [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF] gives 0.2715. As observed in Figure 3(b) and (c) and also confirmed by the results in [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF], the relative velocity uncertainty does matter in this case: Algorithm 2 provides a value similar to that of [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF], of 0.36511. Our 6D method, with d = 2l = 8, provides a PSS volume, whose projection in 3D, for the fixed mean velocity value at TCA is plotted in Figure 3(e). One observes an overestimation for this case. Depending on the h-σ bounding ellipsoid considered as bounding, the results obtained with our method range from 0.50 for 6-σ to 0.371 for 1-σ.

  ) becomes three-dimensional. Secondly, we are only interested in relative positions in the swept-volume X 0 T . Straightforward calculations lead to the following characterization. Proposition 1 (3D Swept-volume): Let the relative dynamics transition matrix Φ(t, t 0 ) be given and denoted by blocks by Φ := Φ 11 Φ 12 Φ 21 Φ 22 .

) Proposition 3 (

 3 Convergence of LMI hierarchy): For each fixed d ∈ N, the value of Problem (17) converges to w * d,K , as ℓ → ∞ and moreover, for any 2ℓ d, the solution p * 2ℓ,d,K of Problem (17) satisfies the constraints of Problem (15) i.e., PSS p * 2ℓ,d,K is a PSS approximation of K.

Algorithm 2

 2 BFVT(τ N , Φ(t, t 0 ), B, R, M, P I ) Input: A time grid τN , transition matrix Φ(t, t0), bounding box B ∈ R 6 , hardbody radius R, Gaussian samples no. M and corresponding covariance matrix PI . Output: approximative value of collision probability 1: Generate the Gaussian samples x 0 r j=1,...,M ∈ R 6 from the covariance matrix PI ; 2: C ← Cardinality of {x 0 r j ∈ B : ∃ i ∈ {1, . . . , N } s.t. gi(x 0 r j ) 0} 3: return C/M

Fig. 2 :

 2 Fig. 2: (a) Miss distance (m.) (b) Instantaneous probability of collision P(t) with (blue) or without velocity uncertainties (red) at TCA (c) 3D Swept-volume shape (no velocity uncertainty) (d) PSS approximation of degree d = 4.

  (d), the exact swept-volume (union of 40 ellipsoids obtained on the time interval T = [-10800, 10800]) is plotted. Algorithm 1, applied with d = 2l = 8, requires about 10 seconds (Matlab execution on a classical workstation) for obtaining the degree-8 PSS, plotted in Figure 3(e).

Fig. 3 :

 3 Fig. 3: (a) Miss distance (m) (b) Instantaneous collision probability P(t) with (c) and without (d) velocity uncertainty (d) 3D Swept-volume shape (no velocity uncertainty) (e) PSS approximation d = 8 (f) Projection of 6D PSS approx, d = 8, for a fixed mean velocity.

  In the implementation, a scaling of B, to the unit box [-1, 1] n is important for the numerical quality of the results. Also, in line with Remark 2, from a theoretical perspective, working with the Lebesgue or the Gaussian measure is similar. In practice, computing the Lebesgue moments on [-1, 1] n , offered the best quality numerical results. Secondly, the constraints can be recast in terms of Linear Matrix Inequalities (LMIs); this is already a classical strategy and several software tools are available to model problems of the form above, like for instance the Matlab Toolbox YALMIP

		,
	which is a linear function of the coefficients p i of the
	polynomial p, requires the computation of the Lebesgue
	moments m i :=	x i dx, which is straightforward.
	B Remark 3 (Box scaling):

Implicit means defined by an inequality constraint, as opposed to an explicit parametric representation of each point of the set.

-R 2 0}. (3)

VII. CONCLUSION

By using models from polynomial optimization, we have provided (i) a theoretical framework and (ii), practical algorithms to obtain approximate closed-form descriptions of the set of collision-prone states (and an approximation of their Gaussian measure). This allows efficient approximations of non-convex shapes which can be used effectively for long-term and repeated conjunctions, generalizing both Chan's and Coppola's formulations. We are confident that visual accurate outer-approximation of the swept volume can provide further insight on classifying the types of encounter occurring in practice. Along these lines, a further complexity and numerical analysis on the provided algorithms would help to better assess and design more tuned and efficient algorithms for special practical cases.

This work was supported by CNES D.


Pelin Sözen

Jean-Claude Daran

Eric Manoury

Funda Demirhan

Rinaldo Poli
email: rinaldo.poli@lcc-toulouse.fr

Reaction of [Cp*2W2O5] with mercaptocarboxylic acids: addition rather than reduction. Isolation and characterization of Cp*WO2(SCH2CH2COOH)

Keywords: Tungsten, oxido ligands, high-oxidation state organometallics, alkylsulfido ligands, pentamethylcyclopentadienyl ligands

The reaction of Cp*2W2O5 with HS(CH2)nCOOH (n = 1, 2) in MeOH or in CH2Cl2 solutions at room temperature proceeds in a slightly different way: for n = 2, the reaction selectively yields compound Cp*WO2(SCH2CH2CO2H), which has been isolated and characterized by elemental analysis, NMR and single crystal X-ray diffraction. The reaction is equilibrated, being shifted to the product by absorption of water by anhydrous Na2SO4 in CH2Cl2, and to the reactants by addition of water. Contrary to the Mo analogue, no products resulting from metal reduction are obtained. The corresponding reaction for n = 1 occurs similarly at low substrate/W ratios (<0.5), but proceeds further to several uncharacterized products for greater substrate amounts. The primary product could not be isolated, but its 1 H NMR spectrum suggests a different, asymmetric structure.

Introduction

High oxidation state organometallic chemistry has long attracted attention, particularly for its potential in oxidation catalysis [1][2][3][4]. We have explored for some time the chemistry of half-sandwich complexes of the heavier Group 6 metals Mo and W in high oxidation states [5][6][START_REF] Collange | Perspectives in Organometallic Chemistry[END_REF][START_REF] Gun | [END_REF][9][10][11][12][13] and recently reported on the activity of the dinuclear oxido complexes Cp*2M2O5 (M = Mo, W) as precatalysts in the oxidation of thiophene derivatives to sulfoxides and sulfones [14], and in cyclooctene epoxidation [15,16]. Most of the studies have been centered on the molybdenum system because of its more straightforward preparation, but we have recently developed an improved and convenient synthesis of the tungsten compound [17] and have thus renewed our attention to this system, since its catalytic performance in oxidation processes appears to be far superior to that of the Mo congener, at least under certain conditions [14,16].

A few years ago, we have reported on the interaction between [Cp*2Mo2O5] and mercaptocarboxylic acids, HS(CH2)nCOOH (n = 1, mercaptoacetic or thioglycolic acid [18]; n = 2, mercaptopropionic acid [19]), which resulted in the isolation and characterization of dinuclear Mo IV products, [Cp*2Mo2(-S)(-SCH2COO)2] in the first case and [Cp*2Mo2(-O)(-SCH2CH2COO)2] in the latter. One of the reasons for investigating the interaction of these substrates, in particular the thioglycolic acid, with the Mo VI complex was as model for the action of cysteine in molybdenum hydroxylase enzymes [20]. In these reactions, the mercaptocarboxylic acid exerts three simultaneous functions: reducing agent, ligand, and (limited to the n = 1 case) substrate for the reductive cleavage of the S-C bond, leading to the incorporation of the sulphur atom in the isolated product as a bridging sulfido ligand. We have now extended these studies to the analogous W system, since the bioinorganic chemistry of W VI with sulphur donor ligands is also of interest [21], and report the results of this investigation in the present contribution.

Experimental Section

General procedures. All preparations and manipulations were carried out with Schlenk techniques under an oxygen-free argon atmosphere. All glassware was oven-dried at 120°C. Solvents were dried by standard procedures and distilled under dinitrogen prior to use. 1 H and 13 C{ 1 H} NMR spectra were recorded on Bruker DPX 300 and ARX 250 instruments.

Chemical shifts are expressed in ppm downfield from Me4Si. Coupling constants are given in Hertz. The starting compound, [Cp*2W2O5], was prepared as described in the literature [17].

The compounds 3-mercaptopropionic acid (99%, ACROS) and 2-mercaptoacetic acid (98%, Fluka) were used as received.

Preparation of [Cp*WO2(SCH2CH2COOH)].

To a solution of Cp*2W2O5 (100 mg, 0.14 mmol) in 10 mL MeOH was added 3-mercaptopropionic acid (23 µL, 0.28 mmol). The solution stirred at room temperature for 24 hours. The reaction mixture was then evaporated to dryness leaving an oily pale-yellow residue, which was washed with 5x3 mL of pentane.

The product was then crystallized from CH2Cl2/pentane as white crystals. Yield: 96 mg (81. (b) In CH2Cl2. In a Schlenk tube, compound Cp*2W2O5 (177 mg, 0.246 mmol) was dissolved in CH2Cl2 (15 mL). To this solution was added anhydrous Na2SO4 (1.5 g) and subsequently 3-mercaptopropionic acid (40 L, 0.49 mmol). The solution was stirred at room temperature for 18 h, then the solid Na2SO4 was filtered off and the solution dried under vacuum. The residue was used for an NMR investigation in CDCl3, showing the complete conversion to compound Cp*WO2(SCH2CH2COOH) (NMR properties identical to the authentic sample described above). To this solution in the NMR tube, a drop of D2O was subsequently added and the solution monitored by 1 H NMR (see Results and Discussion section). (0.5 mL). Then thioglycolic acid was added in five incremental portions of 0.5 L (0,71•10 -2 mmol). After each addition, the progress of the reaction was monitored by 1 H NMR.

(b) In CDCl3. In a NMR tube, Cp*2W2O5 (13.1 mg, 1.82•10 -2 mmol) was dissolved in CDCl3 (0.5 mL) and to the resulting solution was added thioglycolic acid (2.5 L, 3.64•10 -2 mmol). 1 H NMR spectra of the resulting solution were recorded immediately, after 1 h 45 min and after 4 h (see Results and Discussion section).

X-ray diffraction study.

A single crystal of the compound was mounted under inert perfluoropolyether on the tip of glass fibre and cooled in the cryostream of the Oxford-Diffraction XCALIBUR CCD diffractometer. Data were collected using the monochromatic MoK radiation (= 0.71073). The structure was solved by direct methods (SIR97) [22] and refined by least-squares procedures on F 2 using SHELXL-97 [START_REF] Sheldrick | SHELXL97. Program for Crystal Structure refinement[END_REF]. All H atoms attached to carbon and oxygen (OH) were introduced in calculation in idealised positions and treated as riding on their parent atoms. The value of the Flack's parameter, 0.25 (1), indicates the occurrence of a twin by inversion. The drawing of the molecules was realised with the help of ORTEP32 [START_REF] Farrugia | [END_REF]. Crystal data and refinement parameters are shown in Table 1.

Crystallographic data (excluding structure factors) have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC 795729. Copies of the data can be obtained free of charge on application to the Director, CCDC, 12 Union Road, Cambridge CB2 1EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk). A similar NMR monitoring was also carried out in CDCl3, where an equilibrium conversion of ca. 40% was reached in 3 days. This shows a slight solvent effect both on the kinetics and the thermodynamics of the reaction. The reaction is a bit slower and the equilibrium slightly less displaced toward the product in CDCl3. The isolation of the pure product from the synthesis in MeOH in ca. 80% yield may be attributed to the less hydrated nature of the solvent relative to the CD3OD used in the NMR experiment.

In order to displace the equilibrium toward the product, anhydrous Na2SO4 was added to the reaction mixture in CH2Cl2. Indeed, a complete conversion could be achieved under these conditions according to NMR. Subsequent addition of water to the solution after filtration of Na2SO4 slowly regenerated the starting materials, see Figure 2. It is important to underline that the solution color during all these manipulation remained pale yellow, consistent with the fact that the metal maintains its d 0 configuration for the oxidation state +6. During the analogous reaction of the Mo congener, on the other hand, the metal reduction to Mo IV was accompanied by a color change to deep red [19]. NMR about its presence in the product.

Final confirmation of the binding mode was obtained from the X-ray study, see Figure 3. Besides this confirmation, the most interesting structural feature is the presence of an intermolecular H-bond between the carboxylic acid of one molecule and atom O2 of a second molecule, building infinite zig-zag chain parallel to the a axis. This is also reflected in the fact that the W-O2 bond is slightly longer than the W-O1 bond. According to the Cambridge Data Base, this is the first reported structure where W is surrounded by a "CpO2(SR)" ligand set, whereas only one is available for molybdenum, Cp*MoO2(StBu) [25].

(b) Thioglycolic acid

The analogous reaction of mercaptoacetic (or thioglycolic) acid with Cp*2W2O5 in a 2:1 ratio (S/W ratio of 1) revealed interesting differences relative to that of mercaptopropionic acid described in the previous section. The NMR monitoring in CDCl3 did not exhibit any major change, leaving a substantial amount of starting materials unreacted but revealing also a number of new Cp* peaks and a slight color deepening. This suggests an equilibrated and unselective process. The shape of the NMR spectrum was the same after immediate recording following mixing, and several hours later, showing that the equilibration is rapid. Since the related mercaptopropionic acid addition was found to be favored by methanol (see previous section), a second experiment was conducted in this solvent and found to lead to complete consumption of the starting materials, but once again at least 4 different Cp*-containing products were obtained (resonances at  2.55, 2.52, 2.34, 2.30 (major peak), and 2.25) and the solution color darkened.

In order to learn more about this transformation, the 1 H NMR monitoring experiment in CD3OD was repeated with stepwise addition of substoichiometric amounts of substrate (total S/W ratios of 0.23, 0.46, 0.70, 0.93, 1.16, and 1.40 equivalents, respectively), see results in Figure 4. After the first and second addition, only the major product with the Cp* resonance at  2.30 was formed. The amount of this product approximately doubled from the first to the second addition. After the third addition, while the depletion of Cp*2W2O4 continued to be linear, the  2.30 resonance increased less than proportionally and the other by-products started to form. Note that the interpolation of the starting material data cuts the axis at an S/W ratio of 1, indicating the essentially quantitative nature of the transformation in CD3OD. The slope observed for the growth of the major product is identical in value and opposite in sign to the slope of the starting material decrease, showing that this main product has an S/W stoichiometry of 1:1. The lack of disappearance of the starting material beyond the 1:1 stoichiometry indicates that the minor by-products are characterized by a higher S/W stoichiometry. After the addition of 1.4 equivalents of the substrate, the combined minor products remained under 20% of the total according to integration of the Cp* peaks, whereas the main product reached a maximum of 67% and slightly less than 20% of the starting compound remained unreacted. The NMR monitoring was not pursued for higher S/W ratios. However, on the basis of the evidence that the by-products are likely resulting from the reducing action of the substrate (deepening of the solution color, higher S/W stoichiometry)

we run an experiment with a larger excess of thioglycolic acid (10 equivalents) at the methanol reflux temperature. This experiment let to a deep green colored final solution, the 1 H NMR of which showed the complete consumption of the starting material but many products were again observed (same Cp* resonances observed in the NMR titration experiments, see above), with the major product being that with 2.34. The resonance of the primary 1:1 addition product was not visible in this product mixture. Unfortunately, all our efforts to crystallize one or more of the resulting products were unsuccessful. The stronger color that develops, however, suggests that one or more of these by-products derive from a reduction process. Any further speculation on their molecular constitution is unjustified.

Another interesting observation concerns the structure and symmetry of the first product obtained at low S/W ratios. The absence of a visible SH resonance suggests, like for the mercaptopropionic acid reaction described above, that the substrate has reacted with the metal atom via the SH function, presumably to yield a similar product to that obtained with the mercaptopropionic acid. The methylene resonance is downfield shifted by nearly 1 ppm relative to the free substrate, a similar shift also experienced by the mercaptopropionic substrate (see above). However, the CH2 resonance in the product is visible as an AB pattern (JHH = 18.1 Hz) rather than as a singlet (see Figure 5), clearly indicating that the two methylene protons are inequivalent and hence that the structure is asymmetric. Therefore, a structure related to that of Cp*WO2(SCH2CH2COOH) shown in Figure 3 cannot be adopted by this compound. A tentative formulation for this product is proposed in Scheme 2. The aptitude to form a metallacycle would presumably be higher in the thioglycolate derivative because of the formation of a more favored 5-membered ring. Clearly, the conversion of this structure to the symmetric Cp*WO2(SCH2COOH) structure similar to that of the mercaptopropionato derivative, if energetically accessible, must be slow on the NMR time scale in order to maintain the inequivalence of the methylene protons. Unfortunately, all our attempts to crystallize this product, in order to confirm its structure, were also unsuccessful. Another interesting aspect of the reactions reported in this contribution is the fact that a hard ligand, the bridging oxido ligand in Cp*2W2O5, is exchanged by a softer ligand, the thiolate. Although the reaction is not quantitative, it proceeds to high conversions at equilibrium with the production of water. This is an unusual behaviour for the oxophilic W VI center. Other related W VI O2(SR) systems, notably the hydrotris(3,5-dimethylpyrazolyl)borate complex Tp*WO2(SPh) which may be prepared by salt metathesis under strictly anhydrous conditions, reacts with hard donors such as phenol, methanol, and formic acid to yield Tp*WO2(OPh), Tp*WO2(OMe) and Tp*WO2(O2CH), respectively [32]. The latter transformation is particularly relevant, because the reactions reported here implicate the direct competition of SH and COOH functions for addition to W VI , with a clear preference for the former. Hard-for-soft ligand exchange may also account for the moisture sensitivity of other dioxo-W VI S-donor complexes such as (phen)WO2(SPh)2 [26], WO2(S2CNR2)2 [33], and [WO2(bdt)2] 2- [34]. The reason for this switch of preference between a hard and a soft donor ligand is probably related to the presence of the soft Cp* ligand, leading to a "softening" of the tungsten Lewis acidity. Hence, the use of the Cp*W VI system has opened a new and simpler route for the assembly of sulphur-containing W VI complexes that are of potential interest as models of W-containing biological systems.

Conclusions

1 H

 1 NMR monitoring of the reaction between Cp*2W2O5 and HSCH2CH2COOH. (a) In CD3OD. In an NMR tube, compound Cp*2W2O5 (10 mg, 1.4•10 -2 mmol) was dissolved in CD3OD (0.5 mL) and to the resulting solution was added 3-mercaptopropionic acid (2.3 L, 2.8•10 -2 mmol). 1 H NMR spectra of the resulting solution were taken immediately, after 4 hours and after 1 day (see Results and Discussion section).

1 H

 1 NMR monitoring of the reaction between Cp*2W2O5 and HSCH2COOH. (a) In CD3OD. In a NMR tube, Cp*2W2O5 (11.1 mg, 1.54•10 -2 mmol) was dissolved in CD3OD

Figure 1 . 1 H

 11 Scheme 1

Figure 2 . 1 H

 21 Figure 2. 1 H NMR spectra of compound CpWO2(SCH2CH2COOH) in CDCl3: (I) anhydrous solution; (II) immediately after addition of D2O; (III) 24 h after addition of D2O. Refer to Scheme 1 for the peak assignments.

  and 2.607(3) Å[29]. In all the other above-cited molecules, the SR donors are mutually trans. Hence, the trans effect of the Cp* ligand is weaker than those of the SR and O ligands.

Figure 3 .

 3 Figure 3. An ORTEP view of the CpWO2(SCH2CH2COOH) molecule with the numbering scheme used. The ellipsoids are drawn at the 50% probability level. Relevant distances (Å): W-CNT (Cp* ring centroid), 2.094(4); W-S1, 2.3476(15); W-O1, 1.723(4); W-O2, 1.734(2); C3-O31, 1.319(7); C3-O32, 1.208(6). Relevant angles (°): CNT-W-S1, 105.8(6); CNT-W-O1, 119.5(6); CNT-W-O2, 116.22(10); S-W-O1, 103.78(15); S-W-O2, 103.42(18); O1-W-O2, 106.3(2). H-bonding interaction

Figure 4 .

 4 Figure 4. Relative amounts (in moles of W) of starting compound, Cp*2W2O5 (diamonds), main product (squares), and the sum of all other products (triangles) as a function of the S/W ratio.

Figure 5 .

 5 Figure 5. Expansion of the CH2 resonance of the thioglycolate ligand in the major product obtained from Cp*2W2O5 and thioglycolic acid.

Scheme 2 (

 2 Scheme 2

Table 1 .

 1 Selected crystallographic and refinement parameters for compound [Cp*WO2(SCH2CH2COOH)].

	Identification code	Cp*WO2(SCH2CH2COOH)
	Empirical formula	C13 H20 O4 S W
	Formula weight	456.20
	Temperature	180(2) K
	Wavelength	0.71073 Å
	Crystal system	Orthorhombic
	Space group	P n a 21
	Unit cell dimensions	a = 15.223(3) Å	= 90°.
		b = 8.8749(18) Å	= 90°.
		c = 11.337(2) Å	 = 90°.
	Volume	1531.6(5) Å 3
	Z	4
	Density (calculated)	1.978 Mg/m 3
	Absorption coefficient	7.685 mm -1
	F(000)	880
	Crystal size	0.462 x 0.318 x 0.231 mm 3
	Theta range for data collection	2.66 to 30.03°.

Acknowledgements

We thank the Centre National de la Recherche Scientifique (CNRS), the Institut Universitaire de France (IUF) and Celal Bayar University (FEF 2006-114) for support of this work.

Contrary to Cp*2Mo2O5, the W congeners reacts with HS(CH2)nCOOH (n = 1, 2) to yield observable substrate addition products with displacement of water in equilibrated processes. This can certainly be attributed to the lower susceptibility of the heavier metal to undergo reductive processes. Oxo-bridge splitting is more favourable for the S-H function to yield an alkylthiolato derivative than for the COOH function to yield a carboxylato derivative.

The unusual and unexpected preference of Cp*W VI for a soft donor (RS) vs. a hard one (RCOO) may be attributed to a "softening" effect of the Cp* ligand. It is quite possible that the compounds obtained for the W system represent models of the first generated intermediates in the corresponding reaction of the Mo system.

Graphical Abstract:

Pictogram

Synopsis

Contrary to Cp*2Mo2O5, the tungsten analogue is not reduced by the thiol function in mercaptocarboxylic acid reagents, leading to equilibrated addition processes with release of water. The addition of the SH function yielding an alkylthiolato derivative is more favourable than addition of the COOH function yielding a carboxylato derivative.


Prof R Poli
email: rinaldo.poli@lcc-toulouse.fr

Generation and reactivity of [SiP iPr

3]M(N2) complexes.

Ruthenium and Osmium Metalloradicals

Rinaldo Poli*

Keywords: Ruthenium • Osmium • Dinitrogen • Radicals• Non-innocent ligands

With a limited number of well-understood exceptions (the square planar d 8 configuration, complexes with strongly -donating ligands, …), complexes of transition metals that belong to the 4d and 5d series most typically follow the so-called EAN rule by adopting a closed-shell 18-electron configuration. Relative to the 3d metals, those of the 4d and 5d series possess more diffuse and more polarizable d orbitals, leading to greater bond covalence and to lower pairing energies and consequently to lower stability for openshell configurations, especially in the lower oxidation states and especially for the late transition metals. [1] Coordinative saturation is achieved by coordination of a suitable number of 1-and 2-electron ligands and/or by forming metal-metal bonds in oligonuclear clusters. Nevertheless, enforcing mononuclear metalloradical configurations through the use of specially designed ligands can lead to peculiar properties and reactivity. A good example is the use of bulky porphyrins or geometrically constrained alkene-containing ligands leading to stable (porphyrin)Rh II complexes or (alkene)Rh II and (alkene)Ir II complexes that have shown interesting applications in C-H activation and in catalysis. [2,3] For the heavier Group 8 elements ruthenium and osmium, complexes with the formal oxidation state I and a metalloradical character were previously produced, generally by electrochemical methods, only as unstable species or could not be isolated. Pilloni et al. already pointed out in 1977 that the coordination geometry could be an important factor allowing the generation of stable d 7 mononuclear complexes and that the presumed lability of the chloride ligand in complex [Ru(dppp)2Cl] (dppp = 1,3bisdiphenylphosphinopropane) opens decomposition pathways through disproportionation. [4] A longer-living but still not isolable Ru I complex, [Ru(PP3)Cl], [5] was obtained by using the "Sacconitype" tetradentate phosphine ligand PP3 = P(CH2CH2PPh2)3, which is known to stabilize M(L4) moieties either in an octahedral (with two additional cis ligands) or in a trigonal bipyramidal (with one additional axial ligand) geometry. A key to further stabilize the construct could be the incorporation of the anionic function into the tetradentate ligand to yield a M(XL3) moiety. On the basis of this idea, Peters has recently developed tris(phosphino)silyl ligands, the relationship with the Sacconi system being illustrated in Scheme 1, initially showing that they are capable of stabilizing dinitrogen complexes of Fe I , [(SiP R 3)Fe(N2)], with R = Ph or iPr. [6][7][8] Scheme 1. Relationship between Sacconi-type XP3 and Peter's

[SiP R 3] -ligands.
More recently, the same ligand (with R = iPr) has successfully been used to generate the first stable mononuclear Ru I and Os I metalloradicals (see Scheme 2). [9] The precursor complexes [(SiP iPr 3)MCl] were prepared from [( 6 -C6H6)MCl2]2 and HSiP iPr 3 in the presence of Et3N and then converted into [(SiP iPr 3)M(N2)] (M = Ru, 1; Os, 2) by reduction with KC8 under dinitrogen.

Scheme 2. Generation and reactivity of [SiP iPr

3]M(N2) complexes of ruthenium and osmium.

The spectroscopic (IR, NMR, EPR) and structural (single crystal X-ray diffraction, see Figure 1 for 1) characterization leaves no doubt about the formulation as mononuclear M I species. The N2 ligand is labile and could be substituted by PMe3 yielding the corresponding [(SiP iPr 3)M(PMe3)] complexes (M = Ru, 3; Os, 4), which have also been fully characterized. The two dinitrogen complexes were also investigated electrochemically, revealing a reduction and an oxidation. The reduction process is reversible for both compounds and the stable M 0 products could be generated by stoichiometric reductions with KC8, isolated and crystallographically characterized. The oxidation process is reversible only for Ru and the stable product, containing however a very labile N2 ligand, could be obtained by stoichiometric oxidation with [Cp2Fe][B{C6H3(CF3)2-3,5}4]. The radical character at the metal is indicated by the chemical reactivity for compound 1, i.e. by the generation of the Ru II complexes [(SiP iPr 3)RuH(N2)], [(SiP iPr 3)RuI] and [(SiP iPr 3)Ru(SPh)] upon treatment with Bu3SnH, I2 and PhSSPh, respectively. However, two-electron reactivity is also shown, in addition to the above described transformation to 3 and 4, by the reaction with organoazides, leading to imido/nitrene products via elimination of N2 (Scheme 2). The EPR investigation and DFT calculations for compounds 1-4 give strong support to the description of the complexes as bona fide metalloradicals. The EPR spectrum shows significantly anisotropic rhombic tensors with larger (gmax-gmin) values than typically observed for ligand-centered radicals, while the calculations place the majority of the spin density (76% for 1, 69% for 2, 84% for 3 and 79% for 4) on the metal and also a non negligible fraction at the P nuclei (Figure 1). The observed P hyperfine splitting is consistent with the unpaired electron being located in the equatorial plane. Another recent study by Berry [10] has shown that the Ru 0 complexes ( 6 -toluene)Ru[N3] ([N3] = 2,6-diiminopyridine), where [N3] is only coordinated in a bidentate fashion, are converted under dinitrogen to the diamagnetic dinuclear {[N3]Ru}2(-N2) complexes 5-7, Scheme 3, where the coordination geometry of the Ru center is strictly square planar, an unusual observation for a formally Ru 0 center. In light of the known non-innocent character of 2,6diiminopyridine ligands and on the previous recognition that related [N3]FeCl and [N3]Fe(N2)2 complexes are best described as Fe II complexes with transfer of one or two electrons to the ligand, [11,12] the electronic structure of compounds 3-5 has been carefully analyzed. Structural evidence (ligand imino C=N distances) and an unusual temperature independent shift of the imine Me proton resonance in the 1 H NMR spectrum suggest a certain degree of ligand non-innocence. Replacement of the N2 ligand in compound 5 by other neutral donors (C2H4, PMe3, CO) yields 5-coordinate complexes [N3 xyl ]RuLL' (Scheme 3) that seem adequately described as Ru 0 derivatives. On the other hand, the addition of H2 affords an unusual dinuclear Ru hydride complex {[N3 xyl ]RuH}2(-N2), 8 (see Scheme 3), the structure of which shows two square pyramidal Ru centers with an axial hydrido ligand, linked by a linear N2 bridge. [13] This is formally a Ru I species and is indeed paramagnetic as expected for one unpaired electron per [N3 xyl ]Ru(H) unit, but displays no tendency to further add H2 to yield a dihydrido Ru II product. Broken symmetry DFT calculations on a mononuclear [N3 xyl ]Ru(H)(N2) model complex reveal a large contribution of the Ru II /[N3] -formalism, with 73% SOMO localization on the [N3] ligands and only 19% on the metals. In conclusion, the one-electron reactivity of 5 with H2 appears more consistent with a formal transformation of Ru I to Ru II rather than Ru 0 to Ru I .

In conclusion, the two above-highlighted elegant studies by Peters and Berry have underlined for the first time that relatively stable complexes of ruthenium and osmium in the oxidation state I may be obtained by judicious choice of the supporting ligand system, but also that internal electronic rearrangements are possible when the ligand has the ability to play a non innocent role. They have clearly shown the need to carefully apply a vast array of experimental and theoretical characterization methods for achieving an appropriate description of the electronic structure for these interesting new systems. [9,10,13] The preliminary reactivity studies reported in these contributions already announce that they are amenable to the development of interesting new chemistry. Note that a recent review on ruthenium complexes with non innocent ligands has appeared, [14] although it makes no mention of systems best described as having a Ru center with a formal oxidation state of I.

Figure 1 .

 1 Figure 1. X-ray structure and spin density plot for compound 1.

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Metalloradicals

Unprecedented stability for ruthenium(I) and osmium(I) complexes with metalloradical character has been achieved by use of judiciously chosen supporting ligands based on the tris(phosphino)silyl motif, but pronounced radical character on Ru I is also enforced by non-innocent ligands of diiminopyridine type.

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