Alexandre Stathopulos
email: alexandre.stathopulos@cea.fr

Stefan Skupin

Luc Bergé

A Raman-delayed nonlinearity for elliptically polarized ultrashort optical pulses

Keywords: nonlinear optics, ultrashort laser pulses, terahertz pulse generation

Introduction

The propagation of femtosecond, intense laser pulses in air has important applications for active or passive remote sensing, ultrafast Lidar or artificial lightning. Ultrashort laser pulses with high enough peak power undergo unique interactions with the atmosphere in which both linear and nonlinear mechanisms play a key role. In particular, laser filaments produced by ultrashort light pulses proceed from the dynamic balance between Kerr self-focusing and plasma defocusing [START_REF] Chin | The propagation of powerful femtosecond laser pulses in optical media: Physics, applications, and new challenges[END_REF][START_REF] Bergé | Optical ultrashort filaments in weakly-ionized, optically-transparent media[END_REF][START_REF] Kasparian | Physics and applications of atmospheric nonlinear optics and filamentation[END_REF]. The interplay of these nonlinear effects broadens the pulse spectrum, promotes self-compression [START_REF] Bergé | Few-cycle light bullets created by femtosecond filaments[END_REF] and may lead to the well-known self-guiding of intense light waves over long (meter-range) distances [START_REF] Kasparian | White-light filaments for atmospheric analysis[END_REF]. Filamentation of pulses composed of more than one carrier frequency has moreover been proposed as an innovative way to downconvert optical radiation into the THz range [START_REF] Wang | High energy terahertz emission from two-color laser-induced filamentation in air with pump pulse duration control[END_REF] and create broadband THz sources remotely [START_REF] Liu | Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases[END_REF].

Interest in terahertz (THz) radiation is motivated by its numerous applications in the fields of physics, chemistry, biology and engineering. Since the advent of ultrafast light sources, laser-induced plasmas have been one of the most promising means to provide ultrabroadband, intense THz waves. However, water vapor considerably attenuates THz wave propagation in air, which limits the efficiency for atmospheric remote sensing. A current challenge for air THz photonics is thus to overcome the strong absorption caused by humidity over meter-range distances. Therefore, the demand for intense THz radiation sources, able to deliver electric fields reaching the GV/m level, is in constant increase. An important constraint for absorption spectroscopy is also to have broadband spectra in order to measure as many spectral signatures as possible [START_REF] Bergé | Terahertz spectroscopy from air plasmas created by two-color femtosecond laser pulses: The ALTESSE project[END_REF].

An efficient method to generate intense and broadband THz radiations consists in focusing two-color ultrashort infrared light pulses composed of fundamental and second harmonic frequencies into the air. If the laser intensity exceeds the medium ionization threshold ∼ 10 13-14 W cm -2 , an electron plasma is created and behaves like a nonlinear frequency converter that produces, through photocurrents, low-frequency emissions belonging to the THz domain [START_REF] Kim | Coherent control of terahertz supercontinuum generation in ultrafast lasergas interactions[END_REF][START_REF] Thomson | Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications[END_REF][START_REF] Babushkin | Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases[END_REF]. Operated in the filamentation regime, extended two-color plasma channels can help produce THz pulses remotely [START_REF] Bergé | 3D numerical simulations of THz generation by two-color laser filaments[END_REF], offering a way to overcome water absorption in air. Because THz radiation is also emitted by optical rectification through four-wave mixing [START_REF] Cook | Intense terahertz pulses by four-wave rectification in air[END_REF], Kerr self-focusing may contribute to the overall THz yield [START_REF] Borodin | Transformation of terahertz spectra emitted from dual-frequency femtosecond pulse interaction in gases[END_REF][START_REF] Andreeva | Ultrabroad terahertz spectrum generation from an air-based filament plasma[END_REF]. At clamping intensities > 50 TW/cm 2 from which Kerr self-focusing is stopped by plasma defocusing, THz emission is, nonetheless, dominated by photocurrents and a peak in the THz frequency range appears in the pulse spectrum.

Several techniques exist to increase the THz yield by photocurrents, such as using longer FH wavelengths [START_REF] Clerici | Wavelength scaling of terahertz generation by gas ionization[END_REF][START_REF] Nguyen | Wavelength scaling of terahertz pulse energies delivered by two-color air plasmas[END_REF], increasing the number of colors [START_REF] De Alaiza Martínez | Boosting terahertz generation in laser-field ionized gases using a sawtooth wave shape[END_REF][START_REF] Vaičaitis | Terahertz radiation generation by three-color laser pulses in air filament[END_REF] or even modifying the polarization state of the FH and SH components [START_REF] Dai | Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma[END_REF][START_REF] Meng | Enhancement of terahertz radiation by using circularly polarized two-color laser fields[END_REF]. In this respect, Ref. [START_REF] Meng | Enhancement of terahertz radiation by using circularly polarized two-color laser fields[END_REF] experimentally evidenced from helium gas jets that circularly polarized two-color pulses with same helicity (CP-S) could deliver ∼ 5 times higher THz powers than their linearly polarized, parallel (LP-P) counterparts. This increase in the THz power was attributed in [START_REF] Tailliez | Terahertz pulse generation by twocolor laser fields with circular polarization[END_REF] to a direct dependency of the drift velocity acquired by photo-ionized electrons on the dominant FH pump amplitude and longer ionization sequences.

Several important issues in the underlying physics still need to be addressed, such as the role of the Raman-delayed part of the Kerr nonlinearity, which arises due to the excitation of transitions in the molecular constituents of air [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF]. To our knowledge, there is no formulation for such Raman-delayed response suitable for laser pulses with arbitrary polarization. Stimulated rotational Raman scattering (SRRS) of a laser pulse in air is a quantum mechanical process involving the excitation of the rotational states of air molecules by the laser pulse [START_REF]Nonlinear Optics[END_REF]. It can be characterized as an instability that scatters laser energy into multiple Stokes and anti-Stokes frequency bands which, because of the dispersive properties of air, propagate at different velocities and at large angles with respect to the input beam, causing a severe distortion of the laser envelope [START_REF] Champeaux | Postionization regimes of femtosecond laser pulses selfchanneling in air[END_REF].

The present paper aims at clarifying the impact of the Kerr nonlinearities, instantaneous and delayed, on the filament dynamics for twocolor pump pulses being either linearly or circularly polarized as well as on THz pulse generation. After deriving the expression of the complete nonlinear polarization for vectorial laser fields, we display numerical evidence that SRRS decreases the THz generation process over long propagation distances [START_REF] Nguyen | Spectral dynamics of thz pulses generated by two-color laser filaments in air: the role of kerr nonlinearities and pump wavelength[END_REF], even in filamentation regimes where the photocurrent mechanism is the dominating THz emitter. Compared with LP-P pulses, CP-S pulses are found to drive an efficient THz gain over meter distances after the onset of selffocusing while keeping lower clamping intensity and plasma density in the presence of SRRS. These results are confirmed by direct simulations employing the unidirectional pulse propagation equation (UPPE) [START_REF] Kolesik | Nonlinear optical pulse propagation simulation: From Maxwell's to unidirectional equations[END_REF] and they are justified by means of the local current (LC) model [START_REF] Babushkin | Tailoring terahertz radiation by controling tunnel photoionization events in gases[END_REF].

The paper is organized as follows: Section 2 specifies the input laser-gas parameters under consideration and recalls the vectorial model integrated by our unidirectional solver. This section also presents the derivation for the nonlinear Raman response of air molecules based on a microscopic approach. Section 3 addresses our numerical results and discusses the role of SRRS in laser filamentation for linearly or circularly polarized pulses in the presence of a vectorial Ramandelayed nonlinearity. The changes in the laserto-terahertz efficiency generated by photocurrents versus the optical nonlinearities for different pump pulse polarizations are justified through LC evaluations.

The nonlinear responses

We consider two-color input Gaussian pulses with elliptical polarization composed of fundamental (FH, ω 0 ) and second harmonic (SH, 2ω 0 ),

⃗ E L (t) = E 0 exp -2 ln 2 t 2 τ 2 0 - x 2 + y 2 w 2 0 (1) ×   √ 1-r √ 1+ρ 2 sin(ω 0 t) + √ r √ 1+ρ 2 sin(2ω 0 t + ϕ) -ρ √ 1-r √ 1+ρ 2 cos(ω 0 t) -ρ √ r √ 1+ρ 2 cos(2ω 0 t + ϕ)   ,
with FWHM pulse duration τ 0 = 50 fs, spatial width w 0 = 0.5 mm, FH wavelength λ 0 = 2πc/ω 0 = 1.6 µm; r ≃ 0.2 denotes the fraction of second harmonic, ϕ is the phase difference between the two colors and ρ is the ellipticity comprised between 0 and 1. When ρ = 0, the laser is linearly polarized with both components parallel (LP-P); when ρ = 1 the pump is circularly polarized with same helicity in both colors (CP-S) [START_REF] Meng | Enhancement of terahertz radiation by using circularly polarized two-color laser fields[END_REF].

To understand the effect of the pump polarization onto laser filaments, we review the nonlinear source terms of the medium. It is well-known that femtosecond filaments proceed from local balances between Kerr self-focusing and defocusing due to a tenuous self-generated electron plasma. The plasma response of weakly ionized gases with neutral density N a is classically modeled by the free electron rate equation

∂ t N e = W (E)(N a -N e ), (2)
where N e denotes the electron density, W (E) is the ionization rate depending on the modulus of the vectorial electric field. Triggering sufficient ionization requires high enough intensity, typically larger than 50-100 TW/cm 2 , fostering thus the tunnel ionization regime. This rate is ussually given by the instantaneous rate from Ammosov-Delone-Krainov (ADK) theory [START_REF] Thomson | Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications[END_REF][START_REF] Ammosov | Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field[END_REF] that reduces for hydrogenoid atoms to the wellknown quasi-static tunneling (QST) rate [START_REF] Landau | Quantum Mechanics[END_REF]:

W (E) = 4(U i U -1 H ) 5/2 ν au |E(t)E -1 au | e -2 3
(U i U -1 H ) 3/2 |E(t)E -1 au | , (3)
where U H = 13.6 eV denotes the ionization potential of hydrogen; E au = 514 GV/m and ν au = 41.3 PHz. Here, we assume that only oxygen is ionized starting from the neutral density N a = 5.4 × 10 18 cm -3 (20% of air compounds) and having the lowest ionization potential U i = 12.1 eV. At moderate intensities < 10 15 W/cm 2 , the electron current density ⃗ J is given by the cold-plasma kinetic equation [START_REF] Kim | Coherent control of terahertz supercontinuum generation in ultrafast lasergas interactions[END_REF]:

∂ t ⃗ J + ν c ⃗ J = e 2 m e N e ⃗ E, (4)
where e and m e are the electron charge and mass, respectively, and ν c = 2.85 ps -1 denotes the electron-neutral collision rate.

The optical polarization vector contains linear ( ⃗ P L ) and nonlinear ( ⃗ P NL ) contributions. The Fourier transform of ⃗ P L involves the first-order frequency-dependent susceptibility χ (1) (ω) entering the optical linear index n(ω) = [1+χ (1) (ω)] 1/2 . For centro-symmetric media, ⃗ P NL involves the third-order susceptibility χ (3) responsible for four-wave mixing. For molecular gases, the Kerr response admits a fraction x K of delayed contribution due to Raman scattering by rotational molecular transitions. Because the laser is not resonant with the transition frequencies [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF], SRRS affects the total time-dependent refraction index of the medium through a delayed-Kerr nonlinearity. The instantaneous Kerr nonlinearity

⃗ P inst (t) = ϵ 0 χ (3) E 2 (t) ⃗ E(t)
, where ϵ 0 is the permittivity in vacuum, is associated with self-phase modulation and nonlinear self-focusing. SRRS, by contrast, leads to the generation of multiple Stokes and anti-Stokes waves that scatter the laser energy. Along this process, a portion of the laser energy is absorbed to excite ro-vibrational transitions of atmospheric molecules, mainly dinitrogen.

In the following we generalized the derivation proposed in [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF][START_REF] Sprangle | Propagation of intense short laser pulses in the atmosphere[END_REF] to the case of a vector laser pulse. Details can be found in appendix A. Our expression of SRRS for vectorial pulses reads as

⃗ P R (t) = ϵ 0 χ (3) x K 2 +∞ 0 R(τ ) ⃗ F ( ⃗ E) dτ , (5)
⃗ F ( ⃗ E) = [3E 2 x (t -τ ) + E 2 y (t -τ )]E x (t) [3E 2 y (t -τ ) + E 2 x (t -τ )]E y (t) , (6)
where 0 ≤ x K ≤ 1 denotes the SRRS fraction with respect to the instantaneous Kerr response and R(τ ) is the delayed response function

R(τ ) = sin τ τ 1 τ 2 1 + τ 2 2 τ 1 τ 2 2 e -τ τ 2 . (7)
Here, τ 1 and τ 2 represent the rotational Raman and dipole dephasing times, respectively. In air those quantities take the values τ 1 ≈ 62.5 fs and τ 2 ≈ 77 fs [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF][START_REF] Pitts | Propagation of self-focusing laser pulses in atmosphere: experiment versus numerical simulation[END_REF]. The overall nonlinear polarization thus expresses as

⃗ P NL (t) = ϵ 0 χ (3) (1 -x K )E 2 (t) ⃗ E(t) (8)
+ x K 2 +∞ 0 R(τ ) ⃗ F ( ⃗ E) dτ .
In our UPPE code, the Raman polarization is recast as

⃗ P R (t) = ϵ 0 χ (3) x K 2 τ 2 1 + τ 2 2 τ 2 1 τ 2 2 (3Q x + Q y )E x (3Q y + Q x )E y , (9)
with Q x and Q y satisfy the differential equations

d 2 Q x dt 2 + 2 τ 2 dQ x dt + τ 2 1 + τ 2 2 τ 2 1 τ 2 2 Q x = E 2 x (t) , ( 10
)
d 2 Q y dt 2 + 2 τ 2 dQ y dt + τ 2 1 + τ 2 2 τ 2 1 τ 2 2 Q y = E 2 y (t) . ( 11
)
Note that the temporal variations of the second members E 2

x and E 2 y are much faster than those caused by the natural frequency of the equations, because ω 0 ≫ 2π/τ 1 . Therefore, the fast oscillations of the laser field are averaged out during a period of slow variation of Q x and Q y . The Raman polarization is thus expected to be mainly highfrequency, and in particular barely contributing to the THz part of the spectrum.

UPPE Simulations

We perform (3 + 1)-dimensional numerical computations based on a vectorial version of the unidirectional pulse propagation equation (UPPE) that governs the forward-propagating transverse electric field components E x , E y of elliptically polarized pulses. Following [START_REF] Kolesik | Nonlinear optical pulse propagation simulation: From Maxwell's to unidirectional equations[END_REF][START_REF] Kolesik | Unidirectional optical pulse propagation equation[END_REF], our nonlinear propagation model reads as (see also [START_REF] Tailliez | Terahertz pulse generation by twocolor laser fields with circular polarization[END_REF])

∂ z ⃗ E = i k 2 (ω) -k 2 ⊥ ⃗ E + i µ 0 ω 2 2k(ω) ⃗ F NL , (12)
where ⃗ E(k x , k y , z, ω) is the Fourier transform of the transverse laser electric field components with respect to x, y, and t, and k ⊥ = k 2

x + k 2 y . The first term on the right-hand side of Eq. ( 12) describes linear dispersion and diffraction of the pulse. The term

⃗ F NL = ⃗ P NL + i ⃗ J/ω + i ⃗ J loss /ω
contains the third-order nonlinear polarization ⃗ P NL given by Eq. ( 8) with Kerr index n 2 ≃ 3χ (3) /4cϵ 0 , the electron current ⃗ J following Eq. ( 4), and a loss term [START_REF] Bergé | Optical ultrashort filaments in weakly-ionized, optically-transparent media[END_REF][START_REF] Bergé | Few-cycle light bullets created by femtosecond filaments[END_REF][START_REF] Tailliez | Terahertz pulse generation by twocolor laser fields with circular polarization[END_REF]. Simulations of two-color laser filaments have been performed employing LP-P or CP-S pump pulses, with or without Raman-delayed nonlinear polarization. Our input laser pulses are given by Eq. ( 1), with an initial phase difference ϕ = π. Dispersion of air is accounted for by the refractive optical index n(ω) taken from [START_REF] Peck | Dispersion of air[END_REF]. With n 2 = 3.8 × 10 -19 cm 2 /W [START_REF] Zahedpour | Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths[END_REF], the critical power for self-focusing, defined by P cr ≃ λ 2 0 /2πn 2 , is P cr = 10.7 GW at 1.6 µm. With 4.43 mJ pulse energy, our beam thus contains 15.5 critical powers. As shown in [START_REF] Bergé | 3D numerical simulations of THz generation by two-color laser filaments[END_REF], purely (unperturbed) Gaussian pulses do not directly decay into multiple filaments even for input powers much larger than critical. When SRRS is taken into account, we apply the fraction x K ≃ 0.8 as measured in [START_REF] Zahedpour | Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths[END_REF]. The radiated field operating in the THz domain is extracted from a frequency window with cutoff frequency ν co = 90 THz. Our simulations use a time window of 0.8 ps (1.25 THz spectral resolution) and a transverse numerical box of 3 × 3 mm 2 . The numerical resolution steps are ∆t = 98 attoseconds in time and ∆x = ∆y = 12 µm in transverse space. Such a large spatial step is acceptable at 1.6 µm as the filament width increases like λ 0 , and the plasma channel width like √ λ 0 [START_REF] Champeaux | Postionization regimes of femtosecond laser pulses selfchanneling in air[END_REF]. The maximum laser intensity has input value I 0 ≈ 30 TW/cm 2 in LP-P configuration and I 0 ≈ 20 TW/cm 2 in CP-S.

⃗ J loss = [W (E)(N a - N e )U i /E 2 ] ⃗ E due to ionization

Results

Figure 1(a) illustrates the peak electron density reached along meter-range distances in two-color filamentation regime for LP-P (blue curves) or CP-S (red curves) polarization whenever an instantaneous nonlinearity only (x K = 0, dashed curves) or the Raman-delayed nonlinearity (x K = 0.8, solid curves) is included. With a purely instantaneous nonlinearity, laser filaments selffocus at about z c ≈ 10 cm where they reach ∼ 150 TW/cm 2 clamping intensity before relaxing to ∼ 50 -100 TW/cm 2 from z ≥ 50 cm (not shown). With SRRS, the self-focusing distance is delayed to z c ≃ 25 -30 cm from which the clamping intensity does not exceed 50 -60 TW/cm 2 . The electron yield follows the same tendency as shown by Fig. 1(a) and exhibits comparable dynamics for both polarization states along the filament range. From this figure it is clear that SRRS, by weakening the contribution of the instantaneous Kerr response, results in a longer self-focusing distance and an enhanced self-guiding range.

Figure 1(b) details the THz energy yield. SRRS, by promoting lower plasma densities, preserves better the laser energy. LP-P pulses undergo the strongest losses through photoionization. Without SRRS (dashed curves), due to a sharp ionization stage in early propagation, the LP-P pulse produces the larger THz energy, which is later followed by a high THz burst promoted by the CP-S pulse as the latter reaches similar ionization levels. Note that the THz energy can escape early our numerical box, which explains why the THz energy curves drop at larger propagation distances. With a smoother self-focusing response due to SRRS (solid curves), the expected hierarchy CP-S > LP-P in terms of THz gains [START_REF] Meng | Enhancement of terahertz radiation by using circularly polarized two-color laser fields[END_REF][START_REF] Tailliez | Terahertz pulse generation by twocolor laser fields with circular polarization[END_REF] is clearly refound over the first meter of filamentation. We thus confirm the superiority in the THz yield, by a factor close to 2, in the filamentation regime when using circularly polarized pulses compared to linearly polarized ones in realistic air accounting for SRRS, up to fluctuations in the peak electron density at the end of the filament range.

Figure 2 depicts the evolution of the optical and THz spectra along the filamentary propagation. With no SRRS, supercontinuum generation through which all optical frequencies merge to form a plateau of harmonics starts around z = 10 cm and is achieved 40 cm further. For comparison, a similar plateau is formed with SRRS from z > 180 cm. One can also observe that SRRS tends to inhibit the growth of high-order harmonics in circular polarization. By contrast, THz emission triggered by CP-S pulses takes over that produced with LP-P pulses from z ≥ 50 cm, in agreement with Fig. 1

(bottom).

Finally Fig. 3 displays some temporal profiles of laser fields undergoing plasma-induced distortions along propagation. Typically, the absence of SRRS favors sharper defocusing of the trailing edge of the pulse.

LC estimates

To understand the above THz gain performances, we can use the Local Current (LC) model [START_REF] Tailliez | Terahertz pulse generation by twocolor laser fields with circular polarization[END_REF][START_REF] Babushkin | Tailoring terahertz radiation by controling tunnel photoionization events in gases[END_REF], based on the approximation that the nonlinear currents are produced locally, ignoring thus propagation effects. According to [START_REF] Jefimenko | Electricity and Magnetism: An Introduction to the Theory of Electric and Magnetic Fields[END_REF] the radiated field is then proportional to

⃗ E rad ∝ ∂ ⃗ J ∂t + ∂ 2 ⃗ P NL ∂t 2 . ( 13
)
In order to quantify the suitability of a laser pulse to produce terahertz radiation, we estimate the terahertz energy yield as

η THz = +∞ -∞ E 2 THz (t) dt , (14)
where E THz is the radiated field taken in the THz frequency window ν < ν co = 90 THz. Figure 4(a) illustrates the LC contribution of the current density ⃗ J (thus photocurrents) to the THz yield at constant ionization level versus the pump ellipticity ρ, cf. Eq. ( 1). The most efficient polarization configuration is, as expected, that corresponding to CP-S pulses (ρ = 1). For comparison Fig. 4(b) shows the LC contribution of the nonlinear polarization ⃗ P NL when the Raman response Eq. ( 5) is taken into account or not. As expected (see also [START_REF] Nguyen | Spectral dynamics of thz pulses generated by two-color laser filaments in air: the role of kerr nonlinearities and pump wavelength[END_REF]), the influence of the nonlinear polarization (and SRRS) on the THz energy yield is negligible, whatever the pulse polarization may be. Its only relevant role remains in strongly modifying the filamentation dynamics and related spectral evolution through which THz pulse generation may be kept efficient over longer propagation distances.

Conclusion

In summary we have derived an expression for the stimulated rotational Raman scattering applied to elliptical (vectorial) laser pulses. The main influence of SRRS is to retard the occurrence of self-focusing and thereby plasma generation. With circularly polarized pulses, plasma is shifted to larger z distances compared with linearly polarized pumps, as circular polarisation promotes lower maximum laser fields (by a factor 1/ √ 2). The present work evaluates the efficiency of linear and circular polarization in the generation of THz radiation along the filamentation regime for a realistic air response taking SRRS into account. It appears that, although the THz Fig. 4 THz energy yields as function of ρ computed from the LC approach Eqs. ( 13) and ( 14) for (a) the photocurrent source and (b) the nonlinear optical response with a purely instantaneous Kerr nonlinearity (x K = 0, blue curve) and with the Raman-delayed Kerr effect (x K = 0.8, black curve). Note the six-order-of magnitude difference between the yields in (a) and (b). The two-color laser field is characterized by r = 0.2, ϕ = π and 100 TW/cm 2 intensity for ρ = 0. With increasing ρ, the intensity is adjusted such that the ionization yield remains constant.

energy yield delivered by laser filaments is lowered by Raman scattering in air, the longer filament range reached by smoothing the entire Kerr response allows an efficient THz wave generation over longer propagation ranges. Our simulations demonstrate that THz wave generation in filamentation regime is still reinforced by the use of circularly polarized pulses that promote distortions in the pulse spectrum to the benefit of the THz bandwidth.

Appendix

The microscopic Raman response

Here we derive the nonlinear polarization vector due to Raman scattering in air molecules, assumed to be only due to the rotations of non-interacting dinitrogen molecules. We shall suppose that N 2 molecules have only three energy levels W 1 , W 2 and W 3 . The first two levels are rotational states attainable by Stockes scattering, and the third is a virtual level of energy much higher than the other two, such that it cannot be directly excited by the laser, i.e., W 3 ≫ W 2 -W 1 . We note also Ω 1 , Ω 2 and Ω 3 the respective natural frequencies of these states and Ω nm = Ω n -Ω m . In view of the above hypotheses, one has Ω 21 ≪ ω 0 ≪ Ω 32 .

Let us define ψ(t, ⃗ r) the molecular wave function of dinitrogen. We also note H 0 the unperturbed Hamiltonian and

V (t, ⃗ r) = -⃗ µ • ⃗ E = -µ x E x -µ y E y = V x + V y
the perturbation operator of the Hamiltonian, with ⃗ µ = -e⃗ r the dipole moment vector of N 2 molecules. The Schrödinger equation is therefore written:

(H 0 + V x + V y )ψ = iℏ ∂ψ ∂t . ( 15
)
The molecular wave function ψ is decomposed into the sum of eigenstates of H 0 , u n (⃗ r), n being an integer between 1 and 3, ℏΩ n their associated eigenvalues, and ⟨u m |u n ⟩ = δ mn , with δ mn being the Kronecker symbol. Following [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF], we substitute

ψ(t, ⃗ r) = 3 n=1 c n (t)e -iΩnt u n (⃗ r) (16)
into Eq. ( 15) and treat separately each of its contributions defining (⃗ µ • ⃗ E) mn ≡ µ x mn E x + µ y mn E y with µ x,y mn = ⟨u m |µ x,y |u n ⟩. The time dependent coefficients c n must then satisfy

∂c m (t) ∂t = i ℏ 3 n=1 c n (t)(⃗ µ • ⃗ E) mn e -iΩnmt . (17)
The microscopic polarization vector ⃗ p is defined as

⃗ p(t) = ⟨ψ|⃗ µ|ψ⟩ = 3 m=1 3 n=1 ρ nm e iΩmnt µ x mn µ y mn , (18)
with ρ mn = c m c * n . By noticing that ρ * nm = ρ mn , µ x,y mn = (µ x,y nm ) * and (⃗ µ• ⃗ E) mn = (⃗ µ• ⃗ E) * nm , one gets

∂ρ nm ∂t = i ℏ 3 j=1 ρ jm (⃗ µ • ⃗ E) nj e -iΩjnt -ρ nj (⃗ µ • ⃗ E) jm e -iΩmj t . (19)
Similar assumptions as in [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF], among which ρ 33 = µ x,y 12 = µ x,y ii = 0 (1 ≤ i ≤ 3), give that the non-zero µ x mn (resp. µ y mn ) are all equal and assimilate all the coefficients (⃗ µ

• ⃗ E) mn to (⃗ µ • ⃗ E).
Let us now calculate the elements participating in the macroscopic dipole moment. By using the relation Eq. ( 18), we obtain ⃗ p(t) = (ρ 13 e iΩ31t + ρ 32 e -iΩ32t )⃗ µ + c.c.

where ρ 13 and ρ 32 are governed by Eq. [START_REF] Vaičaitis | Terahertz radiation generation by three-color laser pulses in air filament[END_REF].

To integrate the two equations for ρ 13 and ρ 32 , we neglect the time variations of ρ

∂ρ 12 ∂t = - i(⃗ µ • ⃗ E) 2 ℏ 2 Ω 31 e -iΩ21t . (23)
Following [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF], we add an heuristic population decrease of ρ 12 , related to a decay term with lifetime τ 2 . Introducing τ 1 = Ω -1 21 , we get

∂ρ 12 ∂t = - ρ 12 τ 2 - i(⃗ µ • ⃗ E) 2 ℏ 2 Ω 31 e -i t τ 1 . (24)
Using these expressions, we can simplify the expression for the nonlinear part of the polarization vector oriented along μ = ⃗ µ/µ to

⃗ p R (t) = 2µ 2 ℏΩ 31 (μ • ⃗ E)ρ 12 e iΩ21t μ + c.c. , (25)
and finally obtain after solving Eq. ( 24)

⃗ p R (t) = 1 ℏ 2µ 2 ℏΩ 31 2 [μ • ⃗ E(t)] (26)
× t -∞ [μ • ⃗ E(t ′ )] 2 sin t -t ′ τ 1 e t ′ -t τ 2 dt ′ μ .
Finally let ξ be the angle between ⃗ µ and ⃗ e x . To obtain a macroscopic expression of the Raman polarization, it is necessary to average the above expression over all angles ξ and multiply the expression by the molecular density of the medium N a . For isotropic air, the angle ξ is distributed uniformly between 0 and 2π. The macroscopic Raman polarization is thus given by

⃗ P R (t) = N a 2π 2π 0 ⃗ p R (t, ξ)dξ . (27)
After integration we obtain the final form of the Raman polarization Eq. ( 5) for a vector laser field. Note that if we set E y = 0, we recover the standard one-dimensional expression of [START_REF] Peñano | Stimulated raman scattering of intense laser pulses in air[END_REF] for a linearly polarized laser pulse.

Fig. 1

 1 Fig. 1 (a) Peak electron density without SRRS (x K = 0) (dashed curves) and with SRRS (x K = 0.8) (solid curves) for two-color Gaussian pulses in linear polarization (blue curves) or in circular polarization (red curves). (b) Corresponding THz energy in the numerical box (see text).

Fig. 2

 2 Fig. 2 Optical spectra with THz spectra as insets along different propagation distances: (a) z = 10 cm, (b) z = 50 cm and (c) z = 2 m. Blue (red) curves refer to LP-P (resp. CP-S) pulses. Solid (dashed) refer to accounting for (resp. discarding) SRRS.

Fig. 3

 3 Fig. 3 Optical electric fields Ex (blue curves, left-hand side axis) and Ey (red curves, left-hand side axis) with corresponding density profiles (black curves, right-hand side axis) along time at z = 50 cm for (a,b) LP-P and (c,d) CP-S pulses: (a,c) without and (b,d) with Raman effect.

  11 , ρ 12 , ρ 21 and ρ 22 compared to those of ρ 13 and ρ 32 , because Ω 31 ≫ Ω 21 and Ω 31 ∼ Ω 32 . With initially ρ 13 (t = 0) = ρ 32 (t = 0) = 0, this integration results in ℏΩ 31 ρ 12 e iΩ31t + ρ 22 e iΩ32t . (22) Since ρ 11 + ρ 22 ≃ 1 and state 2 remains sparsely populated compared to state 1, we can also assume ρ 11 -ρ 22 ≃ 1. ρ 12 is thus given by Eq. (19) as

	ρ 13 (t) =	(⃗ µ • ⃗ E) ℏΩ 31	ρ 11 e -iΩ31t + ρ 12 e -iΩ32t , (21)
	ρ 32 (t) =	(⃗ µ • ⃗ E)

Acknowledgments. The authors acknowledge the Agence Nationale de la Recherche (ANR -ALTESSE: ANR-19-ASMA-0007). Numerical simulations were performed using resources at Grand Équipement National De Calcul Intensif (GENCI) (A0100507594). S. Skupin thanks the Qatar National Research Fund (NPRP 12S-0205-190047) for support.


Jessy Abou Nakad

Remy Rajapaksha

Kai Chung Szeto

Aimery De Mallmann

Mostafa Taoufik
email: mostafa.taoufik@univ-lyon1.fr

Selective Non-Oxidative Dehydrogenation of Propane

published or not. The documents may come

Introduction

Propylene is an important intermediate in the chemical industry and is used for the synthesis of several key products such as polypropylene, acrylonitrile and propylene oxide. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF][START_REF] Hu | State-of-the-Art Catalysts for Direct Dehydrogenation of Propane to Propylene[END_REF] It is mainly produced by steam cracking (SC) and fluid catalytic cracking (FCC) of hydrocarbon feedstocks. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF][START_REF] Akah | Maximizing Propylene Production via FCC Technology[END_REF] . With the increasing demand for propylene globally, SC and FCC will be unable to satisfy the market need.

Hence, alternative routes such as on-purpose technologies have been developed, like olefin metathesis, [START_REF] Mazoyer | Production of Propylene from 1-Butene on Highly Active "Bi-Functional Single Active Site" Catalyst: Tungsten Carbene-Hydride Supported on Alumina[END_REF][START_REF] Mazoyer | High Selectivity Production of Propylene from 2-Butene: Non-Degenerate Pathways to Convert Symmetric Olefins via Olefin Metathesis[END_REF][START_REF] Mazoyer | Study of Ethylene/2-Butene Cross-Metathesis over W-H/Al2O3 for Propylene Production: Effect of the Temperature and Reactant Ratios on the Productivity and Deactivation[END_REF] methanol to olefin process, [START_REF] Svelle | Conversion of Methanol to Alkenes over Medium-and Large-Pore Acidic Zeolites: Steric Manipulation of the Reaction Intermediates Governs the Ethene/Propene Product Selectivity[END_REF] and more recently, the direct CO2 conversion to light olefins [START_REF] Liu | Selective Transformation of Carbon Dioxide into Lower Olefins with a Bifunctional Catalyst Composed of ZnGa2O4 and SAPO-34[END_REF][START_REF] Pawelec | Catalysts for the Conversion of CO2 to Low Molecular Weight Olefins-A Review[END_REF] as well as catalytic propane dehydrogenation. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF][START_REF] Nawaz | Light Alkane Dehydrogenation to Light Olefin Technologies: A Comprehensive Review[END_REF] The latter uses catalysts that can operate under oxidative or non-oxidative conditions. The main issue of the oxidative dehydrogenation is the low selectivity to olefins as CO2 can easily be produced from a total oxidation of light hydrocarbons.

Non-oxidative dehydrogenation of propane (PDH) is widely used at industrial scale as it offers many advantages, such as a high selectivity toward the desired alkene (propylene) and the production of valuable dihydrogen. This reaction is highly endothermic (ΔH° = 124 kJmol -1 ) and requires high operating temperatures (above 550 °C), resulting in undesired side-reactions (cracking) and deactivation of the catalyst due to the formation of coke and degradation/restructuration of the active phase. Two PDH processes dominate the market, known as Oleflex and Catofin. The first one employs supported hetero-bimetallic particles based on a noble metal Pt, alloyed with toxic Sn as a promoter, to enhance the selectivity to propylene, and to avoid cracking reactions and the fast deactivation of the catalyst at high temperature. However, despite the presence of promoters, high operating temperatures lead to a slow deactivation as a function of time on stream, which requires frequent regeneration treatments in order to maintain the catalytic performances. The Catofin technology employs a cheap catalyst based on chromium oxides (typically 18-20 wt% Cr) supported on alumina with working temperatures ranging from 575 to 625 °C at reduced pressures (0.2 to 0.5 bar). The high temperature required to improve catalytic activity toward the desired product, leads inevitably to the formation of coke. Consequently, the regeneration cycles occurring every 15-30 min, decrease the activity, and therefore a replacement of the catalyst every two years is needed. In addition to coke formation, the process conditions lead to aging of the catalyst. This causes: i) a decrease of the dispersion of chromium species on the surface; ii) an incorporation of the Cr active species within alumina, resulting in an increase of the catalyst's acidity that reduces propylene selectivity; iii) an increase of the surface Cr 6+ /Cr 3+ ratio after each regeneration cycle by calcination. [START_REF] Fridman | Deactivation Studies of the CrOx/Al2O3 Dehydrogenation Catalysts under Cyclic Redox Conditions[END_REF] To overcome these challenges, other metal oxides such as VOx, GaOx, InOx or MoOx have been widely explored in the literature for PDH. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF] Supported vanadium oxides catalysts have been intensively studied for the oxidative dehydrogenation of propane and showed promising results, while few examples were reported in the case of the non-oxidative propane dehydrogenation. Vanadium(V) oxide supported on silica, alumina or titanium oxide shows promising activities for the catalytic dehydrogenation of light alkane. [START_REF] Chalupka | Mononuclear Pseudo-Tetrahedral V Species of VSiBEA Zeolite as the Active Sites of the Selective Oxidative Dehydrogenation of Propane[END_REF][START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF][START_REF] Rossetti | Catalysts Prepared by Flame Pyrolysis for the Oxidative Dehydrogenation of Propane to Propylene[END_REF][START_REF] Cheng | Structure-Activity Relationships for Propane Oxidative Dehydrogenation by Anatase-Supported Vanadium Oxide Monomers and Dimers[END_REF][START_REF] Karakoulia | Propane Oxidative Dehydrogenation over Vanadia Catalysts Supported on Mesoporous Silicas with Varying Pore Structure and Size[END_REF][START_REF] Routray | Oxidative Dehydrogenation of Propane on V2O5/Al2O3 and V2O5/TiO2 Catalysts: Understanding the Effect of Support by Parameter Estimation[END_REF] The conventional synthesis methods of these catalysts described in literature involve various processes such as aqueous impregnation, sol-gel, chemical vapor deposition, to name a few. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF] However, depending on the nature of the vanadium precursor and the support used, these synthesis methods of supported vanadium oxo complexes, lead to a distribution of monomeric, dimeric and oligomeric VOx species on the surface or even to particles of oxide. [START_REF] Carrero | Critical Literature Review of the Kinetics for the Oxidative Dehydrogenation of Propane over Well-Defined Supported Vanadium Oxide Catalysts[END_REF][START_REF] Wegener | Design Strategies for the Molecular Level Synthesis of Supported Catalysts[END_REF] Several studies in this area have been reported and suggest that the structure of the supported precursor of the active species, for example onto silica, is postulated as an isolated vanadium center bearing an oxo and three siloxy ligands. In the case of vanadium systems, it is reasonable to propose that the active species is V 3+ trisiloxy (Scheme 1) obtained during the initiation step, after an activation of propane on V 5+ oxo tripodal species [START_REF] Wu | Effective Synthesis of Vanadium-Doped Mesoporous Silica Nanospheres by Sol-Gel Method for Propane Dehydrogenation Reaction[END_REF] , similar to chromium and gallium catalytic systems. Scheme 1. Proposed tripodal V (III) active species for PDH Based on DFT calculations and Operando studies, the mechanism of propane dehydrogenation in the case of metal oxides has been proposed to occur via a heterolytic C-H bond cleavage via activation on a surface M-O-bond, leading to a coordinated alkyl, M-R and a surface hydroxyl, -OH or via a σ-bond metathesis on a M-H intermediate. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF][START_REF] Kaphan | Mechanistic Aspects of a Surface Organovanadium(III) Catalyst for Hydrocarbon Hydrogenation and Dehydrogenation[END_REF] The low concentration of active sites hampers their characterization and thus limited information is available on their role in catalysis and the understanding of the elementary steps. In fact, many attempts to characterize the surface-active sites of supported VOx species, prepared by conventional methods, using spectroscopic techniques for bulk systems such as XAFS and [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V MAS-NMR led to inconsistent results. [START_REF] Muylaert | Supported Vanadium Oxide in Heterogeneous Catalysis: Elucidating the Structure-Activity Relationship with Spectroscopy[END_REF][START_REF] Nielsen | The Complete 51V MAS NMR Spectrum of Surface Vanadia Nanoparticles on Anatase (TiO2): Vanadia Surface Structure of a DeNOx Catalyst[END_REF][START_REF] Izumi | Structure of Low Concentrations of Vanadium on TiO2 Determined by XANES and Ab Initio Calculations[END_REF][START_REF] Kozlowski | X-Ray Absorption Fine Structure Investigation of Vanadium(V) Oxide-Titanium(IV) Oxide Catalysts. 2. The Vanadium Oxide Active Phase[END_REF][START_REF] Wolke | Surface Organometallic Chemistry of Vanadium(V): Reactivity of (ButN )VNp3 towards Silica[END_REF][START_REF] Ruitenbeek | The Structure of Vanadium Oxide Species on γ-Alumina; an in Situ X-Ray Absorption Study during Catalytic Oxidation[END_REF][START_REF] Weckhuysen | Situ Raman Spectroscopy of Supported Transition Metal Oxide Catalysts: 18O2-16O2 Isotopic Labeling Studies[END_REF][START_REF] Tanaka | X-Ray Absorption (EXAFS/XANES) Study of Supported Vanadium Oxide Catalysts. Structure of Surface Vanadium Oxide Species on Silica and γ-Alumina at a Low Level of Vanadium Loading[END_REF] An obvious strategy to access to the real model of the active species and to increase the activity in PDH, is to control the number of active sites and to eliminate presumably inactive oligomeric VOx sites. Regarding the preparation of well-defined functional materials at a molecular scale, especially single-site catalysts for PDH, surface organometallic chemistry has proven to be a powerful approach. This methodology involves a controlled grafting of a reactive organometallic precursor on a pretreated support. With a judicious choice of the precursor in combination with the pre-treatment of the support, supported species with a desired podality, oxidation state and coordination environment around the metal center can be designed. [START_REF] Pelletier | Catalysis by Design: Well-Defined Single-Site Heterogeneous Catalysts[END_REF] This approach has resulted in highly active catalysts associated with the propylene chemistry, mainly through olefin metathesis, [START_REF] Mazoyer | Production of Propylene from 1-Butene on Highly Active "Bi-Functional Single Active Site" Catalyst: Tungsten Carbene-Hydride Supported on Alumina[END_REF][START_REF] Mazoyer | High Selectivity Production of Propylene from 2-Butene: Non-Degenerate Pathways to Convert Symmetric Olefins via Olefin Metathesis[END_REF][START_REF] Mazoyer | Study of Ethylene/2-Butene Cross-Metathesis over W-H/Al2O3 for Propylene Production: Effect of the Temperature and Reactant Ratios on the Productivity and Deactivation[END_REF][START_REF] Popoff | Expanding the Scope of Metathesis: A Survey of Polyfunctional, Single-Site Supported Tungsten Systems for Hydrocarbon Valorization[END_REF][START_REF] Mazoyer | Development of the First Well-Defined Tungsten Oxo Alkyl Derivatives Supported on Silica by SOMC: Towards a Model of WO3/SiO2 Olefin Metathesis Catalyst[END_REF][START_REF] Bouhoute | Well-Defined Supported Mononuclear Tungsten Oxo Species as Olefin Metathesis Pre-Catalysts[END_REF][START_REF] Espinas | Preparation and Characterization of Metallacalixarenes Anchored to a Mesoporous Silica SBA-15 LP as Potential Catalysts[END_REF][START_REF] Garron | Improved Direct Production of 2,3-Dimethylbutenes and 3,3-Dimethylbutene from 2-Methylpropene on Tungsten Hydride Based Catalysts[END_REF][START_REF] Popoff | Design and Application of a Hybrid Material Featuring Well-Defined, Tuneable Grafting Sites for Supported Catalysis[END_REF][START_REF] Popoff | Enforcing Z-Selectivity in Olefin Metathesis through Use of Catalysts Grafted on Well-Defined Phenolic Hybrid Material[END_REF][START_REF] Szeto | Metallacyclobutane Substitution and Its Effect on Alkene Metathesis for Propylene Production over W-H/Al2O3: Case of Isobutene/2-Butene Cross-Metathesis[END_REF] but also for PDH. [START_REF] Szeto | A Strong Support Effect in Selective Propane Dehydrogenation Catalyzed by Ga(i-Bu)3 Grafted onto γ-Alumina and Silica[END_REF][START_REF] Szeto | Vanadium Oxo Organometallic Species Supported on Silica for the Selective Non-Oxidative Dehydrogenation of Propane[END_REF][START_REF] Love | Elucidation of Anchoring and Restructuring Steps during Synthesis of Silica-Supported Vanadium Oxide Catalysts[END_REF][START_REF] Högerl | SOMC Grafting of Vanadium Oxytriisopropoxide (VO(O i Pr) 3 ) on Dehydroxylated Silica; Analysis of Surface Complexes and Thermal Restructuring Mechanism[END_REF][START_REF] Barman | Single-Site VOx Moieties Generated on Silica by Surface Organometallic Chemistry: A Way To Enhance the Catalytic Activity in the Oxidative Dehydrogenation of Propane[END_REF][START_REF] Schweitzer | Propylene Hydrogenation and Propane Dehydrogenation by a Single-Site Zn2+ on Silica Catalyst[END_REF][START_REF] Hu | Isolated FeII on Silica As a Selective Propane Dehydrogenation Catalyst[END_REF] Therefore, we intend to apply surface organometallic chemistry for the synthesis of a silica material containing supported trisiloxy vanadium oxo species as pre-catalyst for propane dehydrogenation reaction. We recently developed a new synthetic method for the preparation of supported active sites, by using a molecular complex of group VI bearing tert-butoxide ligands [M(O)(OtBu)4] (M = Mo or W). After a grafting reaction of these complexes onto silica followed by a thermolysis, [(≡SiO)2M(O)2] species were obtained and showed high activities for the metathetic oxidation or the metathesis of olefins. [START_REF] Quéméner | Metathetic Oxidation of 2-Butenes to Acetaldehyde by Molecular Oxygen Using the Single-Site Olefin Metathesis Catalyst (≡SiO)2Mo(═O)2[END_REF][START_REF] Larabi | New Synthetic Approach towards Well-Defined Silica Supported Tungsten Bis-Oxo, Active Catalysts for Olefin Metathesis[END_REF] In the case of vanadium, we expect that this strategy would result in the formation of a well-defined tripodal vanadium oxo species after the grafting of [V(=O)(OtBu)3], (1), onto silica supports (dehydroxylated at 200 and 700 °C). A thermal treatment at high temperature of the resulting surface species, bearing one or two tert-butoxide ligands, leads to the targeted active species by a -H elimination of isobutene from V-OtBu, resulting in a hydroxy ligand σ-coordinated to vanadium, V-OH. This reaction can be followed by a ring opening of a neighboring siloxane (≡Si-O-Si≡) which forms a new V-OSi≡ bond and a surface hydroxyl group, ≡Si-OH. These well-defined surface species have been further characterized by elemental analysis and different spectroscopic techniques such as DRIFT, solid state NMR of different nuclei ( 1 H, [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C, 51 V), UV-Vis, XAS and studied as catalysts for the non-oxidative dehydrogenation of propane in a continuous flow reactor.

Experimental Section

General Procedure.

All experiments were carried out using standard air-free methodology within an argon-filled glovebox, on a Schlenk line or in a Schlenk-type apparatus interfaced to a high-vacuum line (10 -5 mbar). Solvents were purified and dried according to standard procedures. [V(O)(OtBu)3] was synthesized in one step followed by purification. Elemental analyses were performed at Mikroanalytisches Labor Pascher.

Diffuse reflectance infrared spectra were collected with a Nicolet 6700 FT-IR spectrophotometer, in 4 cm -1 resolution with a SMART module. An air-tight IR cell equipped with CaF2 windows was employed and the final spectra comprise 64 scans. Diffuse reflectance UV-Vis spectra in the range of 200-850 nm were recorded on a Perkin Elmer 1050 UV-Vis-NIR spectrophotometer adapted with the Praying Mantis optical unit provided by Harrick. The samples were diluted in dry BaSO4 and the spectra were recorded against a BaSO4 baseline. An air-tight cell equipped with quartz windows was used.

Solution NMR spectra were recorded on an Avance-300 Bruker spectrometer. All chemical shifts were measured relative to residual 1 H or [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C resonance of the deuterated C6D6 solvent: δ 7.15 ppm for 1 H and 128 ppm for [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C. Solid-state NMR spectra were acquired with a Bruker Avance 500 spectrometer ( 1 H: 500.1 MHz, 13 C: 125.7 MHz). For 1 H experiments, the spinning frequency was 10 kHz, the recycle delay was 5 s and 8 scans were collected using a 90° pulse excitation of 3 μs. The 13 C CP MAS spectra were obtained at a spinning frequency of 10 kHz, with a recycle delay of 5 s and 12288 scans were collected. Chemical shifts were given in ppm with respect to TMS as external reference for 1 H and [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C NMR. The 51 V MAS NMR spectrum was acquired in a Bruker DSX 300 spectrometer ( 51 V: 78.9 MHz) using a single pulse excitation. The signal was optimized with crystalline V2O5 (= -609 ppm) as external standard with a pulse angle of /2. Two spinning rates were used in order to determine the isotropic chemical shift (10 kHz and 12 kHz).

X-ray absorption spectra were acquired at ESRF on BM23 beam-line, at room temperature at the vanadium K edge. A double Si(111) crystal was used as monochromator and a system based on a total reflection through a double X-ray mirror with an incidence angle variable from 2 to 5 mrad allowed harmonic rejection to a level better than 10 -5 . [START_REF] Mathon | The Time-Resolved and Extreme Conditions XAS (TEXAS) Facility at the European Synchrotron Radiation Facility: The General-Purpose EXAFS Bending-Magnet Beamline BM23[END_REF] The spectra were recorded in the transmission mode between 5.3 to 6.44 KeV at room temperature (295 K). Each data set was collected simultaneously with a V metal foil reference and was later aligned according to that reference (first maximum of the derivative of the V foil spectrum set at 5465.1 eV). The molecular complexes and vanadyl sulfate were homogeneously mixed with dry BN and the supported V samples were packaged as pellets within an argon filled glove-box in a double air-tight sample holder equipped with kapton windows. The data analyses were performed by standard procedures using in particular the program "Athena". [START_REF] Ravel | HEPHAESTUS: Data Analysis for X-Ray Absorption Spectroscopy Using IFEFFIT[END_REF]

Preparation of the complex:

The vanadium precursor [V(O)(OtBu)3], (1), was synthesized by the reaction of V2O5 (Aldrich) with tBuOH in benzene. The entire preparation procedure was carried out in an inert atmosphere using a Schlenk line. 20 grams of V2O5 was first placed in a 1-L round-bottom flask and connected to a Dean-Stark trap in order to capture the water from the reaction, coming through the reflux condenser. After adding 300-mL of dry benzene, 165 ml of dry tBuOH was added to the benzene suspension of V2O5.

The temperature was increased to 100 °C and maintained for 48 hours. The solution was filtered away from unreacted V2O5. After removing the solvent by distillation, white crystals were observed. This product was further dried overnight under reduced pressure. In order to purify the freshly formed product, [V(O)(OtBu)3], a sublimation was conducted under vacuum at 40 °C. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF]

H NMR (C6D6):

1.43 ppm (CH3); [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C{H} NMR (C6D6)  31.4 ppm (CH3) and 86 ppm (C-CH3).

Preparation and characterization of (2), (2a), (3) and (3a) materials.

A mixture of [V(O)(OtBu)3] (in excess with respect to total amount of Si-OH groups, 0.6 g in this case) and SiO2-(200) (2 g) in pentane (10 mL) was stirred at 25 °C for 4 hours. After filtration, the solid was washed 4 times with pentane and the liquid was transferred into another reactor in order to quantify the tBuOH evolved during the grafting reaction by GC and using an internal standard. The resulting material, (2), is a white powder that was dried under vacuum (10 -5 mbar). Elemental analysis: V 1.6 wt% C 1.5 wt%. The catalyst was characterized by DRIFT, Solid-State NMR, UV-Vis and XANES.

The same procedure was employed with SiO2-700 to yield (3), which was also characterized as (2).

Elemental analysis: V 1.12 wt%, C 2.23 wt%.

Both materials, (2) and (3) were heated to form tripodal V species leading to the synthesis of (2a) and (3a) materials, respectively. These species were characterized by DRIFT, Solid-State NMR UV-Vis, and XANES and their catalytic performances evaluated for propane dehydrogenation.

Catalysts evaluation. Catalytic propane conversion was carried out in a stainless-steel continuous flow reactor (Ptotal = 1 bar, T = 540 °C, total flow rate = 5 mL.min -1 : 20% propane in argon; mass of the catalyst = 500 mg diluted in 2.5 of CSi). The gases were purified with a column of molecular sieves and activated Cu2O/Al2O3 and controlled by Brooks mass flow controllers. The catalyst was charged in the glovebox. A 4-way valve allowed the isolation of the charged catalyst in the reactor from the environment and extensive purging of the tubes before the reaction. The products were determined by an online HP 6890 GC equipped with a 50 m KCl/Al2O3 column and a FID detector.

Results and Discussions Preparation and characterization of [V(O)(OtBu)3], (1)

The precursor, [V(O)(OtBu)3], was easily synthesized using commercially available compounds V2O5 and tBuOH in benzene at 100 °C for 48 h. After purification, this complex was identified by intense 1 H NMR signal at 1.43 ppm, attributed to proton of methyl fragments and [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C NMR signals atppm, corresponding to carbons of methyl groups, CH3, and at 86 ppm attributed to the quaternary C of the tert-butyl ligand (Figure S1). In addition, [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V NMR showed a signal at 642ppm, corresponding to a tetrahedral vanadium supported complex, bearing an oxo and three alkoxy ligands (Figure S2). [START_REF] Högerl | SOMC Grafting of Vanadium Oxytriisopropoxide (VO(O i Pr) 3 ) on Dehydroxylated Silica; Analysis of Surface Complexes and Thermal Restructuring Mechanism[END_REF]

Reactivity of [V(O)(OtBu)3], (1),with a silica dehydroxylated at 200 °C, synthesis of material (2)

Amorphous silica, Aerosil 200 from Degussa was partially dehydroxylated at 200 °C under high vacuum (10 -5 mbar) to afford SiO2-200 with a density of silanols ca. 2.5 OH per nm 2 (0.86 mmol OH.g - 1 ). [START_REF] Szeto | Vanadium Oxo Organometallic Species Supported on Silica for the Selective Non-Oxidative Dehydrogenation of Propane[END_REF] The grafting of [V(O)(OtBu)3] on SiO2-200 was carried out by an anhydrous impregnation in pentane at 25 °C. Then, the powder was washed four times with dry pentane in order to remove the excess of complex affording V(O)(OtBu)3/SiO2-200 (2) as a colorless material. The reaction of [V(O)(OtBu)3] proceeds with the consumption of almost all the surface free (or "isolated") silanols of SiO2-200, as shown from the irreversible disappearance of the νO-H stretching band at 3747 cm -1 (Figure 1, Left). Importantly, the intensity of the other large νO-H band centered at 3600 cm -1 , attributed to vicinal silanols, did not decrease after reaction with the complex due to the low reactivity of these vicinal silanols. This phenomenon was already observed for the reaction of SiO2-200, with several complexes of metal of groups IV, V and VI. [START_REF] Basset | Preparation of Single Site Catalysts on Oxides and Metals Prepared via Surface Organometallic Chemistry[END_REF] Concomitantly, two groups of bands appear in the 3200-2700 cm -1 and 1500-1300 cm -1 region, assigned respectively to (CH) and (CH) vibrations of the methyl groups of the perhydrocarbyl ligands. The amounts of vanadium and carbon, as quantified by elemental analysis of V(O)(OtBu)3/SiO2-200 material, (2), are 1.6 wt% V and 1.5 wt % C, which accounts for an atomic ratio of ca. 4.0 C/V. The vanadium loading of the material is thus 0.315 mmol.g -1 and since the concentration of silanols is ca.

0.86 mmol OH/g on SiO2-200, [START_REF] Szeto | Vanadium Oxo Organometallic Species Supported on Silica for the Selective Non-Oxidative Dehydrogenation of Propane[END_REF] this gives a V/OH ratio of 0.365. The quantification of the tBuOH evolved during the grafting reaction, was carried out by GC analysis, using octane as an internal standard. It leads to a release of 1.8 tBuOH/V. Thus, the results of these characterizations are in line with the formation of a major bipodal species bearing one tert-butoxide ligand, [(SiO)2V(O)(OtBu)].

Further spectroscopic characterizations provided additional information on the grafting of [V(O)(OtBu)3] on SiO2-200. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF] H MAS NMR shows one signal at 1.4 ppm assigned to CH3 protons (Figure S3) and one signal in [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C CP MAS NMR at 28 ppm attributed to the carbon of the methyl groups (Figure 2). The lack of a significant signal for the quaternary carbon of the tert-butoxide ligand is most probably due to the larger chemical shift anisotropy, owing to a lower mobility of the surface species compared to the case of SiO2-700 support (material (3)), as previously observed. [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF][START_REF] Bini | Surface Organometallic Chemistry of Titanium: Synthesis, Characterization, and Reactivity of (⋮Si-O)NTi(CH2C(CH3)3)4-n (n = 1, 2) Grafted on Aerosil Silica and MCM-41[END_REF] No improvement for the [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] C signal of the quaternary carbon was shown under the 13 C{ 1 H} MAS mode. [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V NMR is a powerful technique that brings additional information on the coordination sphere and geometry of the vanadium centers in supported catalysts. [START_REF] Hu | Investigation of the Structure and Active Sites of TiO2 Nanorod Supported VOx Catalysts by High-Field and Fast-Spinning 51V MAS NMR[END_REF][START_REF] Mcgregor | Solid-State NMR of Oxidation Catalysts[END_REF] The 51 V MAS NMR of (2) reveals a signal with an isotropic chemical shift of -693 ppm (Figure 3, top), which represents an upfield shift of 51 ppm in comparison to [V(O)(OtBu)3] molecular complex, (1), which shows a signal at -642 ppm (Figure S2). This result is consistent with the formation of a bipodal surface species on silica-200. In fact, the upfield shift was also reported in literature, for a [V(O)(OiPr)3] complex with a [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V NMR signal at -630 ppm, that shifts to -650 ppm when two OiPr ligands of this complex are exchanged by two siloxy groups from the silica surface, leading to [(≡SiO)2V(O)(OiPr)] supported species. 55

( 51 V) / ppm

The UV-Vis spectrum of (2), comprises absorption bands starting from 200-400 nm (Figure 4). The absence in the spectrum of any absorption in the d  d transition part, suggests that the supported vanadium is in its highest oxidation state. [START_REF] Catana | Supported Vanadium Oxide Catalysts: Quantitative Spectroscopy, Preferential Adsorption of V4+/5+, and Al2O3 Coating of Zeolite Y[END_REF] The broad peak in the UV region (max = 267 nm) can be assigned to a ligand to metal charge transfer (LMCT) originated from the tert-butoxide ligands. The energy edge (Eg) derived from the OV charge transfer peak (determined from the intercept of the [F(R)h] 2 vs h plot in the low energy rise) reflects the degree of vanadia polymerization. [START_REF] Gao | Investigation of Surface Structures of Supported Vanadium Oxide Catalysts by UV-vis-NIR Diffuse Reflectance Spectroscopy[END_REF] This value is extrapolated to be 3.58 eV for (2), suggesting that the vanadium sites are present as monomer on the surface. 58 Additional information on the vanadium coordination geometry, its structural environment and oxidation state in the case of the molecular complex, (1), and materials (2) and (3) were given by Xray absorption spectroscopy. Normalized V K-edge XANES spectra of the molecular complex, [V(O)(OtBu)3], of the presumed [(≡SiO)2V(O)(OtBu)] complex in material (2), obtained from the grafting of [V(O)(OtBu)3] onto SiO2-200 are shown in Figure 5. In the two spectra (a and b), there is a prominent pre-edge peak. This pre-edge feature arises from electronic transitions from 1s core levels to the empty 3d levels, more or less 4p hybridized by the ligands of vanadium. These electronic transitions become electric-dipolar allowed in the absence of an inversion center. In this case, the loss of symmetry allows partial overlapping and mixing of the unfilled d states of the metal with its 4p orbitals. Several studies of the V K-edge XANES features have considered the pre-edge intensity as well as its absolute position to determine both the vanadium oxidation state and its symmetry. [START_REF] Wong | K$-Edge Absorption Spectra of Selected Vanadium Compounds[END_REF][START_REF] Nabavi | Xanes and 51V Nmr Study of Vanadium-Oxygen Compounds[END_REF][START_REF] Chaurand | New Methodological Approach for the Vanadium K-Edge X-Ray Absorption Near-Edge Structure Interpretation: Application to the Speciation of Vanadium in Oxide Phases from Steel Slag[END_REF] The position of the pre-edge peak on its own is generally insufficient to evaluate accurately the vanadium oxidation state. The position and intensity of the main pre-edge peak along with the position of the edge of different vanadium compounds studied are reported in Table 1. For the molecular complex, (1) and material (2

)
referenced above, the position of the pre-edge peak at ca. 5470.4 eV and its high intensity (0.79 to 0.85) are consistent with those observed in other complexes or materials containing V (V) in a tetrahedral symmetry. [START_REF] Nabavi | Xanes and 51V Nmr Study of Vanadium-Oxygen Compounds[END_REF] Moreover, the position of the K-edge of the three samples, shifted by 16 to more than 18 eV to higher energy relative to the metal foil, are within the range of K-edge energies reported for known V (V) compounds. 59

Reactivity of [V(O)(OtBu)3] with a silica dehydroxylated at 700 °C, material (3) synthesis

The same amorphous silica support was partially dehydroxylated at 700 °C (SiO2-700) under high vacuum (10 -5 mbar) in order to compare the structure of the supported species on the resulting material,

(3), to those obtained on material (2), after the grafting of [V(O)(OtBu)3] on SiO2-200. SiO2-700 presents a density of ca. 0.70 OH per nm 2 corresponding to 0.27 mmol OH.g -1 . [START_REF] Szeto | A Strong Support Effect in Selective Propane Dehydrogenation Catalyzed by Ga(i-Bu)3 Grafted onto γ-Alumina and Silica[END_REF] The impregnation of [V(O)(OtBu)3] on SiO2-700 in a pentane solution occurred at room temperature for 4h. After an intensive washing with pentane to remove the excess of the complex and a drying under high vacuum, a white powder was obtained (3). The infrared spectroscopy of V(O)(OtBu)3/SiO2-700 (3) revealed the consumption of all the isolated silanols initially present on SiO2-(700), shown by the disappearance of the νO-H band at 3747 cm -1 (Figure 1, Right). Simultaneously, alkyl bands in the range of 3000-2800 cm -1 (νC-H) and 1500-1300 cm -1 (δCH 2 and δCH 3 ) appeared, confirming the grafting of the vanadium complex on the silica surface. Elemental analysis of material (3) indicates V and C wt% contents of 1.12 and 2.23 wt%, respectively, corresponding to an atomic ratio of 7.9 C/V. The V/OH ratio is ca.

0.815, since 0.220 mmol of V/g of silica are grafted on the surface. GC analysis of tBuOH released during the grafting revealed a release of 0.88 tBuOH/V after reaction. The comparison of the mass balance analyses obtained after the grafting of [V(O)(OtBu)3] on SiO2-200 and SiO2-700 are resumed in Table 2. It is obvious from these results that the structure of the resulting species in materials (2) and

(3) depends on the dehydroxylation temperature of the support. Thus, the characterization results suggest the formation of a major monopodal species, [(≡SiO)V(O)(OtBu)2] for (3), from SiO2-700, and a bipodal species [(≡SiO)2V(O)(OtBu)] for (2), from SiO2-200.

Table 2. Mass balance analysis for the grafting of [V(O)(OtBu)3], (1), onto silica.

c The error on V is < 5 %. d The error on the C elemental analyses is ±1 a.m.u.

The characterization of material (3) by 1 H MAS NMR shows a similar result to the one observed for material (2). The spectrum also consists in a main signal at 1.4 ppm corresponding to CH3 protons of tert-butoxide ligands (Figure S3). SiO2-700 1.12 2.23 7.9 0.88 at 86 ppm is visible compared to (2). This is either due to the presence of two tert-butoxide ligands on the surface (higher concentration), compared to the main surface complex of (2), or to a higher mobility of the surface species in (3), leading to a narrower chemical anisotropy. A signal centered at -687 ppm appears in the [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V MAS NMR spectrum, attributed to the monopodal complex [(≡SiO)V(O)(OtBu)2] (Figure 3, down). This chemical shift corresponds to an up-field shift of -45 ppm compared to the precursor [V(O)(OtBu)3] (-642 ppm). Nevertheless, the observed chemical shift is rather similar to the bipodal [(≡SiO)2V(O)(OtBu)] (-693 ppm). [START_REF] Högerl | SOMC Grafting of Vanadium Oxytriisopropoxide (VO(O i Pr) 3 ) on Dehydroxylated Silica; Analysis of Surface Complexes and Thermal Restructuring Mechanism[END_REF][START_REF] Rice | Characterization of Silica-Supported Vanadium(V) Complexes Derived from Molecular Precursors and Their Ligand Exchange Reactions[END_REF] Taking into consideration the peak broadening (FWHM ≈ 35 ppm), the podality of supported [V(O)(OtBu)3] is indistinguishable by [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V MAS NMR spectroscopy. The UV-vis spectrum of [(≡SiO)V(O)(OtBu)2] consists in an absorption band between 200 and 400 nm with a max = 262 nm, corresponding to ligand to metal charge transfer (LMCT O 2-→ V 5+ ) (Figure 4). Similarly to the bipodal species, no absorption band at 650 nm was observed, hence, the vanadium adsorbed on the surface has the highest oxidation state (+V). [START_REF] Catana | Supported Vanadium Oxide Catalysts: Quantitative Spectroscopy, Preferential Adsorption of V4+/5+, and Al2O3 Coating of Zeolite Y[END_REF] The energy edge (Eg) is 3.6 eV (compared to 3.58 eV in the case of V(O)(OtBu)3/SiO2-200), which is in line with the formation of monomers vanadium sites on the surface. [START_REF] Wegener | Precursor Nuclearity Effects in Supported Vanadium Oxides Prepared by Organometallic Grafting[END_REF] XANES analysis of the molecular complex on SiO2-700.

The normalized V K-edge XANES spectra of the molecular complex, [V(O)(OtBu)3], of [(≡SiO)V(O)(OtBu)2] complex, obtained from the grafting of [V(O)(OtBu)3] onto SiO2-700 are shown in Figure 6. The position and intensity of the main pre-edge peak along with the position of the edge of these vanadium compounds are also reported in Table 1. For V(O)(OtBu)3/SiO2-700 supported species, the position of the pre-edge peak (5470.3 eV), its relative intensity (0.85) and the position of the edge (5483.2 eV) are also indicative of V(V) tetrahedral species.

Tripodal species

Controlled thermolyses of [(≡SiO)V(O)(OtBu)2], (2) from 25 to 300 °C (for 2 h), 400 °C (for 2 h) as well as 500 °C (for 2 h) were conducted under vacuum in a reactor equipped with a cold trap (liquid N2) in order to collect the gas phase evolved during the process. The analysis of the gas phase by GC after thermal treatment at 300 °C revealed the formation 0.88 isobutene/V for (2), and some traces of tBuOH were observed. As reported in the case of silica supported Mo and W complexes bearing oxo tert-butoxide ligands, [START_REF] Quéméner | Metathetic Oxidation of 2-Butenes to Acetaldehyde by Molecular Oxygen Using the Single-Site Olefin Metathesis Catalyst (≡SiO)2Mo(═O)2[END_REF][START_REF] Larabi | New Synthetic Approach towards Well-Defined Silica Supported Tungsten Bis-Oxo, Active Catalysts for Olefin Metathesis[END_REF] the formation of isobutene is triggered by the elimination from a methyl group of a H in β-position of the oxygen coordinated to the metal, M-O-C(Me)2-CH2-H, to form a hydroxy intermediate, M-OH. This step can be assisted by the presence of an oxo ligand in the coordination sphere of vanadium.

The DRIFT analysis of the resulting materials from bipodal [(≡SiO)2V(O)(OtBu)] species after heating at 300 °C revealed the total disappearance of the alkyl vibrational bands in the range of 3000-2800 cm -1 and 1500-1300 cm -1 , assigned respectively to (CH), (CH) vibrations from the perhydrocarbyl ligands (Figure 7, Left). At 300 °C (after 2h), we observed the reappearance of isolated silanol groups at 3747 cm -1 as well as the appearance of a new band at 3660 cm -1 attributed to VO-H. [START_REF] Keller | Hydration Effects on the Molecular Structure of Silica-Supported Vanadium Oxide Catalysts: A Combined IR, Raman, UV-Vis and EXAFS Study[END_REF][START_REF] Nguyen | Study of New Catalysts Based on Vanadium Oxide Supported on Mesoporous Silica for the Partial Oxidation of Methane to Formaldehyde: Catalytic Properties and Reaction Mechanism[END_REF] At 400 °C, the intensity of the band characteristic of SiO-H increases while the one of VO-Hband decreases. At 500 °C, we observed the absence of the vibration band attributed to VO-H elongation vibration, accompanied by the decrease in intensity of the ≡SiO-H silanols vibration band. This phenomenon could be explained by a partial condensation of V-OH with the remaining silanols, leading to the formation of water and V-OSi bonds and thus tripodal V oxo surface species (Scheme S1). The same procedure including a thermal treatment under vacuum at progressively increasing temperatures, has been applied to surface species of (3). This leads to similar observations from DRIFT and 1 H MAS NMR data (right parts of Figure 7 and Figure 8, respectively), allowing to the formation of a tripodal vanadium oxo on SiO2-700 (3a) with a release of 1.9 isobutene/V. The resulting vanadium surface species (2a) and (3a) were further characterized by XANES, UV-vis and 51 V MAS NMR.

XANES of the tripodal species

Both tripodal vanadium systems obtained on silica were characterized by XANES spectroscopy to provide additional details on the vanadium structural environment after thermal treatment of the grafted species. Normalized V K-edge XANES spectra of tripodal systems on SiO2-200 (2a) and SiO2-700 (3a), in contrast with [V(O)(OtBu)3] and species (2) and (3) are shown in Figure 5,c and Figure 6,c. The position and intensity of the main pre-edge peak along with the position of the edge of targeted species

[(≡SiO)3V(O)] is compared to molecular complex [V(O)(OtBu)3], surface species (2)
[(≡SiO)2V(O)(OtBu)] and (3) [(≡SiO)V(O)(OtBu)2] (Table 1). Similar to the complex and to the grafted species, the XANES spectra of tripodal oxo species (2a) and (3a) displayed a pre-edge peak located at ca. 5470.4 eV. The relative intensities of these pre-edges are high and range from 0.85 (for grafted species (2) and (3)) to 0.81 for tripodal species (2a) and (3a). These results are consistent with those observed for other complexes or materials containing V(V) in a tetrahedral symmetry. [START_REF] Nabavi | Xanes and 51V Nmr Study of Vanadium-Oxygen Compounds[END_REF] Moreover, the position of the K-edge of the samples, shifted from 16 to more than 18 eV to higher energy relative to the metal foil, are within the range of K-edge energies reported for known V(V) compounds. [START_REF] Wong | K$-Edge Absorption Spectra of Selected Vanadium Compounds[END_REF] Additional characterizations of the surface tripodal oxo species (2a) and (3a) were given by UV-vis spectroscopy (Figure 9). This method allows the detection of isolated V(IV), V(V) and vanadium oligomeric species. The UV-vis spectra of (2a), [(≡SiO)3V(O)] species on SiO2-200 exhibited a band at 253 nm along with a shoulder at 286 nm attributed to π(t2)d(e) and π(t1)d(e) oxygen-totetrahedral V(V) charge transfer transitions, involving oxygen in bridging (V-O-Si) and terminal (VO) positions, respectively, in agreement with reported literature. [START_REF] Lewandowska | Different Kinds of Tetrahedral V Species in Vanadium-Containing Zeolites Evidenced by Diffuse Reflectance UV-vis, Raman, and Periodic Density Functional Theory[END_REF] A similar result was obtained in the UV-vis spectrum of (3a), [(≡SiO)3V(O)] on SiO2-700 with two bands at 256 and 287 nm. The absence of (d-d) transitions in the 600-800 nm range indicates that V(IV) species are not significantly present. Indeed, the energy edge (Eg) of the tripodal V species on SiO2-200 and SiO2-700 are ca. 3.50 and 3.35 eV, respectively (compared to 3.58 and 3.6 eV in the case of (2), V(O)(OtBu)3/SiO2-200 and (3), V(=O)(OtBu)3/SiO2-700, respectively). The slight red-shift in the energy edge when transforming the grafted complexes into the tripodal species may be due to the creation of supplementary restrains around the deformed tetrahedral vanadium centers. These results are in agreement with the formation of monomeric vanadium sites on the surface, as supported by XANES and 51 V MAS NMR. [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V MAS NMR is a complementary technique to IR, 1 H MAS NMR, XANES and UV-vis used to identify the surface vanadia species on the supported catalysts. Due to the importance in catalysis of the isolated tetrahedral vanadium complexes on various supports, the intensive 51 V solid-state NMR characterization of these types of catalysts by several groups showed that the chemical shift is generally in the range of -460 to -700 ppm. [START_REF] Hu | Investigation of the Structure and Active Sites of TiO2 Nanorod Supported VOx Catalysts by High-Field and Fast-Spinning 51V MAS NMR[END_REF][START_REF] Liu | Vanadium Oxide Supported on Mesoporous SBA-15 as Highly Selective Catalysts in the Oxidative Dehydrogenation of Propane[END_REF][START_REF] Luca | Spectroscopic Investigation of Vanadium Speciation Invanadium-Doped Nanocrystalline Anatase[END_REF][START_REF] Keränen | Characterization and Activity Testing of Vanadia Catalysts Deposited onto Silica and Alumina Supports by Atomic Layer Deposition[END_REF] The attribution of the signals in the [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V NMR spectrum of these materials remains challenging due to the lack of supporting models (such as (Ph3SiO)3V(O) and vanadium oxo containing POMSS) [START_REF] Das | Bonding States of Surface Vanadium (V) Oxide Phases on Silica: Structural Characterization by Vanadium-51 NMR and Raman Spectroscopy[END_REF][START_REF] Feher | Synthesis and Characterization of Vanadium-Containing Silsesquioxanes[END_REF] and to the variety of experimental conditions during the NMR acquisition. [START_REF] Jaegers | Investigation of Silica-Supported Vanadium Oxide Catalysts by High-Field 51V Magic-Angle Spinning NMR[END_REF][START_REF] Miller | V2O5 Catalysts Supported on Al2O3-SiO2 Mixed Oxide: 51V, 1H MAS Solid-State NMR, DRIFTS and Methanol Oxidation Studies[END_REF] Grant et al. [START_REF] Grant | Enhanced Two-Dimensional Dispersion of Group V Metal Oxides on Silica[END_REF] have studied by [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V MAS NMR silica-supported vanadia catalysts prepared by impregnation with a surface coverage between 1.3 to 15.4 V•nm -2 . In the [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V NMR spectra obtained, the authors observed two different isotropic shifts, depending on the surface coverage of vanadium. For the materials prepared with the highest vanadium content (13.4 and 15.4 V•nm -2 ), a sharp peak appeared at -614 ppm, corresponding to the distorted trigonal bipyramidal geometry of crystalline V2O5, which is in line with previous work revealing close isotropic shift of crystalline V2O5 at -612 ppm [START_REF] Mcgregor | Solid-State NMR of Oxidation Catalysts[END_REF] or -610 ppm. [START_REF] Hu | Investigation of the Structure and Active Sites of TiO2 Nanorod Supported VOx Catalysts by High-Field and Fast-Spinning 51V MAS NMR[END_REF] In parallel, another signal was also observed at -675 ppm in all samples, but its intensity in the case of catalysts prepared with lower coverage of vanadium, in particular for 1.3 V•nm -2 , was very low. This signal is attributed to dispersed monomeric species, [(≡SiO)3V(O)]. [START_REF] Das | Bonding states of surface vanadium(V) oxide phases on silica: structural characterization by vanadium-51 NMR and Raman spectroscopy[END_REF] Recently, Jaegers. et al., [START_REF] Jaegers | Investigation of Silica-Supported Vanadium Oxide Catalysts by High-Field 51V Magic-Angle Spinning NMR[END_REF] confirmed these MAS NMR attributions, by employing faster spinning rates (above 30 kHz) at a higher field (19.975 T), and reported a better resolution of the signal assigned to tripodal V species (at -675 ppm), even at low V content (0.6 V.nm -2 ). In the case of the supported species of (2a) and (3a), with low V coverages, ca. 1.1 and 0.8 V.nm -2 , respectively, [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V MAS NMR revealed a similar signal with a chemical shift of -670 ppm for both samples (Figure 10), attributed to [(≡SiO)3V(O)] isolated vanadium surface species, on SiO2-200 and SiO2-700 (Scheme 3). The mechanism of the formation of these active species would be similar on SiO2-200 and SiO2-700. It has been proposed that a hydrogen transfer from a methyl group of a tert-butoxy ligand to the vanadyl oxygen occurs, leading to a V-OH fragment. The latter reacts with a siloxane bridge from the support by a Si-O-Si ring opening, leading to the formation of new isolated silanols as described in Scheme 3 (e.g on SiO2-200). The hydrogen transfer to the vanadyl oxygen can occur either by a transfer of one hydrogen to the oxo ligand, or by direct migration of one hydrogen from a methyl group to the oxygen of a tert-butoxide ligand, thus releasing a tBuOH molecule. It was indeed demonstrated that the thermal treatment of metal derivatives of -OSi(OtBu)3 led to the direct elimination of isobutylene, even at low temperatures (100 -150 °C) to form silanols without any assistance of other ligands.

Catalytic properties

The catalytic behavior of the tripodal V species supported on SiO2-200 and SiO2-700 was evaluated in the non-oxidative dehydrogenation of propane. A continuous fixed-bed type reactor was used, under the following operating conditions: θ = 540 °C, P = 1 bar and a total flow of 5 mL.min -1 (20 % of C3H8 in Ar). The reactor was heated from room temperature to 540 °C under a flow of 4 mL.min -1 of argon and then propane was introduced at 540 °C.

When propane was fed to the catalytic bed of tripodal V obtained on SiO2-200 (2a), a conversion of 19

% was reached at early times on stream and slightly decreased to 17 % after 1000 min (Figure 11). At that time, the selectivity toward propene was 84.5 %, and the by-products included methane (6.8 %), ethene (4.7 %) and ethane (2.5 %) as well as traces of C4 (< 2%) (Figure 12, Left). On the other hand, tripodal V on SiO2-700 (3a) exhibited a different behavior in function of time on stream. A lower initial conversion of 15% followed by a gradual deactivation was observed, leading to 10.4% after 1000 min.

Initially, the products comprised propene (88.4%), methane (6.8%), ethane (2.4%) and ethane (2.4 %).

However, after 1000 min, the selectivity toward methane increased to 11.4% and propene decreased to 81.7% while ethane and ethene slightly increased to 3.6% and 3.2%, respectively. The activity and selectivity observed with (2a), composed of tripodal V species on SiO2-200 seems better than those with (3a), composed of tripodal V species on SiO2-700. Such tendency was previously obtained in the case of vanadium species resulting from the grafting of OV(mesityl)3 complex on SiO2-200 and on SiO2-700. [START_REF] Szeto | Vanadium Oxo Organometallic Species Supported on Silica for the Selective Non-Oxidative Dehydrogenation of Propane[END_REF] This is probably due to the difference in the local structure of the tripodal V on a silica pretreated at different temperature. This difference in terms of structure has been also observed in the [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V solid-state NMR of the tripodal species [(≡SiO)3V(O)] obtained on SiO2-200 and SiO2-700 after a thermal treatment. In fact, the larger signal observed in the case of vanadium supported on SiO2-700 (3a), is attributed to the presence of different tetrahedral vanadium species, with the same coordination sphere but different local environment, which could explain the difference in terms of catalytic activity. The present catalytic results can be directly associated to the vanadium tripodal species obtained through SOMC. These silica supported vanadium species exhibit different behavior in the propane dehydrogenation reaction with respect to conventional V2O5/SiO2 catalysts prepared by aqueous impregnation. [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] This is potentially due to the presence of different vanadia species (from monomeric to oligomeric) on the surface of the latter catalyst. The determination of the mechanism and of the nature of the active species in each step is of high interest. However, these key-issues cannot be addressed with our current ex-situ characterization techniques. As reported for chromium, it is reasonable to assume that the structure of the active species in PDH, is a tripodal V (III) , [(≡SiO)3V]. In fact, at the initiation step, the first hydrogen abstraction of propane occurs on VO ligand of [(≡SiO)3V(O)], yielding an intermediate V (V) bearing an alkyl and a hydroxy ligand. This latter species would then undergo a -H elimination of the V coordinated propyl ligand, leading to the formation of V hydrido hydroxyl species and the release of one propylene. The reductive elimination of the water from the V hydrido hydroxyl species, would then lead to the formation of the active species, [(≡SiO)3V (III) ], during propane dehydrogenation (Scheme S2).

The full catalytic cycle can be described by two mechanistic pathways. Both involve an initialization of a V-O bond of [(≡SiO)3V] by a C-H activation of propane, giving V-propyl species and an adjacent SiOH group. The next step is comprised by a β-H transfer to give a V-H and a coordinated propylene.

Following a new activation of C-H of propane, the products (propylene and hydrogen) are released.

The latter reaction scheme has previously been reported for supported metal hydride catalysts. [START_REF] Niccolai | Primary Selectivity in the Activation of the Carbon-Hydrogen Bonds of Propane by Silica Supported Zirconium Hydride[END_REF][START_REF] Szeto | Selective Conversion of Butane into Liquid Hydrocarbon Fuels on Alkane Metathesis Catalysts[END_REF] The second mechanistic pathway may involve cleavage and reconstruction of the V-H species with propylene coordinated, resulting in direct product release and regeneration of the tripodal species, as described for the chromium(III) system. [START_REF] Olsbye | Mechanistic Insight in the Ethane Dehydrogenation Reaction over Cr/Al2O3 Catalysts[END_REF] This mechanism has been earlier proposed in several reports.

Firstly, Harlin et al. who observed that the reduced V2O5 based catalyst has an enhanced activity in non-oxidative dehydrogenation of n-butane and i-butane, suggested that the reduced states of vanadium (V (IV) or V (III) ) are more active. [START_REF] Harlin | Alumina-Supported Vanadium Oxide in the Dehydrogenation of Butanes[END_REF] Moreover, Chen et al. have revealed the presence of V (III) as a V2O3

phase by XRD after the dehydrogenation of ethylbenzene under nitrogen at 550 °C. 80 In addition, Takahara et al. have studied the hydrogen treatment of silica supported V2O5, and suggested that a cleavage of V-O bonds has occurred with the formation of V-O-H sites adjacent to the V (IV) /V (III) , possible hydrides. [START_REF] Takahara | Dehydrogenation of Propane over a Silica-Supported Vanadium Oxide Catalyst[END_REF] Hence, the V (IV) /V (III) species are most likely responsible for the dehydrogenation whereas VO-H acidic sites are believed to trigger the formation of carbonaceous deposit which increases with time on stream. Advanced operando techniques and theoretical calculations should offer additional insights on the reaction mechanism (Scheme S3).

Conclusion

In conclusion, supported oxovanadium(V) complexes were prepared by surface organometallic chemistry from the grafting of the molecular complex [V(O)(OtBu via a thermal treatment of the grafted species, triggering -H transfer and isobutene release. In addition to the characterization of all these tripodal species by mass balance analysis, IR, UV-Vis, solid-state NMR ( 1 H) and XANES spectroscopy, the characterization by [START_REF] Conte | State of the Art of CPMAS 13C-NMR Spectroscopy Applied to Natural Organic Matter[END_REF] V MAS NMR allowed the confirmation of the chemical isotropic shift of tripodal V at -670 ppm, which is in agreement with recently reported results. The catalytic behavior of the resulting tripodal species was evaluated in the direct dehydrogenation of propane to propene, exhibiting high activity and propene selectivity. Importantly, the presented catalytic results are exclusively originated from the isolated tripodal species. Regarding the gradual deactivation of the catalyst generally reported and attributed to the formation of carbonaceous species, this phenomenon was barely observed in our case compared to the conventional V2O5 supported on silica catalysts, which presents V-O-H acidic sites responsible for the coke formation.

Figure 1 .

 1 Figure 1. Left: IR-spectra of a) SiO2-200 and b) after grafting of [V(O)(OtBu)3] on SiO2-200 (2), and Right: IR-spectra of a) SiO2-700 and b) after grafting of [V(O)(OtBu)3] on SiO2-700 (3)

Figure 2 .

 2 Figure 2. 13 CP CP MAS NMR of Left: V(O)(OtBu)3/SiO2-200 and Right: V(O)(OtBu)3/SiO2-700 (11.8 T, 10 kHz, d1= 5 s)

Figure 3 .

 3 Figure 3. 51 V MAS NMR spectra of: V(O)(OtBu)3/SiO2-200, (2) (top) (7.1 T, 10 kHz, d1= 1 s) and V(O)(OtBu)3/SiO2-700, (3) (down) (7.1 T, 10 kHz, d1= 1 s)

Figure 4 .

 4 Figure 4. UV-Vis spectrum of (2) (left) and (3) (right)

Figure 5 .

 5 Figure 5. XANES: of (a) [V(O)(OtBu)3] molecular complex, (1); (b) V(O)(OtBu)3/SiO2-200 (2); (c) V(O)(OtBu)3/SiO2-200 heated at 500°C (2a).

  The 13 C CP MAS spectrum of V(O)(OtBu)3/SiO2-700, (3), shows two signals at 86 ppm and 28 ppm, assigned to quaternary carbon and CH3 terminal groups, respectively (Figure 2, Right). It is important to note that the signal attributed for the quaternary carbon Supports %wtV c %wt C d C/V ratio tBuOH/V

Figure 6 .Scheme 2 .

 62 Figure 6. XANES of (a) [V(O)(OtBu)3] molecular complex (1); (b) V(O)(OtBu)3/SiO2-700, (3); (c) V(O)(OtBu)3/SiO2-700 heated at 500°C, (3a).

Figure 7 .

 7 Figure 7. Left. IR-spectra of a) V(O)(OtBu)3/SiO2-200 (2), and then thermally treated at: b) 300 °C, c) 400 °C, and d) 500 °C; Right. IR-spectra of a) V(O)(OtBu)3/SiO2-700 (3), and then thermally treated at: b) 300 °C, c) 400 °C and d) 500 °C

Figure 8 .

 8 Figure 8. Left. 1 H NMR spectra of a) V(O)(OtBu)3/SiO2-200 (2), and then thermally treated at: b) 300 °C, c) 400 °C, and d) 500 °C; Right. 1 H NMR spectra of a) V(O)(OtBu)3/SiO2-700 (3), and then thermally treated at: b) 300 °C, c) 400 °C and d) 500 °C

Figure 9 .

 9 Figure 9. UV-vis of Left: (2a) and Right: (3a)

Figure 10 .

 10 Figure 10. Solid-state 51 V NMR spectra of: Left: V(=O)(OtBu)3/SiO2-200 (2) (7.1 T, 10 kHz, d1= 1 s) (top) and [(≡SiO)3V(O)] (2a) (down) (7.1 T, 10 kHz, d1= 1 s), and Right: V(O)(OtBu)3/SiO2-700 (3) (7.1 T, 10 kHz, d1= 1 s) (top) and [(≡SiO)3V(O)] (3a) (down) (7.1 T, 12 kHz, d1= 1 s). Grafted species in (2) and (3) revealed a major peak at -693 and -687 ppm respectively. Tripodal species in (2a) and (3a) exhibited a major peak at -670 ppm.

75

 75

Scheme 3 .

 3 Scheme 3. Proposed mechanism for the formation of [(≡SiO)3V(O)] species

Figure 11 .Figure 12 .

 1112 Figure 11. Propane conversion catalyzed by vanadium tripodal species supported on (2a) and on SiO2-700, (3a). Conditions: P= 1 bar; θ= 540 °C; Total flow = 5 mL.min -1 (20 % C3H8 in Ar)

) 3 ], ( 1 )

 31 , onto SiO2-200 and SiO2-700, leading to materials exhibiting bipodal species [(SiO)2V(O)(OtBu)] on SiO2-200 (2) and monopodal species [(SiO)V(O)(OtBu)2] on SiO2-700 (3). They were fully characterized by elemental analysis, IR, UV-Vis, solid state NMR ( 1 H, 13 C, 51 V) and XANES. A new original method has been developed in this work for the preparation of supported tetrahedral vanadium oxo tripodal species [(SiO)3V(O)]

Table 1 .

 1 Positions and intensities of the pre-edge signal and edge positions in the V K-edge XANES of several vanadium samples.

	Sample	Pre-edge signal position a	Pre-edge signal	Edge position a,c
		(eV)	intensity b (a.u.)	(eV)
	V (0) (metal foil)			5465.1
	[V III (Mes)3•THF]	5466.6	0.55	5474.2
	[V IV (O)SO4(H2O)x]	5469.8	0.48	5476.5
	[V V (O)(OtBu)3] (1)	5470.4	0.79	5481.9
	V(O)(OtBu)3/SiO2-200 (2)	5470.3	0.85	5483.4
	V(O)(OtBu)3 /SiO2-200	5470.5	0.81	5483.5
	heated at 500°C (2a)
	V(O)(OtBu)3 /SiO2-700 (3)	5470.3	0.85	5483.2
	V(O)(OtBu)3 /SiO2-700	5470.4	0.81	5483.6
	heated at 500°C (3a)

a Relative to a calibration with a vanadium foil (5465.1 eV); A ± 0.3 eV error is estimated on the energy. b Intensity normalized to the step height of the K-edge. c Defined as the first inflexion point after the pre-edge peak.

Acknowledgements

This work was carried out as a part of the BIZEOLCAT project, which has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 814671.

In particular, J. A. is grateful for her PhD grant. AdM thanks Olivier Mathon and Kiril Lomachenko for their help during the recording of the X-ray absorption spectra at ESRF, on beam line BM23.

Note

The authors declare no competing financial interests.

Supporting information

NMR spectra and proposed reaction schemes.

Supporting information