Chad Povey

Mileno Guillorel-Obregon

Oleg G Ivanov

Franck Buzykin

Thibault

Dr Adriana Predoi-Cross
email: adriana.predoicross@uleth.ca

Sergey V Ivanov

Oleg G Buzykin

Franck Thibault

Low pressure line shape study of nitrogen perturbed acetylene transitions in the ν 1 +ν 3 band over a range of temperatures

Keywords: Temperature variable cell, spectroscopy, absorption, control software, laser spectroscopy, acetylene, nitrogen, spectral line shapes, dicke narrowing, pressure broadening, infrared spectroscopy, remote sensing, spectral line parameters

Low pressure line shape study of nitrogen-perturbed acetylene transitions in the ν1 + ν3 band over a range of temperatures

ABSTRACT

Six nitrogen perturbed transitions of acetylene within the ν 1 +ν 3 absorption band have been recorded using a 3-channel diode laser spectrometer. The goal being an improved understanding of both broadening and narrowing effects of low pressure acetylene under the influence of nitrogen gas. To this end we have examined C 2 H 2 spectra using a hard collision Rautian profile over a range of five temperatures (213 K-333 K) and five pressures . From these fits we have obtained the N 2 -broadening and narrowing coefficients of C 2 H 2 and examined their temperature dependence. The experimentally measured narrowing coefficients have been used to estimate the nitrogen diffusion coefficients (D 12 ) and are presented within. The broadening coefficients and corresponding temperature dependence exponents have also been compared to that of calculations completed using a classical impact approach on an ab initio potential energy surface. We have observed a good agreement between our theoretical and experimental results. Accurate detailed knowledge of individual line parameters of the spectrum of acetylene is critical for the correct interpretation and modeling of many planetary atmospheres containing hydrocarbon molecules. The motivation behind this comes from our understanding that acetylene can be either a product or a reactant in many processes that are observed in planetary atmospheres [1][2][3][4][5][6][7][8][9]. Not only do we require good information about the self-interacting molecule but we also require an intimate understanding of perturbed interactions. One of the most important perturbers we can examine is nitrogen as this is a major constituent of many atmospheres such as Earth and Titan [7].

Table 1 shows a summary of the studies previously reporting on nitrogen perturbed acetylene to date. From the table it is easy to see that most of our information has come from studies focused on room temperature measurements [10][11][12][13][14][15][16][17][18][19][20]. However, there have been a handful of studies that examined other temperature ranges [21][22][23][24][25][26][27][28][29]. It is the examination of these temperature dependencies that are vital to improving our interpretation of atmospheric data, as most regions of the atmosphere are not at a constant temperature. Therefore by obtaining temperature dependence exponents for broadening, shift and narrowing we can improve radiative transfer models used to examine data collected from remote sensing observations.

Our choice of the particular line shape of interest we want to examine is also vital to improving our understanding of these transitions. It has been found that the Voigt profile, thought easy to fit does not always obtain the most accurate results [12,14,16,24,25,[27][28][29][30]. This is even more important when considering low pressure gas when collisional narrowing of the Doppler profile known as "Dicke narrowing" takes place [31]. The work presented here adopts the use of the hard collision Rautian line shape profile [32] to examine the temperature dependence of the N 2 -broadening and narrowing coefficients.

Present experimental measurements are compared to theoretical calculation of the collisional linewidths. The calculations presented here are the first ones performed for this system on a true potential energy surface (PES). Indeed, very recently [33] a four dimensional ab initio PES has been determined for the C 2 H 2 -N 2 system. This PES has been computed using a supermolecular method in which the interaction energies are expressed as the difference between the energy of the complex and the energies of the monomers. Namely, the coupled cluster method at a single and double excitations level with triple excitations included perturbatively, i.e. CCSD(T), has been used.

EXPERIMENTAL DETAILS

All of the experimental data presented here were recorded using a 3-channel tunable diode laser spectrometer at the University of Lethbridge. A description of the setup and its functionality can be found in Ref [27]. The New Focus Velocity laser system provides a tunable range of 1.5 to 1.57 µm and can provide a signal-to-noise ratio better than 1000. This high S/N F o r R e v i e w O n l y has proven to be very useful when examining low pressure spectra [29]. The radiation from the laser is directed through two sample chambers. The first is a temperature and pressure controlled chamber with a path length of 1.54 cm where the mixture of N 2 and C 2 H 2 was filled. The second is a slightly smaller chamber with the same path length. This chamber was filled with pure gas at room temperature and used as a calibration source.

The spectra of six transitions P (21), P (19), P (16), R (21), R (19), and R( 16) belonging to the ν 1 +ν 3 absorption band of acetylene were recorded. The gas used for the reference chamber was pure C 2 H 2 provided by Praxair with a quoted concentration of 99.6%. The gas samples kept inside the temperature control chamber was also provided by Praxair and was quoted as being 9.94% C 2 H 2 with the remaining being N 2 . Low pressure tests on the P(11) transition in the ν 1 +ν 3 band showed that the concentration was within 0.5% of the quoted value.

The spectra were recorded at the following temperatures 213, 253, 296, 313, and 333 K. For each set temperature we recorded spectra at 40, 30, 20, 10, and 5 Torr. The temperature of the temperature controlled chamber was monitored by an 8-channel Lakeshore (model 218) temperature monitor and controlled by a NesLab ULT 80 chiller. The gas samples pressures were monitored by two MKS Baratron capacitance manometers with full scale reading of 10 Torr and 100 Torr. The 10 Torr gauge was used for monitoring the gas pressures inside the reference chamber and the 100 Torr gauge was used to monitor the pressures inside the temperature controlled chamber. Once thermal equilibrium was reached the sample temperatures and pressures were stable to within ±0.3 K and ±0.1% during the course of each scan [27,29]. Table 2 shows the experimental details for the R(21) transition with the other five transitions having very similar pressures and temperatures.

The recorded spectra were fit using a multispectral fitting routine [34]. This routine allows for simultaneous fitting of spectra at several different pressures for one set temperature. It also allows for the inclusion of an instrumental line shape (ILS) which is convolved with the spectral line shape of choice. In order to minimize any errors in our results we have used the same ILS as was previously discussed and verified in Ref [29]. The experimental errors associated with the measured broadening and narrowing coefficients are assumed to have the same values as those discussed in Ref [29], where we had found there to be an error no greater than 1% and 3% for our determined broadening and narrowing parameters respectively. Initial parameters for both self-and N 2 -broadening, shift, and narrowing were obtained from Refs [28,29]. The HITRAN database [35] was used for all line positions and intensities. Though not directly observed at the pressures measured here the effects due to line mixing were included within the fitting routine and were set to the values reported in Ref [27].

SPECTROSCOPIC LINE SHAPE ANALYSIS

When fitting recorded laboratory spectra it becomes very important to choose the appropriate line shape function that will mimic the physical conditions of the gas under examination. Typically for remote sensing purposes we use a Voigt line shape function which is

F o r R e v i e w O n l y
the convolution of a Lorentz and Doppler line shape function. The Voigt profile is a fast and reliable way to examine spectra of this nature. However, with the ever improving advances in signal-to-noise ratio for the new remote sensing instruments being developed the more subtle effects that were not noticeable in the past now influencing the errors in our results.

One such effect is known as "Dicke narrowing". This narrowing effect occurs at intermediate pressures when the mean free path of the molecules becomes equal to or less than the wavelength of the incident radiation. When this condition is met a simple Voigt profile will result in the retrieval of broadening coefficients that are smaller than what we would expect to observe. In order to account for this narrowing we can adopt a line shape function that includes the mechanisms of the Dicke narrowing effect.

To take into account for such an effect the soft and hard collision models are often used [32,36]. Despite the physics leading to these models being very different, it was found for the case of self-perturbed spectrum of C 2 H 2 [29] that there was no substantial difference in the residuals produced from fits of these two line shape functions to that of the experimental lineshape even though they did produce slightly different narrowing parameters. Generally, the choice of which line shape function to fit to the experimental lineshape comes down to the masses of the perturber and radiator, as was seen for the self-perturbed case [29] neither lineshape function was preferable. For the case of N 2 perturbed spectrum of C 2 H 2 it was decided to only make use of an un-correlated hard collision Rautian line shape function as the mass of N 2 and C 2 H 2 are very similar. Therefore fitting of the experimental lineshape function using the Galatry lineshape function was assumed to add no extra value to our analysis. Figure 1 shows the spectra of the P(16) line of the ν 1 +ν 3 band measured at 40 Torr. In the figure one can see that when fitting with a Voigt profile we get a 'w'-type residual which is indicative of the broadening in the Dicke narrowing regime. It is clearly shown that once we fit the data using a diffusion profile such as the Rautian we get an improved result. It was found that for the spectra recorded at 333 K the percent difference between the obtained broadening coefficients using the Voigt and Rautian lineshape functions was as high as 6%. This difference in retrieved values diminished along with the temperature as we observed that at lower temperatures (i.e. 213 K) we had only 3% difference.

Below we present the expressions used to retrieve the N 2 -broadened half-width and narrowing coefficients temperature dependences:

              +       - = 2 1 2 ) , ( ) 1 )( , ( ) , ( n o o o o Self n o o o o N T T T p T T T p p T p χ γ χ γ γ (1)               +       - = 2 1 2 ' ' ) , ( ) 1 )( , ( ) , ( n o o o o Self n o o o o N T T T p T T T p p T p χ β χ β β (2) F o r R e v i e w O n l y
In equations ( 1) and ( 2) the reference pressure and temperature are p o = 1 atm and T o =296 K, respectively. γ o N 2 and β o N 2 are the retrieved N 2 -broadened half-width and narrowing coefficients at the reference pressure p o (1 atm) and reference temperature T o (296 K). γ and β are the measured broadened half-width and narrowing coefficients of the spectral line at the total sample pressure of p and temperature T. The total sample pressure p = p N 2 +p self and χ is the ratio of p self to p. n and n' are the temperature dependence exponents of the broadened half-width and narrowing coefficients. The use of equation (2) for examining the temperature dependence of the narrowing coefficient has been justified by the work of both Li et al. [37,38] and Povey et al. [29]. It is important to note that equations ( 1) and ( 2) are merely approximations that will become less valid with increasing temperatures [39]. Using a hard collision Rautian line shape model we have examined the N 2 -broadening and narrowing parameters for six transitions in the ν 1 +ν 3 band of C 2 H 2 . The line parameters were obtained by using a weighted multispectrum analysis software [34]. For one set temperature multispectrum fits were performed on the set of spectra recorded for different pressures. The software uses a non-linear fitting routine to optimize the line parameters by minimizing the residual. For each line examined a simultaneous fit of the intensity, N 2 -broadening and N 2narrowing was completed. The shift was also fit simultaneously however due to the low pressures of the sample gas the results for the shift were not ideal for publication.

Modeled N 2 -Broadend Line Width

In this work we calculate the linewidths of C 2 H 2 infrared spectral lines (electric dipole absorption) broadened by N 2 in the frames of the classical impact approach of Gordon [40,41] for isolated lines (no pressure-induced interference). To model translational and rotational motion of C 2 H 2 and N 2 molecules we use 3D exact classical equations of motion which describe the collision of two rigid linear molecules.

The dynamics were performed on the four dimensional ab initio potential energy surface of Thibault et al [33]. For technical (and computing) reasons this PES has been developed, from 2 to 21 Å, over 85 bispherical harmonics [42,43]. Such a large expansion is required because of the steepness of the repulsive part of the potential at short range: a region of major importance for the calculations of pressure broadening coefficients for temperature greater than about 100 K.

Classical calculations were done for three temperatures of the C 2 H 2 -N 2 mixture: 213 K, 296 K and 333 K and are shown in Figure 2. Exact classical 3D dynamics of C 2 H 2 -N 2 collisions was described by 17 first-order Hamilton differential equations in body-fixed coordinates [44]. Spin statistical weights of perturbing N 2 molecules were taken into account in usual way. Initial intermolecular distance was set large enough (15 Å) to exclude starting interaction between molecules. Other collision parameters (initial conditions for trajectory calculations) were selected via a Monte-Carlo procedure. Uniform sampling was applied to the directions of rotation and initial orientations of both molecules in 3D space. A very effective algorithm of impact parameter b sampling [45] was used in Monte-Carlo calculations which reduces the

N 2 -Broadening of Acetylene Transitions

Table shows the N 2 -broadening coefficients measured for all five temperatures. A comparison of our experimental results to those of the theoretically determined line widths from section 3.1 can be seen in Figure 3. From the figure it can be seen that both sets of data agree with each other quite well with a maximum relative difference no greater than 7%.

Figure 4 shows the results of our room temperature N 2 -broadening coefficients and a comparisons with previous measurements reported in the literature. As can be seen in the figure the linewidths that were retrieved in this study agree reasonably well with most of the previous work completed to date.

Using Equation (1) we have calculated the temperature dependence of the N 2 -broadening coefficients. Both experimentally and theoretically determined N 2 -broadening temperature dependence coefficients n are shown in Table 4 and plotted in Figure 5. In Figure 5 we compare our current results with those found by Rozario et al. [28] and Campbell et al. [26]. Our coefficients of line broadening temperature dependence n reasonably match previous work done for larger values of the quantum number |m|. We immediately notice that as |m| becomes smaller we have more deviation from both the theoretical work presented in this paper and that of Campbell et al. [26]. However, it is important to note that the work done by Campbell et al. [26] where they have smoothed out their results agrees quite well with the proposed theoretical calculations presented here. From Figure 5 it can be seen that for the R-branch transitions we have much better agreement than we get for the P-branch transitions. It is our belief that the initial parameters used for the self-narrowing obtained from Ref [29] have allowed for less error to be introduced into the results obtained for the R-branch transitions. However, for the P-branch transitions we had to make the assumption that initial self-narrowing coefficients were similar to those obtained for the R-branch. At this moment we don't know of any published work showing both the m and temperature dependence of the self-narrowing coefficients for P-branch transitions. In our previous study [29] we determined that only transitions recorded at pressures below 40 Torr were strictly in the Dicke narrowing regime. Therefore only spectra recorded at 40 Torr and below were recorded for this analysis. Figure 6 shows the relationship of the pressure dependent narrowing coefficient for the P( 16) line of the ν 1 +ν 3 band at 296 K plotted against the N 2 partial pressure of the chamber. It is clear from the figure that we have a linear relationship with pressure. This justifies our examination of the narrowing effects in this pressure region.

N 2 -Narrowing of Acetylene Transitions

Table 5 shows the N 2 -narrowing coefficients obtained from fitting the lineshape with the hard collision Rautian profile. The room temperature N 2 -narrowing coefficients from Table 5 are compared in Figure 7 to the values obtained for both self-and N 2 -narrowing by Povey et al. [29], McRaven et al. [14], Dhyne et al. [24,25] and Valipour and Zimmermann [16]. From the figure we see that all studies report similar results. Each of the previous studies that examined both self-and N 2 -narrowing [14,16] report the N 2 -narrowing coefficients to be smaller than the self-narrowing coefficients. This is what we also observe when we compare our previous work [29] with the present results.

The exponents n 1 ' for temperature dependence of N 2 -narrowing are given in Table 5. These temperature dependence coefficients are plotted in Figure 8 along with self-narrowing temperature dependence coefficients from Ref [29]. From the figure it can be seen that the obtained temperature dependence is very similar and for the most part they agree to within one standard deviation of each other.

Diffusion Coefficients

Based on our previous study [29] we have equated the narrowing parameter β o N 2 to the dynamic friction coefficient β o diff in an attempt to estimate the nitrogen diffusion coefficient and its temperature dependence. The dynamic friction coefficient β o diff can be expressed by the following equation [37,46]:

, 2 12 mcD T k B o diff π β = (3)
Here k B is the Boltzmann constant, T is the temperature in kelvin, m is the mass of acetylene molecule, c is the speed of light and D 12 is the nitrogen diffusion coefficient for acetylene.

The mass diffusion coefficient for a gas mixture can be estimated in a general sense using the following equation [47,48]:

2 1 2 1 3 12 ) 1 , 1 ( 12
2 12 2 3 12 2 ) ( ) ( M M M M T T p k D B D + Ω = σ π ε (4) F o r R e v i e w O n l y
where M 1 and M 2 are the molecular weights of C 2 H 2 and N 2 respectively, T is the temperature in kelvin, p is the pressure in atmospheres, ε D = 3/8 as shown in [47]; T 12 is a reduced temperature equal to k B T/ε 12 ; ε 12 is the depth of the isotropic potential and σ 12 is the finite distance at which the intermolecular isotropic potential is zero. From the ab initio PES [33] values of ε 12 = 88.8 cm -1 and σ 12 = 4.015 Å were deduced and used. Finally, Ω 12 (1,1) is a dimensionless reduced collision integral, a function of the reduced temperature; values are tabulated in [47].

Using our obtained β o N 2 narrowing coefficients in equation ( 3) we calculated D 12 for all the temperatures examined and have presented them in Table 6. The temperature dependence of the nitrogen diffusion coefficient was determined using equation ( 4) and was plotted in Figure 9. A comparison of the experimentally estimated, through equation ( 3), nitrogen diffusion coefficients to those obtained by equation ( 4) shows a similar trend in the temperature dependence of the nitrogen diffusion coefficients. However, unlike in our previous work [29] where it was determined that our diffusion coefficients were slightly lower than those obtained theoretically, we see that now our values are slightly larger than what is found using equation ( 4). This result indicates that β o diff > β o N 2 which implies that we found less narrowing as expected on the basis

β o diff = β o N 2
.Such a result was also found in the work done by Wehr et al [49] for the CO-Ar system. We also should expect to see that for the P-and R-branch transitions the diffusion coefficients are similar (for a given m or J" value). In Figure 9 we do in fact see this for the P(21) and R( 21) transition. The fact that we don't see as close agreement between the P(16) and R (16) as well as the P (19) and R (19) may in fact be due to neighboring spectral features that are not properly accounted for as there are many weak transitions in this spectral region. However, it does indicate the need for a closer examination of both self-and N 2narrowing coefficients for both P-and R-branch transitions on a larger scale than what is presented here.

Moreover, our comparison between the binary diffusion coefficients as given by equation (4) and our effective (or optical) diffusion coefficient retrieved by our lineshape analysis cannot be completely satisfactory for the following reasons. First, but this is not the main reason, equation ( 4) is a first order approximation [47,[START_REF] Mccourt | Non equilibrium phenomena in polyatomic gases[END_REF]. Secondly, paraphrasing Varghese and Hanson [START_REF] Varghese | [END_REF] "the two parameters (the mass diffusion constant and the effective diffusion coefficient) appear in different contexts in the theory and hence do not have the same physical significance…". More recently, Wehr et al [49] have also pointed out that both coefficients are related to (or take into account) different collisional mechanism (in addition, many studies found large discrepancies between these two kinds of coefficients, see Refs. in [49]). Therefore, one can only expect a rough agreement between these coefficients and such spectroscopic analysis can only provide an order of magnitude of the true binary diffusion coefficient.

CONCLUSION

Using a 3-channel tunable diode laser system we have for the first time examined the temperature dependence of N 2 -narrowing coefficients for six transitions in the ν 1 +ν 3 absorption band of acetylene. The N 2 -broadening that was obtained while examining the hard collision Rautian line shape profile agrees very well with the theoretical calculations presented here. The full classical values calculated on the very first complete four-dimension ab initio PES [33] for the C 2 H 2 -N 2 system are very encouraging and will allow us to compare the methods for determining pressure broadening coefficients for this system like was recently done for similar systems (N 2 -H 2 , N 2 -N 2 , C 2 H 2 -H 2 , see Refs [52][53][54]). In addition, careful comparisons of both our N 2 -broadening and narrowing coefficients with previous studies have shown strong agreement with each other.

For a more physical interpretation of the narrowing effects in our low pressure spectra we should consider two main effects. The first one is the Dicke narrowing effect discussed in detail in this study. The second contribution to line narrowing arises from the effects of molecular speed-dependence on collisional line broadening. Using our software we were unable to quantify individual contributions to line narrowing. Any analysis that takes into account both Dicke narrowing and speed-dependent collisional broadening would have to be done carefully as to avoid any fitting error due to the highly correlated nature of these two effects. Figure 6: The pressure dependent narrowing coefficient β N 2 for the P(16) and R(16) transitions at 296 K as they vary with the partial N 2 pressure.

  half. The impact parameter was bracketed within the range of (0-12) Å. All the calculations were made using the approximation of mean thermal velocity (MTHV) which greatly reduces CPU time. In MTHV approximation all initial relative velocities of colliding molecules are set equal to their mean thermal relative velocity 8 B being the Boltzmann constant and µ = 13.487, µ being the reduced mass of the C 2 H 2 -N 2 pair). The statistical error of calculated linewidths (i.e. RMS error of Monte-Carlo averaging over initial conditions of collisions) was kept for different values of temperature T and rotational quantum number J" generally at the level less than 1%.

  L. Gomez, S.V. Ivanov, O.G. Buzykin, F. Thibault, Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1942-1949. [54] F. Thibault, L. Gomez, S.V. Ivanov, O.G. Buzykin, C. Boulet, Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 1887-1897.

Figure 1 :

 1 Figure captions

Figure 2 :

 2 Figure 1: Observed and calculated spectra for the P(16) transition of the ν 1 +ν 3 absorption band of C 2 H 2 at a pressure of 40 Torr and temperature of 296 K. The residuals have been shifted and magnified for easier comparison.Figure 2: Classically calculated 3D pressure broadening coefficients at the 3 temperatures studied.Figure 3: Comparison of classically calculated and experimentally determined N 2broadening coefficients for three different temperatures. In each panel the line depicts the theoretically determined broadening coefficients. Figure 4: Panel A shows a comparison of the present N 2 -broadening coefficients with previous studies. Panel B shows a smaller region between m=10 to m=25 where only values obtained from a narrowing model are compared. Figure 5: Temperature dependence exponents n 1 obtained from equation (1).Figure6: The pressure dependent narrowing coefficient β N 2 for the P(16) and R(16) transitions at 296 K as they vary with the partial N 2 pressure. Figure 7: Comparison of the N 2 -Narrowing coefficients obtained at 296 K to previous work.Figure 8: Narrowing temperature dependence coefficient n 1 ' obtained from equation (2). The diamonds show the results from our previous work [29] with selfnarrowing coefficients.Figure 9: Estimated diffusion coefficients obtained from retrieved experimental β o

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 1: Observed and calculated spectra for the P(16) transition of the ν 1 +ν 3 absorption band of C 2 H 2 at a pressure of 40 Torr and temperature of 296 K. The residuals have been shifted and magnified for easier comparison.Figure 2: Classically calculated 3D pressure broadening coefficients at the 3 temperatures studied.Figure 3: Comparison of classically calculated and experimentally determined N 2broadening coefficients for three different temperatures. In each panel the line depicts the theoretically determined broadening coefficients. Figure 4: Panel A shows a comparison of the present N 2 -broadening coefficients with previous studies. Panel B shows a smaller region between m=10 to m=25 where only values obtained from a narrowing model are compared. Figure 5: Temperature dependence exponents n 1 obtained from equation (1).Figure6: The pressure dependent narrowing coefficient β N 2 for the P(16) and R(16) transitions at 296 K as they vary with the partial N 2 pressure. Figure 7: Comparison of the N 2 -Narrowing coefficients obtained at 296 K to previous work.Figure 8: Narrowing temperature dependence coefficient n 1 ' obtained from equation (2). The diamonds show the results from our previous work [29] with selfnarrowing coefficients.Figure 9: Estimated diffusion coefficients obtained from retrieved experimental β o

Figure 7 :

 7 Figure 1: Observed and calculated spectra for the P(16) transition of the ν 1 +ν 3 absorption band of C 2 H 2 at a pressure of 40 Torr and temperature of 296 K. The residuals have been shifted and magnified for easier comparison.Figure 2: Classically calculated 3D pressure broadening coefficients at the 3 temperatures studied.Figure 3: Comparison of classically calculated and experimentally determined N 2broadening coefficients for three different temperatures. In each panel the line depicts the theoretically determined broadening coefficients. Figure 4: Panel A shows a comparison of the present N 2 -broadening coefficients with previous studies. Panel B shows a smaller region between m=10 to m=25 where only values obtained from a narrowing model are compared. Figure 5: Temperature dependence exponents n 1 obtained from equation (1).Figure6: The pressure dependent narrowing coefficient β N 2 for the P(16) and R(16) transitions at 296 K as they vary with the partial N 2 pressure. Figure 7: Comparison of the N 2 -Narrowing coefficients obtained at 296 K to previous work.Figure 8: Narrowing temperature dependence coefficient n 1 ' obtained from equation (2). The diamonds show the results from our previous work [29] with selfnarrowing coefficients.Figure 9: Estimated diffusion coefficients obtained from retrieved experimental β o

Figure 8 :

 8 Figure 1: Observed and calculated spectra for the P(16) transition of the ν 1 +ν 3 absorption band of C 2 H 2 at a pressure of 40 Torr and temperature of 296 K. The residuals have been shifted and magnified for easier comparison.Figure 2: Classically calculated 3D pressure broadening coefficients at the 3 temperatures studied.Figure 3: Comparison of classically calculated and experimentally determined N 2broadening coefficients for three different temperatures. In each panel the line depicts the theoretically determined broadening coefficients. Figure 4: Panel A shows a comparison of the present N 2 -broadening coefficients with previous studies. Panel B shows a smaller region between m=10 to m=25 where only values obtained from a narrowing model are compared. Figure 5: Temperature dependence exponents n 1 obtained from equation (1).Figure6: The pressure dependent narrowing coefficient β N 2 for the P(16) and R(16) transitions at 296 K as they vary with the partial N 2 pressure. Figure 7: Comparison of the N 2 -Narrowing coefficients obtained at 296 K to previous work.Figure 8: Narrowing temperature dependence coefficient n 1 ' obtained from equation (2). The diamonds show the results from our previous work [29] with selfnarrowing coefficients.Figure 9: Estimated diffusion coefficients obtained from retrieved experimental β o

Figure 9 : 2 and equation ( 4 )( 3

 9243 Figure 1: Observed and calculated spectra for the P(16) transition of the ν 1 +ν 3 absorption band of C 2 H 2 at a pressure of 40 Torr and temperature of 296 K. The residuals have been shifted and magnified for easier comparison.Figure 2: Classically calculated 3D pressure broadening coefficients at the 3 temperatures studied.Figure 3: Comparison of classically calculated and experimentally determined N 2broadening coefficients for three different temperatures. In each panel the line depicts the theoretically determined broadening coefficients. Figure 4: Panel A shows a comparison of the present N 2 -broadening coefficients with previous studies. Panel B shows a smaller region between m=10 to m=25 where only values obtained from a narrowing model are compared. Figure 5: Temperature dependence exponents n 1 obtained from equation (1).Figure6: The pressure dependent narrowing coefficient β N 2 for the P(16) and R(16) transitions at 296 K as they vary with the partial N 2 pressure. Figure 7: Comparison of the N 2 -Narrowing coefficients obtained at 296 K to previous work.Figure 8: Narrowing temperature dependence coefficient n 1 ' obtained from equation (2). The diamonds show the results from our previous work [29] with selfnarrowing coefficients.Figure 9: Estimated diffusion coefficients obtained from retrieved experimental β o N 2

Table captions Table 1 : Summary of previous studies examining acetylene broadened by N 2 . Table 2: Sample of experimental conditions for the R(21) transition. Table 3: N 2 -Broadening coefficients for five different temperatures obtained using an uncorrelated hard collision Rautian line shape model. The normal convention of m = -J" for the P branch and m = J"+1 for the R branch is used. Table 4: Comparison of theoretical and experimental results obtained for N 2 - broadening coefficients at 296 K and temperature dependence exponents n 1 . Table 5: N 2 -Narrowing coefficients for the five recorded temperatures using an un- correlated hard collision Rautian line shape model. The last column shows the temperature dependence terms obtained from the data in the table and equation (2). Table 6: Estimated N 2 -Diffusion coefficients obtained from equation (3).

 captions1

-Broadening Coefficients (cm -1 atm -1 ) at 296 K and Determined Temperature Dependence Exponents

  ).

	Vibrational Band ν 5 ν 5 ν 5 ν 5 ν 5 ν 4 +ν 5 -ν 4 0.0693(7) Sample of Experimental Conditions for the R(21) Broadener Temperatures (K) Assignment range N 2 , O 2 297 P(29)-R(25) N 2 , O 2 297 P(35)-R(34) H 2 , N 2 ,He, Ar 147-295 P(8)-R(21) H 2 , N 2 ,He, Ar 296 R(3)-R(34) N 2 173.4 P(29)-R(28) N 2 ,He 296 Q(1)-Q(35) 42.371(42) 3 0.0717 3.37% 0.55(5) Transition Line Temperature (K) Pressure (Torr) # of Recorded 213.9(3) 5.214(5) 4 10.214(10) 4 20.270(20) 4 30.408(30) 4 P(19) -19 R(21) 22 6507.3982 0.0947(9) 0.0832(8) 0.0769(8) 0.0719(7) 0.0704(7) Ref. [10] [17] [21] [18] [22] 0.57 3.35% [19] P(16) -16 6515.7516 R(16) 17 6592.5030 R(19) 20 6598.0774 R(21) 22 6601.6617 R(19) 20 0.0739(7) 0.0757 2.35% 0.66(4) 0.60 7.97% 0.0859(9) 0.0768(8) 0.0693(7) 0.0689(7) 0.0676(7) R(16) 17 0.0752(8) 0.0798 5.73% 0.77(2) 0.67 12.94% 0.0913(9) 0.0823(8) 0.0739(7) 0.0725(7) 0.0673(7) P(16) -16 0.0774(8) 0.0815 5.02% 0.78(4) 0.67 13.66% 0.0985(10) 0.085(9) 0.0752(8) 0.0727(7) P(19) -19 0.0769(8) 0.0775 0.80% 0.67(4) 0.60 9.30% 0.1017(10) 0.0889(9) 0.0774(8) 0.0766(8) 0.0714(7) P(21) -21 0.0708(7) 0.0736 3.77% 0.66(2) 0.57 12.87% Spectra Retrieved N 2 -Broadening Coefficients γ γ γ γ o N 2 (cm -1 atm -1 ) Line m Line Position (cm -1 ) 213 K 253 K 296 K 313 K 333 K P(21) -21 6501.7047 0.0888(9) 0.0798(8) 0.0708(7) 0.0692(7) 0.0665(7) Rautian γ γ γ γ o N 2 Calculated γ γ γ γ o N 2 % Difference Rautian n 1 Calculated n 1 % Difference Retrieved N 2 Line m Retrieved N 2
	F o r R e v i e w 2ν 5 -ν 5 N 2 ,He 296 ν 4 +ν 5 -ν 4 N 2 ,He 183.2,198.2 2ν 5 -ν 5 N 2 ,He 183.2,198.2 F o 253.4(3) 5.267(5) 10.668(11) 20.131(20) F o F o F o ν 4 +ν 5 N 2 , air 296 r 30.185(30) r r r ν 4 +ν 5 N 2 173.2-273.2 ν 4 +ν 5 N 2 298 ν 4 +ν 5 N 2 173.2-298.2 ν 1 +ν 3 H 2 , N 2 ,D 2 , air 295 ν 1 +ν 3 C 2 H 2 , N 2 296 ν 1 +ν 3 N 2 195, 373, 473 R e v i e w R(21) 40.409(40) 295.8(3) 5.264(5) 10.324(10) 19.839(20) 30.013(30) 40.364(40) 5.170(5) 10.250(10) R e v i e w R e v i e w R e v i e w	Q(1)-Q(35) Q(1)-Q(35) Q(1)-Q(35) 4 4 4 P(31)-R(20) 4 P(1)-R(23) P(17)-R(22) R(11)-P(23) P(31)-R(27) P(11) P(25)-R(25) 4 4 4 4 4 4 4 4	[19] [23] [23] [11] [24] [12] [25] [13] [14] [26]
	ν 1 +ν 3 ν 1 +ν 3 ν 1 +ν 5	N 2 , O 2 , CO 2 N 2 C 2 H 2 , N 2 , Ar 312.7(3) N 2 , O 2 ,He,	O n l y P(26)-P(22) P(31)-R(33) O n 19.910(20) 296 213-333 296 4 30.087(30) 4 39.286(39) 4 O n O n O n R(0)-R(7),Q(7)-Q(29) l y 5.499(5) 4 10.898(11) 4 l y l y l y	[20] [28] [15]
	ν 1 +3ν 3	Ar, Ne, 332.5(3)	298 21.153(21)	P(17)-R(22) 4	[16]
		Kr, Xe	29.983(30)	4
			40.022(40)	2

-Narrowing Coefficients β o N 2 (cm -1 atm -1 ) and Temperature Dependence n 1 '

			Retrieved N 2
	Line	m	213 K	253 K	296 K	313 K	333 K	n 1 '
	P(21) -21 0.0279(9) 0.0225(7) 0.0182(6) 0.0177(6) 0.017(5) 1.146(78)
	P(19) -19 0.0273(9) 0.0232(7) 0.0184(6) 0.018(6) 0.0173(6) 1.076(81)
	P(16) -16 0.0256(8) 0.0239(8) 0.0217(7) 0.021(7) 0.0203(6)
	R(16) 17	0.0212(7) 0.0186(6) 0.0165(5) 0.0157(5) 0.015(5) 0.773(9)
	R(19) 20	0.0211(7) 0.0188(6) 0.017(5) 0.016(5) 0.0153(5) 0.721(29)
	R(21) 22	0.0225(7) 0.0209(7) 0.0187(6) 0.0173(6) 0.0163(5) 0.719(82)
			F o
			r
			R e v i e w
					O n
						l y

-Diffusion Coefficients D 12 (cm 2 sec -1 )

	Line	m 213 K	253 K	296 K % Difference 313 K	333 K
	P(21)	-21 0.129(4) 0.190(6) 0.275(9)	43.37%	0.300(10) 0.331(11)
	P(19)	-19 0.132(4) 0.185(6) 0.272(9)	42.74%	0.326(10)
	P(16)	-16 0.141(5) 0.179(6) 0.232(7)	32.77%	0.253(8) 0.278(9)
	R(16)	17 0.170(5) 0.231(7) 0.304(10)	48.73%	0.338(11)
	R(19)	20 0.171(5) 0.228(7) 0.295(9)	47.22%	0.332(11) 0.370(12)
	R(21)	22 0.160(5) 0.206(7) 0.268(9)	41.82%	0.307(10) 0.345(11)
		F o F o F o F o F o F o F o F o F o
		r r r r r r r r r
		R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w
					O n O n O n O n O n O n O n O n O n
					l y l y l y l y l y l y l y l y l y

http://mc.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics

Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada. We would like to further thank the AMETHYST program at the University of Lethbridge for their continued funding and academic support. We are grateful to Dr. Daniel R. Hurtmans of Universite Libre de Bruxelles for making available to us his line shape fitting software. Finally Chad Povey would like to thank his father, Rod Povey (who passed away during the writing of this document) for his long-time support and guidance.


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

Christian Boulet

Qiancheng Ma

LINE COUPLING EFFECTS IN THE ISOTROPIC RAMAN SPECTRA OF N 2 : A QUANTUM CALCULATION AT ROOM TEMPERATURE

Abstract

We present quantum calculations of the relaxation matrix for the Q branch of N 2 at room temperature using a recently proposed N 2 -N 2 rigid rotor potential. Close coupling (CC) calculations were complemented by coupled states (CS) studies at high energies and provide about 10200 two-body state-to state cross sections from which the needed one-body crosssections may be obtained. For such temperatures, convergence has to be thoroughly analyzed since such conditions are close to the limit of current computational feasibility. This has been done using complementary calculations based on the energy corrected sudden (ECS) formalism. Agreement of these quantum predictions with experimental data is good, but the main goal of this work is to provide a benchmark relaxation matrix for testing more approximate methods which remain of a great utility for complex molecular systems at room (and higher) temperatures.

I. INTRODUCTION

Collision-induced rotational energy transfer in molecular gases plays an important role in a variety of gas phase processes, such as pressure broadening and shifting and/or mixing of spectral lines. [START_REF] Hartmann | Collisional effects on Molecular spectra[END_REF] The most accurate calculation of the corresponding cross-sections involves solving sets of coupled differential equations using the close coupling (CC) method. [START_REF] Green | [END_REF] However, even with present day computers, such calculations remain prohibitively expensive for complex molecular systems (e.g. a mixture of polyatomic molecules). As a result of these limitations, various approximate models have been developed. They include fitting laws and more accurate scaling approaches; amongst the latter, those derived from the energy corrected sudden approximation (ECS) seem to be very powerful. [START_REF] Hartmann | Collisional effects on Molecular spectra[END_REF] Alternatively, some researchers have developed purely classical 3 or semi-classical 4,5 calculations of the corresponding relaxation matrices.

In a recent paper, 6 some of us have shown that it is possible to introduce line coupling effects within the semi-classical Robert-Bonamy formalism, 7 leading to a new method for calculating the off -diagonal elements of the relaxation matrix. We have also shown in that paper that considering line coupling leads to better agreement between semi-classical and fully quantum linewidths (diagonal elements of the relaxation matrix W), starting from the same intermolecular potential for the N 2 -N 2 pair. That system was chosen for benchmarking purposes: Thibault et al. 8,9 had previously reported detailed fully quantum calculations of the half-widths of the isotropic Raman Q lines, based on a very recent potential energy surface (PES) proposed by Gomez et al. 10 Before investigating the validity of this new semi-classical approach for the calculation of the off -diagonal elements of W, it is necessary to dispose of fully quantum non diagonal elements which will be used as a reference for testing the semiclassical results. Such CC calculations were previously reported by Green and Huo, 11 but based on the PES of van der Avoird, 12 which has been shown to need some refinements. 8,9,13 More recently, Fonfria et al. 14 have reported CC two -body cross-sections at total energies too low to obtain converged results for temperatures higher than 50 K. It is therefore necessary to considerably extend these calculations in order to work at room temperature. This is the purpose of the present paper; the comparison with the semi-classical results obtained from the method of Ref. 6 will be presented in a forthcoming paper.

In the following section, an outline of the quantum formalism and numerical calculations is given together with a brief summary of the ECS formalism. Sec. III compares the current results with experimental data and previous calculations and provides a brief discussion of the convergence of the calculations, before a brief summary of our findings.

II. THEORY

A. Some definitions

Within the impact and binary collisions approximations, the effects of molecular collisions on isotropic Raman spectra is described by a relaxation matrix W(j' 1 , j 1 ; T) whose rows and columns are labeled by spectral lines [START_REF] Hartmann | Collisional effects on Molecular spectra[END_REF]15,16 (for simplicity, j 1 means here Q(j 1 ); j 1 and j' 1 refer to the spectrally active molecule). At low pressures, only diagonal elements are important since they define the half-widths of non-overlapping lines (   ( , ) , ; j T W j j T  

).

At higher pressures, off-diagonal elements become important and describe line mixing transfer of intensity between overlapping lines. When the PES does not contain any vibrational dependence, W is real and may be written in terms of one -body cross-sections which are, for off-diagonal elements, the negative of ordinary state-to-state cross-sections for rotational excitation (or deexcitation):

    ' 0 ' 2 2 1 1 1 1 1 1 , ; , ; ( ' ; ) 2 2 n v n v W j j T j j T j j T c c         (1)
where n 2 is the perturber density and ̅ is the mean relative velocity. These cross-sections may be written as a weighted sum of two body rotational state to state cross-sections:

    2 2 ' ' 1 1 2 1 2 1 2 ' ;
( ) , , ';

j j j j T j j j j j T         (2)
where j 2 and j' 2 refer to the bath molecule and ρ(j 2 ) is the normalized rotational population of level j 2 (including the spin factor). Finally, we recall that the two -body cross-sections, at a given temperature T, are obtained as a thermal average over the initial relative kinetic energy E kin of inelastic cross-sections ,; )

' ' 1 2 1 2 ( ,
kin j j j j E   :   ' ' ' ' 1 2 1 2 1 2 1 2 2 1 ( , , ; ) ( , , ; ) exp( / )
s kin kin B kin kin E B j j j j T j j j j E E k T E dE k T         (3)
E S is the minimum kinetic energy for the levels j' 1 and j' 2 to become accessible.

In cases where the collision dynamics do not depend on vibrational motion, a sum rule exists which says that, for each line, the diagonal elements of W are equal to the negative of the sum of the off-diagonal elements 16 :

1 1 1 ' ( ' , ; ) 0 j W j j T   (4-a) or   1 1 0 1 1 1 1 ' ( '; ) , ; j j j j T j j T       (4-b)
Another important relation is the detailed balance relationship:

        ' 1 1 1 1 1 1 ; ' ' ; j j j T j j j T        (5)
which results, at a more elementary level, from the detailed balance for the two -body cross-sections:

            ' 1 2 1 2 1 2 1 2 1 2 1 2 , , '; ' ' ', ' , ; j j j j j j T j j j j j j T          (6)
the latter resulting from the microscopic reversibility, at a more elementary level:

                            ' ' ' ' ' ' 1 2 1 2 1 2 1 2 1 1 2 2 ' ' 1 2 1 2 1 1 2 2 2 1 2 1 , , ; 2 1 2 1
' , ' , ;

k k k k j j E j j j j E j j E E j E j E j E j j j j j E E j E j E j E j                   . (7)

B. Quantum dynamical calculations

Quantum dynamical calculations were performed on the four dimensional ab initio potential energy surface of Ref. 10 determined by using symmetry adapted perturbation theory. This interaction potential, between rigid rotors, is expected to be the most accurate to date. [8][9][10]17 In particular, it provides accurate self-broadening coefficients over a wide range of temperatures (77 -1700 K) for Raman isotropic Q lines. 8,9 Close-coupling and coupled states calculations were performed with the MOLSCAT code and its parallel version. [START_REF] Hutson | MOLSCAT version 14[END_REF] Technical details handling the PES and the dynamical calculations can be found in Ref. [8,9]. Both variants of N 2 have been considered, ortho (oN 2 ; even j's) and para (pN 2 ; odd j's). The PES does not allow interconversion between ortho and para species. Following previous work, 11,14,16 the calculations were performed assuming that the two colliding molecules are distinguishable, even for oN 2 -oN 2 or pN 2 -pN 2 collisions, thus neglecting (small) resonance-exchange effects. Moreover, since the experimental results we are comparing with [START_REF] Sitz | [END_REF] involve vibrational excitation of one of the two N 2 molecules, such a contribution will be much smaller than in the case of two molecules in the same vibrational state. Dynamical calculations were performed at various total energies E T , and thus at various kinetic energies

1 2 ( , ) kin T rot E E E j j  
for a given total internal rotational energy

given by E rot (j 1 ,j 2 )=Bj 1 (j 1 +1)+Bj 2 (j 2 +1) with the rotational constant B=1.998 cm -1 . Here we only present results for an ortho active molecule (j 1 , j' 1 even). for rotational energy levels higher are fully CS. In order to illustrate the progress of the computational power, note that the CC calculations of Ref. 11 were limited to total energies up to 200 cm -1 while CS calculations were obtained up to 700 cm -1 .

Because of the limitations of our grid in kinetic energies, for oN 2 -pN 2 and oN 2 -oN 2 we have only retained the cross-sections   ' ' 1 2 1 2 ; j j j j T   respectively between the very first 70 two-body rotational levels (i.e. up to j 1 =10, j 2 =15 associated with a rotational energy of about 700 cm -1 ) and those between the very first 75 two-body rotational levels (i.e. up to j 1 =2, j 2 =18 associated with a rotational energy of about 695 cm -1 ). However experiments measure effective one particle cross sections 14,[START_REF] Sitz | [END_REF]   (2). In evaluating this double sum, it is necessary to include all the significantly populated bath level j 2 , as well as all the significant two-body cross sections (i.e. those with a small inelasticity). The most populated level at 300 K corresponds to j 6-8. We include all the levels up to j 2 =18, which corresponds to 98% of the population and the sum over j' 2 incorporates contributions up to j' 2 =j 2 +20. Therefore, due to the limits of our CC/CS calculations, it was necessary to check the convergence of our CC/CS one body crosssections. We have therefore developed a method inspired by the previous work of Green and Huo 11 : we have used the ECS formalism to generate, when necessary, extrapolated twobody cross-sections (i.e. those not obtained from our CC/CS calculations). If the ECS contribution to Eq. ( 2) remains small, then the one body cross-section may be reasonably considered as a CC/CS result; otherwise it will not be retained as a reference value. It is therefore necessary to recall briefly some basic ECS relations allowing the calculation of all the two body state to state cross sections from a limited set of "fundamental" ones:

1 2 1 2 ( , ) ( , 0,0; ) Q L L L L T    . ( 8
)
Note here that our definition of these basic cross-sections is different from that of Green and Huo 11,16 who used

1 2 1 2 ( , ) (0,0 , ) GH Q L L L L   
, so that the two sets are related by the detailed balance Eq. ( 6).

C. ECS formalism

Within the approach of Green 16 , a non-diagonal two -body cross-section can be deduced from the following scaling relation :

                   1 2 ' ' ' ' 1 2 1 2 1 2 2 2 2 2 ' ' 1 2 1 1 1 2 2 2 1 2 1 2 2 , 1 2 , , ; 2 1 2 1 24 , 2 1 2 1 , 0 0 0 0 0 0 24 , c L L c j j j j T j j L L j L j j L j L L Q L L j j                                   (9)
where ⋯ ⋯ is a 3j-symbol. Eq. ( 9) is applied only for energetically downward collisions (positive transfer from rotation to translation) while upward cross-sections are obtained from detailed balance (Eq. ( 6)). According to De Pristo et al. 20 , the adiabaticity factor

        2 2 1 2 2 1 2 24 , 24 , c c L L j j              
which accounts for the non -resonant character of the collisions, is defined in terms of an average collision duration

/ c c l v  
where l c is an adjustable scaling length. The frequency factor represents the dominant inelasticity for level j 1 ,j 2 and it can be defined in various ways. Following the previous analysis of Green 16 , we used:

  1 2 1 2 , 2 ( ) j j B j j    . ( 10
)

III. RESULTS AND DISCUSSION

In the following, results will be indifferently presented as relaxation matrix elements W j',j in units of 10 -3 cm -1 atm -1 (Eq. ( 1)) or as one body excitation rates 2 1 1 ( '; ) n v j j T   , in units of μs -1 Torr -1 which were adopted by Sitz and Farrow [START_REF] Sitz | [END_REF] to present their experimental data.

A. Determination of the ECS scaling length

As may be seen from Eq. ( 8), the basic

1 2
( , ) Q L L are also given by the CC/CS calculations since they simply correspond to the usual inelastic cross-sections. They are available up to (L 1 =18; L 2 =18) and are given in Table I. Therefore the next step is the determination of the only unknown parameter: l c . This was determined via a least-square fit of Eq. ( 9) to the complete set of 10232 available CC/CS cross-sections. The best fit was obtained for c l .= 2.4 Å. Among the 10232 cross sections, 72% of the ECS predictions were in error by less than a factor of 2, with a root mean square relative error of 0.12; 28% were in error by more, with an extremely large rms relative error of 2.3, reflecting very large errors in a very small number of tiny cross-sections, mostly corresponding to very large inelasticity.

Before continuing, it is worthwhile comparing these results with those previously obtained by Huo and Green (LSQ method in Ref. 11a) with a slightly different procedure : they fitted both the scaling length and the fundamental , Q L L , via the detailed balance), at least for the lowest (L 1 ,L 2 ) couples. It is therefore not surprising that the two methods converge towards very similar solutions for the effective one-body rate constants.

B. Determination of the

(last column).

When the ECS correction is much smaller than the pure CC/CS contribution (less than about a few per cent), the final result may be considered as a confident result: "quite a pure CC/CS cross-section". This is the case for the  plotted as function of j 2 in Fig. 1a. As it appears the most important differences between CC/CS and ECS results appear mainly for high j 2 values which are weighted by small populations in Eq. ( 2). Consequently, as shown in Fig. 2a, the two calculations of Eq. ( 2) lead to values differing only by 20%. One may also observe the good convergence of the CC/CS sum.

Things are more complex for higher j 1 values, as may be seen from Table II-b, which corresponds to j 1 =14. As expected, the accuracy of the ECS predictions is not as good for such a high j 1 value. Moreover, while the ECS supplement remains small (when compared to the CC/CS sum) up to j' 1 =12, it becomes important for j' 1 =16-18 (and higher j' 1 ) and is probably underestimated. This may be easily understood from a similar analysis as above.

We consider the case 14 → ′ 16. As shown in Fig. 1b

C. Comparison with experimental data

As mentioned previously, rotational excitation rates for N 2 (v=1)-N 2 (v=0) collisions have been measured by Sitz and Farrow [START_REF] Sitz | [END_REF] at 298 K by a pump-probe technique and may be used to check the consistency of our final results. Indeed as shown by Huo and Green 11 , rotational collisional rates do not strongly depend on the different vibrational states of the two colliding partners. As can be seen from Table III, there is a good agreement overall between experiment and theory: most of our CC/CS final results are inside or very close to the error bars of the experimental data. Unfortunately, there are large uncertainties on some of the experimental results and further experimental investigation would be of great interest before any new analysis of the remaining discrepancy between theory and experiment. As outlined previously, 11,14 some of the differences may be due to inaccuracy in the PES and/or to the neglect of exchange contributions. While interesting, such problems are beyond the scope of the present paper. , j j ), analyzing at that step the convergence and performing then the thermal average and finally summing over j 2 the partial pressure broadening cross-sections  . Nevertheless, the collisional halfwidths obtained summing the off-diagonal elements of table IV (Eq.(4b)) reproduce the main trend with j 1 of the diagonal elements 9 and lead to very close values to the latter at least for the first few values. Due to the lack of off-diagonal terms for j 1 (j' 1 ) >12 the agreement strongly deteriorates for j 1 >10.

IV. CONCLUSION

To summarize, using the most recent rigid rotor N 2 -N 2 potential reported so far, we have carried out, at the CC and CS levels, room temperature calculations of the relaxation matrix for the isotropic Raman Q branch of N 2 . Such a temperature gives rise numerical problems associated with the convergence of the quantum calculations. However, we have shown that it is possible to quantify the convergence by using the ECS approximation. We then demonstrated that for moderate values of the rotational quantum number 12 j  , the convergence of our CC/CS cross sections has been practically achieved, since ECS complements are limited to a few percent. This fully quantum relaxation matrix is expected to be a benchmark for testing the validity of more approximate methods using the same PES, in particular the semi-classical approach proposed in Ref. 6.

Figure captions

FIG. 1



  that are obtained from Eq.

  the sum rule (Eq. (4-b)) cannot be labeled as a CC/CS result.

  Results labeled ECScorrespond to the same sum except that we used ECS two body cross-sections. Comparison of these two columns gives an idea of the accuracy of the ECS predictions for the (highly averaged) one body cross-sections. The third column gives an ECS estimation of the missing CC/CS terms in the double sum of Eq. (2) (those set to zero). Finally, the CC/CS results were supplemented by adding the ECS correction to the CC/CS results, leading to the final results

	1 σ j (		1 j' T cross-sections ; )

On this basis it was possible to calculate one body cross-sections in different ways, from Eq. (

2

). Some results are given in Table

II

. Results labeled CC/CS used only available CC/CS two body cross-sections, setting all missing ones to zero.

  TableIVsummarizes the main result of the present work: the fully quantum relaxation matrix at room temperature for the isotropic Raman Q branch of N 2 . Note that the diagonal elements are those of Ref. 9. Indeed, since the rule in the present work is to only consider "quite pure CC/CS results" it is not possible to obtain fully converged

	linewidths from Eq. (4b) for all Q(j 1 ) lines. The convergence of the diagonal elements was
	considered in a different way in Ref. 9, by summing first over the post-collisional rotational
	quantum numbers ( ' ' 1 2

  .

	Fig. 2. Convergence of the sum  	1 j		' 1 j T ;			2 max 2 0 j j  		2 ( ) j	2 '  j 		1 j j 2 ,		' 1 j j T 2 , ';		as function
	 of j 2 max . (a) for j 1 =4, j' 1 =6; (b) for j 1 =14, j' 1 =16.  2 ' 1 2 1 2 ' , , '; j j j j j T    (in 10 -3 cm -1 atm -1 ) as function of j 2 . (a) for j 1 =4 ; j' 1 =6; (b) for
	j 1 =14 ; j' 1 =16.

TABLE I .

 I Basic rates for N 2 -N 2 treated as distinguishable particles at 298 K, in μs -1 Torr -1 .

	Column (1) corresponds to the	, → 00 as given by our CC/CS calculations. Column
	(2) gives the corresponding		(00		1 L L 2 , )	obtained from column (1) via the detailed
	balance (Eq. (6)). To be compared with column labeled LSQ in Table III of Ref. 11a.
							6 6	8.14 10 -3	6.14 10 -1
							6 8	3.84 10 -3	2.84 10 -1
	L 1 L 2 0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 16 0 18 2 2 2 4 2 6 2 8 2 10 2 12 2 14 2 16 2 18 4 4 4 6 4 8 4 10 4 12 4 14 4 16	(1) 4.26 10 -1 6.14 10 -2 2.4 10 -2 1.182 10 -2 6.39 10 -3 3.98 10 -3 2.42 10 -3 1.32 10 -3 6.79 10 -4 1.65 10 -1 3.651 10 -2 1.11 10 -2 5.4 10 -3 2.72 10 -3 1.55 10 -3 8.49 10 -4 4.46 10 -4 2.24 10 -4 2.324 10 -2 8.51 10 -3 3.52 10 -3 2.09 10 -3 1.074 10 -3 5.72 10 -4 3.02 10 -4			(2) 2.011 0.456 0.208 0.1 4.67 10 -2 2.22 10 -2 9.35 10 -3 3.2 10 -3 9.4 10 -4 3.673 1.28 0.456 0.217 9.38 10 -2 4.09 10 -2 1.55 10 -2 5.09 10 -3 1.46 10 -3 1.282 5.49 10 -1 2.23 10 -1 1.13 10 -1 4.454 10 -2 1.64 10 -2 5.42 10 -3	6 10 6 12 6 14 6 16 6 18 8 8 8 10 8 12 8 14 8 16 8 18 10 10 10 12 10 14 10 16 10 18 12 12 12 14 12 16 12 18 14 14 14 16 14 18 16 16 16 18 18 18	1.45 10 -3 7.11 10 -4 3.59 10 -4 1.94 10 -4 9.64 10 -5 3.464 10 -3 1.185 10 -3 3.7 10 -4 2.29 10 -4 1.154 10 -4 5.82 10 -5 9.19 10 -4 2.76 10 -4 1.04 10 -4 5.29 10 -5 3.57 10 -5 2.76 10 -4 7.4 10 -5 4.48 10 -5 2.14 10 -5 2.125 10 -5 1.86 10 -5 1.3 10 -5 1.71 10 -5 7.9 10 -6 4.8 10 -6	9.22 10 -2 3.45 10 -2 1.2 10 -2 4.07 10 -3 1.16 10 -3 2.51 10 -1 7.365 10 -2 1.76 10 -2 7.52 10 -3 2.38 10 -3 6.86 10 -4 4.9 10 -2 1.125 10 -2 2.92 10 -3 9.34 10 -4 3.61 10 -4 8.61 10 -3 1.6 10 -3 6.05 10 -4 1.66 10 -4 3.16 10 -4 1.74 10 -4 6.93 10 -5 1.0 10 -4 2.64 10 -5 9.21 10 -6
	4 18	1.58 10 -4			1.62 10 -3

TABLE II .

 II One body rate constants for j 1 → j '1 transitions at 298 K, in 10 -3 cm -1 atm -1 , calculated from Eq. (2). Column labeled CC/CS: sum of the available CC/CS two body rates. Column labeled Equivalent ECS: sum of the same two body rates but obtained from the ECS scaling relation (Eq. (9)). Column labeled ECS complement: sum of the two body rates missing in the CC/CS calculation and calculated from the ECS scaling relation. Final result corresponds to column (2)+column(4). II-a : for j 1 =4; II-b : for j 1 =14.

TABLE II

 II

	-a
	j' 1 CC/CS Equivalent	ECS	Final result
		ECS	complement
	0 2.02	1.88	0.04	2.06
	2 11.43	10.34	0.18	11.61
	6 14.85	12.0	0.24	15.09
	8 8.37	6.17	0.18	8.55
	10 4.5	3.3	0.15	4.65
	12 2.12	1.54	0.12	2.24
	14 0.8	0.58	0.083	0.883
	16 0.2	0.17	0.05	0.25
	18 0.008	0.045	0.037	0.045
	TABLE II-b
	j' 1 CC/CS Equivalent	ECS	Final result
		ECS	complement
	0 0.168 0.098	0.016	0.184
	2 0.873 0.55	0.09	0.963
	4 1.55	1.11	0.18	1.73
	6 2.3	1.79	0.26	2.56
	8 3.21	2.54	0.35	3.56
	10 5.0	3.34	0.47	5.47
	12 10.6	4.93	0.74	11.34
	16 3.13	2.78	0.81	-
	18 0.1	0.99	0.79	-

TABLE III .

 III One body rate constants for j 1 → '1 transitions at 298K, in μs -1 Torr -1 . Comparison of our CC/CS results, completed by ECS corrections, with the experimental data ofSitz and

	Farrow. 19
	j 1 j' 1 experiment CC /CS final results
	0 2 6.64 1.18	7.23
	0 4 3.76 0.83	3.78
	0 6 2.73 0.61	2.58
	0 8 0.86 0.51	1.69
	0 10 0.64 0.12	0.94
	0 12 0.29 0.06	0.45
	0 14 0.22 0.04	0.175
	2 4 5.13 0.58	4.52
	2 6 2.40 0.44	2.83
	2 8 1.52 0.34	1.81
	2 10 0.97 0.22	1.01
	2 12 0.28 0.14	0.49
	2 14 0.12 0.04	0.19
	4 6	4.7 0.6	3.74
	4 8	2.2 0.4	2.12
	4 10	1.4 0.2	1.15
	4 12 0.54 0.13	0.56
	4 14 0.21 0.04	0.22
	6 8 3.37 0.47	3.18
	6 10 2.22 0.29	1.51
	6 12 0.71 0.14	0.77
	6 14 0.23 0.05	0.28
	8 10 2.52 0.41	2.65
	8 12 1.12 0.22	1.07
	8 14 0.29 0.07	0.39
	10 12 2.68 0.43	2.26
	10 14 1.04 0.13	0.72
	12 14 1.83 0.26	1.96


Christian Boulet
email: christian.boulet@u-psud.fr

Qiancheng Ma

Franck Thibault

Line interference effects using a refined Robert-Bonamy formalism: The test case of the isotropic

come

Line interference effects using a refined Robert-Bonamy formalism: The test case of the isotropic Raman spectra of autoperturbed N2

I. INTRODUCTION

As is well known, within the binary collision and impact approximations, the spectral density may be written in terms of a relaxation matrix W, which contains all the dynamics of the active molecule-bath interactions and which is generally not diagonal within the line space. [START_REF] Hartmann | Collisional effects on Molecular spectra[END_REF] At low perturber pressures, when the isolated line approximation is applicable, only the diagonal elements are important; these define the Lorentzian line widths ( k  ) and shifts.

At higher pressures (depending on the distance between adjacent components), the lines overlap and the off-diagonal elements of the W matrix can no longer be neglected since they lead to line mixing effects, i.e. transfer of intensity among the various lines.

Therefore, determination of the whole relaxation matrix W (i.e. including its nondiagonal elements) is essential. Most of the theoretical analyses have been based on fitting or scaling laws, and calculations from first principles starting from a given potential energy surface appeared only recently, thanks to progress in computing power. Of course, a full quantum theory, at the Close Coupling (CC) level is the most accurate approach. However the very large number of coupled channels involved render CC calculations unfeasible for complex molecular systems except for those consisting of two diatomic molecules (in limited situations [START_REF] Fonfria | [END_REF]3 ) or even simpler ones. [4][5][6] In an attempt to overcome this difficulty, alternative approximate theories have been developed, either purely classical 7 or semi-classical. [8][9][10] However, it can be reasonably claimed that the calculation of the whole relaxation matrix for complex molecular systems, starting from the knowledge of the intermolecular potential remains an open problem.

If one considers only the diagonal elements of W, until recently, it was believed that the ultimate refinement of the Anderson-Tsao-Curnutte theory, known as the Robert-Bonamy formalism, 11 can treat molecule-molecule systems reasonably well. However a recent series of papers have shown that this can been achieved only if the intermolecular potential is adjusted. 12,13 When there is no room to adjust the potential (as in those cases where the potentials are accurately known), the RB formalism significantly overestimates the halfwidths, at least for the systems investigated in these works. In a recent paper, 14 we have shown that part of that deficiency was due to an oversimplification of the RB formalism: the neglect of the non diagonality (within the line space) of the matrix elements of the cumulant expansion of the Liouville scattering operator . By removing this approximation, i.e. by including line coupling into the formalism one obtains better results, at least for the calculated halfwidths of the Raman Q lines of the N 2 -N 2 pair, for which benchmarking CC results were available. 12,15 Moreover it becomes possible, with this new formalism to calculate not only the diagonal elements of W but also its off-diagonal elements as well, giving us an opportunity to propose a method allowing line mixing effects to be considered for complex molecular systems (like a mixture of polyatomic molecules). In the present paper, we will still test the new formalism on the Raman isotropic spectra of N 2 which is now possible after recent fully quantum calculations of W. 16 Sec. II gives a brief summary of the symmetrized refined RB formalism which is then applied to the N 2 -N 2 system with a particular emphasis on the various rules that must be verified by the non-diagonal elements. As will be shown, the semi-classical frame leads to some deficiencies in the formalism. Sec. III and IV propose a number of correction schemes in order to obtain a very reasonable agreement with the quantum data.

II. THEORY

A. General expressions

As is well known, within the binary collision and impact approximations, the spectral density can be written as: [START_REF] Hartmann | Collisional effects on Molecular spectra[END_REF] , 0

1 1 ( ) Im l k k k l F d l kd L iW         (1) where (2 1
)

i k i E k k j e Z     
is the population of the initial level of the transition ≡ → , . Moreover, in the following, we will consider a symmetrized version of the formalism previously developed in Ref. 14. With a symmetrization of the density matrix, 18 Eq. (1) can also be written as:

d
, 0 1 1 ( ) Im l l k k k l F d l k d L iW               (2)
The correspondence between the two conventions may be quickly retrieved as shown in Appendix A. In the following, numerical calculations will be made starting from Eq. ( 2), , l k W  matrix elements will be then transformed into , elements in order to compare with the quantum results of Ref. 16.

We have shown in Ref. 14 that a correct way of applying the cumulant expansion, based on a new choice of the implied average, allows line coupling effects to be taken into account.

Following Ben-Reuven 17 we define a basic vector of the line space by:

     - | , ( -1)
( , - ) | i f i i j m J f i f iJ fi m m fi JM C j j J m m M fm m k i . ( 3
)
Then, a matrix element of the relaxation matrix may be expressed in terms of the average of the Liouville scattering operator over the internal degrees of the bath molecule:

,min ' ', , 2 0 1 2 ( ) { , | | , }. 2 ( ) kin c E b f i fi l k kin kin c i i f f J J B c r n v db W W dE E e b dr f i JM S fi JM c k T dr                        (4)
As detailed in appendices A and B of Ref. 14, Ŝ   is expressed via a second order cumulant expansion and a refined definition of the average <…>. This definition must be adapted to the symmetrized version of the formalism. We give this in appendix B together with some elements allowing to build the symmetrized formalism by following the procedures detailed in Ref. 14.

B. Application to the autoperturbed N 2 isotropic Raman Q branch

Recently, we have reported a quantum calculation of the corresponding W matrix based on the potential energy surface (PES) of Ref. 19. Using the same potential, we can check the accuracy of the new formalism by comparing our results with those of Ref. 16. Since the PES of Ref. 19 does not contain any vibrational dependences, S 1 disappears. Moreover for isotropic Raman Q branches (labelled here by a single quantum number since j i = j f ≡ ), within the rigid rotor approximation, S 2 is purely real, so that a matrix element of may be simply written as:

             ' ' ' 2 1 1 1 1 1 1 1 1 , ; ( ) ( ) ' , ; 2 n v W j j T Q j W Q j j W j j j T c ( 5
)
where the cross-section , ; includes a Boltzmann average over the initial relative kinetic energy:

  1 1 1 1 2 0 1 ( ' , ; ) ( ' , ; ) exp( / ) kin kin B kin kin B j j T j j E E k T E dE k T         (6)
Cross-sections were calculated over a large grid of kinetic energies , E kin /k B : 50, 100, 150, 200, 250, 296, 300, 350, 400, 450, 500, 600, 800, 1000, 1200, 2080 and 2400 K.

For a given kinetic energy, the cross-section contains an average over all collisional trajectories labelled by the distances of closest approach r c :

1 1 ( ' , ; ) kin j j E              2 1 1 , ( , ) ' 1 1 2 ( ) ' | | c k in c min S r E c jj r c db dr b j e j dr . (7)
As detailed in the previous paper, 14 after all the matrix elements of 2 S e  within the line space are available, it is easy to calculate the whole relaxation matrix. The intermolecular potential, and consequently

2 S e  do not allow interconversion between ortho and para species. As a result one can divide the whole line space into two independent subspaces corresponding respectively to even and odd j 1 lines. By setting their limits respectively to j 1 = 30 and j 1 = 31

we have calculated two 16x16 sub-matrices of W  . Here we only present matrices corresponding to ortho-N 2 . Table 1 gives the matrix elements As expected the matrix is symmetric. Consequently, after averaging over the kinetic energy (cf. Eq. ( 6)), the non -diagonal elements will verify the detailed balance relation (Eq.

(A-10) in the symmetrized form). As may be shown by comparing with the initial (unsymmetrized) version of the model, 14 this is a consequence of the use of a symmetrized formalism, which automatically provides a matrix verifying the detailed balance principle.

We present in Fig. 1 a comparison between some semi-classical off-diagonal crosssections and the CC/CS results of Ref. 16 for some given values of j 1 . Note here that Eq. (A-7) which gives the correspondence between the conventions of Gordon and Ben Reuven has to be written within the semi-classical approximation. Indeed, our semi-classical formalism is a "classical path" formalism where the rotational degrees of freedom are treated quantum mechanically while translation is treated classically, neglecting any exchange of energy between translation and rotation, by assuming that:

1 1 1 ' 1 ( ' ) ( ) j j B j E E k T j        1. (8)
Inserting Eq. ( 8) into Eq. (A-7) gives:

1 1 1 1 1 1 (2 ' 1) ( ' , ; ) ( ' , ; ) (2 1) SC j j j T j j T j       . (9)
From Fig. 1, it appears that even if the rotational distribution is reproduced reasonably well, the agreement is not so good, particularly for upward cross-sections ( ' 1 1 j j  ), which correspond precisely to the greater inelasticity. They contribute mainly to the overestimation of the linewidths. For such upward transitions, the cross-sections must vanish if the inelasticity is greater than the available kinetic energy, and as is known, an obvious failure of the semi-classical scheme is the prediction of non-zero value for such processes. Moreover, as recalled above, the semi-classical formalism neglects any exchange of energy between translation and rotation. As expected, such an approximation fails for high inelasticity, leading to important deviations in the calculation of the corresponding cross-sections.

We now consider the sum rule (Eq. (A-9) or (A-12)). In the particular case of isotropic Raman spectra and when the collision dynamics do not depend on vibrational motion, this rule holds. 16 Starting from Table 1, one can easily establish that the semi-classical crosssections verify the following sum rule:

1 1 1 1 1 1 1 1 1 1 1 ' ' 1 (2 ' 1) ( , ; ) ( ' , ; ) ( ' , ; ) ( 2 1
)

kin kin SC kin j j j j j j j E j j E j j E j                . (10)
After averaging over E kin , one obtains:

1 1 1 1 1 1 1 ' 1 (2 ' 1) ( , ; ) ( ' , ; ) (2 1) j j j j j T j j T j           (11)
which must be compared to Eq. (A-12). This result was not unexpected since it is also a consequence of the semi-classical approximation (Eq. ( 8)). In other words, the semi-classical W matrix elements verify the semi-classical approximation of the exact quantum sum rule, since instead of Eq. (A-12) one has:

(

)

lk l kk k l k k j W W j           . ( 12
)
This is also a clear indication of one of the weaknesses remaining in our refined formalism, even after including line coupling effects. Indeed, our semi-classical formalism still overestimates the halfwidths, although by a smaller amount than the former Robert-Bonamy formalism. Our results (black squares) are plotted in Fig. 2 and compared with the CC/CS results (blue triangles).

From the present analysis, it appears that part of the remaining differences may be a consequence of the semi-classical approximation. Indeed, since the half-width is connected to the off-diagonal elements via the sum rule, an overestimation of the amplitude of these off-diagonal elements (which are negative) leads automatically to an overestimation of the linewidth.

It will be therefore of interest to try to overcome this limit, i.e. to take into account, at least in a simple way, the exchange of energy between translation and rotation during inelastic processes. This is the goal of the following sections which present possible directions of improvement, following previous works on this topic.

III. HOW TO IMPROVE THE SEMI-CLASSICAL FORMALISM

In the semi-classical scheme of the RB formalism, the single classical trajectory is driven by the isotropic part of the potential, and consequently the kinetic energy is conserved (and not the total energy). Therefore, following Billing, 20 we will consider that the constant kinetic energy U which drives the trajectory is some average kinetic energy and not the initial one. We assume that it can be defined in terms of an effective velocity

' 2 eff v v v   equal to
the arithmetic mean of the relative velocities before (v) and after (v') the inelastic collision:

  2 ' 1 1 1 ' ( ) 2 kin E E j v E j     (13-a)
so that:

      2 2 ' 1 1 1 1 2 4 eff kin kin U v E E E j E j       . ( 13
-b)
is the reduced mass of the colliding pair). Eq. ( 7) then becomes:

1 1 ( ' , ; ) kin j j E      2 1 1 , ( , ) ' 1 1 2 ( ) ' | | c c min S r U c jj c r db dr b j e j dr         (14-a)
and, in order to work within the Gordon's convention, one also introduces:

1 1 1 1 1 1 (2 ' 1) ( ' , ; ) ( ' , ; ) ( 2 1
)

kin kin j j j E j j E j       . (14-b)
As is known, 20 this has the advantage of forcing Following Davis and Boggs 21 and McCann and Flannery, 22 an additional correction has been added, interpreted in Ref. 22 as a "counting of trajectories". Instead of Eq. (14-a), we use therefore:

1 1 ( ' , ; )

kin j j E          2 1 1 , ' 1 1 ( , ) ' 1 1 ( ) 2 ( ) ' | | c c min kin S r U c jj kin c r E E j E j db dr b j e j E d r           . ( 15
)
Finally, the T dependent cross -sections are obtained from the average over the Boltzmann distribution of kinetic energies. Some results are given in Fig. 3 and compared with the CC/CS results of Ref. 16.

Comparing Fig. 1 and3, it appears that these new results are, on the average, in better agreement with the fully quantum results, particularly for the upward transitions ( ' 1 1 j j  ) but not for the downward ones at low j 1 (see for instance j 1 =4). As a consequence the sum rule, when evaluated with the new off-diagonal elements leads to halfwidths which still strongly differ both from the (unchanged) semi-classical diagonal elements and from the CC/CS data, as seen in Fig. 2 (red dots). If the agreement with the benchmark data is better for high j 1 , the halfwidths are conversely smaller than the CC/CS references at low j 1 .

At this stage, it appears that the semi-classical scheme, while leading to a reasonable description of the relaxation matrix elements with off diagonal elements verifying the detailed balance, still fails to provide, via the sum rule, diagonal elements in good agreement with the CC/CS references. It is clear that further work is needed in order to introduce exchange of translation and rotation energies and angular momentum in a more accurate way in the formalism since the very simple method introduced here, while going into the right direction, appears to be too crude.

In some sense, this situation is similar to that encountered in the application of the ECS formalism to the calculation of the relaxation matrix. [START_REF] Hartmann | Collisional effects on Molecular spectra[END_REF]23 In this method, any non-diagonal element of W may be expressed in terms of a set of "fundamental" basic cross-sections ( ,0; ) L T  and an adjustable scaling length l c . This set and l c are then obtained from the observed line widths by a least square fit based on the sum rule. Of course, with such a procedure, residuals subsist between observed and ECS optimized widths (generally around 15%), which may lead to important errors in the calculation of the spectral line shape in regions extremely sensitive to the accuracy of the sum rule (for instance the wings; see Ref.

23).

To overcome this difficulty, Niro et al. 23 have proposed a renormalization procedure, forcing the off-diagonal elements to reproduce exactly the observed widths via the sum rule, while still satisfying the detailed balance. Such a method can be also applied here.

IV. RENORMALIZATION PROCEDURE

The reader will find in Ref. 23 a detailed description of the renormalization procedure.

Here we impose the diagonal elements to be equal to the CC/CS data. Then we renormalize results of Sec. III according to the method of Ref. 23. Comparisons between "renormalized" and CC/CS off-diagonal elements are illustrated in Fig. 4, showing that, in most cases, differences are now small. Although empirical, this procedure offers the possibility of applying the improved Robert-Bonamy formalism to practical situations encountered in various fields (atmospheric physics, combustion diagnostic,..) where one has to calculate a "realistic" relaxation matrix for complex molecular systems. By "realistic" we mean here that the W relaxation matrix, as calculated by the refined RB method, can be normalized to observed widths as given for instance in spectroscopic data bank.

V. CONCLUSION

This study has demonstrated that the refined RB formalism can provide semiquantitative information on the relaxation matrix for complex molecular systems. The approach includes rotational levels of the perturber (which we do not assume to be an effective atomic perturber), and consequently resonance effects that may modify the intensity of the coupling.

Of course, the refined RB method remains a semi-classical formalism in which energy exchange between translation and rotation is neglected. Consequently the relaxation matrix elements cannot verify at the same time both detailed balance and the sum rule. Following Davis and Boggs, 21 we have tried to introduce such exchanges into the formalism in a very simple way. However these corrections, while going in the right direction, were too approximate to provide linewidths via the sum rule in good agreement with the CC/CS references. In our opinion, attention should now focus on the development of techniques for taking into account more accurately the exchange of rotational and translational energies and angular momentum during the collisions. Meanwhile, we have proposed a method to circumvent this difficulty, forcing the offdiagonal elements of the RB-refined formalism to exactly reproduce a given set of line widths. This procedure, when applied to the isotropic Raman spectra of N 2 , gives good results. It may be also applied to more complex molecular systems encountered in various fields of applications. Some such systems were discussed in the conclusion of Ref. 14. In a forthcoming paper, we will consider another test case: the infrared spectra of C 2 H 2 perturbed by N 2 for which a new model potential has recently become available. 26 One could apply the present method to calculate the relaxation matrix W of this system and to analyze the intrabranch (R-R; P-P) as well as interbranch (R-P) coupling.

F dd l k L iW         (A-1) where ( 2 1) (2 1)
i k i E i k k k k j e j Z          , (A-2) with 1 i k E k e Z      . (A-3) k
 is the population of the initial level of the k th transition, while k   is the population exclusive of the degeneracy factor (for simplicity, we omit here all spin factors).

Here however we use a different convention derived from Ben Reuven's one 17 writing the spectral density in a symmetrized form as:

, 0 1 1 ( ) Im l l k k k l F d l k d L iW               . (A-4)
We recall here how to retrieve quickly the correspondence.

Property 1: the intensity of an isolated line is constant and does not depend on the convention used:

2 2 k k k k d d      so that ( 2 1
)

k k k d j d    . (A-5)
Property 2: in the wing, the profile is given by: ,

lk lk k l k l k k l k l k l k l k l W W d d d d                        . (A-6) , ( )( ) ( )( )
Identifying term by term and using the previous results, one obtains:

l lk lk k W W     . (A-7)

About the detailed balance and the sum rule:

In Gordon's convention: Case of isotropic Raman spectra:

lk k kl l W W    (A-
In that case, d k is constant and Eq. (A-11) may be also written as:

(

)

lk lk l l l kk k l k l k k k k j W W W j                      . (A-12)
It should be remembered that lk W  means ( ) lk W T  and contains an average over a Boltzmann distribution of kinetic energy (cf. Eq. ( 4)).

Appendix B

How to derive the symmetrized version of the refined Robert-Bonamy formalism.

The symmetrized formalism may be easily built by following the procedures detailed in Ref.

b i i m i m i m         (B-1)
In terms of this vector, the average < > defined in applying the cumulant expansion is now defined as:

ˆ| | , b b O O          (B-2)
where Ô is a Liouville operator of interest. It is obvious that the normalization condition ˆ1 b I   is satisfied. Then, using Eq. (B-2) a matrix element of < Ô > can be explicitly written as:

                                             2 2 2 2 2 2 ' ( ) 2 2 2 2 2 2 2 2 ˆ' , | | , ' , ; | | , ; 1 ( 1) ( , ) 2 1 ( , ) ' ' , | | , , i i i i J J J b J b j m j m i i f i f i J i m i m m f i f i J f i f i f i J M O fi J M f i J M O f i J M C j j J m m M J C j j J m m M f m i m i m i m O fm im i m i m (B-3)
where a summary notation (m) means summations over all magnetic quantum numbers associated with the absorber molecule and a summation over M J as well. Then, by following the procedures of Ref. 14, one is able to obtain all corresponding formulas in the current symmetrized version. For example, expressions for the diagonal and off-diagonal matrix elements of S 2,middle are given by:

J L i f i f i i f f L L L L i i i i S r j j W j j j j JL C j j L C j j L i i C ii L H                       i i ' (B-4) and 1 1 2 2 2 12 2 2 2 2 ' ', 2, 1 1 1 1 2 2 2 22 2 ( ) 2 ( 1) (2 1)(2 1)(2 1)( 2 1)
{ 16 for selected off-diagonal matrix elements (in 10 -3 cm -1 atm -1 ) at T=298 K; j 1 =4 (a), 6 (b) and 8 (c). (J stands for j 1 and J' stands for '

( ; ) ( , 000) ( ' , 000)} (2 1)(2 1) ( , 000) ( ). i i j j f i fi middle c i f i f J L i f i f f f i i L L L L i i i i i i S r j j j j W j j j j JL C j j L C j j L i i C ii L H                                    i i ' (B ( 1)

  k the reduced matrix element of the tensor coupling matter and light (assumed to be real) and L 0 the diagonal matrix of transition frequencies. Eq. (1) which was used in the quantum study presented in the Ref.16 is based on the Gordon's convention4 since the population k  appears in that equation. Conversely, the formalism developed in Ref. 14 is based on the Ben-Reuven's convention 17 and uses, for example, the matrix element of the density operator, exclusive of the degeneracy factor /

  threshold of excitation. But it is therefore necessary to re-establish the microscopic reversibility (and consequently the detailed balance after averaging over the kinetic energy).

  A-11) corresponds, within that convention, to the double sum rule defined by Filippov and Tonkov24 or Kouzov.25

14 .

 14 We first introduce a symmetrized density vector | b

5 )

 5 respectively. Recall that Eq. (B-5) is applicable only for Q lines since one has used the relation simplify the expression.

FIG. 2 .

 2 FIG.2. Comparison between calculated halfwidths (in 10 -3 cm -1 atm -1 ) for N 2 -N 2 at T=298K; semi-classical diagonal elements of TableI(black squares), SC diagonal elements making use of Eqs. (15) and (A-12) (red disks) and CC/CS results (blue triangles).

FIG. 4 .

 4 FIG.4. Comparison of room temperature selected renormalized semi-classical off-diagonal elements (in 10 -3 cm -1 atm -1 ) with the quantum data of Ref.16; j 1 =4 (a), 6 (b) and 8 (c). (J stands for j 1 and J' stands for ' 1 j ).

Fig

  Fig. 1a Boulet et al

Fig

  Fig. 1b Boulet et al

Fig

  Fig. 1c Boulet et al

Fig

  Fig. 3c Boulet et al

Fig

  Fig. 4a Boulet et al

Fig

  Fig. 4b Boulet et al

Fig. 4c

 4c Fig. 4c Boulet et al

TABLE I

 I Comparison of the present semi-classical calculations with the quantum data of Ref.

	. Matrix elements Figure captions		1 1 ( ' , ; j j E kin	)	for E kin /k B = 377 K (in Å 2 ) and even j's.
	86.44 -11.05 -4.55 -3.11 -2.41 -1.84 -1.34 -0.92 -0.57 -0.32 -0.15 -0.06 -0.02 FIG. 1.
	-11.06 71.59 -12.51 -7.38 -5.59 -4.24 -3.09 -2.11 -1.32 -0.74 -0.35 -0.13 -0.04
	-4.55 -12.51 66.12 -12.05 -8.19 -6.07 -4.39 -3.01 -1.90 -1.07 -0.52 -0.20 -0.06
	-3.12 -7.38 -12.05 62.91 -12.56 -8.27 -5.85 -3.99 -2.53 -1.44 -0.71 -0.29 -0.09
	-2.41 -5.59 -8.19 -12.56 61.32 -12.71 -7.95 -5.30 -3.37 -1.95 -0.98 -0.41 -0.13
	-1.84 -4.24 -6.07 -8.27 -12.71 59.74 -12.67 -7.47 -4.66 -2.72 -1.41 -0.61 -0.21
	-1.35 -3.09 -4.40 -5.85 -7.95 -12.67 57.78 -12.62 -6.99 -4.02 -2.11 -0.95 -0.34
	-0.92 -2.11 -3.01 -3.99 -5.30 -7.48 -12.62 55.29 -12.59 -6.50 -3.39 -1.56 -0.58
	-0.58 -1.32 -1.90 -2.53 -3.37 -4.66 -6.99 -12.59 52.19 -12.56 -5.97 -2.75 -1.07
	-0.32 -0.74 -1.07 -1.44 -1.95 -2.72 -4.02 -6.50 -12.56 48.46 -12.48 -5.34 -2.12
	-0.15 -0.35 -0.52 -0.72 -0.99 -1.41 -2.11 -3.39 -5.97 -12.48 44.18 -12.23 -4.59
	-0.06 -0.14 -0.20 -0.29 -0.41 -0.61 -0.95 -1.56 -2.75 -5.34 -12.23 39.43 -11.75
	-0.02 -0.04 -0.06 -0.09 -0.13 -0.21 -0.34 -0.58 -1.07 -2.12 -4.59 -11.75 34.37

Acknowledgments

Q. Ma acknowledges financial support from the National Science Foundation under grant 1228861. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The authors thank Dr Kevin Dunseath (Rennes) for his careful reading of the manuscript.

Appendix A

Comparison between different conventions. In Gordon's convention, 4 which is used by Thibault et al., 16 the spectral density is written as: