J.-B Trebbia

C L Garrido Alzar

R Cornelussen

C I Westbrook

I Bouchoule

Roughness suppression via rapid current modulation on an atom chip

Keywords: numbers: 39, 25, +k, 03, 75, Be

We present a method to suppress the roughness of the potential of a wire-based, magnetic atom guide: modulating the wire current at a few tens of kHz, the potential roughness, which is proportional to the wire current, averages to zero. Using ultra-cold 87 Rb clouds, we show experimentally that modulation reduces the roughness by at least of a factor five without measurable heating or atom loss. This roughness suppression results in a dramatic reduction of the damping of center of mass oscillations.

Atom chips, devices which trap and guide atoms with micro-fabricated structures on a substrate, hold enormous promise for the applications of cold atoms and for the exploration of new physical regimes of degenerate gases [1,2]. For applications, their small size facilitates the design of a large variety of structures and functionalities [3,4,5,6,7]. For fundamental studies, the strong confinement possible on an atom chip permits, e.g. the study of low dimensional gases. In particular nearly one dimensional (1D) gases have been studied in the weakly interacting regime on atom chips [8,9], and the attainment of the strongly interacting or Girardeau regime [10,11,12,13,14] on a chip is attracting significant experimental effort [15].

To exploit the full potential of these devices, atoms must often be close to the material structures they contain. This proximity however, renders atom chips highly sensitive to defects [16,17] which produce roughness in the trapping potential. In the case of current carrying wires, most of these defects are due to the wires themselves [18,19,20,21,22]. Advances in fabrication procedures have steadily improved wire quality [23,24], but the roughness of the trapping potential remains a problem. For example, it is a serious hindrance for the attainment of the Girardeau regime, where the atoms need to be tightly confined in the transverse direction and weakly confined in the longitudinal direction.

In this paper we demonstrate a method, first suggested in [18], which nearly eliminates the longitudinal potential roughness of a magnetic wire trap by rapid current modulation. This idea is reminiscent of the TOP trap [25] in the sense that the atoms see a time averaged potential. The wire configuration here is such that the rough potential component rapidly oscillates and averages to zero. We compare the potential roughness for unmodulated (DC) and modulated (AC) guides (with the same current amplitudes) and demonstrate a reduction factor of at least 5 without noticeable atom loss or heating. This roughness suppression results in a significant reduction of the damping of the center of mass oscillations.

A simple atomic guide can be made by a current carrying wire and a homogeneous field B bias perpendicular to the wire [1]: atoms are confined transversely and guided parallel to the wire. Atoms, of magnetic moment µ in the low field seeking state, feel a potential proportional to the absolute value of the magnetic field. The guide is centered on a line where the transverse field vanishes and the longitudinal potential V (z) is given by µ|B 0 z + δB w z (z)|, where B 0 z is an external homogeneous field, typically of one Gauss, and δB w z (z) is the small, spatially fluctuating field (few mG) created by current flow deformations inside the micro-wires. Since |B 0 z | > δB w z (z) and δB w z (z) is proportional to the micro-wire current, current modulation around zero results in a longitudinal time-averaged potential in which the effect of δB w z (z) disappears. The instantaneous position of the guide is determined by both the wire current and the transverse bias field B bias . If only the wire current is modulated, the atom experiences strong forces that eliminate the transverse confinement. One must therefore also modulate B bias . In our experiment, this latter field is also produced by on-chip micro wires whose low inductance permits high modulation frequencies.

We use a two layer atom chip as sketched in Fig. 1. The lower silicon wafer has a H structure of 15 µm thick wires which carry the currents I z1 and I z2 . Atoms are initially loaded in an initial magnetic trap (Z-trap) [1], which is created by setting I z1 = 3 A, I z2 = 0 A, and applying a transverse field (in the x -y plane). After evaporative cooling, we have a few times 10 4 trapped 87 Rb atoms in the |F = 2, m F = 2 state at a temperature of the order of 1 µK. Next, we transfer the atomic cloud into a second trap (five-wire trap), where we perform the potential roughness measurements for both DC and AC cases. This five-wire trap is a combination of a transverse magnetic guide and a weak longitudinal confinement. The guide is realized with five parallel micro-wires (700 × 700 nm 2 and 2 mm long in the z direction) deposited on the upper silicon wafer. The wire geometry is described in [24], and shown in Fig. 1. By applying the currents I b = ±15 mA in the two outer pairs of wires and I c = ±13 mA in the central wire, the resulting magnetic field vanishes on a line (parallel to the z axis), 7 µm above the central wire.

To prevent Majorana losses, we add a constant and homogenous longitudinal field B 0 z = 1.8 G. The weak longitudinal confinement is provided by the H-structure on the lower chip with I z1 = I z2 = 0.4 A (no current flows in the central wire of the H).

First, we measured the roughness of the trapping potential for positive and negative DC currents. After the transfer to the five-wire trap, we wait 600 ms to let the sample thermalize in the presence of an evaporative knife leading to about 5 × 10 3 atoms at around 280 nK. To ensure that the two DC traps have the same transverse position to within 500 nm, we compensate the transverse magnetic fields to within 0.1 Gauss. Figures 2(a) and 2(b) show absorption images, averaged over 50 runs, of the trapped cloud at thermal equilibrium for the two DC traps. In both situations the cloud becomes fragmented, revealing the presence of potential roughness. Moreover, the density maxima and minima are inverted when the currents are reversed, as previously observed in [18]. The time of flight (tof) for imaging is 1.5 ms. This delay allows the atoms to leave the vicinity of the five-wire structure where diffraction of the imaging beam produces systematic errors in the atomic density profile. The 1.5 ms tof of the atoms partially washes out the density modulation. This effect and the optical resolution contribute to a resolution function with a rms width of 8 µm modeled by a gaussian function. Convolution with this gaussian gives a contrast reduction of 40% for a wavelength of 60 µm, the typical spatial scale observed in Fig. 2. The 10 % modulation in the figure is thus an underestimate of the actual potential roughness. This smearing does not affect the relative measurements we present below, except insofar as to reduce our signal to noise ratio.

To extract the potential V (z) along the guide center from the linear atomic density n(z), we use the Maxwell-Boltzmann distribution V (z) = -k B T ln[n(z)/n 0 ]. This formula is valid because the transverse oscillation frequency is independent of z (to within 1% over the cloud extent) and provided that the phase space density is much smaller than unity. We deduce the temperature from the transverse velocity distribution measured by the standard tof technique.

The longitudinal confinement, produced by the Hshaped wire, is expected to be harmonic within 6% over the extent of the cloud. A harmonic fit of V (z) gives ω z,DC /2π = 7.1 Hz. Subtracting this potential from V (z), we obtain the potential roughness plotted in Fig. 2(d) for both positive and negative DC currents. The observed rms roughness amplitudes are 39 nK and 22 nK for negative and positive currents, respectively. This asymmetry in the potential roughness may be due to residual transverse magnetic fields which change the trap location, or to residual noise in the imaging system.

We now turn to the study of the AC trap. To give a more quantitative analysis, we begin with a theoretical description of the AC trap. Since ω m is much larger than all the characteristic frequencies associated with the atomic motion, the transverse and the longitudinal dynamics are described by the one cycle averaged potential V (r, t) . Because B 0 z is large compared to any other magnetic field, V (r, t) can be approximated by

V (r, t) = µ B |B z (r, t)| + µ B 2B 0 z |B w ⊥ (r, t) + B ⊥ (r)| 2 • (1)
Here, µ B is the Bohr magneton, B z (r, t) is the longitudinal magnetic field, B w ⊥ (r, t) is the transverse magnetic field of the five wires, and B ⊥ (r) denotes the transverse field created by the H wires plus any uncompensated, homogeneous, time-independent, transverse field. Since δB w z oscillates, its contribution to |B z (r, t)| averages to zero.

The cross term B w ⊥ (r, t) • B ⊥ (r) vanishes, so that B ⊥ (r) only produces a position dependent energy shift µ B |B ⊥ (r)| 2 /2B 0 z . It has a negligible effect on the transverse confinement since the confinement created by the five-wires is very strong. On the other hand, |B ⊥ (r)| 2 affects the weak longitudinal confinement. The length scale on which B ⊥ (r) varies is much larger than the cloud extent, because it is produced by elements located at least a millimeter away from the atomic cloud. It thus only contributes to a curvature without introducing additional roughness. The transverse magnetic field created by the H-shaped wire is B ⊥ (z) = B ′ H z u y , where u y is the unit vector pointing along y. This field increases the longitudinal frequency according to

ω 2 z,AC = ω 2 z,DC + µ B B ′ 2
H /(mB 0 z ). On the other hand, the transverse frequency is reduced by √ 2 compared to the DC case. This factor is confirmed by our transverse frequency measurements which give (1.2 ± 0.2) kHz for the AC trap, (2.2 ± 0.1) kHz and (2.1 ± 0.1) kHz for DC negative and positive currents, respectively.

We extract the potential in the AC trap from the measured longitudinal profile for a sample at T = 180 nK. From a harmonic fit we obtain a longitudinal frequency of 11.3 Hz, slightly larger than the one in the DC trap, as expected. In Fig. 2(d), we compare the measured roughness of the AC trap to those obtained in the DC traps. In the AC configuration, the roughness rms amplitude is reduced by a factor 4 and 7 with respect to the positive and negative DC cases. The residual apparent roughness in the AC case is consistent with the noise in the imaging system and so the mean reduction factor of 5 is a lower limit.

We also tested the reduction of the potential roughness by looking at the evolution of the longitudinal center of mass oscillations (CMO) as shown in Fig. 3. In a harmonic trap, the CMO of an atomic cloud are undamped. In the presence of a potential roughness, the oscillation period depends on the amplitude and the de- phasing between trajectories of different particles results in a damping of the CMO. Figure 3 shows the dramatic contrast between the AC and DC traps. We fit the oscillations to the function Z(t) = Z 0 + Z 1 e -t 2 /τ 2 cos(ωt + φ), where Z 0 , Z 1 , τ , ω and φ are fitting parameters. This function ensures the expected quadratic decrease of the oscillation amplitude at small times for a damping due to trajectory dephasing. The fitted damping time in the AC case, τ = (1.9 ± 0.1) s, is ten times larger than its value in the DC case. In fact, the observed damping in the AC guide can be explained by the anharmonicity of the H-shaped wire confinement alone. Thus, the data are consistent with the absence of roughness.

To estimate an upper bound on the ratio of the AC to DC roughness amplitudes, we performed one dimensional classical Monte Carlo calculations for non-interacting atoms experiencing a rough potential superimposed on a harmonic confinement. The calculations are averaged over potential roughness realizations with the spectral density expected for a single wire having white noise border deformations [22,26]. We adjust the wire border noise in order to recover the measured damping time in the DC trap (it corresponds to an rms roughness of 80 nK). We find that the observed increase of the damping time by a factor 10 requires a reduction of the potential roughness by about 14. A drawback of the model is that it does not take into account atomic collisions, happening at a rate on the order of the damping time. However, as collisions are present in both the DC and AC configurations, we assume that the above roughness reduction deduced from the calculation is still relevant.

The modulation technique does not correct for all guide defects. By considering only the time-averaged potential we neglect the micro-motion of the atoms [START_REF] Landau | Mechanics, Fourth Edition, Course of Theoretical physics[END_REF] whose energy is responsible for a residual roughness. This roughness is 10 -8 times smaller than the one present in the DC guides for our experimental parameters. In fact, other imperfections are more important. First, the transverse component of the rough micro-wire field produces meanders of the guide equipotentials (displacement of about 2 nm) and a z-dependent modification of the transverse trapping frequency (approximately 0.3%). These two effects are not averaged out in the AC guide. Second, the rough longitudinal field δB w z appears in the expansion of V (r, t) to orders that are neglected in Eq. ( 1). The resulting guide imperfections are one order of magnitude smaller than those produced by the transverse component of the rough field.

We also investigated the dependence of the trap lifetime on ω m as shown in Fig. 4. For modulation frequencies above 15 kHz, the lifetime of the sample is the same as in the DC case. For lower modulation frequencies, we observe a decrease in lifetime that we attribute to instabilities of the atomic trajectories in the modulated potential. For zero unmodulated transverse magnetic field B ⊥ (r), the transverse atomic motion close to the guide center is governed by the Mathieu equation [START_REF] Landau | Mechanics, Fourth Edition, Course of Theoretical physics[END_REF] and no instabilities are expected for ω m > 0.87ω ⊥,DC , where 0.87ω ⊥,DC /2π = 1.82 kHz which is significantly below 15 kHz. We attribute this discrepancy to the presence of a non vanishing transverse field. Classical trajectory Monte Carlo simulations, with fields of 150 mG in both x and y directions, reproduce well the measured lifetime as shown in Fig. 4 (dashed line). This value is of the same order as the accuracy of the residual transverse field compensation. Within the explored range of frequencies (up to 50 kHz), we have not identified any upper bound for the modulation frequency. The atomic spin should adiabatically follow the magnetic field orientation to avoid spin flip losses. This condition is fulfilled provided that ω m /2π ≪ µ B B 0 z /4π , where µ B B 0 z /4π ≃ 1.3 MHz is the Larmor frequency.

The temporal modulation of the rough potential can lead to a heating [START_REF] Bouchoule | [END_REF]. In our trap however, the expected heating rate is very small. This is confirmed by our measured heating rate of about 160 nK/s which is close to the technical heating, also observed in the DC trap. We have also loaded a Bose-Einstein condensate in the AC trap without noticeable fragmentation.

The method presented here will enable new experimental explorations on quantum gases using atom chips. We have already mentioned the study of a 1D Bose gas in the Girardeau regime. Note that the reduction of transverse trapping frequency by √ 2 can be overcome by going closer to the wire. Second, the relative insensitivity of the time-averaged potential to residual transverse magnetic fields will enable us to revisit proposals of double well potentials in static magnetic fields, such as [19,24], which are very sensitive to residual magnetic fields. Finally, this method may also allow the study of atom dynamics in disordered potentials [START_REF] Clément | [END_REF]30,31] since the roughness strength can be tuned.

FIG. 1 :

 1 FIG.1: Side (a) and top (b) views of the atom-chip. The first wafer (dark grey) has 15 µm thick wires (grey). On the second wafer (white), a five wire structure creates the atomic guide whose roughness is studied (d = 2.5 µm). Lengths not to scale.

FIG. 2 :

 2 FIG. 2: (Color on line). Absorption images of a thermal cloud for negative DC (a), positive DC (b), and AC currents (c), respectively. Plotted is the number of atoms per pixel (pixel size is 6 × 6 µm 2 ). In (d), the potential roughnesses are extracted from longitudinal profiles using Maxwell-Boltzmann's distribution.

  Using phaselocked oscillators at a frequency ω m /2π = 30 kHz, we produce the five-wire currents I c (t) = I c cos(ω m t) and I b (t) = I b cos(ω m t), where I c = 13 mA and I b = 15 mA are the same currents as those used in the DC cases. The field B 0 z and the currents I z1 and I z2 are not modulated. Loading a thermal cloud in the AC trap, the fragmentation is no longer visible in the absorption image Fig. 2(c).

FIG. 3 :

 3 FIG. 3: Center of mass position vs time for negative DC current (a) and AC current (b) (ωm/2π = 30 kHz). The solid line is a fit to the function Z(t) given in the text. The dramatic increase in damping time in the AC trap illustrates the strong reduction of the roughness.

FIG. 4 :

 4 FIG.4: Lifetime vs. ωm in the AC trap (dots). The dashed line represents the result of Monte Carlo calculations with a transverse magnetic field of 150 mG for both x and y directions.

The authors thank A. Aspect, T. Schumm and J. Estève for fruitful discussions. The Atom Optics group of Laboratoire Charles Fabry is a member of the IFRAF Institute. This work was supported by the EU under grants No. MRTN-CT-2003-505032 and No. IP-CT-015714.


Franck Thibault
email: franck.thibault@univ-rennes1.fr

Benoît Corretja

Alexandra Viel

Dionisio Bermejo

Raúl Z Martínez

Béatrice Bussery-Honvault

Linewidths of C 2 H 2 perturbed by H 2 : experiments and calculations from an ab initio potential

In this work we present a theoretical and experimental study of the acetylene -hydrogen system.

A potential surface considering rigid monomers has been obtained by ab initio quantum chemistry methods. This 4-dimensional potential is further employed to compute using the close-clouping approach and the coupled-states approximation pressure broadening coefficients of C 2 H 2 isotropic Raman Q lines over a temperature range of 77 to 2000 K. Experimental data for the acetylene ν 2 Raman lines broadened by molecular hydrogen are obtained using stimulated Raman spectroscopy.

The comparison at 143 K of theoretical values with experimental data is promising. Approximations to increase computational efficiency are proposed.

I. INTRODUCTION

The acetylene-hydrogen system is of interest for astrophysical, atmospherical, and combustion applications. It is well known that acetylene is a minor constituent of the hydrogendominant atmospheres of Jupiter and Saturn and of the nitrogen-dominant atmosphere, but also containing hydrogen, Saturn's moon Titan. Traces of acetylene are also found in the atmosphere of the giant Uranus and Neptune planets. It has been shown that even if C 2 H 2 is present as a trace constituent, it plays an important role in the photochemistry of these atmospheres [refs. 1-6 and references therein]. Acetylene is also a trace constituent of the Earth's atmosphere, mainly produced by anthropogenic sources (see refs in [1][2][3][4][5][6].

From a more applied point of view, acetylene is present in a number of processes of industrial interest: hot-filament-activated C 2 H 2 -H 2 gas mixtures are used to grow diamond carbon surfaces [7][8][9] or carbon nanotubes. [10] It is known that acetylene also plays a role as an intermediate species in the process of soot formation often observed in combustion research [START_REF] Warnatz | Combustion[END_REF][START_REF] Lamprecht | [END_REF]. Finally, acetylene is frequently used as a target molecule to measure the temperature of combustion mixtures through the use of rotational or vibrational coherent anti-Stokes Raman spectroscopy (CARS) [Ref. 13 and references therein]. This application takes advantage of the relative simplicity of C 2 H 2 Raman spectrum and the associated large scattering cross section.

To get a better insight into these processes, a detailed knowledge of the collisional broadening of acetylene lines with various gases is desired. A number of studies have been devoted to rare gas-broadening [1][2][3][4][5][6][14][15][16][17] and some of us have recently presented a series of papers devoted to the acetylene and its interactions with rare gases. [1][2][3][4][5] These weakly bound complexes have received much attention in recent years because their intermolecular potential is mainly affected by the interaction of the rare gas atom with the triple bond of acetylene. [3][4][5] Works have been dedicated to the calculation of the collisional broadening effects of infrared (IR) or Raman C 2 H 2 lines. In the present paper, we extend these studies to the C 2 H 2 -H 2 system.

Much less attention has been paid to the molecular-broadening of acetylene lines. Experimentally, N 2 -, O 2 -, H 2 -pressure broadening [6,[15][16][17][18][19][20][21][22] and self broadening pressure coefficients [6,15,23] have been measured at various temperatures between 147 and 300 K.

Before this work, the theoretical studies of molecular partners [18][19][20][21][22][23] were typically per-formed on the basis of the Robert-Bonamy semi-classical formalism [24] using semi-empirical potential energy surfaces (PES). Only two of these works dealt with hydrogen. [21,22] Quantum dynamical studies of collisional line broadening in the case of a linear molecule colliding with another one (eventually the same) are much more laborious than in the case of atomicmolecule systems. The corresponding computer cost is much higher than for semi-classical computations, and thus such quantum calculations are quite rare. [25][26][27][28][29][30] However they are valuable benchmark calculations to evaluate the accuracy of more approximate methods.

Important information on molecular relaxation phenomena can be obtained from the pressure broadening coefficients for different perturbing gases at different temperatures. This feature has induced numerous experimental and theoretical studies of spectral lineshapes.

Pressure broadening is closely linked to state-to-state rate coefficients, especially for isotropic Raman lines. An improved understanding and modeling of planetary atmospheres can be achieved with a better knowledge of H 2 collision-induced lines broadening of acetylene. The pressure broadening data for C 2 H 2 -H 2 have not yet been analyzed in term of an interaction potential for this system. Since no ab initio PES for the title system has been reported, one objective of this study is to build such a surface from first principles.

Over the past decades there has been considerable interest in the weakly-bound complexes of small molecules. [31] Among those, only a small number of studies [32][33][34][35][36][37][38] focused on complexes formed by acetylene with various small molecules. Significant progress has been made in the ab initio computation of intermolecular interactions. Methods like coupled cluster or symmetry adapted perturbation theory (SAPT) [39][40][41] together with its more recent extension SAPT(DFT) based on the density functional theory [42] enable the determination of van der Waals intermolecular potentials between closed-shell systems with spectroscopic accuracy. After comparative tests between the available ab initio methods, we retain the CCSD(T) method for the determination of the C 2 H 2 -H 2 PES. While the supermolecular CCSD(T) method has to be corrected for basis set superposition error (BSSE) [43], it is competitive in accuracy with the perturbative and BSSE-free SAPT method (see also Ref. 26 for additional tests).

The accuracy of the surface is tested by a quantum determination of pressure broadening coefficients which are compared with current experimental measurements of isotropic Raman lines of the acetylene ν 2 band at 143 K. The experimental technique used for this purpose is stimulated Raman loss spectroscopy [1,44]. Such experimental data are the first ones reported for the title collisional system.

In the next section we briefly describe the experimental setup used in this study. In section III we present the method employed for the ab initio determination of the PES. Section IV is devoted to the quantum close-coupling calculations of the collisional dynamics. Results and discussions are given in section V, and finally, concluding remarks and perspectives are summarized in section VI.

II. EXPERIMENT

The spectra recorded and analyzed for this work have been obtained with the stimulated Raman spectrometer of Instituto de Estructura de la Materia in Madrid. The spectrometer has been thoroughly described in a number of previous works [1,44] and only a brief description of the technique emphasizing the details specific to the present experiment will be given here. In a typical stimulated Raman experiment the sample is simultaneously illuminated with two laser beams while the frequency of one of the beams is scanned. When the frequency difference of the two beams matches the frequency of a Raman transition of the sample under study, a transfer of energy can be observed between the high-frequency beam, which loses intensity, and the low-frequency (Stokes) beam, which gains intensity.

Every photon of the high-frequency beam involved in the process produces one photon at the Stokes frequency and one quantum of molecular excitation. Despite the underlying theoretical foundation being relatively complex, stimulated Raman spectra can be seen as the Raman analog of stimulated emission: when the frequency condition is met, the presence of Stokes photons provided by the second laser beam stimulates the generation of more Stokes photons with the same frequency, phase, polarization and directional properties. The process can thus be detected as a net increase in the energy of the Stokes beam and a net loss in the energy of the high-frequency beam. The intensity of either beam can be monitored against the frequency to provide a Raman spectrum of the sample. The main strength of this type of experiment (by comparison with linear Raman spectroscopy) is the fact that the instrumental resolution is not limited by dispersive or interferometric elements used to analyze the scattered radiation, but only by the linewidths of the lasers used.

The Ar + laser that generates the probe beam is operated at 529 nm and in a single-mode configuration by means of a temperature-stabilized intra-cavity étalon. The laser frequency is electronically locked to the transmission fringe of an external Fabry-Perot interferometer, reducing the fast frequency jitter to 500 kHz. The frequency of the transmission fringe is in turn locked to the frequency of a molecular reference ( 130 Te 2 for these experiments) by means of a secondary stabilization loop that uses a sub-Doppler polarization setup. In this way, the laser frequency is stabilized against both short and long-term drifts and is also known to high accuracy. It must be noted that the frequencies of the spectral lines measured in the course of this work are well known, but since the present measurements rely on accurate fitting of line profiles, it is important to maintain proper laser stability and to avoid frequency drift that could alter the profiles while they are being recorded. The cw ring dye laser that generates the seed for the tunable pump beam is operated with Rhodamine 590, and this seed is later amplified in a pulsed dye amplifier operated with Kiton Red. The amplifier is pumped by the second harmonic of a pulsed, single-mode Nd:YAG laser. The resulting pump pulses have typical energies of ∼3 mJ and a nearly Gaussian temporal profile of 12 ns FWHM, which translates into a Gaussian spectral profile of ∼70 MHz FWHM.

The two beams are overlapped with a dichroic mirror, focused into the sample cell using an f=500 mm planoconvex lens and then recollimated by a similar lens at the exit. In order to improve the S/N ratio of the experiment, two additional broadband mirrors are used to arrange a triple pass through the cell. After the triple pass the laser beams are separated and the probe beam is taken to a fast PIN silicon photodiode. The Raman loss signals thus detected are then amplified and processed by a boxcar integrator. The output of this integrator is sent to a computer where the representation of the Raman loss signal versus the frequency scan of the ring dye laser provides the high resolution Raman spectrum of the sample.

The sample cell is a variation of a design already in use in our laboratory for lowtemperature measurements. [1] The system is based on the controlled evaporation of liquid nitrogen to generate a continuous flow of cold nitrogen vapor that is circulated through one of the jackets of the sample cell and cools down the sample. The final temperature can be adjusted by controlling the rate of evaporation. The cell is a tubular pyrex design of approximately 95 cm in length and 10 cm outer diameter, and consists of three internal concentric chambers also tubular in shape and totally isolated from each other. The innermost chamber is the sample chamber, and brewster angle fused silica windows have been installed on both ends so that the laser beams can travel through the cell. The middle chamber is the one through which the cold nitrogen vapor flows. Finally, the outer chamber is used as isolation chamber to prevent heating of the two inner chambers by the lab atmosphere.

To this end, the chamber is pumped to a moderately high vacuum (∼10 -5 torr) and its outer wall lined with mylar foil. Each one of the chambers has its own gas inlet/outlet and valve. For the middle chamber, special consideration had to be paid to the possibility of temperature gradients forming along the cell, since the cold nitrogen vapor warms up as it travels through it. To minimize this effect the nitrogen enters the chamber through a central inlet and exits through two outlets symmetrically located near both ends of the cell. A thermocouple installed in the inner chamber is used to measure the sample temperature at the center of this chamber, where the largest contribution to the Raman signal is generated. The nitrogen evaporation system consists of a 50-liter dewar in which a ceramic resistor has been submerged. The power dissipated by the resistor, and thus the amount of nitrogen evaporated by unit of time, is controlled with a variac. The vapor generated is conducted, through thermally isolated tubing, to the inlet in the sample cell. In this way, changes in the sample temperature can be accurately induced/controlled by changing the variac voltage. With this system, sample temperatures as low as ∼120 K have been reached in our laboratory. The present experiments are conducted at a temperature of 143 K, slightly above the point of deposition of 12 C 2 H 2 for the highest of the partial pressures used.

The samples for the experiments consist of diluted mixtures of 12 Raman spectra of the Q-branch of the ν 2 band of 12 C 2 H 2 are recorded up to j = 17 for all pressures. Beyond this line, the S/N ratio at the lowest pressures is too poor at this temperature to obtain accurate linewidth data. The spectral lines are analyzed by fitting them to Voigt profiles, either individually or simultaneously when partial overlapping of the lines makes it necessary, as it is the case with the first lines of the band. The Gaussian part of the Voigt profile results from the convolution of the Gaussian Doppler width at 143 K and the Gaussian apparatus function. This part of the Voigt profile can thus be pre-calculated and kept fixed during the fit of the profiles. In this way the Lorentzian contribution to the Voigt profile of each spectral line could be extracted.

To obtain the pressure broadening coefficients, a linear fit of Lorentzian width versus pressure is carried out for each spectral line. The slope of the fit provides the pressure broadening coefficient of that line.

III. A C 2 H 2 -H 2 INTERMOLECULAR POTENTIAL A. Ab initio calculations
Different methods are available to treat van der Waals interaction potentials between dimers with an accuracy close to one wavenumber. In order to reach such an accuracy, we have tested different methods and basis sets to evaluate the interaction potential of C 2 H 2 -H 2 .

Convergence with respect to the size of the basis set has thus been tested. With van der Waals interactions, binding energies are small and thus, relative variation among them can be large. To perform the calculations, we have used the Molpro package [45] which permits the use of ab initio methods such as: the Hartree-Fock (HF) method, the second order Møller-Plesset perturbation theory (MP2), the coupled cluster method at a level of single and double excitations with the triple excitations included perturbatively, i.e. CCSD(T), and the symmetry adapted perturbation method based on monomer orbitals built up with the density functional theory (SAPT(DFT)) [42]. Several atomic basis sets have been tested: the augmented -correlation consistent -polarized triple and quadruple zeta basis sets (augcc-pVXZ with X=T and Q) and the polarization optimized Sadlej basis set [46]. We add to this Sadlej set a polarization orbital "d" for the H atoms taken from the aug-cc-pVTZ basis set to obtain [3s2p1d] contracted Gaussian functions for H atoms and [5s3p2d] for C atoms. In all cases, mid-bond functions as proposed by Tao and Pan [47] have been included to improve the convergence of the calculation of the dispersion energy without increasing too much the number of basis functions of the whole system. This adds (3s3p2d1f) functions (with exponents s,p: 0.90, 0.30, 0.10 ; d: 0.60, 0.20 ; f: 0.30) in the basis set.

To test the influence of the choice of the ab initio method and of the basis set, we have conducted potential energy calculations for specific conformations (R, θ A , θ B , φ) of the C 2 H 2 -H 2 dimer where (R, θ A , θ B , φ) are the four angular Jacobi coordinates which describe the relative orientation of the two monomers. R is the distance between the two monomer centers of mass and defines the intermolecular axis, θ {A,B} defines the bending angle of the monomer axis relative to the intermolecular axis (the "A" index refers to C 2 H 2 while "B" refers to H 2 ) and φ is the rotational angle around R of one monomer relative to the other. So, specific conformations can be defined as (θ A , θ B , φ) =(90, 90, 90) (noted X), (0, 0, 0) (L), (90, 90, 0) (H) , (90, 0, 0) (T a ) and (0, 90, 0) (T b ) (see pictorial representations in fig 1 of Ref 48).

All the above selected methods are not equivalent to treat van der Waals interactions as some, like CCSD(T), are supermolecular while others, like SAPT(DFT), are based on a perturbative treatment. So, the former ones will give values incorporating basis set superposition errors (BSSE) while the later ones will be free of BSSE. [49] This error is a consequence of the unphysical lowering of the monomer energies due to the presence of the basis functions at the other site. In order to compare both methods, we have used the Boys and Bernardi approach [43] to correct the CCSD(T) energies for BSSE. This approach, also called counterpoise (CP) recipe, suggests to compute the interaction energy as the difference of the "supermolecule" AB energy with the monomer energies calculated in the full dimer basis. [49] For that, we define E X (Y ) as the energy of the system X in the basis set Y . The BSSE-free interaction energy is obtained as

E CP AB = E AB (AB) -E A (AB) -E B (AB) while the BSSE correction is evaluated by: E A (A) + E B (B) -E A (AB) -E B (AB).
In view of all these tests, we concluded that the energies evaluated with the CCSD(T) method and the Sadlej basis set give converged energies with a sufficient accuracy keeping a good ratio of computer time required relative to the accuracy. As may be seen in Fig. 1, the results at the CCSD(T) level compare well to the SAPT(DFT) levels for three geometries.

For the ab initio determination of the interaction potential, both monomers are treated as rigid and fixed at their vibrationally ground equilibrium geometry, i.e. An electronic file containing the ab initio energies is available upon request to the authors.

The current study is performed within a reduced dimensionality approach in which the two molecules are rigid linear rotors. The range of the error corresponding to the rigid treatment can be estimated from a static study including geometrical modifications of C 2 H 2 . The lowest frequency mode of C 2 H 2 being for the bending motions, a study of the energy change when a CH bond of C 2 H 2 is bent has been investigated. The results show that conformations which take this bending modification into account are more stable only at very short intermolecular distance (<5.5 bohr). The energy difference with or without C-H bending is negligible (∼ 2 cm -1 for the H geometry near 5 bohr). Therefore, we conclude that treating C 2 H 2 as a rigid rotor is a good approximation for the temperature regime of interest.

B. Representation of the potential energy surface

In order to perform the quantum dynamical study of the collision, an amenable form of the C 2 H 2 -H 2 interaction potential is necessary. A development onto bispherical harmonics [26,50,51] has been used. This expansion,

V (R, θ A , θ B , φ) = L A ,L B ,L V L A ,L B ,L (R) A L A ,L B ,L (θ A , θ B , φ) (1)
is restricted to even L A , L B and L because the colliding partners are homonuclear molecules.

As usual, the relation

|L A -L B | ≤ L ≤ |L A + L B |
has to be fulfilled in eq. ( 1). In the case of two linear molecules, the angular functions A L A ,L B ,L (θ A , θ B , φ) are defined by normalized products of spherical harmonics for monomers A and B,

A L A ,L B ,L (θ A , θ B , φ = φ A -φ B ) = 2L + 1 4π 1/2 m L A mL B m|L0 Y m L A (θ A , φ A ) Y -m L B (θ B , φ B ) (2)
where

Y m L A and Y -m L B are spherical harmonics, • • • | • • • Clebsch-Gordan coefficients, and |m| ≤ min (L A , L B ).
The development in eq. ( 1) has been limited to 30 angular functions as described in the work of A. van der Avoird et al on the N 2 -N 2 PES (see the first column of table II of ref. [51]). This development allows the ab initio points to be represented to better than 2% at long range and better than 8% at shorter intermolecular distances. A further restriction to only 23 terms (thus removing all the terms with L B = 6 and the V 6,4,10 term), does not affect the value of the pressure broadening (PB) cross sections, even if the ab initio points are less well reproduced. The radial coefficients, V L A ,L B ,L (R), for the tabulated R values, are obtained using Gauss-Legendre quadratures over θ A and θ B , and a Chebyshev quadrature over φ [26,[START_REF] Press | Numerical Recipes in Fortran[END_REF]. A standard interpolation method [START_REF] Press | Numerical Recipes in Fortran[END_REF] is then used to compute the radial coefficients for R values needed in the dynamical treatment. This representation of the PES has the expansion most compatible for efficient use with the MOLSCAT code [START_REF] Hutson | MOLSCAT version[END_REF], thus saving CPU time.

IV. CALCULATIONS OF PRESSURE BROADENING CROSS SECTIONS

Pressure broadening, thus the modification of the spectral lineshapes of a molecule induced by the presence of nearby particles, is sensitive to the molecular interaction. Comparison of computed and measured PB coefficients is thus a test of the validity of the interaction potential we propose. The method of calculations is very similar to that used for our N 2 -H 2 study [26] and thus only the salient details will be given here.

The collisional broadening cross-sections, using the impact approximation [START_REF] Ben-Reuven | [END_REF]55], can be expressed as a function of binary diffusion S-matrix elements [25,56] obtained from the MOLSCAT quantum dynamical code. To solve the coupled equations with MOLSCAT code, the propagation is carried out with the diabatic modified log-derivative method from a minimum distance of 4.5 bohr to an intermediate one of 17 bohr and with the Airy method up to a maximum intermolecular distance R = 21 bohr. Convergence in the cross sections is typically reached for total angular momentum J ( J = j AB + ℓ, in which j AB is the composed angular momentum formed by the two molecules and ℓ is associated with the relative motion of the colliding pair) varying approximately from 45 to 60 for kinetic energies between 150 cm -1 and 700 cm -1 .

All energetically open rotational levels (j A , j B ) and the asymptotically closed levels lying within approximately 100 cm -1 above the total energy are included in the rotational basis to perform the scattering calculations for each total energy. The total energy is defined within the rigid rotor approximation by

E T = E kin + B A j A (j A + 1) + B B j B (j B + 1) with B A = 1.1766 cm -1 (for C 2 H 2 ) and B B = 60. cm -1 (for H 2 ).
It is assumed that both acetylene and hydrogen molecules are composed of two noninterconverting species in our experiment. Natural nH 2 is a 3:1 mixture of ortho and para hydrogen (oH 2 and pH 2 ) species with odd and even j 2 , respectively. Acetylene is also a mixture of ortho and para species.

Since, in the rigid rotor approximation, the PB cross section of an isotropic Raman line is simply the sum of two-body rotational state to state cross sections [27], we have performed four types of calculations of state-to-state cross sections: calculations for ortho-ortho, paraortho, ortho-para and para-para species. For each of these pairs, we performed more than 190 computations for total energies E T between 7 and 1600 cm -1 using the close coupling scheme and 4 additional calculations at 2000, 2500, 3000 and 4000 cm -1 using the coupled states method (for a review of these quantum dynamical methods see for instance Ref 57), all using the MOLSCAT package.

Defining [26,27] a partial pressure broadening cross section as

σ (j A , j B , E kin ) = j ′ A ,j ′ B σ j A j B → j ′ A j ′ B ; E kin (3)
with j ′ A = j A , and using the Maxwell Boltzmann thermal average based on the kinetic energies,

σ(j A , j B , T ) = 1 (k B T ) 2 σ(j A , j B , E kin )E kin exp (-E kin /k B T ) dE kin , (4)
the PB cross section at temperature T of a Q(j A ) line is a sum of 2 contributions:

σ (j A , T ) = 1 4 σ pH 2 (j A , T ) + 3 4 σ oH 2 (j A , T ) (5a)
with

σ pH 2 (j A , T ) = j B even ρ j B σ (j A , j B , T ) (5b)
and

σ oH 2 (j A , T ) = j B odd ρ j B σ (j A , j B , T ) . ( 5c
)
Finally, the PB coefficient (in cm -1 ) is given by [56] γ

(j A ) = n H 2 v 2πc σ (j A , T ) . (6)
This quantity can be compared with the measurements of experiments performed at a given temperature T with nH 2 and nC 2 H 2 . In eqs. ( 5), ρ j B is the H 2 (unit) normalized rotational populations for each H 2 species, n H 2 the density of hydrogen and v is the mean relative speed v = 8k B T πµ , with µ = 1.865 u being the reduced mass of the colliding pair. The computation of pressure broadening coefficients of infrared R and P lines relies on a more elaborate theory and requires more computational power [25,[START_REF] Ben-Reuven | [END_REF][55][56] since both even and odd angular momenta of C 2 H 2 need to be included at the same time. The PB coefficients may be expressed as [58][59][60] γ j, j

′ = 1 2 γ(j) + γ(j ′ ) + γ elastic (7)
with j ′ = j ± 1. The first term contains the inelastic contributions and the last term is due to elastic effects. Within the "random phase approximation"(RPA) [58] in which the elastic contribution is neglected, the PB coefficient of such a line is given as the half sum of two isotropic Raman Q-lines.

V. RESULTS AND DISCUSSIONS

A. Ab initio PES calculation Fig. 1 displays angular cuts of the CCSD(T) PES for various geometries. The linear geometry is purely repulsive while the T a geometry is the most attractive. The repulsive part of the L geometry approach occurs at quite large intermolecular distances due to the long size of C 2 H 2 given the definition of R from center of masses. The strong core-core repulsion is thus seen at larger R values than for the other geometries. For similar reason, the core-core repulsion is seen at lower R distance in the T a approach where H 2 is pointing towards C 2 H 2 than in the T b one where C 2 H 2 is pointing towards H 2 . The attractive part of the PES is found to be due to the quadrupole-quadrupole interaction both in the T a and T b geometries. In the L approach, the quadrupole-quadrupole interaction is repulsive. The global minimum value of -141 cm -1 is found at an intermolecular distance of 6.7 bohr. The T b geometry has a minimum of about -122 cm -1 located at R = 8 bohr and the H geometry is only very slightly attractive around 7.6 bohr. Therefore this PES is quite anisotropic in the well region and in its repulsive part. This last point is easily understandable bearing in mind that this repulsive part is very sensitive to the steric effect induced by the "long" linear acetylene molecule.

The anisotropy of the PES can be also seen on Fig. 2 which presents a selection of radial coefficients V L A ,L B ,L of the potential development. The most relevant components are the V 202 which represents [48] the interaction of the acetylene quadrupole with the spherical part of the H 2 molecule, the V 224 which resumes to the quadrupole -quadrupole interaction at large distances, the V 426 (hexadecapole-quadrupole) and the V 022 (spherical part of the acetylene with the quadrupole moment of hydrogen). The contribution of the other terms is significant only at short distances where the anisotropy is high.

The only ab initio study on the C 2 H 2 -H 2 system has been done by Sapse et al. [38] for which a single geometry configuration has been computed, thus limiting the comparison with our work. Yang and Watts [37] determined an empirical PES but considering the H 2 molecule as a spherical partner using measured total differential cross sections. Nonetheless, we can compare the C 2 H 2 -H 2 PES with the N 2 -H 2 one. [26,61] The C 2 H 2 -H 2 potential is deeper and more anisotropic than the N 2 -H 2 potential. We infer that this is due to FIG. 2: Main anisotropic components of the PES (see eq. ( 1)).

the fact that the magnitude of the quadrupole moment of the acetylene is roughly 4.5 times [62,63] the quadrupole of N 2 , [61] and that the polarizability as well as the anisotropic polarizability of the acetylene are larger for acetylene (roughly twice) than for nitrogen. [61,63] The anisotropy at short range is greater for C

B. Pressure broadening cross sections and coefficients

Because quantum studies of pressure broadening cross sections for linear -linear molecules are quite scarce [26][27][28][29][30] we will first discuss the present theoretical results. Fig. 3 gives details of a selection of partial PB cross sections. It provides the contribution, defined by eq. ( 3), FIG. 3: Selected partial pressure broadening cross section σ (j A , j B ; E kin ) [in Å2 ] as a function of the relative kinetic energy [in cm -1 ].

from various rotational quantum number of H 2 (j B = 0, 1, 2, 3) to the PB cross sections (eq. ( 5)) for j A = 0, 1, 6, 13. The cross sections for populated larger j B (4 and 5) are not shown for clarity reasons but they are very close to the ones with j B = 2 or 3, especially for kinetic energies greater than about 500 cm -1 . On this basis, the calculations may be simplified and CPU time reduced.

In examining Fig. 3, it is to be noted that all the coupling terms V 0,L B ,L of the potential do not enter in the partial PB cross section (eqn.( 3)) because such terms induce a rotationally elastic transition for the optically active molecule (see eq.( 9) of Ref. 50). In addition, based on the SAPT(DFT) study which provides the different contributions (electrostatic, dispersion, exchange, ...) to the potential, we can assert that the electrostatic interactions dominate only at intermolecular distances larger than about 9 bohr.

The PB cross sections σ (j A , 0; E kin ) are the smallest regardless of the value of j A because the coupling via the permanent multipolar (essentially quadrupole) moments of the H 2 molecule vanishes and the anisotropy which induces inelastic transitions is the smallest. In other words, in the entrance collisional channel the H 2 molecule is effectively spherical and the V 224 term has a small influence because it implies a post-collisional value j ′ B = 2 with a weak probability transition, especially at low kinetic energy. On the contrary, for j B > 0, the selection rules on the final j ′ B value allow for inelastic as well as elastic transitions for the H 2 molecule. Therefore the cross sections for j B > 0 are larger than the j B = 0 ones. At high energy (> 2000 cm -1 ) the cross sections with j B = 0 become similar to the ones with j B > 0 because collisions are sensitive primarily to the repulsive part of the PES where the V 224 no longer dominates over the other terms (see Fig. 2) which induce more transitions.

In the low or very low kinetic energy regime, which essentially probes the long range part of the potential, we note the very quick decrease of the partial PB cross sections for j A = 0 and 1 (especially for j B = 0). In these levels, the acetylene molecule cannot relax. Moreover, the PB cross sections for j A = 1 is smaller than for j A = 0. This may have two reasons. First, this may be due to a steric effect, indeed the apparent cross section is smaller in j A = 1 than in j A = 0 because in the former the acetylene molecule has preferential orientations.

Second, the energy required to make an upward transition starting from j A = 1 to j ′ A = 3 is nearly twice that required for j A = 0 to j ′ A = 2. Finally, we observe that these partial cross sections for low j A are larger than cross sections for high j A values. This behavior is the same as observed for atomic perturbers. [1][2][3][4][5] We also observe that these quantities have slow and smooth variations with increasing kinetic energy, therefore the calculation of the thermal average is no longer necessary for high temperatures. The thermally averaged cross section in eq. ( 6) can thus be replaced with the value obtained at the single kinetic energy Ēkin = 4 π k B T /hc associated to the mean speed at the temperature T :

σ (j A , T ) = σ(j A , Ēkin ). ( 8
)
Table I provides thermally averaged (partial) PB cross sections of C 2 H 2 perturbed by H 2 for T = 295 and 1500 K. We confirm that the PB cross sections for j B = 0 are the smallest, but more importantly we note that the cross sections for j B = 1, 2, 3 are very close to each other. Thus it is not crucial, especially at high temperatures where the duration of the calculations drastically increases, to compute such quantities for j B > 3. We have verified and ( 4))

T=295 K T=1500 K this using j B values up to 5. To economize CPU time one can replace σ (j A , j B > 3; T ) in eq. ( 5b) or (5c) by a mean value deduced from the first j B values. At 295 K roughly half of the pH 2 molecules are in j B = 0 and that this population quickly decreases as the temperature increases. Moreover (Fig. 3 and Table I) the partial PB cross sections become identical to each other as T increases.

j A j B = 0 j B = 1 j B = 2 j B = 3 j B = 0 j B = 1 j B = 2 j B =
Using the data given in Table I, it can be easily shown that at room temperature the para cross section defined in eq. (5b) is about 9% lower than the ortho cross section eq. (5c) for all j A values while at 1500 K the difference is only about 3%. Thus at high temperature it is not essential to compute the pH 2 cross sections. Instead it may be assumed that σ pH 2 = σ oH 2 . Such a behavior is also observed [65] for N 2 -N 2 but at lower temperatures because the rotational constant for N 2 is much smaller than the one for H 2 and thus the number of populated states increases quickly with the energy.

The grid used for the kinetic energies allows the evaluation of the PB coefficients at various temperatures between 77 and 2000 K. This is of interest for future use in studies related to planetology, circumstellar envelope or combustion. The PB coefficients at any temperature can be retrieved using the following semi-empirical relation,

γ j (T ) = γ j (T 0 ) T 0 T n j , (9)
where the reference temperature T 0 has been chosen at 295 K. Because such a relation cannot provide [66,67] the temperature dependence of the half width at half maximum (HWHM) over all this range of temperature, table II gives the n-exponent for T < 295 K and T > 295 K. Note that n is smaller for the higher temperatures and tends to the classical limit of 0.5 (hard sphere partners).

C. Comparison with experimental data

The results of our calculations are now compared with experimental results. Experimental PB coefficients of isotropic Raman lines, at 143 K, are tabulated in Table III and compared with calculations in Fig. 4.

Taking into account the experimental error bars, a good agreement between the current experimental data and our calculations is observed.

Moreover, there is no substantial disagreement even using eq. ( 6) without performing the thermal average over the relative kinetic energies (eq. ( 8)). For instance for the Q(1) line the relative difference between the proper thermally averaged value and the non averaged value is only about 4%.

We can also compare our theoretical results for isotropic Raman lines with measurements performed in IR bands. [6,21,22] essentially inelastic collisions. [59,60,66,68,69] We also note that the RPA (see eq. ( 7))

would only improve the agreement for the very lowest j values but would not significantly change other values (not shown). As expected, [2-5, 26, 59, 60, 66] the agreement between thermally averaged values and non averaged values improves as the kinetic energies (and thus the temperature) increase. Finally, we observe that quite good agreement is obtained with the experimental values for j greater than about 10, especially at room temperature.

FIG. 6: Comparison between experimental [6,21] pressure broadening coefficients of R lines and calculated (same as Fig. 4) values for Q lines at 295 K.

VI. CONCLUSION

We have determined an acetylene -hydrogen PES at the CCSD(T) level on a four dimensional grid with the monomers taken as rigid rotors. Using this potential, we have calculated pressure broadening cross sections on a wide range of energies allowing the determination of

C 2 H 2 (

 22 5% in volume)in H 2 at different pressures(20,40,60, 80 and 100 mb approximately). All the samples are prepared at least 24 hours in advance in order to ensure proper mixture of the two gases.

FIG. 1 :

 1 FIG. 1: Selected angular cuts of the PES as a function of the intermolecular distance R. Symbols represent the ab initio values. CCSD(T) ab initio energies are compared with SAPT(DFT) values for three angular approaches.

Figs. 5 FIG. 5 :

 55 FIG. 4: Comparison between experimental and calculated PB coefficients of isotropic Raman lines.Displayed theoretical values are the ones obtained through the thermal average of the PB cross section and the ones at the single kinetic energy4 π k B T /hc associated to the mean speed at 143 K.

  pressure broadening coefficients up to 2000 K. Because close coupling calculations are quite time consuming, we discerned means of improving computational efficiency. Results of our calculations are in good agreement with experimental pressure broadening coefficients of the acetylene ν 2 Raman lines measured in a bath of H 2 at 143 K. Additional comparisons of theoretical Raman Q linewidths with pressure broadening coefficients of IR acetylene lines measured at 173 and 295 K are also encouraging. The computation of the IR pressure line-broadening coefficients is currently in progress in the group and will be compared to available experimental data.

FIG. 1 :FIG. 2 :FIG. 3 :FIG. 4 :FIG. 5 :FIG. 6 :

 123456 FIG. 1: Selected angular cuts of the PES as a function of the intermolecular distance R. Symbols represent the ab initio values. CCSD(T) ab initio energies are compared with SAPT(DFT) values for three angular approaches.

  r CC = 2.283 bohr for C-C bond and r CH = 1.999 bohr for C-H bond of C 2 H 2 , r HH = 1.449 bohr for the H-H bond of H 2 . Computations are performed for a large number of geometries located on a (R, θ A , θ B , φ) grid. The product grid includes 20 values of R (bohr) = [5.00, 5.50, 6.00, 6.25, 6.50, 6.75, 7.00, 7.25, 7.50, 7.75, 8.00, 8.25, 8.50, 8.75, 9.00, 9.50, 10.00, 12.00, 15.00, 20.00], 9 values of θ A and θ B (degree)= [0.00, 11.25, 22.50, 33.75, 45.00, 56.25, 67.50, 78.75, 90.00], 17 values of φ (degree)= [0.00, 11.25, 22.50, 33.75, 45.00, 56.25, 67.50, 78.75, 90.00, 101.25, 112.50, 123.75, 135.00, 146.25, 157.50, 168.75, 180] and results in 27540 geometries.

  2 H 2 -H 2 because the acetylene molecule is longer than the nitrogen molecule. Moreover, we observe stronger binding energies for C 2 H 2 -H 2 relatively to C 2 H 2 -He[64] resulting from the stronger (4 times) H 2 polarizability relatively to the He's one, and from a more favorable electron exchange of C 2 H 2 with H 2 due to the diatomic character of H 2 .

TABLE I :

 I Selected thermally averaged partial PB cross sections at 295 and 1500 K (see eqs.(3)

TABLE II :

 II Temperature dependence of the PB coefficients (see eq.(9))

	j	γ j (295 K)	n T<295 K	n T>295 K
	0	106.66	0.63	0.56
	1	82.74	0.51	0.51
	2	83.67	0.58	0.51
	3	83.97	0.60	0.54
	4	85.37	0.61	0.54
	5	85.96	0.61	0.56
	6	86.87	0.62	0.55
	7	87.11	0.61	0.57
	8	87.61	0.61	0.55
	9	87.69	0.61	0.57
	10	87.89	0.61	0.55
	11	87.88	0.61	0.58
	12	87.55	0.61	0.55
	13	87.57	0.61	0.57
	14	86.98	0.61	0.54
	15	86.83	0.61	0.56
	16	86.33	0.61	0.54
	17	84.93	0.61	0.55
	18	85.17	0.61	0.55
	19	83.90	0.62	0.55
	20	84.28	0.63	0.54
	21	83.29	0.63	0.54
	22	81.67	0.63	0.55
	23	80.93	0.65	0.54
	24	80.61	0.61	0.53

TABLE III :

 III Comparison between measured and calculated pressure broadening coefficients, in 10 -3 cm -1 /Atm, of isotropic Raman lines at 143 K.

	j	theory	exp
	0	174.25
	1	125.0
	2	130.2	139.6 ± 28
	3	135.4	134.7 ± 8.
	4	139.15	144.4 ± 8.
	5	141.30	135.5 ± 6
	6	142.35	149. ± 11
	7	143.7	141.1 ± 8
	8	143.9	144.4 ± 16
	9	143.8	142.9 ± 12
	10	141.9	142.2 ± 17
	11	139.9	140.2 ± 4
	12	138.05	130.2 ± 7
	13	136.15	141.1 ± 9
	14	134.0	121.9 ± 36
	15	136.05	135.3 ± 15
	16	131.85	120.6 ± 27
	17	130.7	121.9 ± 9
	18	130.9
	19	128.55
	20	127.55
	21	122.55
	22	124.05
	23	120.95
	24	122.50
	25	118.6

Acknowledgments

The authors thank Margot Mandy, University of Northern British Columbia, Canada, for her careful reading of this manuscript.


Jean-Louis Clerc

Conformally covariant differential operators for the diagonal action of O(p, q) on real quadrics

Conformally covariant differential operators for the diagonal action of O(p, q) on real quadrics

Introduction

Let S = S n be the sphere of dimension n, equipped with its standard Riemannian structure. The group G = O(1, n + 1) acts conformally on S. For λ ∈ C, let

H λ = {f (x)(dx) λ n , f ∈ C ∞ (S)}
be the space of smooth λ n -densities. The space C ∞ (S) correspond to λ = 0, whereas the space of measures on S having a smooth density with respect to the Lebesgue measure dx on S corresponds to λ = n. The natural action of G on H λ induces a (smooth) representation π λ of G on H λ . The family (π λ ) λ∈C is known in semisimple harmonic analysis as the scalar principal series of representations of G.

Now let G act diagonally on S × S. The tensor product π λ ⊗ π µ of two representations of the principal series has a natural realization on a space H λ,µ of sections of a certain line bundle over S × S. In [START_REF] Beckmann | Singular invariant trilinear forms and covariant (bi)-differential operators under the conformal group[END_REF], R. Beckmann and the present author constructed a family of differential operators on S × S, depending on two complex parameters (λ, µ), which are covariant with respect to (π λ ⊗ π µ , π λ+1 ⊗ π µ+1 ). The construction of these operators uses the heavy machinery of Knapp-Stein intertwining operators (see [START_REF] Knapp | Representation Theory of Semisimple Groups, an overview based on examples[END_REF] for a general presentation). Whereas the covariance property of the operators is intrinsic to their definition, the fact that they are differential operators is 1 much more involved. The problem is transferred (by using a stereographic projection) to the non-compact picture or flat model R n × R n , and is solved through a long computation, using the Fourier transform on R n . See also [START_REF] Clerc | Singular conformally invariant trilinear forms, II The higher multiplicity case, Transformation Groups[END_REF] Section 11 for a slightly different presentation of these results. This procedure was generalized recently to the geometric framework of completion of simple real Jordan algebras (see [START_REF] Ben Saïd | Conformally covariant bidifferential operators on a simple real Jordan algebra[END_REF]).

The present paper gives a more elementary construction of these operators in the geometric setting of the real quadrics. The philosophy behind the present construction is based on the following observation. Let X be a real quadric, and let G be its group of conformal transformations. Then G has an open dense orbit in its diagonal action on X × X which is a reductive symmetric space (see Proposition 3.1 for a more explicit statement). This rich underlying geometric structure explains that it is easy to construct G-covariant differential operators on this open orbit. The next question is to study wether such a differential operator can be smoothly extended to X × X.

The real quadric X is realized as the projective variety associated to the isotropic cone Ξ of the ambient space (V, Q), where Q is a quadratic form on a real vector V . The group of conformal transformations of X is O(Q), acting projectively on X. To construct covariant differential operators on X (or on X × X), it is wise to start with a homogenous G-invariant differential operator on V (or V × V ) and try to induce a differential operator on Ξ (or Ξ × Ξ). This is possible only if the operator on V is "tangential along Ξ"(or along Ξ × Ξ). The corresponding verification is obtained through computations in the Weyl algebra (= algebra of differential operators with polynomial coefficients) of V .

To finish this introduction, let us mention an application of these operators, which is not developed in this article. By restriction to the diagonal, they provide covariant bi-differential operators from X × X to X. As it is possible to compose (appropriate) covariant differential operators on X × X, the restriction process also yields higher order covariant bi-differential operators. These bi-differential operators are generalizations of the classical Rankin-Cohen brackets (see [START_REF] Beckmann | Singular invariant trilinear forms and covariant (bi)-differential operators under the conformal group[END_REF] Theorem 3.4 or [START_REF] Ben Saïd | Conformally covariant bidifferential operators on a simple real Jordan algebra[END_REF] Section 8). A similar approach for Juhl's conformally covariant differential operators from S n to S n-1 was proposed in [START_REF] Clerc | Another approach to Juhl's conformally covariant differential operators from S n to S n-1[END_REF]. [START_REF] Beckmann | Singular invariant trilinear forms and covariant (bi)-differential operators under the conformal group[END_REF] The real quadric and a series of representations of O(p, q)

Let V be a real vector space of dimension n = p + q where p, q are natural integers such that p, q ≥ 1, p + q ≥ 3, and let Q be a quadratic form on V of signature (p, q). Choose a basis e 1 , e 2 , . . . , e p , e p+1 , . . . , e n such that the quadratic form Q is given by

Q(v) = Q(x 1 , x 2 , . . . , x p , x p+1 , . . . , x n ) = x 2 1 + x 2 2 + • • • + x 2 p -x 2 p+1 -• • • -x 2 n .
The corresponding symmetric bilinear form will be denoted also by Q, namely for v = (x 1 , x . . . , x p , x p+1 , . . . , x n ) and w = (y 1 , . . . , y p , y p+1 , . . . , y n )

Q(v, w) = x 1 y 1 + • • • + x p y p -x p+1 y p+1 -• • • -x n y n .
For v ∈ V, v = 0, let [v] = R * v be its corresponding element in the projective space P(V ).

Consider the proper isotropic cone

Ξ = {v ∈ V, v = 0, Q(v) = 0} . For v = 0, the differential dQ(v) = 2Q(v, .
) is = 0 and hence Q = 0 is a regular equation of Ξ near any point of Ξ. The projective quotient X = Ξ/R * is a real quadric.

The group G = O(Q) ≃ O(p, q) preserves Ξ. As the action of G commutes with the dilations, the group G acts naturally on X. As a consequence of Witt theorem, this action is transitive.

An open subset

O (resp. Ω) of V {0} (resp. Ξ) is said to be conical if O (resp. Ω) is stable by all dilations v → rv, r ∈ R * .
For λ ∈ C, ǫ ∈ {±} and for r ∈ R * , let

r λ,ǫ = |r| λ if ǫ = + sgn(r)|r| λ if ǫ = - . Let O be a conical open subset of V {0}, and let (λ, ǫ) ∈ C × {±}. Set F λ,ǫ (O) = {F ∈ C ∞ (O), F (rv) = r -λ,ǫ F (v) ∀r ∈ R * , v ∈ O} .
Similarly, for Ω a conical open subset of Ξ, let

H λ,ǫ (Ω) = {F ∈ C ∞ (Ω), F (rv) = r -λ,ǫ F (v), ∀r ∈ R * , v ∈ Ω},
and simply let H λ,ǫ = H λ,ǫ (Ξ), equipped with its natural Fréchet topology.

For g ∈ G, and F ∈ H λ,ǫ , let

π λ,ǫ (g)F = F • g -1 .
Then π λ,ǫ (g)F belongs to H λ,ǫ and this defines a (smooth) representation π λ,ǫ of G on H λ,ǫ . Homogenous functions on Ξ are interpreted as sections of a corresponding line bundle on X, and conversely, differential operators for these line bundles over X are viewed as differential operators acting on homogenous functions on Ξ. These identifications are tacitly used in the sequel.

The covariant differential operator

Let q ∈ C[V ] be a polynomial on V . There is a unique constant coefficients differential operator, denoted by q ∂ ∂x such that for any

y ∈ V q ∂ ∂x e Q(x,y) = q(y)e Q(x,y) . The operator q ∂ ∂x is G-invariant (i.e. commutes with the action of G) if and only if q is a G-invariant polynomial on V . Choosing q = Q, this yields the d'Alembertian operator = Q ∂ ∂x , which in the coordinates (x 1 , x 2 , . . . , x n ) reads = ∂ 2 ∂x 2 1 + • • • + ∂ 2 ∂x 2 p - ∂ 2 ∂x 2 p+1 -• • • - ∂ 2 ∂x 2 n .
The Weyl algebra is the algebra of differential operators on V having polynomial coefficients. For q ∈ C[V ], the multiplication operator by q is simply denoted by q or q(x), depending on the context. The composition of operators in the Weyl algebra is usually denoted by •. However when multiplication by a polynomial is performed after a constant coefficient differential operators, the symbol • may be omitted. The construction of covariant differential operators on the quadric X is well-known (see e.g. [START_REF] Eelbode | Conformally invariant powers of the Dirac operator in Clifford analysis[END_REF]) and is recalled here, as it is used and serves as a model for the more elaborate constructions to come. Lemma 2.1.

• Q = 2n + 4E + Q ,
where E is the Euler operator given by E = E x,

∂ ∂x = n j=1 x j ∂ ∂x j . Proof. Straightforward computation. Let F ∈ H λ,ǫ . It is possible to extend F to a function F ∈ F λ,ǫ (O) for O a conical neighborhood of Ξ in V {0}.
The restriction of F to Ξ belongs to H λ+2,ǫ . However, the extension is not unique and the restriction of F to Ξ usually depends of the extension.

Proposition 2.1. Let F ∈ H n 2 -2,ǫ .
Let F an extension of F to a conical neighborhood of Ξ as above.Then the restriction of F to Ξ only depends on the values of F on Ξ.

Proof. It is enough to show that if F vanishes on Ξ, then F vanishes on Ξ. But such a function can be written as F = QG, where G is defined in a conical neighborhood O of Ξ and satisfies

G(rx) = r -n 2 ,ǫ G(x) for all r ∈ R * and x ∈ O. But differentiating this relation at r = 1 yields EG = -n 2 G. Hence by Lemma 2.1 (QG) = Q G, which implies for x ∈ Ξ F (x) = Q(x) G(x) = 0 . Proposition 2.1 defines a differential operator on X : F -→ F -→ F -→ F |Ξ , mapping H n 2 -2,ǫ into H n 2 ,ǫ
. Moreover, as commutes with the natural action of G on functions, the operator intertwines π n 2 -2,ǫ and π n 2 ,ǫ .

3 The operators D (λ,ǫ),(µ,η) on (X × X) × .

Let (V × V ) × = {(x, y) ∈ V × V, Q(x, y) = 0}. Clearly, (V × V ) × is a conical dense open subset of V × V , which is invariant under the diagonal action of G on V × V . Similarly, let (Ξ × Ξ) × = {(x, y) ∈ Ξ × Ξ, Q(x, y) = 0} .
Consider the corresponding projective situation, i.e. let

(X × X) × = (Ξ × Ξ) × /(R * × R * ) .
The diagonal action of G on (V × V ) × induces an action on (X × X) × . The next proposition will not be used in the sequel, but, as commented in the introduction, it is in the background of the construction of the covariant differential operators on X × X.

Proposition 3.1. The group G acts transitively on (X ×X) × and the stabilizer of a generic element of (X ×X) × is a subgroup of index 2 in a reductive symmetric subgroup of G.

Proof. Let (x, y) be a representative of an element in (X × X) × . Without loosing any generality, it is possible to assume that Q(x, y) = 1. Similarly, let [x ′ , y ′ ] be a representative of another element of (X × X) × , such that

Q(x ′ , y ′ ) = 1. Recall that Q(x) = Q(x ′ ) = 0 and Q(y) = Q(y ′ ) = 0.
By Witt theorem, there exists an isometry g of (V, Q) such that g(x) = x ′ , g(y) = y ′ , thus proving the first part of the proposition Next, let (x, y)

∈ (V × V ) × . The restriction of Q to the 2-subspace Rx ⊕ Ry is of signature (1, -1). Hence V = (Rx ⊕ Ry) ⊕ (Rx ⊕ Ry) ⊥ ,
and the restriction of Q to (Rx ⊕ Ry) ⊥ is of signature (p -1, q -1). Let σ be the transformation which is +1 on Rx ⊕ Ry and -1 on (Rx ⊕ Ry) ⊥ . If g ∈ G stabilizes both [x] and [y], then g stabilizes Rx⊕Ry and its orthogonal subspace (Rx ⊕ Ry) ⊥ , so that σ

• g • σ = g. Let H = {g ∈ G, σ • g • σ} .
Then H ≃ O(1, -1) × O(p -1, q -1) is a symmetric reductive subgroup of G, and the stabiliser G

[x],[y] of ([x], [y] in G is the subgroup of H of index 2, isomorphic to R * × O(p -1, q -1). Let (λ, ǫ), (µ, η) ∈ C × {±}. For O a conical open set of (V × V ) × , let F (λ,ǫ)(µ,η) (O) = {F ∈ C ∞ (O), F (rx, sy) = r -λ,ǫ s -µ,η F (x, y)}
for all (x, y) ∈ O and r, s ∈ R * . Similarly, for Ω a conical open subset of (Ξ × Ξ) × let H (λ,ǫ)(µ,η) (Ω) be the space of all functions F ∈ C ∞ (Ω) such that F (rx, sy) = r -λ,ǫ s -µ,η F (x, y), for all (x, y) ∈ Ω and r, s ∈ R * .

The space corresponding to Ω = (Ξ × Ξ) × is denoted by H × (λ,ǫ),(µ,η) . The diagonal action of G on Ξ × Ξ induces a representation of G on H × (λ,ǫ),(µ,η) .

Consider the differential operator on (V × V ) × given by

E λ,µ = sgn(Q(x, y)) × |Q(x, y)| + n 2 -µ-1 • ∂ ∂y • |Q(x, y)| -λ+µ • ∂ ∂x • |Q(x, y)| -n 2 +2+λ
.

The operator E λ,µ is well defined on C ∞ (V × V ) × , and commutes with the diagonal action of

G on (V × V ) × . Let F ∈ H × (λ,ǫ),(µ,η) . Extend it to a function F ∈ F (λ,ǫ),(µ,η) (O) where O is conical neigborhood of (Ξ × Ξ) × in (V × V ) × .
For y fixed, the function

G y : x -→ |Q(x, y)| -n 2 +2+λ F (x, y)
is defined and smooth on a conical neigborhood of Ξ y = {x ∈ Ξ, Q(x, y) = 0} and homogenous of degree -n 2 + 2. Hence, by (a localized version of) Proposition 2.1, the restriction to Ξ y of ∂ ∂x G y depends only on the values of F on Ξ y .

Similarly, for x fixed the function

H x : y -→ |Q(x, y)| -λ+µ ∂ ∂x |Q(x, y)| -n 2 +2+λ F (x, y)
is defined and smooth on a conical neighborhood of Ξ x and homogeneous of degree -n 2 + 2. Hence, by the same argument as above, the restriction to Ξ x of the function ∂ ∂y H x depends only of the values of F on Ξ x . These observations and some elementary verifications about the homogeneity and the action of G yields the following proposition.

Proposition 3.2. The operator E λ,µ induces a differential operator

D (λ,ǫ),(µ,η) : H × (λ,ǫ),(µ,η) -→ H × (λ+1,-ǫ),(µ+1,-η) .
The induced operator commutes with the natural actions of G on each of the fonction spaces involved.

To have a better understanding of the behavior of the operator E λ,µ near the singular set where Q(x, y) = 0, a more explicit expression of the operator E λ,µ is needed.

Proposition 3.3. The following identity holds on

(V × V ) × E λ,µ = (- n 2 + 2 + λ)(- n 2 + 1 + λ)(- n 2 + µ)(- n 2 + µ -1)Q(x, y) -3 Q(x)Q(y) (I) +2(- n 2 + 2 + λ)(- n 2 + 1 + λ)Q(x, y) -2 Q(y) •   n j=1 x j ∂ ∂y j   (II) +(- n 2 + 2 + λ)(- n 2 + 1 + λ)(2n -4 + 4µ) Q(x, y) -1 (III) +4(- n 2 + 2 + λ)(- n 2 + 1 + λ) Q(x, y) -1 •   n j=1 y j ∂ ∂y j   (IV ) +(- n 2 + 2 + λ)(- n 2 + 1 + λ)Q(x, y) -1 Q(y) Q ∂ ∂y (V ) +2(- n 2 + 2 + λ)(- n 2 + 1 + µ)(- n 2 + µ) Q(x)Q(x, y) -2   n j=1 y j ∂ ∂x j   (V I) +2(- n 2 + 2 + λ)(- n 2 + 1 + µ)Q(x, y) -1   n j=1 x j ∂ ∂x j   (V II) +2(- n 2 + 2 + λ)(- n 2 + 1 + µ) Q(x, y) -1   n j=1 n k=1 x j y k ∂ 2 ∂x k y j   (V III) +2(- n 2 + 2 + λ)   -2 n j=1 ∂ 2 ∂x j ∂y j + n k=1 y k ∂ ∂x k • Q ∂ ∂y   (IX) +(- n 2 + 2 + µ)((- n 2 + 1 + µ) Q(x, y) -1 Q(x) Q ∂ ∂x (X) +2(- n 2 + 2 + µ)   n j=1 x j ∂ ∂y j   • Q ∂ ∂x (XI) +Q(x, y)Q ∂ ∂y Q ∂ ∂x (XII) .
Proof. Computations are first made on {(x, y) ∈ V × V, Q(x, y) > 0} and it will be indicated at the end how to handle the situation when Q(x, y) < 0. With this extra assumption, it is possible, for ρ any complex number, to replace |Q(x, y)| ρ by simply Q(x, y) ρ . An intermediate calculation yields

Q(x, y) -λ+µ • Q ∂ ∂x • Q(x, y) -n 2 +2+λ = (- n 2 + 2 + λ)(- n 2 + 1 + λ)Q(x, y) -n 2 +µ • Q(y) +2(- n 2 + 2 + λ)Q(x, y) -n 2 +1+µ • n j=1 y j ∂ ∂x j +Q(x, y) -n 2 +2+µ • Q ∂ ∂x .
After a long but straightforward computation, the formula of Proposition 3.3 is obtained.

To finish the proof, it is enough to justify that the same formula is valid on the domain where Q(x, y) < 0. To see this, let Q ′ = -Q be the opposite quadratic form, and let E ′ λ,µ be the differential operator obtained from Q ′ by the same procedure as for obtaining E λ,µ from Q. If Q(x, y) < 0, Q ′ (x, y) > 0, so that the previous computation can be used to evaluate E ′ λ,µ using Q ′ instead of Q. Now each term (from (I) to (XII) corresponding to the explicit expression of E ′ λ,µ , can be rewritten using Q = -Q ′ . But each of the twelve terms labeled from (I) to (XII) is easily seen to be changed to its opposite when changing Q ′ to Q = -Q ′ . The conclusion follows as it is easily seen directly from their definition that E ′ λ,µ = -E λ,µ .

. [START_REF] Clerc | Another approach to Juhl's conformally covariant differential operators from S n to S n-1[END_REF] The operators D reg (λ,ǫ)(µ,η) on X × X For reasons to be explicited later, the term labeled (V III) has to be written differently. Let

E (V III) = Q(x, y) -1 n j=1 n k=1 x j y k ∂ 2 ∂x k ∂y j , and
E x, ∂ ∂y = n j=1 x j ∂ ∂y j , E y, ∂ ∂x = n j=1 y j ∂ ∂x j .
The following elementary commutation turns out to be the key for the sequel.

Lemma 4.1.

E (V III) , Q(x) = 2E x, ∂ ∂y (1)
Q ∂ ∂x , ∂ ∂y , Q(x) = 2E x, ∂ ∂y . (2)
Consider the differential operator

F = E (V III) -Q ∂ ∂x , ∂ ∂y Proposition 4.1. F • Q(x) = Q(x) • F, F • Q(y) = Q(y) • F . (3)
Proof. Use Lemma 4.1, the second half of the statement being obtained by exchanging x and y.

Proposition 4.2. Let f be a smooth function on (Ξ×Ξ) × . Let f be a smooth extension of f to a conical neighborhood O of (Ξ × Ξ) × in (V × V ) × . The restriction of F(f ) to (Ξ × Ξ) × depends only on f and not of the particular extension used.

Proof. It is enough to prove that if f vanishes on (Ξ × Ξ) × , then F(f ) vanishes on (Ξ × Ξ) × . Now such a function f can be written (in a conical neighborhood O of a given ray R * (x 0 , y 0 ) ⊂ (Ξ × Ξ) × ) as f (x, y) = Q(x)g(x, y) + Q(y)h(x, y) .

where g and h are smooth functions on O. From Lemma 4.1 follows

[F , Q(x)] = [F , Q(y)] = 0 and hence F(f ) = Q(x)F(g) + Q(y)F(h) .
When (x, y) belongs to (Ξ×Ξ) × , then Q(x) = Q(y) = 0 and the proposition follows.

Consider now the following decomposition

E λ,µ = E sing λ,µ + E reg λ,µ
where

E sing λ,µ = (I) + • • • + (V II) + 2(- n 2 + 1 + λ)(- n 2 + µ) F + (X) E reg λ,µ = 2(- n 2 + 1 + λ)(- n 2 + µ) Q ∂ ∂x , ∂ ∂y + (IX) + (XI) + (XII) Proposition 4.3.
Let O be a conical neighborhood of (Ξ × Ξ) × , and let f ∈ F (λ,ǫ),(µ,η) (O). Then the restriction of E reg λ,µ f to (Ξ × Ξ) reg only depends on the restriction of f to (Ξ × Ξ) reg .

Proof. From Proposition 3.2, it is equivalent to prove the similar statement for E sing λ,µ . As already argued above, it is suffisant to prove that if f vanishes on (Ξ×Ξ) × , then E sing λ,µ f vanishes on (Ξ×Ξ) × . In the expression of E sing λ,µ (f ), terms corresponding to factors (I), (II), (V), (VI), (X) vanish on (Ξ × Ξ) reg as they contain either a factor Q(x) or Q(y). Terms (III), (IV) and (VII) when evaluated on f vanish on (Ξ × Ξ) reg . This is trivially true for term (III). For terms (IV) and (VII), use the homogeneity condition of f to justify the statement. The extra term in the definition of E sing λ,µ is proportional to F. But F(f ) vanishes on (Ξ × Ξ) × as a consequence of Proposition 4.2. This achieves the proof.

Theorem 4.1. The operator E reg λ,µ induces a map from H (λ,ǫ),(µ,η) into H (λ+1,-ǫ),(µ+1,-η) . Viewed as an operator on sections of line bundles over Ξ × Ξ, it is a differential operator D reg (λ,ǫ),(µ,η) which intertwines the representations π λ,ǫ ⊗ π µ,η and π λ+1,-ǫ ⊗ π µ+1,-η .

Proof. Let f ∈ H (λ,ǫ),(µ,η) , extend it, respecting the homogeneities, to a neighborhood of Ξ × Ξ, still denoted by f . By Proposition 4.3 the value of E reg λ,µ f on (Ξ × Ξ) reg depends only on the values of f on (Ξ × Ξ) reg . The operator E reg λ,µ has polynomial coefficients on V × V . Hence by continuity, the value of E reg λ,µ f at a point in (Ξ × Ξ) is well defined and depends only on the values of f on Ξ × Ξ. The invariance property of E reg λ,µ with respect to the diagonal action of G on V × V follows immediately from the definition of E reg λ,µ and implies the intertwining relation for D reg (λ,ǫ),(µ,η) . Now notice that the term (IX) can be rewritten using

n k=1 y n ∂ ∂x n = n k=1 (y k -x k ) ∂ ∂x k + E x, ∂ ∂x . (4)
A similar modification is possible for the term (XI). Define the operator

F λ,µ = Q(x, y) • Q ∂ ∂y • Q ∂ ∂x +2(- n 2 + 1 + µ)   n j=1 (x k -y k ) ∂ ∂y k   • Q ∂ ∂x +2(- n 2 + 1 + λ)   n j=1 (y k -x k ) ∂ ∂x k   • Q ∂ ∂y -2µ(- n 2 + 1 + µ) Q ∂ ∂x -2λ(- n 2 + 1 + λ) Q ∂ ∂y +4(- n 2 + 1 + λ)(- n 2 + 1 + µ) Q ∂ ∂x , ∂ ∂y .
Proposition 4.4. Let f ∈ H (λ,ǫ),(µ,η) . Extend f to a neighborhood of Ξ × Ξ, preserving the homogeneities. Then the restriction of F λ,µ f to Ξ × Ξ does not depend on the particular extension of f and is equal to D reg (λ,ǫ),(µ,η) f .

Proof. Use (4) and the homogeneities of the extension of f to prove that F λ,µ f coincides with E reg λ,µ f .

Remark. The operator F λ,µ exhibits symmetry with respect to the couples (x ↔ y), (λ ↔ µ) which did not exist for the initial operator E λ,µ . In which is not equal to E λ,µ . However, the process of regularization produces the same differential operator D reg (λ,ǫ),(µ,η) on X × X.

  particular, it is possible to start with the operatorsgn(Q(x, y)) × |Q(x, y)| + n 2 -λ-1 • ∂ ∂x • |Q(x, y)| λ-µ • ∂ ∂y • |Q(x, y)| -n 2 +2+µ ,


Hoa T Le

Christophe Cerisara

Alexandre Denis

Report Transfer Learning of Deep Convolutional Network on Twitter

This report aims at showing the capacity of transfering a deep neural network on char-level on massive dataset Twitter, using distant supervision. We showed that more data could help for the Stanford140 dataset. The best overal result observed is 84% of transfer learning for two sentiment polarity classes (positivenegative) from 16M emoticons subjective SESAMm dataset to small SemEval 2013 dataset. Other learning of three classes (with neutral) or nine classes based on emoticons (happy, laughing, kisswink, playful, sad, horror, shock, annoyed, hesitated) didn't show any advantages yet in the study.

Motivation and Previous Work

The recent succes of deep learning methods comes with a heavy dependence on the massive label data set [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], [START_REF] Hinton | Deep neural networks for acoustic modeling in speech recognition[END_REF]. However, there are a lot of domains where the need of human supervised data could become exponentially expensive. As for the case of Twitter, the number of human-labeled data could never catch up the speed of new creating tweets every day. Distant supervision thus gradually become a prominent method for many twitter application tasks. To profit the scale of big data set, this method collect automatically a large number of tweets via some pre-defined rules (hashtag, lexicons, emoticons,...) and transfer to another smaller specific tasks. For sentiment analysis, [START_REF] Go | Twitter sentiment classification using distant supervision[END_REF] [START_REF] Go | Twitter sentiment classification using distant supervision[END_REF] show that emoticons is the most reliable and effective choice. Indeed, many people have followed this stategy and end up winning Semeval competition for many years ( [START_REF] Tang | Coooolll: A deep learning system for twitter sentiment classification[END_REF], [START_REF] Deriu | Swisscheese at semeval-2016 task 4: Sentiment classification using an ensemble of convolutional neural networks with distant supervision[END_REF]). In these works, they transfer only a shallow convolutional networks [START_REF] Kim | Convolutional neural networks for sentence classification[END_REF] on word-level but achieve considerable result.

Recently, there is a line of work [START_REF] Severyn | Twitter sentiment analysis with deep convolutional neural networks[END_REF] showing that learning directly on char-level on twitter could be a better choice because twitter normally contain a lot of slang, elongated words, contiguous sequences of exclamation marks, abbreviations, hashtags,... For this end, inspiring from computer vision ( [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], [START_REF] He | Deep residual learning for image recognition[END_REF]) [START_REF] Zhang | Character-level convolutional networks for text classification[END_REF] [START_REF] Zhang | Character-level convolutional networks for text classification[END_REF], [START_REF] Conneau | Very deep convolutional networks for natural language processing[END_REF] [START_REF] Conneau | Very deep convolutional networks for natural language processing[END_REF] developped a deep network (6 and 29 convolutional layers respectively) to learn directly on atom-level. As realising a deep network could leverage better with a corresponding massive dataset, we want to explore if it could beat the traditional transfer learning of a shallow network on word-level [START_REF] Go | Twitter sentiment classification using distant supervision[END_REF], [START_REF] Glorot | Domain adaptation for large-scale sentiment classification: A deep learning approach[END_REF].

Experimental results

Sentiment Analysis Data

Stanford140: this consists 1.6 millions tweets, automatically collected in 2009 on a wide range of topics by Twitter Search API. The dataset is balanced with 800k tweets with positive emoticons :) and 800k tweets with negative emoticons :(. The training dataset is already pre-processesd: emoticons are stripped off; any tweet containing both positive and negative emoticons are removed; retweets, duplicated tweets and tweets with ":P" are also removed to reduce the most possible the bias in the dataset (more further details of preprocessing could be followed in [START_REF] Go | Twitter sentiment classification using distant supervision[END_REF]). SESAMm data1 : this consists 3 billion subjective tweets, collected automatically from the period 2014-2016, without any particular filter of preprocessing emoticons. From this pool, we filter out 16M balanced positive/negative emoticons tweets as the rules of [START_REF] Go | Twitter sentiment classification using distant supervision[END_REF], for the ease of comparison. The preprocessing procedure also follow the description above.

Deep Neural Network on Twitter

Deep Neural Network

For all the experiements, we explore the structure of deep model on char-level as described in Conneau et al., 2016 [START_REF] Conneau | Very deep convolutional networks for natural language processing[END_REF]. Because the length of the sequence in twitter is shorter than amazon or movie reviews [START_REF] Conneau | Very deep convolutional networks for natural language processing[END_REF], we study only two structures: (1-1-1) and (2-2-1) double-convolutional blocks with (256-256-128) kernel window size correspondingly. Besides, we use five fully connected layer with the configuration (5632-4096-2048-512-128) neurons respectively to observe better the transfer's process. The hyperparameters are followed [START_REF] Conneau | Very deep convolutional networks for natural language processing[END_REF]: Adam Optimizer with an initial learning rate of 0.001, 128 batch size, no L2 regularization and drop-out. Each character is represented as a one-hot encoding vector where the dictionary contains the following 69 tokens: "abcdef ghijklmnopqrstuvwxyz0123456789-, ; .!? : "/| #%ˆ& ˜ + =<> ()[]". The maximum sequence length is 176; smaller texts are padded with 0 while larger texts are truncated. The convolutional layers are initialized following [START_REF] He | Deep residual learning for image recognition[END_REF]. Training is iterated per steps and we evaluate it at every 1000 steps on test set.

Stanford Emoticon Twitter data

We first apply the model (1-1-1) double convolutional blocks to Stanford140 dataset. To see the benefit of data scale, we evaluate the model on data level: 10k, 60k and 1.6M samples. The results are given in the figure 1 As we see in figure 1, it took 14k steps to converge to 80% on both training and test set (note that the shaded line is the real realisation and the bold line is the fitted average, outputted by TensorBoard). This is a good balance between the depth and the quantity of the data. If we use less data (10k and 60k as in figure 8), the model can overfit easily. We have tried dropout on the last fully connected layer but it didn't help. Also, as we observed, taking into account more convolutional blocks (model (2-2-1)) with this size of dataset didn't gain any more performance. We hypothese that a much deeper network will need a massive corresponding dataset to observe the effect, which is our subject of study in section 2.2.3 and 2.2.4.

Augmenting Stanford Emoticon Twitter data

As Stanford Emoticon Twitter data contains very general samples which will help to generalise better, we're interested in augmenting the quantity of the data to see if the model still can get benefit. We've collected by Twitter Streaming 343k positive and negative tweets (by the same emoticons :) :( as Stanford data) from January to March 2017. The result of this augmented data and bigger model (2-2-1) is a gain of 5% more on the test set, which is precisely 85% overall (figure 2). The model still show good compromise between learning and generalization without too much overfitting. The model is then used to compare with model learnt on big (but subjective) SESAMm dataset in section 2.2.4.

SESAMm data

By the constraint of time for collecting new tweets, we can not explore quickly the model on a large scale. However, SESAMm possess already in hands a very large dataset. We're interested if it still can learn some useful patterns differently than the general case and possibly could help our classifier become more discriminative regards each sentiment polarity. The result show interestingly a level 85% accuracy performance on test set, as the augmented Stanford dataset case. We suspect that this is nearly the best performance we could achieve with distant supervision for two classes. It seems that there's some threshold to determine if putting more data in could help. Generally, the first part of data (the first few millions observations) could set up for the whole performance and the later (dozen millions of observations) could barely or couldn't help at all.

Transfer Learning

In this section, we carry out the transfer learning for two case: augmenting Stanford dataset (2.2.3) and SESAMm data (2.2.4). The transfer was done on the Semeval 2013 training and test set.

The transfer from augmenting Stanford data result in a 75% on Semeval 2013 test set (figure 3) if only fine-tune the last layer. As the training set of Semeval is small, we see that the learning could overfit easily without using dropout. Fine-tune more fully connected layers (2 or 3 last layers) degrade more and more the performance (figure 9). This is coherent with what is observed in the vision domain ( [START_REF] Yosinski | How transferable are features in deep neural networks?[END_REF]), where the very last layer contains very specific feature and the first layers contain very generic feature. [START_REF] Yosinski | How transferable are features in deep neural networks?[END_REF] [START_REF] Yosinski | How transferable are features in deep neural networks?[END_REF] suggested that if the target transfer data is small and the domain is close, one should reuse the whole structure and re-train only the last linear classifier on top. As the nature of SESAMm data is different than those of Stanford data, we're intested in examining the capacity of transfer learning in this case, with dropout on Semeval test set to prevent overfitting. Figure 4 show a net amelioration of 84% on test set. The transfer is showed clearly as the starting point of training and test curve are very high (60% and 75% correspondingly) comparing to without the transfer case. Fine-tune more layer (figure 10) could only lead to overfitting and degrade the performance as observed above.

Qualitative analysis and other approaches

To verify the effectiveness of chosen hyperparemters (maximum 176 characters and alphabet following [START_REF] Conneau | Very deep convolutional networks for natural language processing[END_REF]), we plot the frequency of tweet's length and each characters of the alphabet as below (figure 5). There are 67 characters on average in each tweets, 239 tweets exceed our limit maximum 176 characters lengths (which will be truncated). Most of these tokens are characters and numbers; there is only a small number of punctuations. There is a general worry that if the limit size of the sequence is too large (more than two times of the mean or median), the result of the model will be less precise because there are too many harmful useless padding 0 (on the right side). We've thus tried 160, 140 respectively as maximum sequence length and/or use only character and number in the alphabet but the performance didn't show better. The effect is thus not seen in this data. Beside of learning on two polarity sentiment classes, we evaluated on three classes (with neutral). We consider 800k tweets without emoticons from SESAMm as an approximate for neutral tweets and put it in the same pool of 1.6M positive and negative Stanford data. Though this is not really a good approximation but it works quite well in the litteratures ( [START_REF] Go | Twitter sentiment classification using distant supervision[END_REF], [START_REF] Tang | Coooolll: A deep learning system for twitter sentiment classification[END_REF]). The loss of the precision is hoped to become less important when the data is scaled up massively. The results are shown in figure 6 and 11. In terms of accuracy, we observed only 65% on three classes while it is 55% in precision positive-negative ( [START_REF] Nakov | Semeval-2016 task 4: Sentiment analysis in twitter[END_REF]). This is far from the level that we've seen in two classes and state-of-the-art results (63.3% in precision positive-negative for three classes on Semeval data). This result demonstrate how the task of recognizing just one more class (neutral phrases) is difficult (opposing to vision when more classes could help guide the gradient better). Also, this poor result suggest that the choice of neutral tweets from SESAMm may be not a good choice as subjective tweets tends to have more sentiment than a general tweets. This evoke the same difficulty of constructing the neutral sentences automatically in general (prolific sentences without any sentiment). To alleviate the vague notion of neural class in the source data above, we inpsect a broader learning on 9 classes based on emoticons (happy, laughing, kisswink, playful, sad, horror, shock, annoyed, hesitated) on SESAMm dataset. 2 . We hope that this will help the classifier discriminate better and thus generalize well. The learning on test set is as good as on two classes with 85% accuracy performance (figure 7), however result of transfering to 3 classes didn't get any better. It's hard to know if it does not really help or just because we don't have the real hand-labeled samples from each classes (because the emoticons data are somewhat noisy).

Conclusion

Transfer learning emerge recently as an important theme of research after the success of deep learning. How to profit a lot of data while still require only a little effort of human-labeling is an interesting subject of study. In this work, we've conducted some preliminary experiments of a deep network on char-level specifically for twitter, where unlabled data is abundant. The distant supervision works pretty well for two classes but it still has a lot of limitations when expanding to three or higher number of classes. Because the constraint of the resource in this study, further works could examine more dataset of emoticons to see if it can have better effect. Also, there's still not an effective way to collect automatically neutral sentences in the litterature. Thus, thinking of new method to solve this problem is an important question for the future research.

Acknowledgement

Part of the experiments realized in this work have been realized on two GPU clusters: Grid5000 (Inria/Loria Nancy) and Romeo Reims (France). The subjective dataset is supported generously by SESAMm. We are very grateful to their owners for giving us access to these resources.

Figure 1 :

 1 Figure 1: Results on 1.6M training and test set

  below (and figure 8 in the appendix) (a) Accuracy on 1.943M training and test set (b) Loss on 1.943M training and test set

Figure 2 :

 2 Figure 2: Result of augmented Stanford Emoticon Twitter data

Figure 3 :

 3 Figure 3: Accuracy of transfer learning from augmenting Stanford Emoticon Twitter data to SemEval 2013 training and test set. Fine tune only last fully connected layer, without dropout

Figure 4 :Figure 5 :

 45 Figure 4: Accuracy of transfer learning from 16M tweets SESAMm data to SemEval 2013 training and test set. Fine tune only last fully connected layer, with dropout

Figure 6 :

 6 Figure 6: Accuracy of learning on 3 classes

Figure 7 :

 7 Figure 7: Accuracy of learning on 9 classes

http : //www.sesamm.com/

The classes are chosen from Wikipedia's list: https : //en.wikipedia.org/wiki/List of emoticons