Line intensities of acetylene: Measurements in the 2.5-μm spectral region and global modeling in the Δp=4 and 6 series

Line intensities of acetylene: Measurements in the 2.5-μm spectral region and global modeling in the Δp=4 and 6 series

ARTICLE IN PRESS Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523 www.elsevier.com/locate/jqsrt Line intensities of acet...
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ARTICLE IN PRESS

Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523 www.elsevier.com/locate/jqsrt

Line intensities of acetylene: Measurements in the 2.5-mm spectral region and global modeling in the Dp ¼ 4 and 6 series O.M. Lyulina, V.I. Perevalova,, J.-Y. Mandinb, V. Danab, F. Gueyeb, X. Thomasc, P. Von der Heydenc, D. De´catoirec, L. Re´galia-Jarlotc, D. Jacquemartd, N. Lacomed a

Laboratory of Theoretical Spectroscopy, Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, 1, Akademicheskii av., 634055 Tomsk, Russia b Laboratoire de Physique Mole´culaire pour l’Atmosphe`re et l’Astrophysique, Universite´ Pierre et Marie Curie-Paris 6; CNRS, UMR 7092, Case courrier 76, 4, place Jussieu, 75252Paris Cedex 05, France c Groupe de Spectrome´trie Mole´culaire et Atmosphe´rique, CNRS, UMR 6089, Universite´ de Reims-Champagne-Ardenne, Faculte´ des Sciences, BP 1039, 51687 Reims Cedex 2, France d Universite´ Pierre et Marie Curie-Paris 6, Laboratoire de Dynamique, Interactions et Re´activite´; CNRS, UMR 7075, Case courrier 49, Baˆt F 74, 4, place Jussieu, 75252 Paris Cedex 05, France Received 1 June 2006; received in revised form 27 June 2006; accepted 3 July 2006

Abstract More than 670 line intensities of nine perpendicular bands of acetylene are measured in the 2.5-mm spectral region using a step-by-step interferometer. Absolute values of line intensities are obtained with an average accuracy of 5%. Vibrational transition dipole moment and Herman–Wallis coefficients are determined for each studied band. These measured line intensities, and those previously measured in the 3.8-mm region [Jacquemart D, Lacome N, Mandin JY, Dana V, Lyulin OM, Perevalov VI. Multispectrum fitting of line parameters for 12C2H2 in the 3.8-mm spectral region. JQSRT, submitted for publication], are treated simultaneously within the framework of the effective operators approach. The sets of effective dipole moment parameters obtained reproduce the observed line intensities within the experimental uncertainty. The good predictive ability of the modele is demonstrated. r 2006 Elsevier Ltd. All rights reserved. Keywords: Acetylene; Infrared spectroscopy; Vibro-rotational transitions; Line intensities; Transition dipole moment; Effective operators; Global treatment

1. Introduction This paper is aimed to pursue and improve the modelling of the line intensities of the acetylene molecule, within the framework of the method of effective operators [1–3] on the basis of the observed line intensities. To do that, substantial and accurate sets of individual line intensities are needed. In a recent work [4], about 250 line intensities were measured in the 3.8-mm spectral region, enriching noticeably our knowledge of acetylene Corresponding author. Tel.: +7 3822 491794; fax: +7 3822 492086.

E-mail address: [email protected] (V.I. Perevalov). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.07.002

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intensities. According to notations adapted to a global treatment [1–3], this spectral domain concerns the series of vibrational transitions DP ¼ 4, with P the pseudo-quantum number P ¼ 5V 1 þ 3V 2 þ 5V 3 þ V 4 þ V 5 , where the V i’s are the usual vibrational quantum numbers. This series involved interacting vibrational states belonging to the polyads {4n5} through two cold bands, and {5n5} through three hot bands. To extend the knowledge of line intensities, we have studied the acetylene spectrum between 3750 and 4200 cm1. About 670 line intensities, from nine bands, were measured in this spectral region, with an average accuracy of 5%. The concerned transitions belong to the series DP ¼ 6, involving the polyads {6n5} through four cold bands, and {7n5} through five hot bands. In the first part of the paper, the spectra and the methodology used to retrieve line positions and intensities will be presented, and the vibrational transition dipole moments and empirical Herman–Wallis coefficients, deduced from the usual empirical data reduction, will be reported. In the second part of the paper, the results of the simultaneous fittings of the newly measured line intensities in the series DP ¼ 6 and those in the series DP ¼ 4, performed within the framework of the method of effective operators, will be presented. 2. Line intensity measurements in the 2.5-lm spectral region of acetylene 2.1. The acetylene spectrum at 2.5 mm and experimental details To have an idea of the spectrum of acetylene around 2.5 mm, we have gathered the main spectroscopic information in Table 1. In this part of the paper, the vibrational notations of Plı´ va [5,6] are used, in order to facilitate comparisons with previous works who also used the same notations. Thus, vibrational levels are noted V1 V2 V3 (V4 V5)‘7r, with ‘ ¼j ‘4 þ ‘5 j, ‘t being the vibrational angular momentum quantum number associated with the degenerate bending mode t, 7 being the symmetry type for S vibrational states (‘ ¼ 0), and r a roman numeral indicating the rank of the level, by decreasing energy value (r ¼ I for the highest energy level), inside the set of states having the same vibrational symmetry, and coupled by ‘-type resonances. In Table 1, we have recalled the upper vibrational state of each band, and the polyad of interacting states to which it belongs. The nine bands studied in this work are particularly interesting for practical applications (see, e.g., [7]), since they are perpendicular bands exhibiting strong Q-branches. Note that two of the observed hot bands are of the D P type, and therefore are ‘-type doubled. Original information on the 2.5-mm bands of acetylene comes from the Wiggins et al.’s paper [8] in 1961. Later on, Palmer et al. [9] gave vibro-rotational assignments in the main bands n2+(2n4+n5)1 II and n3+n14. (Note that their original vibrational assignments have been interchanged in the present work according to the principle values of the expansion coefficients of the eigenfunctions obtained using our set of effective Hamiltonian parameters [2].) The seven remaining bands were studied by D’Cunha et al. [10]. As far as line intensities are concerned, to our knowledge only integrated intensities of Q-branches or band systems were published by Rinsland et al. [7] and Koops et al. [11]. As Q-branches are observed in the present work, let us Table 1 Bands of acetylene

12

C2H2 studied around 2.5 mm (DP ¼ 6 series of vibrational transitions)

Band

Centera

Ref.b

Upper level

Upper level polyad

Symmetry

n2+(2n4+n5)1 IIc n3+n1c 4 n1+n15 n2+3n15 n1+2n05n15 n1+2n25n15 n1+(n4+n5)0+n14 n1+(n4+n5)0n14 n1+(n4+n5)2n14

3882.41 3898.34 4092.35 4140.06 4069.80 4083.87 4060.76 4075.97 4079.19

Palmer et al. [9] Palmer et al. [9] D’Cunha et al. [10] D’Cunha et al. [10] D’Cunha et al. [10] D’Cunha et al. [10] D’Cunha et al. [10] D’Cunha et al. [10] D’Cunha et al. [10]

010(21)1 II 001(10)1 100(01)1 010(03)1 100(02)0+ 100(02)2 100(11)0+ 100(11)0 100(11)2

{6n5} {6n5} {6n5} {6n5} {7n5} {7n5} {7n5} {7n5} {7n5}

Pu’S+g Pu’S+g Pu’S+g Pu’S+g S+g’Pu Dg’Pu S+u’Pg Su’Pg Du’Pg

a

Rough values of band centers (in cm1) are reported only as a guide. Main references giving vibro–rotational assignments and experimental line positions. c The Palmer et al.’s [9] original vibrational assignments have been interchanged for these two bands. b

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quote some line-mixing studies: line-mixing studies in the Q-branches of three bands of the 2.5-mm region, by Pine and Looney [12], and line profile studies in the Q-branch of the n1+n15 band by Pine [13]. Experimental conditions and characteristics of the nine spectra obtained with the GSMA step-by-step interferometer [14,15], and used to study C2H2 around 2.5 mm, have been gathered in Table 2. This apparatus was used with a globar as radiation source, SiO2 splitting and mixing plates, InSb detector, and CaF2 cell windows. The recording time of an interferogram is about 7 h 20 min. Pressures were measured with a MKS Baratron 100-Torr manometer with an accuracy of 0.5%. On the spectra, H2O lines arise due to traces of water vapor in the evacuated tank of the interferometer. Those lines are useful to calibrate the wavenumber scale. 2.2. Measurement procedure and results The multispectrum procedure [16] used to derive line positions and intensities from the spectra has already been detailed in previous works (see, e.g., [17,18]), so that only what is important is recalled here. First of all, to accurately take into account the throughput in the calculation of the apparatus function, an effective value of the iris radius had to be determined for each spectrum (see Table 2). For that, a number of isolated C2H2 lines, well spread along the studied spectral domain, were chosen. Then, empirical linear laws of the phase error, vs. wavenumber, were obtained for each spectrum, allowing a precise determination of line positions. For an absolute calibration of the wavenumber scale, HITRAN positions of H2O lines were used between 3750 and 4200 cm1 [19]. The zero transmission level was checked with the aid of saturated lines, and it appeared that it did not need to be corrected. Finally, as the maximum pressure was relatively low (see Table 2), a Voigt profile was used to calculate the absorption coefficient of the lines, and the self-broadening Table 2 Experimental conditions and characteristics of the spectra recorded in the 2.5-mm region using the stepping-mode interferometer in Reims Commercial sample (Air Liquide Alphagaz) 97.760% of 12C2H2 Natural C2H2 Stated purity 99.55% Maximum path difference 106.6 cm (spectra 1 and 2) 108.4 cm (spectra 3–9) Unapodized FWHM resolution E 4.7  103 cm1 Spectral step after post-zero filling 0.75  103 cm1 (spectra 1 and 2) 1.08  103 cm1 (spectra 3–9) SNR E500 Collimator focal length 1040 mm Nominal iris radius 2.25 mm Free spectral range 3160–4740 cm1 (spectra 1 and 2) 3385–4515 cm1 (spectra 3–9) Studied spectral domain 3750–4200 cm1 Spectrum number

Effective iris radius (mm)a

Total pressure 70.5%b (Torr)c

Absorbing path 71 cmb

Temperature 70.5 Kb

1 2 3 4 5 6 7 8 9

2.21(7) 2.20(4) 2.23(2) 2.23(2) 2.21(3) 2.17(3) 2.21(3) 2.22(1) 2.19(4)

29.98 3.001 10.02 2.999 1.000 10.00 3.000 1.000 1.034

3221.6 3221.6 2420.3 2420.3 2420.3 417.2 417.2 417.2 31.186d

294.55 294.45 294.25 294.95 295.25 296.55 296.55 296.85 295.85

a

1 SD between parentheses in unit of the last digit. Absolute uncertainty (excess digits are given only as a guide). c 1 Torr ¼ 1.333 hPa. d Absolute uncertainty 70.1 cm. b

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coefficients were fixed to the values calculated according to Ref. [20]. As far as the self-shifting coefficients are concerned, they were fixed to zero. The spectrum was not numerically apodized. Finally, 674 absolute line intensities have been obtained. Their accuracy is estimated to be 5% on the average. Absolute line positions were measured with an accuracy between 104 and a few 103 cm1, depending upon the line intensity. These results are reported in Tables 3–11. 2.3. Empirical data reduction For each line intensity S(T0) obtained from the multispectrum fitting procedure, in cm molecule1 at the standard temperature T 0 ¼ 296 K, we used the following formula to deduce the transition dipole moment squared jRj2 , in D2 (1 debye ¼ 3.33546  1030 C m) SðT 0 Þ ¼ ð1=4p0 Þð8p3 =3hcÞ½g00 n0 =gV QðT 0 ÞjRj2 LðJ; ‘Þ expðhcE 00 =kT 0 Þ½1  expðhcn0 =kT 0 Þ, 36

3

2

27

(1) 7

where 1/4pe0 ¼ 10 erg cm D ; h is Planck’s constant equal to 6.6260755  10 erg s (1 erg ¼ 10 J); c is vacuum speed of light equal to 2.99792458  1010 cm s1; g00 is the statistical weight due to nuclear spin of the lower level (1 for s-type levels and 3 for a-type levels); n0 is the transition wavenumber in cm1; gV is a weight introduced in case of bands with ‘-type doubling (gV is equal to 2 in such a case, otherwise, it is equal to 1); Q(T0) is the total internal partition function at temperature T0; L(J,‘) is the Ho¨nl-London factor, J being the rotational quantum number of the lower level of the transition, and ‘ its secondary vibrational quantum number (‘ ¼ j‘4 þ ‘5 j); E00 , in cm1, is the energy of the lower level; k is Boltzmann’s constant equal to 1.380658  1016 erg K1. For perpendicular bands (D‘ ¼ 1), the Ho¨nl–London factors are given by LðJ; ‘Þ ¼ ðJ þ 2 þ ‘D‘ÞðJ þ 1 þ ‘D‘Þ=½2ðJ þ 1ÞðRbranchÞ,

(2)

LðJ; ‘Þ ¼ ðJ þ 1 þ ‘D‘ÞðJ  ‘D‘Þð2J þ 1Þ=½2JðJ þ 1ÞðQbranchÞ,

(3)

LðJ; ‘Þ ¼ ðJ  1  ‘D‘ÞðJ  ‘D‘Þ=ð2JÞðPbranchÞ.

(4)

Table 3 Line positions and intensities for the band n2+(2n4+n5)1 II of the

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

%

|R|2obs

Pee43 Pee42 Pee41 Pee39 Pee38 Pee37 Pee36 Pee35 Pee34 Pee33 Pee32 Pee31 Pee30 Pee29 Pee28 Pee27 Pee26 Pee25 Pee24 Pee23 Pee22 Pee21 Pee19 Pee18

3768.35518 3771.28746 3774.20340 3779.98684 3782.85320 3785.70559 3788.54137 3791.36138 3794.16589 3796.95495 3799.72858 3802.48701 3805.23049 3807.95900 3810.67278 3813.37188 3816.05633 3818.72689 3821.38302 3824.02518 3826.65344 3829.26804 3834.45661 3837.03090

4.43E25 2.35E25 1.07E24 2.56E24 1.31E24 5.93E24 2.77E24 1.18E23 5.55E24 2.32E23 1.09E23 4.43E23 2.02E23 8.01E23 3.58E23 1.37E22 5.96E23 2.26E22 9.50E23 3.47E22 1.43E22 5.11E22 7.06E22 2.74E22

4.46E25 2.34E25 1.09E24 2.52E24 1.25E24 5.56E24 2.70E24 1.17E23 5.56E24 2.34E23 1.09E23 4.47E23 2.02E23 8.13E23 3.58E23 1.40E22 6.01E23 2.30E22 9.58E23 3.56E22 1.45E22 5.23E22 7.29E22 2.81E22

0.68 0.43 1.87 1.56 4.58 6.24 2.53 0.85 0.18 0.86 0.00 0.90 0.00 1.50 0.00 2.19 0.84 1.77 0.84 2.59 1.40 2.35 3.26 2.55

8.99E05 8.97E05 8.64E05 8.66E05 8.78E05 8.79E05 8.32E05 8.07E05 7.85E05 7.66E05 7.64E05 7.41E05 7.36E05 7.14E05 7.13E05 6.86E05 6.84E05 6.67E05 6.61E05 6.38E05 6.36E05 6.19E05 5.93E05 5.87E05

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Table 3 (continued ) Line

Position

Sobs

Scalc

%

|R|2obs

Pee17 Pee16 Pee15 Pee14 Pee13 Pee12 Pee11 Pee 9 Pee 8 Pee 7 Pee 6 Pee 5 Pee 4 Ree 0 Ree1 Ree2 Ree3 Ree5 Ree6 Ree7 Ree10 Ree11 Ree12 Ree13 Ree14 Ree15 Ree16 Ree17 Ree18 Ree19 Ree20 Ree21 Ree22 Ree23 Ree24 Ree25 Ree26 Ree27 Ree28 Ree29 Ree30 Ree31 Ree32 Ree33 Ree34 Ree35 Ree36 Ree37 Ree38 Ree39 Ree41

3839.59215 3842.14038 3844.67570 3847.19844 3849.70853 3852.20615 3854.69138 3859.62522 3862.07394 3864.51063 3866.93535 3869.34812 3871.74904 3883.57492 3885.90420 3888.22146 3890.52651 3895.09978 3897.36780 3899.62313 3906.31198 3908.51532 3910.70525 3912.88177 3915.04407 3917.19214 3919.32618 3921.44580 3923.55084 3925.64085 3927.71565 3929.77507 3931.81880 3933.84681 3935.85870 3937.85425 3939.83332 3941.79550 3943.74083 3945.66904 3947.57943 3949.47252 3951.34756 3953.20481 3955.04344 3956.86404 3958.66550 3960.44852 3962.21232 3963.95704 3967.38801

9.34E22 3.54E22 1.16E21 4.23E22 1.33E21 4.72E22 1.42E21 1.40E21 4.53E22 1.23E21 3.74E22 9.30E22 2.51E22 1.72E22 7.44E22 3.13E22 1.08E21 1.38E21 4.93E22 1.47E21 4.75E22 1.31E21 4.05E22 1.14E21 3.43E22 8.96E22 2.59E22 6.46E22 1.85E22 4.64E22 1.30E22 3.15E22 8.27E23 1.96E22 5.16E23 1.19E22 2.98E23 6.65E23 1.67E23 3.68E23 8.56E24 1.86E23 4.29E24 8.92E24 2.02E24 4.24E24 9.26E25 1.80E24 3.75E25 7.93E25 3.02E25

9.58E22 3.57E22 1.18E21 4.27E22 1.37E21 4.76E22 1.46E21 1.44E21 4.58E22 1.26E21 3.75E22 9.48E22 2.47E22 1.70E22 7.41E22 3.17E22 1.13E21 1.38E21 4.82E22 1.47E21 4.59E22 1.29E21 3.99E22 1.09E21 3.25E22 8.57E22 2.49E22 6.37E22 1.80E22 4.48E22 1.23E22 2.99E22 7.98E23 1.89E22 4.92E23 1.14E22 2.89E23 6.49E23 1.60E23 3.53E23 8.56E24 1.83E23 4.34E24 9.10E24 2.10E24 4.29E24 9.66E25 1.94E24 4.27E25 8.35E25 3.44E25

2.57 0.85 1.72 0.95 3.01 0.85 2.82 2.86 1.10 2.44 0.27 1.94 1.59 1.16 0.40 1.28 4.63 0.00 2.23 0.00 3.37 1.53 1.48 4.39 5.25 4.35 3.86 1.39 2.70 3.45 5.38 5.08 3.51 3.57 4.65 4.20 3.02 2.41 4.19 4.08 0.00 1.61 1.17 2.02 3.96 1.18 4.32 7.78 13.87 5.30 13.91

5.77E05 5.77E05 5.63E05 5.57E05 5.37E05 5.38E05 5.19E05 5.02E05 5.01E05 4.85E05 4.88E05 4.71E05 4.80E05 4.38E05 4.27E05 4.13E05 3.92E05 3.97E05 3.99E05 3.82E05 3.77E05 3.63E05 3.58E05 3.63E05 3.60E05 3.51E05 3.44E05 3.30E05 3.30E05 3.27E05 3.29E05 3.23E05 3.13E05 3.09E05 3.08E05 3.03E05 2.95E05 2.89E05 2.90E05 2.87E05 2.72E05 2.73E05 2.63E05 2.58E05 2.51E05 2.55E05 2.45E05 2.35E05 2.21E05 2.37E05 2.16E05

a The quoted line position is the measured one in cm1. Sobs and Scalc are measured and calculated intensities, respectively, for pure C2H2, in cm  molecule1 at 296 K. % is the ratio (ScalcSobs)/Sobs in %. |R|2obs is the experimental transition dipole moment squared value, in D2 (1D ¼ 3.33546  1030 C m), deduced from Sobs. 12

ARTICLE IN PRESS O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523 Table 4 Line positions and intensities for the band n3+n14 of the

501

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

%

|R|2obs

Pee42 Pee41 Pee39 Pee36 Pee35 Pee34 Pee33 Pee32 Pee31 Pee30 Pee29 Pee28 Pee27 Pee25 Pee24 Pee22 Pee21 Pee20 Pee19 Pee18 Pee17 Pee16 Pee15 Pee14 Pee13 Pee12 Pee10 Pee9 Pee8 Pee7 Pee6 Pee5 Pee4 Pee3 Qfe41 Qfe40 Qfe39 Qfe38 Qfe37 Qfe35 Qfe33 Qfe30 Qfe28 Qfe27 Qfe26 Qfe25 Qfe24 Qfe23 Qfe22 Qfe21 Qfe20 Qfe19 Ree0 Ree3 Ree4 Ree5 Ree6

3786.53970 3789.50033 3795.37080 3804.04920 3806.90549 3809.74715 3812.57006 3815.37730 3818.16631 3820.93904 3823.69354 3826.43229 3829.15405 3834.54837 3837.22122 3842.51975 3845.14588 3847.75688 3850.35262 3852.93461 3855.50183 3858.05503 3860.59458 3863.12026 3865.63291 3868.13240 3873.09278 3875.55431 3878.00338 3880.44018 3882.86490 3885.27762 3887.67840 3890.06726 3893.85849 3894.13913 3894.45056 3894.74139 3894.99480 3895.40134 3895.73421 3896.15839 3896.36698 3896.45930 3896.54321 3896.61922 3896.68752 3896.74893 3896.80383 3896.85270 3896.89589 3896.93436 3899.50411 3906.45603 3908.74892 3911.02937 3913.29730

2.13E25 9.14E25 2.03E24 2.41E24 1.09E23 4.60E24 2.10E23 9.78E24 3.77E23 1.75E23 6.91E23 3.02E23 1.16E22 1.91E22 8.21E23 1.24E22 4.46E22 1.81E22 6.78E22 2.47E22 8.34E22 3.17E22 1.06E21 3.86E22 1.21E21 4.28E22 4.78E22 1.29E21 4.22E22 1.15E21 3.50E22 8.60E22 2.32E22 4.67E22

1.97E25 9.21E25 2.15E24 2.32E24 1.01E23 4.80E24 2.03E23 9.44E24 3.90E23 1.76E23 7.10E23 3.14E23 1.23E22 2.02E22 8.48E23 1.28E22 4.66E22 1.84E22 6.51E22 2.51E22 8.56E22 3.20E22 1.06E21 3.84E22 1.23E21 4.29E22 4.42E22 1.30E21 4.14E22 1.15E21 3.40E22 8.58E22 2.24E22 4.59E22

7.51 0.77 5.91 3.73 7.34 4.35 3.33 3.48 3.45 0.57 2.75 3.97 6.03 5.76 3.29 3.23 4.48 1.66 3.98 1.62 2.64 0.95 0.00 -0.52 1.65 0.23 7.53 0.78 1.90 0.00 2.86 0.23 3.45 1.71

8.13E05 7.37E05 6.84E05 7.21E05 7.41E05 6.48E05 6.91E05 6.82E05 6.28E05 6.35E05 6.14E05 5.98E05 5.79E05 5.64E05 5.68E05 5.50E05 5.37E05 5.43E05 5.67E05 5.27E05 5.14E05 5.15E05 5.09E05 5.06E05 4.87E05 4.86E05 5.10E05 4.61E05 4.65E05 4.48E05 4.54E05 4.34E05 4.41E05 4.25E05

1.46E22 6.49E23 2.49E22 1.09E22 4.14E22 1.80E22 6.55E22 3.03E22 1.07E21 1.58E22 1.02E21 3.93E22 1.24E21 4.32E22

1.40E22 6.52E23 2.64E22 1.16E22 4.49E22 1.87E22 6.89E22 2.75E22 9.68E22 1.53E22 1.01E21 3.81E22 1.23E21 4.31E22

4.11 0.46 6.02 6.42 8.45 3.89 5.19 9.24 9.53 3.16 0.98 3.05 0.81 0.23

8.09E06 8.88E06 9.52E06 1.06E05 1.15E05 1.31E05 1.41E05 1.76E05 1.90E05 4.02E05 3.71E05 3.72E05 3.56E05 3.48E05

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Table 4 (continued ) Line

Position

Sobs

Scalc

Ree7 Ree8 Ree9 Ree10 Ree11 Ree12 Ree13 Ree14 Ree15 Ree16 Ree18 Ree20 Ree21 Ree22 Ree23 Ree24 Ree25 Ree26 Ree27 Ree28 Ree29 Ree30 Ree31 Ree32 Ree33 Ree34 Ree35 Ree36 Ree37 Ree39

3915.55220 3917.79422 3920.02298 3922.23823 3924.43970 3926.62711 3928.80015 3930.95854 3933.10189 3935.22991 3939.43858 3943.58192 3945.62824 3947.65692 3949.66818 3951.66162 3953.63650 3955.59280 3957.53052 3959.44895 3961.34828 3963.22816 3965.08848 3966.92940 3968.74981 3970.55162 3972.33114 3974.09170 3975.83249 3979.25402

1.31E21 4.48E22 1.27E21 4.13E22 1.14E21 3.54E22 9.43E22 2.84E22 7.25E22 2.10E22 1.52E22 1.03E22 2.45E22 7.08E23 1.50E22 3.93E23 8.83E23 2.26E23 4.99E23 1.23E23 2.64E23 6.38E24 1.34E23 3.02E24 6.48E24 1.49E24 2.89E24 6.34E25 1.19E24 5.58E25

1.32E21 4.37E22 1.28E21 4.05E22 1.14E21 3.50E22 9.50E22 2.81E22 7.40E22 2.13E22 1.52E22 1.03E22 2.47E22 6.54E23 1.54E22 3.95E23 9.05E23 2.26E23 5.05E23 1.23E23 2.67E23 6.39E24 1.35E23 3.13E24 6.47E24 1.47E24 2.95E24 6.51E25 1.28E24 5.28E25

a

% 0.76 2.46 0.79 1.94 0.00 1.13 0.74 1.06 2.07 1.43 0.00 0.00 0.82 7.63 2.67 0.51 2.49 0.00 1.20 0.00 1.14 0.16 0.75 3.64 0.15 1.34 2.08 2.68 7.56 5.38

|R|2obs 3.39E05 3.42E05 3.25E05 3.27E05 3.13E05 3.11E05 2.99E05 2.98E05 2.83E05 2.79E05 2.70E05 2.58E05 2.50E05 2.67E05 2.35E05 2.34E05 2.24E05 2.24E05 2.16E05 2.13E05 2.05E05 2.02E05 1.95E05 1.85E05 1.87E05 1.84E05 1.73E05 1.67E05 1.55E05 1.66E05

See caption of Table 3. For some Q lines, only the position could be determined.

In Eq. (1), the E00 energy values, for the fundamental state, or for the v4 ¼ 1 or v5 ¼ 1 lower states, have been taken from HITRAN [19]. To calculate the partition function, we used the expansion given by Gamache et al. [21]. To reduce the data, effective parameters can be deduced expanding |R|2 empirically to take into account the rotational dependence RP 2 jRj2 ¼ jR0 j2 ð1 þ ARP 1 m þ A2 m Þ ðP and RbranchesÞ,

(5)

2 jRj2 ¼ jR0 j2 ð1 þ AQ 2 m ÞðQbranchÞ,

(6)

m being equal to J in the P-branch, J+1 in the R-branch, and J in the Q-branch. jR0 j2 is the vibrational transition dipole moment squared, and A1RP, A2RP , and A2Q, Herman–Wallis coefficients. Note that the terms between parenthesis in Eqs. (5,6) are not squared, contrary to what is sometimes done for linear molecules [22]. Transition dipole moment squared values and Herman-Wallis coefficients deduced from an unweighted fit of the experimental jRj2 values are reported in Table 12. Such a fitting process allows, first to remove anomalous intensities corresponding to lines suspected to be strongly blended with other ones, and second to interpolate missing intensities and slightly extrapolate calculated values (no more than the last observed J value increased by about 5). These results are plotted in Figs. 1–9. On these figures, one can see that the agreement between calculated and experimental values is very good, except for a few cases corresponding to very weak or slightly blended lines.

ARTICLE IN PRESS O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523 Table 5 Line positions and intensities for the band n1+n15 of the

503

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

%

|R|2obs

Pee39 Pee38 Pee37 Pee36 Pee34 Pee33 Pee32 Pee31 Pee30 Pee29 Pee28 Pee27 Pee26 Pee25 Pee23 Pee21 Pee20 Pee19 Pee18 Pee17 Pee12 Pee11 Pee10 Pee9 Pee8 Pee4 Pee3 Pee2 Qfe38 Qfe37 Qfe36 Qfe35 Qfe33 Qfe31 Qfe30 Qfe29 Qfe28 Qfe27 Qfe26 Qfe24 Qfe23 Qfe22 Qfe21 Qfe20 Qfe19 Qfe18 Qfe17 Qfe16 Qfe15 Qfe14 Qfe13 Qfe12 Qfe11 Qfe10 Qfe9 Qfe3 Ree0

3989.35232 3992.21088 3995.05688 3997.89037 4003.51951 4006.31508 4009.09786 4011.86796 4014.62513 4017.36954 4020.10086 4022.81937 4025.52488 4028.21736 4033.56300 4038.85605 4041.48279 4044.09611 4046.69620 4049.28289 4062.01442 4064.52010 4067.01197 4069.49019 4071.95490 4081.67507 4084.07039 4086.45143 4087.74254 4087.92329 4088.09777 4088.26757 4088.59223 4088.89669 4089.04134 4089.18105 4089.31584 4089.44571 4089.57038 4089.80634 4089.91689 4090.02245 4090.12340 4090.21943 4090.31098 4090.39768 4090.47978 4090.55715 4090.62994 4090.69807 4090.76160 4090.82045 4090.87485 4090.92453 4090.96974 4091.14617 4093.51156

4.96E24 2.59E24 1.06E23 5.29E24 1.08E23 4.56E23 2.15E23 8.73E23 3.97E23 1.58E22 7.06E23 2.78E22 1.21E22 4.58E22 7.19E22 1.06E21 4.33E22 1.47E21 5.77E22 1.96E21 9.98E22 3.09E21 1.05E21 3.17E21 9.77E22 5.42E22 1.11E21 2.02E22 3.26E24 1.39E23 6.85E24 3.05E23 6.84E23 1.26E22 5.66E23 2.34E22 1.05E22 4.17E22 1.85E22 3.12E22 1.20E21 4.84E22 1.81E21 7.07E22 2.63E21 1.03E21 3.64E21 1.36E21 4.71E21 1.71E21 5.73E21 2.05E21 6.54E21 2.29E21 6.92E21 4.05E21 3.79E22

4.85E24 2.43E24 1.08E23 5.28E24 1.09E23 4.63E23 2.14E23 8.89E23 4.02E23 1.61E22 7.17E23 2.81E22 1.21E22 4.61E22 7.27E22 1.07E21 4.28E22 1.51E21 5.83E22 1.99E21 1.01E21 3.10E21 1.04E21 3.06E21 9.76E22 5.36E22 1.11E21 1.89E22 3.11E24 1.40E23 6.96E24 3.06E23 6.39E23 1.26E22 5.83E23 2.38E22 1.07E22 4.28E22 1.88E22 3.12E22 1.18E21 4.91E22 1.81E21 7.34E22 2.64E21 1.04E21 3.62E21 1.39E21 4.69E21 1.74E21 5.69E21 2.04E21 6.47E21 2.23E21 6.81E21 3.93E21 3.74E22

2.22 6.18 1.89 0.19 0.93 1.54 0.47 1.83 1.26 1.90 1.56 1.08 0.00 0.66 1.11 0.94 1.15 2.72 1.04 1.53 1.20 0.32 0.95 3.47 0.10 1.11 0.00 6.44 4.60 0.72 1.61 0.33 6.58 0.00 3.00 1.71 1.90 2.64 1.62 0.00 1.67 1.45 0.00 3.82 0.38 0.97 0.55 2.21 0.42 1.75 0.70 0.49 1.07 2.62 1.59 2.96 1.32

1.59E04 1.64E04 1.49E04 1.50E04 1.45E04 1.42E04 1.43E04 1.38E04 1.37E04 1.34E04 1.33E04 1.32E04 1.32E04 1.29E04 1.25E04 1.22E04 1.23E04 1.17E04 1.17E04 1.15E04 1.08E04 1.07E04 1.07E04 1.08E04 1.03E04 9.83E05 9.59E05 1.01E04 9.68E05 9.16E05 9.12E05 9.25E05 1.00E04 9.35E05 9.12E05 9.25E05 9.23E05 9.21E05 9.34E05 9.52E05 9.70E05 9.40E05 9.54E05 9.22E05 9.55E05 9.49E05 9.66E05 9.44E05 9.67E05 9.49E05 9.71E05 9.71E05 9.77E05 9.93E05 9.84E05 1.00E04 9.17E05

ARTICLE IN PRESS 504

O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523

Table 5 (continued ) Line

Position

Sobs

Scalc

%

|R|2obs

Ree1 Ree2 Ree3 Ree4 Ree5 Ree6 Ree7 Ree8 Ree9 Ree10 Ree11 Ree12 Ree13 Ree14 Ree15 Ree16 Ree17 Ree18 Ree19 Ree20 Ree22 Ree23 Ree24 Ree25 Ree26 Ree27 Ree28 Ree29 Ree30 Ree31 Ree32 Ree33

4095.83665 4098.14751 4100.44428 4102.72676 4104.99489 4107.24875 4109.48827 4111.71338 4113.92413 4116.12028 4118.30203 4120.46917 4122.62178 4124.75972 4126.88297 4128.99161 4131.08549 4133.16466 4135.22888 4137.27834 4141.33234 4143.33730 4145.32702 4147.30174 4149.26138 4151.20601 4153.13540 4155.04966 4156.94886 4158.83281 4160.70141 4162.55476

1.65E21 7.10E22 2.51E21 9.48E22 3.03E21 1.08E21 3.27E21 1.10E21 3.23E21 1.04E21 2.96E21 9.25E22 2.48E21 7.37E22 1.93E21 5.66E22 1.45E21 4.20E22 1.02E21 2.82E22 1.89E22 4.30E22 1.13E22 2.60E22 6.68E23 1.47E22 3.68E23 8.08E23 1.93E23 4.13E23 9.73E24 2.01E23

1.65E21 7.03E22 2.51E21 9.46E22 3.09E21 1.08E21 3.32E21 1.11E21 3.26E21 1.04E21 2.94E21 9.08E22 2.48E21 7.40E22 1.96E21 5.69E22 1.47E21 4.12E22 1.03E21 2.82E22 1.83E22 4.34E22 1.13E22 2.61E22 6.60E23 1.48E22 3.68E23 8.07E23 1.95E23 4.17E23 9.81E24 2.05E23

0.00 0.99 0.00 0.21 1.98 0.00 1.53 0.91 0.93 0.00 0.68 1.84 0.00 0.41 1.55 0.53 1.38 1.90 0.98 0.00 3.17 0.93 0.00 0.38 1.20 0.68 0.00 0.12 1.04 0.97 0.82 1.99

8.95E05 8.88E05 8.66E05 8.56E05 8.27E05 8.29E05 8.05E05 8.02E05 7.88E05 7.82E05 7.77E05 7.75E05 7.50E05 7.36E05 7.16E05 7.14E05 7.00E05 7.11E05 6.82E05 6.78E05 6.81E05 6.43E05 6.42E05 6.28E05 6.29E05 6.08E05 6.05E05 5.97E05 5.83E05 5.74E05 5.67E05 5.53E05

a

See caption of Table 3.

Table 6 Line positions and intensities for the band n2+3n15 of the

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

Pee21 Pee19 Pee18 Pee16 Pee15 Pee11 Pee10 Pee9 Pee7 Pee5 Pee4 Pee3 Pee2 Qfe1 Qfe2 Qfe5 Qfe6

4087.52420 4092.60718 4095.13194 4100.14768 4102.63975 4112.50685 4114.95008 4117.38247 4122.22125 4127.02415 4129.41207 4131.79158 4134.16222 4138.88768 4138.90811 4139.02942 4139.08958

8.59E24 1.21E23 4.57E24 6.17E24 1.99E23 2.57E23 8.83E24 2.73E23 2.37E23 1.80E23 4.52E24 9.20E24 1.58E24 1.47E23 7.83E24 4.43E23 1.62E23

8.66E24 1.22E23 4.74E24 6.11E24 2.03E23 2.58E23 8.68E24 2.57E23 2.28E23 1.73E23 4.51E24 9.33E24 1.59E24 1.43E23 7.72E24 4.40E23 1.62E23

% 0.81 0.83 3.72 0.97 2.01 0.39 1.70 5.86 3.80 3.89 0.22 1.41 0.63 2.72 1.40 0.68 0.00

|R|2obs 9.74E07 9.56E07 9.19E07 9.42E07 9.03E07 8.79E07 8.87E07 9.15E07 8.73E07 8.53E07 8.11E07 7.88E07 7.84E07 7.90E07 7.76E07 7.64E07 7.58E07

ARTICLE IN PRESS O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523

505

Table 6 (continued ) Line

Position

Sobs

Scalc

%

|R|2obs

Qfe7 Qfe10 Qfe11 Qfe12 Qfe13 Qfe14 Qfe15 Qfe16 Qfe17 Qfe18 Qfe19 Qfe21 Qfe22 Qfe24 Qfe25 Qfe26 Qfe27 Ree0 Ree1 Ree2 Ree3 Ree4 Ree5 Ree6 Ree7 Ree8 Ree9 Ree10 Ree11 Ree12 Ree13 Ree14 Ree15 Ree16 Ree17 Ree18 Ree19 Ree20 Ree21 Ree22 Ree23 Ree24 Ree25 Ree26 Ree27 Ree28 Ree29 Ree30 Ree31

4139.15938 4139.42455 4139.53074 4139.64529 4139.76783 4139.89798 4140.03546 4140.17957 4140.32982 4140.48554 4140.64629 4140.98016 4141.15053 4141.49643 4141.66958 4141.84106 4142.01008 4141.22223 4143.55782 4145.88466 4148.20256 4150.51108 4152.81037 4155.10025 4157.38046 4159.65077 4161.91098 4164.16085 4166.40017 4168.62842 4170.84560 4173.05132 4175.24520 4177.42711 4179.59650 4181.75327 4183.89697 4186.02744 4188.14418 4190.24657 4192.33553 4194.40944 4196.46876 4198.51237 4200.54119 4202.55478 4204.55174 4206.53331 4208.49849

5.13E23 1.75E23 4.93E23 1.55E23 4.25E23 1.30E23 3.42E23 1.01E23 2.61E23 7.21E24 1.86E23 1.25E23 3.32E24 2.06E24 5.14E24 1.22E24 2.62E24 3.08E24 1.33E23 5.99E24 2.24E23 8.01E24 2.62E23 9.66E24 2.73E23 9.21E24 2.68E23 8.58E24 2.39E23 7.39E24 1.99E23 5.90E24 1.57E23 4.55E24 1.16E23 3.28E24 8.12E24 2.18E24 5.29E24 1.48E24 3.34E24 9.71E25 1.94E24 5.15E25 1.09E24 2.96E25 5.85E25 1.33E25 2.83E25

5.14E23 1.73E23 4.98E23 1.56E23 4.32E23 1.31E23 3.49E23 1.02E23 2.64E23 7.48E24 1.88E23 1.26E23 3.36E24 2.06E24 4.75E24 1.20E24 2.67E24 3.16E24 1.39E23 5.94E24 2.12E23 7.99E24 2.60E23 9.12E24 2.79E23 9.29E24 2.72E23 8.69E24 2.45E23 7.52E24 2.05E23 6.10E24 1.61E23 4.63E24 1.18E23 3.32E24 8.24E24 2.24E24 5.41E24 1.43E24 3.36E24 8.67E25 1.98E24 4.97E25 1.10E24 2.69E25 5.84E25 1.39E25 2.93E25

0.19 1.14 1.01 0.65 1.65 0.77 2.05 0.99 1.15 3.74 1.08 0.80 1.20 0.00 7.59 1.64 1.91 2.60 4.51 0.83 5.36 0.25 0.76 5.59 2.20 0.87 1.49 1.28 2.51 1.76 3.02 3.39 2.55 1.76 1.72 1.22 1.48 2.75 2.27 3.38 0.60 10.71 2.06 3.50 0.92 9.12 0.17 4.51 3.53

7.52E07 7.49E07 7.28E07 7.24E07 7.11E07 7.11E07 6.94E07 6.96E07 6.84E07 6.59E07 6.68E07 6.49E07 6.36E07 6.22E07 6.59E07 6.06E07 5.72E07 7.36E07 7.15E07 7.41E07 7.64E07 7.15E07 7.07E07 7.32E07 6.64E07 6.63E07 6.47E07 6.38E07 6.20E07 6.13E07 5.95E07 5.82E07 5.77E07 5.68E07 5.57E07 5.49E07 5.36E07 5.17E07 5.09E07 5.27E07 4.94E07 5.44E07 4.64E07 4.79E07 4.44E07 4.81E07 4.27E07 3.96E07 3.88E07

a

See caption of Table 3.

Q-branches of the n2+(2n4+n5)1 II and n3+n14 bands deserve a short discussion. As far as the n3+n14 band is concerned, Q-lines with J smaller or equal to 18 are too overlapped and could not be treated, whereas lines with J greater or equal to 28 are too weak. Our set of intensities for Q-lines is therefore somewhat reduced for this band (see Table 4 and Fig. 2), however, significant line positions for J up to 41 are given tentatively in

ARTICLE IN PRESS 506

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Table 7 Line positions and intensities for the band n1+2n05n15 of the

12

C2H2 molecule in the 2.5-mm region

a

Line

Position

Sobs

Scalc

%

|R|2obs

Pee32 Pee31 Pee30 Pee29 Pee28 Pee26 Pee25 Pee24 Pee23 Pee21 Pee20 Pee18 Pee17 Pee16 Pee14 Pee15 Pee13 Pee12 Pee10 Pee11 Pee9 Pee8 Pee6 Pee4 Pee3 Pee2 Qef27 Qef25 Qef23 Qef21 Qef19 Qef18 Qef17 Qef16 Qef15 Qef12 Qef11 Qef10 Qef9 Qef8 Qef5 Qef3 Qef2 Qef1 Ree6 Ree9 Ree10 Ree11 Ree12 Ree13 Ree14 Ree15 Ree16 Ree17 Ree18 Ree19 Ree20

3991.85411 3994.53027 3997.18966 3999.83167 4002.45666 4007.65608 4010.23123 4012.78987 4015.33315 4020.37468 4022.87438 4027.83456 4030.29654 4032.74758 4037.61957 4035.18830 4040.04217 4042.45624 4047.26286 4044.86303 4049.65680 4052.04484 4056.80529 4061.54721 4063.91205 4066.27241 4064.66439 4065.67879 4066.58715 4067.39283 4068.09999 4068.41839 4068.71444 4068.98849 4069.24187 4069.88308 4070.06006 4070.21977 4070.36289 4070.49002 4070.78017 4070.90286 4070.94210 4070.96856 4087.33227 4094.25902 4096.55261 4098.83848 4101.11517 4103.38205 4105.63805 4107.88298 4110.11518 4112.33460 4114.53988 4116.73035 4118.90550

3.17E24 1.49E24 6.01E24 2.58E24 1.04E23 1.69E23 7.29E24 2.66E23 1.08E23 1.53E23 5.37E23 7.05E23 2.55E23 8.77E23 1.06E22 3.33E23 3.70E23 1.18E22 1.35E22 4.08E23 4.04E23 1.14E22 1.06E22 7.88E23 2.14E23 5.12E23 4.65E24 8.66E24 1.66E23 3.13E23 4.74E23 1.97E23 7.21E23 2.83E23 1.01E22 4.75E23 1.57E22 5.49E23 1.74E22 5.84E23 1.46E22 1.05E22 2.51E23 4.96E23 7.82E23 2.76E23 8.99E23 2.88E23 8.21E23 2.54E23 7.44E23 2.19E23 6.21E23 1.74E23 4.48E23 1.32E23 3.45E23

3.26E24 1.48E24 5.91E24 2.61E24 1.02E23 1.67E23 6.99E24 2.60E23 1.06E23 1.53E23 5.41E23 7.17E23 2.69E23 9.04E23 1.07E22 3.28E23 3.78E23 1.18E22 1.24E22 4.06E23 4.11E23 1.19E22 1.05E22 8.10E23 2.21E23 5.06E23 3.95E24 8.69E24 1.70E23 3.01E23 4.90E23 2.03E23 7.40E23 2.94E23 1.04E22 4.93E23 1.60E22 5.65E23 1.76E22 5.90E23 1.52E22 1.09E22 2.69E23 4.99E23 7.42E23 2.82E23 8.41E23 2.74E23 7.88E23 2.48E23 6.89E23 2.10E23 5.70E23 1.68E23 4.44E23 1.28E23 3.28E23

2.84 0.67 1.66 1.16 1.92 1.18 4.12 2.26 1.85 0.00 0.74 1.70 5.49 3.08 0.94 1.50 2.16 0.00 8.15 0.49 1.73 4.39 0.94 2.79 3.27 1.17 15.05 0.35 2.41 3.83 3.38 3.05 2.64 3.89 2.97 3.79 1.91 2.91 1.15 1.03 4.11 3.81 7.17 0.60 5.12 2.17 6.45 4.86 4.02 2.36 7.39 4.11 8.21 3.45 0.89 3.03 4.93

2.29E04 2.30E04 2.25E04 2.12E04 2.11E04 1.97E04 1.97E04 1.87E04 1.80E04 1.66E04 1.60E04 1.49E04 1.39E04 1.38E04 1.33E04 1.40E04 1.27E04 1.26E04 1.30E04 1.23E04 1.14E04 1.08E04 1.08E04 9.95E05 9.68E05 9.93E05 3.64E05 3.97E05 4.70E05 5.79E05 6.05E05 6.39E05 6.71E05 6.90E05 7.25E05 7.85E05 8.17E05 8.29E05 8.62E05 8.76E05 8.87E05 9.04E05 8.77E05 9.39E05 9.27E05 8.60E05 9.42E05 9.32E05 9.30E05 9.24E05 9.83E05 9.62E05 1.02E04 9.84E05 9.76E05 1.02E04 1.06E04

ARTICLE IN PRESS O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523

507

Table 7 (continued ) Line

Position

Sobs

Scalc

Ree21 Ree22 Ree26 Ree27

4121.06408 4123.20615 4131.59351 4133.64358

9.20E24 2.23E23 9.27E24 2.36E24

9.29E24 2.33E23 9.94E24 2.60E24

a

% 0.98 4.48 7.23 10.17

|R|2obs 1.02E04 1.01E04 1.09E04 1.09E04

See caption of Table 3.

Table 8 Line positions and intensities for the band n1+2n25n15 of the

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

Pee30 Pee28 Pee26 Pee25 Pee24 Pee23 Pee22 Pee21 Pee20 Pee19 Pee18 Pee17 Pee16 Pee15 Pee14 Pee13 Pee11 Pee10 Pee9 Pee8 Pee6 Qfe23 Qfe22 Qfe20 Qfe19 Qfe18 Qfe16 Qfe15 Qfe13 Qfe11 Qfe10 Qfe8 Ree3 Ree4 Ree5 Ree6 Ree8 Ree9 Ree14 Ree15 Ree17 Ree18 Ree19 Ree20 Ree22

4009.43747 4014.03356 4018.66795 4020.99914 4023.33813 4025.68532 4028.04039 4030.40239 4032.77066 4035.14457 4037.52345 4039.90631 4042.29235 4044.68120 4047.07118 4049.46234 4054.24441 4056.63405 4059.02155 4061.40719 4066.16876 4079.00382 4079.11684 4079.32866 4079.42678 4079.52055 4079.69200 4079.77039 4079.91225 4080.03433 4080.08783 4080.18038 4089.71844 4092.04820 4094.37346 4096.69476 4101.32762 4103.63986 4115.18333 4117.49262 4122.11857 4124.43607 4126.75807 4129.08462 4133.75390

8.84E25 1.96E24 3.97E24 1.91E24 7.08E24 3.28E24 1.23E23 5.20E24 2.02E23 8.41E24 2.99E23 1.20E23 4.05E23 1.47E23 5.17E23 1.90E23 2.11E23 6.34E23 2.04E23 5.92E23 4.46E23 1.09E23 4.05E23 6.16E23 2.51E23 9.20E23 1.19E22 4.59E23 5.44E23 6.03E23 1.87E22 1.87E22 3.45E23 1.15E22 3.94E23 1.19E22 1.10E22 3.78E23 6.12E23 1.62E23 1.09E23 2.99E23 6.45E24 1.60E23 9.08E24

8.87E25 1.98E24 3.99E24 1.83E24 7.37E24 3.27E24 1.27E23 5.38E24 2.01E23 8.23E24 2.98E23 1.17E23 4.08E23 1.55E23 5.16E23 1.88E23 2.08E23 6.29E23 2.05E23 5.80E23 4.44E23 1.11E23 4.15E23 6.18E23 2.47E23 8.72E23 1.16E22 4.35E23 5.26E23 5.95E23 1.84E22 1.83E22 3.75E23 1.18E22 4.05E23 1.22E22 1.17E22 3.69E23 6.25E23 1.77E23 1.18E23 2.78E23 7.18E24 1.62E23 8.40E24

% 0.34 1.02 0.50 4.19 4.10 0.30 3.25 3.46 0.50 2.14 0.33 2.50 0.74 5.44 0.19 1.05 1.42 0.79 0.49 2.03 0.45 1.83 2.47 0.32 1.59 5.22 2.52 5.23 3.31 1.33 1.60 2.14 8.70 2.61 2.79 2.52 6.36 2.38 2.12 9.26 8.26 7.02 11.32 1.25 7.49

|R|2obs 7.55E05 9.20E05 1.08E04 1.21E04 1.18E04 1.30E04 1.31E04 1.37E04 1.48E04 1.56E04 1.58E04 1.66E04 1.65E04 1.62E04 1.74E04 1.79E04 1.85E04 1.86E04 1.86E04 1.92E04 1.91E04 1.84E04 1.83E04 1.87E04 1.91E04 1.98E04 1.92E04 1.98E04 1.94E04 1.90E04 1.91E04 1.92E04 1.65E04 1.72E04 1.68E04 1.65E04 1.51E04 1.60E04 1.25E04 1.11E04 9.84E05 1.06E04 8.14E05 8.11E05 6.94E05

ARTICLE IN PRESS 508

O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523

Table 8 (continued ) Line

Position

Sobs

Scalc

%

|R|2obs

Ree24 Pff30 Pff29 Pff27 Pff26 Pff25 Pff24 Pff23 Pff22 Pff21 Pff20 Pff19 Pff18 Pff15 Pff14 Pff13 Pff12 Pff10 Pff9 Pff8 Pff7 Pff6 Pff5 Qef29 Qef28 Qef26 Qef22 Qef21 Qef17 Qef15 Qef13 Qef11 Qef9 Qef6 Rff5 Rff7 Rff8 Rff9 Rff10 Rff11 Rff12 Rff13 Rff14 Rff15 Rff17 Rff18 Rff19 Rff20 Rff21 Rff22 Rff23 Rff24 Rff25 Rff27 Rff29 Rff30 Rff33

4138.44843 4003.46728 4006.22544 4011.70220 4014.42100 4017.12664 4019.81902 4022.49782 4025.16368 4027.81572 4030.45494 4033.08029 4035.69311 4043.44711 4046.00517 4048.54864 4051.07887 4056.09786 4058.58665 4061.06149 4063.52237 4065.96941 4068.40235 4075.78897 4076.01555 4076.46661 4077.35411 4077.57193 4078.40320 4078.78377 4079.13983 4079.45751 4079.73384 4080.06027 4094.22829 4098.73721 4100.96969 4103.18762 4105.39081 4107.57952 4109.75350 4111.91239 4114.05666 4116.18609 4120.40011 4122.48480 4124.55447 4126.60891 4128.64830 4130.67264 4132.68179 4134.67577 4136.65439 4140.56599 4144.41614 4146.31772 4151.93047

4.90E24 1.07E24 4.22E24 7.62E24 3.10E24 1.20E23 4.96E24 1.75E23 7.75E24 2.86E23 1.10E23 3.81E23 1.41E23 6.01E23 2.30E23 6.99E23 2.43E23 2.36E23 6.82E23 2.11E23 5.56E23 1.57E23 3.62E23 1.43E23 4.85E24 9.07E24 2.09E23 7.27E23 1.33E22 1.74E22 1.83E22 1.97E22 2.09E22 5.65E23 1.22E22 1.25E22 4.02E23 1.14E22 3.72E23 1.01E22 3.11E23 8.28E23 2.48E23 6.33E23 4.73E23 1.36E23 3.28E23 8.95E24 2.14E23 5.81E24 1.35E23 3.27E24 7.94E24 4.44E24 2.36E24 5.52E25 5.28E25

3.70E24 1.04E24 4.18E24 7.27E24 3.14E24 1.20E23 5.03E24 1.87E23 7.65E24 2.76E23 1.09E23 3.84E23 1.48E23 6.18E23 2.22E23 7.05E23 2.43E23 2.43E23 6.98E23 2.15E23 5.71E23 1.58E23 3.60E23 1.28E23 5.57E24 9.10E24 2.09E23 7.49E23 1.32E22 1.62E22 1.87E22 2.02E22 2.02E22 5.67E23 1.21E22 1.22E22 3.96E23 1.14E22 3.60E23 1.01E22 3.07E23 8.25E23 2.45E23 6.40E23 4.69E23 1.31E23 3.26E23 8.81E24 2.13E23 5.65E24 1.33E23 3.42E24 7.84E24 4.44E24 2.35E24 5.64E25 5.79E25

24.49 2.80 0.95 4.59 1.29 0.00 1.41 6.86 1.29 3.50 0.91 0.79 4.96 2.83 3.48 0.86 0.00 2.97 2.35 1.90 2.70 0.64 0.55 10.49 14.85 0.33 0.00 3.03 0.75 6.90 2.19 2.54 3.35 0.35 0.82 2.40 1.49 0.00 3.23 0.00 1.29 0.36 1.21 1.11 0.85 3.68 0.61 1.56 0.47 2.75 1.48 4.59 1.26 0.00 0.42 2.17 9.66

5.94E05 2.81E04 2.72E04 2.76E04 2.57E04 2.58E04 2.51E04 2.36E04 2.52E04 2.55E04 2.44E04 2.38E04 2.26E04 2.22E04 2.34E04 2.21E04 2.20E04 2.08E04 2.07E04 2.05E04 2.01E04 2.02E04 2.02E04 4.01E04 3.02E04 3.24E04 2.86E04 2.70E04 2.49E04 2.52E04 2.18E04 2.08E04 2.12E04 1.95E04 1.74E04 1.71E04 1.67E04 1.62E04 1.65E04 1.58E04 1.57E04 1.53E04 1.52E04 1.46E04 1.44E04 1.46E04 1.39E04 1.38E04 1.34E04 1.35E04 1.31E04 1.21E04 1.26E04 1.20E04 1.16E04 1.11E04 9.75E05

a

See caption of Table 3.

ARTICLE IN PRESS O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523 Table 9 Line positions and intensities for the band n1+(n4+n5)0+n14 of the

509

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

%

|R|2obs

Pee35 Pee33 Pee32 Pee30 Pee29 Pee28 Pee27 Pee26 Pee25 Pee23 Pee22 Pee21 Pee20 Pee19 Pee18 Pee17 Pee16 Pee15 Pee14 Pee13 Pee9 Pee8 Pee6 Pee5 Pee4 Pee3 Pee2 Pee1 Qef28 Qef27 Qef26 Qef24 Qef22 Qef21 Qef16 Qef14 Qef13 Qef12 Qef11 Qef9 Qef8 Qef7 Qef6 Qef5 Qef4 Qef3 Ree3 Ree4 Ree7 Ree8 Ree9 Ree10 Ree11 Ree12 Ree13 Ree17 Ree19

3975.93359 3981.23656 3983.86288 3989.06692 3991.64485 3994.20704 3996.75409 3999.28641 4001.80453 4006.80022 4009.27971 4011.74658 4014.20275 4016.64868 4019.08489 4021.51214 4023.93114 4026.34246 4028.74677 4031.14479 4040.68367 4043.05788 4047.79430 4050.15828 4052.51925 4054.87758 4057.23325 4059.58644 4055.43269 4055.94910 4056.43993 4057.34434 4058.14909 4058.51518 4060.02129 4060.48334 4060.68739 4060.87373 4061.04397 4061.33546 4061.45831 4061.56656 4061.66024 4061.73982 4061.80616 4061.85855 4071.31288 4073.65002 4080.63685 4082.95688 4085.27215 4087.58179 4089.88490 4092.18187 4094.47110 4103.53927 4108.01000

7.53E25 1.48E24 6.99E25 1.25E24 4.96E24 2.19E24 8.37E24 3.58E24 1.35E23 2.05E23 8.90E24 2.86E23 1.20E23 3.96E23 1.51E23 5.07E23 2.05E23 6.30E23 2.25E23 7.14E23 7.68E23 2.59E23 2.38E23 6.01E23 1.75E23 4.31E23 1.09E23 2.19E23 2.93E24 1.40E24 5.77E24 1.00E23 1.86E23 8.06E24 6.41E23 7.99E23 3.05E23 1.01E22 3.50E23 4.17E23 1.21E22 3.95E23 1.16E22 3.54E23 9.48E23 2.49E23 3.02E23 1.35E23 5.77E23 1.90E23 5.74E23 1.93E23 5.20E23 1.78E23 4.96E23 3.45E23 2.44E23

7.62E25 1.49E24 6.85E25 1.24E24 4.95E24 2.16E24 8.36E24 3.55E24 1.35E23 2.05E23 8.26E24 2.96E23 1.17E23 4.06E23 1.55E23 5.27E23 1.96E23 6.46E23 2.34E23 7.46E23 8.12E23 2.64E23 2.33E23 6.27E23 1.80E23 4.44E23 1.13E23 2.27E23 2.82E24 1.35E24 5.70E24 1.06E23 1.86E23 7.93E24 6.42E23 8.43E23 3.15E23 1.03E22 3.70E23 4.01E23 1.21E22 3.94E23 1.12E22 3.42E23 8.94E23 2.44E23 3.00E23 1.26E23 5.31E23 1.83E23 5.60E23 1.85E23 5.40E23 1.71E23 4.84E23 3.25E23 2.45E23

1.20 0.68 2.00 0.80 0.20 1.37 0.12 0.84 0.00 0.00 7.19 3.50 2.50 2.53 2.65 3.94 4.39 2.54 4.00 4.48 5.73 1.93 2.10 4.33 2.86 3.02 3.67 3.65 3.75 3.57 1.21 6.00 0.00 1.61 0.16 5.51 3.28 1.98 5.71 3.84 0.00 0.25 3.45 3.39 5.70 2.01 0.66 6.67 7.97 3.68 2.44 4.15 3.85 3.93 2.42 5.80 0.41

8.97E05 8.51E05 8.51E05 7.93E05 7.67E05 7.54E05 7.23E05 7.07E05 6.83E05 6.43E05 6.72E05 5.85E05 6.06E05 5.58E05 5.41E05 5.19E05 5.48E05 4.98E05 4.78E05 4.62E05 4.11E05 4.16E05 4.14E05 3.79E05 3.76E05 3.68E05 3.60E05 3.53E05 1.72E05 1.85E05 1.94E05 2.03E05 2.39E05 2.53E05 2.94E05 2.93E05 3.06E05 3.15E05 3.10E05 3.51E05 3.42E05 3.47E05 3.61E05 3.64E05 3.76E05 3.64E05 3.43E05 3.61E05 3.59E05 3.41E05 3.37E05 3.42E05 3.17E05 3.43E05 3.40E05 3.66E05 3.53E05

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Table 9 (continued ) Line

Position

Sobs

Scalc

Ree20 Ree21 Ree23 Ree25

4110.22650 4112.42999 4116.79325 4121.09415

6.83E24 1.73E23 1.17E23 7.11E24

6.94E24 1.75E23 1.19E23 7.74E24

a

% 1.61 1.16 1.71 8.86

|R|2obs 3.54E05 3.62E05 3.73E05 3.64E05

See caption of Table 3.

Table 10 Line positions and intensities for the band n1+(n4+n5)0n14 of the

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

%

|R|2obs

Pff36 Pff35 Pff32 Pff30 Pff29 Pff28 Pff27 Pff26 Pff25 Pff24 Pff23 Pff22 Pff21 Pff20 Pff18 Pff17 Pff16 Pff15 Pff14 Pff13 Pff12 Pff11 Pff10 Pff9 Pff8 Pff7 Pff6 Pff4 Pff3 Pff2 Qfe25 Qfe23 Qfe22 Qfe20 Qfe19 Qfe17 Qfe16 Qfe15 Qfe14 Qfe13 Qfe11 Qfe10 Qfe9 Qfe8 Qfe7

3983.49689 3986.32726 3994.74184 4000.29020 4003.04526 4005.78741 4008.51697 4011.23349 4013.93690 4016.62754 4019.30511 4021.96951 4024.62080 4027.25877 4032.49470 4035.09258 4037.67686 4040.24772 4042.80483 4045.34836 4047.87806 4050.39399 4052.89614 4055.38437 4057.85880 4060.31912 4062.76494 4067.61557 4070.01928 4072.40897 4075.88758 4076.07680 4076.16564 4076.33188 4076.40923 4076.55244 4076.61835 4076.68202 4076.73835 4076.79267 4076.88959 4076.93229 4076.97093 4077.00542 4077.03667

5.64E25 2.63E25 2.28E24 4.23E24 1.77E24 6.97E24 3.15E24 1.19E23 5.06E24 1.85E23 7.60E24 2.79E23 1.10E23 3.97E23 5.35E23 2.05E23 6.92E23 2.55E23 8.21E23 2.99E23 9.42E23 3.32E23 9.98E23 3.42E23 9.93E23 3.27E23 9.30E23 6.98E23 2.02E23 4.37E23 1.45E23 2.42E23 1.05E23 1.57E23 5.70E23 7.97E23 3.16E23 1.13E22 4.15E23 1.36E22 1.51E22 5.28E23 1.61E22 5.55E23 1.62E22

5.49E25 2.63E25 2.19E24 4.04E24 1.79E24 7.10E24 3.09E24 1.19E23 5.03E24 1.89E23 7.79E24 2.86E23 1.14E23 4.08E23 5.54E23 2.11E23 7.09E23 2.62E23 8.55E23 3.06E23 9.70E23 3.36E23 1.03E22 3.42E23 1.00E22 3.20E23 8.93E23 6.99E23 1.92E23 4.40E23 1.48E23 2.46E23 1.04E23 1.59E23 5.80E23 8.14E23 3.14E23 1.07E22 4.01E23 1.32E22 1.53E22 5.30E23 1.62E22 5.40E23 1.57E22

2.66 0.00 3.95 4.49 1.13 1.87 1.90 0.00 0.59 2.16 2.50 2.51 3.64 2.77 3.55 2.93 2.46 2.75 4.14 2.34 2.97 1.20 3.21 0.00 0.70 2.14 3.98 0.14 4.95 0.69 2.07 1.65 0.95 1.27 1.75 2.13 0.63 5.31 3.37 2.94 1.32 0.38 0.62 2.70 3.09

1.02E04 9.70E05 9.47E05 9.09E05 8.38E05 8.15E05 8.29E05 7.97E05 7.81E05 7.42E05 7.24E05 7.08E05 6.83E05 6.75E05 6.41E05 6.32E05 6.20E05 6.05E05 5.84E05 5.81E05 5.66E05 5.64E05 5.44E05 5.48E05 5.31E05 5.38E05 5.38E05 4.97E05 5.16E05 4.78E05 3.69E05 3.82E05 4.00E05 4.01E05 4.06E05 4.14E05 4.30E05 4.55E05 4.51E05 4.52E05 4.41E05 4.48E05 4.49E05 4.68E05 4.72E05

ARTICLE IN PRESS O.M. Lyulin et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 496–523

511

Table 10 (continued ) Line

Position

Sobs

Scalc

%

|R|2obs

Qfe6 Qfe5 Rff2 Rff6 Rff7 Rff8 Rff9 Rff10 Rff11 Rff12 Rff14 Rff15 Rff16 Rff17 Rff18 Rff19 Rff20 Rff21 Rff25

4077.06378 4077.08782 4084.14277 4093.26971 4095.51489 4097.74562 4099.96162 4102.16303 4104.34970 4106.52163 4110.82116 4112.94863 4115.06137 4117.15932 4119.24240 4121.31034 4123.36337 4125.40187 4133.40498

5.16E23 1.47E22 2.82E23 6.40E23 2.27E23 7.08E23 2.39E23 6.94E23 2.27E23 6.21E23 5.31E23 1.62E23 4.19E23 1.21E23 3.31E23 8.82E24 2.35E23 6.04E24 2.45E24

4.94E23 1.34E22 2.72E23 6.32E23 2.24E23 6.97E23 2.34E23 6.90E23 2.22E23 6.30E23 5.34E23 1.60E23 4.26E23 1.24E23 3.21E23 9.11E24 2.30E23 6.34E24 2.63E24

4.26 8.84 3.55 1.25 1.32 1.55 2.09 0.58 2.20 1.45 0.56 1.23 1.67 2.48 3.02 3.29 2.13 4.97 7.35

4.80E05 5.04E05 4.61E05 4.29E05 4.24E05 4.22E05 4.21E05 4.11E05 4.15E05 3.98E05 3.98E05 4.04E05 3.92E05 3.89E05 4.11E05 3.86E05 4.09E05 3.82E05 3.82E05

a

See caption of Table 3.

Table 11 Line positions and intensities for the band n1+(n4+n5)2n14 of the

12

C2H2 molecule in the 2.5-mm regiona

Line

Position

Sobs

Scalc

Pee27 Pee25 Pee23 Pee22 Pee21 Pee20 Pee19 Pee18 Pee17 Pee16 Pee15 Pee14 Pee13 Pee11 Pee10 Pee8 Pee7 Qfe27 Qfe24 Qfe22 Qfe20 Qfe19 Qfe18 Qfe17 Qfe15 Qfe13 Qfe11 Ree6 Ree7

4011.90634 4016.54250 4021.21387 4023.56017 4025.91287 4028.27136 4030.63493 4033.00252 4035.37405 4037.74817 4040.12475 4042.50272 4044.88146 4049.63960 4052.01752 4056.76924 4059.14219 4074.20954 4074.52045 4074.70575 4074.87400 4074.95054 4075.02367 4075.09268 4075.21894 4075.32902 4075.42298 4092.02353 4094.34711

2.75E24 4.85E24 8.34E24 3.36E24 1.27E23 5.82E24 1.93E23 7.77E24 2.65E23 1.00E23 3.34E23 1.32E23 4.12E23 4.68E23 1.45E23 1.31E23 3.76E23 1.21E23 7.87E24 1.10E23 1.77E23 5.62E23 2.30E23 7.69E23 9.61E23 1.16E22 1.29E22 2.66E23 8.00E23

2.81E24 5.01E24 8.37E24 3.53E24 1.31E23 5.36E24 1.94E23 7.64E24 2.66E23 1.02E23 3.42E23 1.25E23 4.05E23 4.39E23 1.47E23 1.34E23 3.59E23 1.15E23 7.83E24 1.19E23 1.69E23 5.96E23 2.30E23 7.88E23 9.84E23 1.16E22 1.28E22 2.72E23 8.06E23

% 2.18 3.30 0.36 5.06 3.15 7.90 0.52 1.67 0.38 2.00 2.40 5.30 1.70 6.20 1.38 2.29 4.52 4.96 0.51 8.18 4.52 6.05 0.00 2.47 2.39 0.00 0.78 2.26 0.75

|R|2obs 5.50E05 5.77E05 6.23E05 6.09E05 6.32E05 7.20E05 6.72E05 6.96E05 6.91E05 6.91E05 6.93E05 7.53E05 7.33E05 7.76E05 7.20E05 7.19E05 7.69E05 1.05E04 9.42E05 8.36E05 9.09E05 8.04E05 8.40E05 8.05E05 7.81E05 7.79E05 7.65E05 6.23E05 6.18E05

ARTICLE IN PRESS 512

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Table 11 (continued ) Line

Position

Sobs

Scalc

%

|R|2obs

Ree8 Ree10 Ree11 Ree16 Ree17 Ree19 Ree22 Pff26 Pff24 Pff21 Pff19 Pff18 Pff17 Pff14 Pff13 Pff12 Pff10 Qe25 Qef24 Qef23 Qef22 Qef21 Qef20 Qef19 Qef18 Qef17 Qef14 Qef10 Qef9 Rff2 Rff4 Rff5 Rff6 Rff7 Rff8 Rff10 Rff11 Rff12 Rff15 Rff16 Rff17 Rff18 Rff20 Rff21 Rff22 Rff23 Rff25 Rff26 Rff28

4096.66841 4101.30535 4103.62198 4115.20472 4117.52571 4122.17637 4129.18493 4009.76730 4015.16603 4023.16195 4028.42437 4031.03510 4033.63218 4041.34098 4043.88306 4046.41151 4051.42682 4071.83818 4072.08261 4072.32408 4072.56231 4072.79649 4073.02583 4073.25006 4073.46782 4073.67926 4074.26555 4074.91318 4075.04700 4082.66754 4087.26133 4089.53682 4091.79780 4094.04437 4096.27634 4100.69652 4102.88451 4105.05773 4111.48833 4113.60185 4115.70033 4117.78359 4121.90442 4123.94223 4125.96357 4127.97022 4131.93709 4133.89462 4137.76731

2.51E23 2.26E23 6.25E23 1.03E23 2.48E23 1.65E23 2.66E24 4.80E24 8.06E24 5.09E24 7.47E24 2.63E23 1.02E23 4.10E23 1.50E23 4.63E23 4.69E23 6.91E24 2.47E23 1.02E23 3.82E23 1.56E23 5.45E23 2.12E23 7.45E23 2.82E23 1.12E22 1.35E22 4.54E23 7.05E23 7.94E23 2.66E23 8.21E23 2.68E23 7.86E23 6.95E23 2.13E23 5.76E23 1.33E23 3.37E23 9.35E24 2.36E23 1.49E23 4.25E24 9.83E24 2.53E24 1.52E24 3.00E24 1.64E24

2.60E23 2.30E23 6.34E23 1.04E23 2.55E23 1.63E23 2.37E24 4.98E24 8.21E24 5.22E24 7.43E24 2.61E23 9.99E24 4.07E23 1.45E23 4.54E23 4.60E23 6.85E24 2.55E23 1.05E23 3.82E23 1.53E23 5.41E23 2.11E23 7.29E23 2.76E23 1.11E22 1.33E22 4.41E23 6.94E23 7.86E23 2.71E23 8.23E23 2.72E23 7.96E23 7.14E23 2.20E23 6.00E23 1.36E23 3.48E23 9.72E24 2.42E23 1.58E23 4.15E24 9.65E24 2.47E24 1.38E24 3.03E24 1.53E24

3.59 1.77 1.44 0.97 2.82 1.21 10.90 3.75 1.86 2.55 0.54 0.76 2.06 0.73 3.33 1.94 1.92 0.87 3.24 2.94 0.00 1.92 0.73 0.47 2.15 2.13 0.89 1.48 2.86 1.56 1.01 1.88 0.24 1.49 1.27 2.73 3.29 4.17 2.26 3.26 3.96 2.54 6.04 2.35 1.83 2.37 9.21 1.00 6.71

5.86E05 5.66E05 5.51E05 4.54E05 4.23E05 3.92E05 3.44E05 7.50E05 7.68E05 7.66E05 7.89E05 7.92E05 8.00E05 7.84E05 8.03E05 7.89E05 7.81E05 1.07E04 1.00E04 9.80E05 9.83E05 9.78E05 9.43E05 9.20E05 9.13E05 8.92E05 8.27E05 7.80E05 7.79E05 6.93E05 6.70E05 6.42E05 6.41E05 6.23E05 6.13E05 5.81E05 5.65E05 5.49E05 5.18E05 5.01E05 4.83E05 4.75E05 4.30E05 4.50E05 4.30E05 4.15E05 4.08E05 3.48E05 3.36E05

a

See caption of Table 3.

Table 4. It was not possible to fit simultaneously the P-, R-, and Q-branches. Furthermore, it was not possible to fit the Q-branch alone using Eq. (6), even fixing jR0 j2 . The only way we found to reduce Q-line intensities of the n3+n14 band was to fit them separately, fixing jR0 j2 to the value found for P- and R-lines fitted apart, and

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Table 12 Summary of 12C2H2 experimental vibrational transition dipole moments squared |R0|2 in D2 (1D ¼ 3.33546  1030 C m), and Herman–Wallis coefficients, see Eqs. (5,6), for nine bands in the 2.5-mm spectral regiona Band

|R0|2  105 D2

ARP 1

ARP 2

AQ 2

n2+(2n4+n5)1 II n3+n14

1.746(18)  102 1.818(26)  102

+1.659(80)  104 +7.7(11)  105

b

n2+3n15 n1+2n05n15 n1+2n25n15

4.401(26) 70.22 3.961(32) 70.20 3.961 (fixed) 9.184(42) 70.46 9.708(70) 70.49 0.07669(37) 70.0038 9.462(89) 70.47 18.76(13) 70.94

n1+(n4+n5)0+n14 n1+(n4+n5)0n14 n1+(n4+n5)2n14

3.592(29) 70.18 4.653(32) 70.23 7.075(49) 70.35

n1+n15

A1 2

Q

¼ 2.98(4)  102

c

5

1.436(14)  10

+8.84(70)  10

2

5

1.397(36)  10 1.654(41)  102 ee 7.57(47)  103 ff 1.395(36)  102 1.633(42)  102 1.586(34)  102 ee 9.79(43)  103 Ff 1.110(36)  102

2.84(17)  10 +9.35(30)  104 9.14(23)  104 +3.9(18)  105 +7.80(26)  104 +4.35(20)  104 6.45(25)  104 2.79(20)  104

3.38(10)  105 3.16(17)  104 8.91(34)  104 fe 0 (fixed) ef +1.044(34)  103 6.64(27)  104 2.97(32)  104 fe +5.41(30)  104 ef +7.70(30)  104

Transition dipole moment squared in debye2

a Bands are given in the same order as in Table 1. Confidence intervals (1 SD, in unit of the last quoted digit) are given between parenthesis. For |R0|2 values, the 75% accuracy is added after the 7 signs. b Measured integrated absorption coefficient at unit pressure for the Q-branch: 0.749 7 0.040 cm2 atm1 at 296 K, see text. c 2 First degree AQ 1 effective coefficient, see Eq. (7). The Q-branch has been fitted alone, |R0| being fixed to the value obtained for the Pand R-branches, see text.

8x10-5

6x10-5

4x10-5

2x10-5 -40

-30

-20

-10

0 m

10

20

30

40

Fig. 1. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the P- and R-branches of the band n2+(2n4+n5)1 II . The solid line has been calculated using the constants found in this work (see Table 12).

using the following first degree effective expansion: jRj2 ¼ jR0 j2 ð1 þ AQ 1 mÞ.

(7)

This expansion does not allow extrapolation. It was used only to better show the coherence of Q-line intensities on the plot (see Fig. 2), however, it can be interpolated to get rough intensities for lines with J lower than 19. For the n2+(2n4+n5)1 II band, the Q-branch is very confined, so that overlappings are too strong to allow any treatment. Even line positions could not be retrieved. As this Q-branch extends completely within a narrow spectral domain (about 0.2 cm1 in our spectrum having the largest pressure  absorbing path length product), and as it does not include lines of other bands having noticeable intensity, we could measure its

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514

8x10-5

6x10-5

4x10-5

2x10-5

-40

-30

-20

-10

0 m

10

20

30

40

Transition dipole moment squared in debye2

Fig. 2. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n3+n14 (solid triangles are for P- and R-branches, and open triangles for the Q-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

1.5x10-4

1.0x10-4

5.0x10-5

-40

-30

-20

-10

0

10

20

30

40

m

Fig. 3. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n1+n15 (solid triangles are for P- and R-branches, and open triangles for the Q-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

integrated absorption coefficient at unit pressure, neglecting effects of the apparatus function. We found 0.749 cm2 atm1 (75%) at 296 K. This value is very close to the value reported by Pine and Looney [12] as the sum of calculated Q-line intensities, namely, 0.744 cm2 atm1 at 295 K. For the other bands, P-, R-, and Q-branches were fitted simultaneously, except for the n1+n15 band for which the Q-branch is better recalculated when it is fitted separately (see Fig. 3). Note that for the n1 þ 2n25  n15 band, the A2Q Herman–Wallis coefficient of the Qfe-branch was fixed to zero (see Fig. 6). 3. Simultaneous fitting of line intensities in the series Dp ¼ 4 and 6 3.1. Theoretical approach Contrary to the empirical data reduction performed in the preceding section, where each band is studied separately, in this section we perform a simultaneous treatment of all bands lying in the 2.5 and 3.8-mm regions, taking into account the resonance interactions between them in explicit form.

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1x10-6 9x10-7 8x10-7 7x10-7 6x10-7 5x10-7 4x10-7 3x10-7 -20

-10

0

10

20

30

m

Transition dipole moment squared in debye2

Fig. 4. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n2+3n15 (solid triangles are for P- and R-branches, and open triangles for the Q-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

2.5x10-4 2.0x10-4 1.5x10-4 1.0x10-4 5.0x10-5

-30

-20

-10

0

10

20

30

m

Fig. 5. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n1+2n05n15 (solid triangles are for P- and R-branches, and open triangles for the Q-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

The theoretical approach used in this section has been presented in details in our previous paper [3]. For convenience, we recall it briefly here. In a general form, the intensity S V 0 J 0 0 VJ (expressed in cm1/ (molecule cm2) at temperature T in K) of an absorption line corresponding to the transition V 0 J 0 0 VJ, where V and J are, respectively, the vibrational index and the angular momentum quantum number, and where  ¼ 1 is the parity, is related to the transition moment squared W V 0 J 0 0 VJ by the well known equation S V 0 J 0 0

VJ ðTÞ

¼

8p3 CgVJ nV 0 J 0 0 3hc

VJ

expðhcE VJ =kTÞ  1  expðhcnV 0 J 0 0 QðTÞ



VJ =kT Þ

W V 0 J 0 0

VJ ,

(8)

where E VJ is the energy of the lower state, nV 0 J 0 0 VJ the wavenumber of the line, Q(T) is the total internal partition function at temperature T, C is the isotopic abundance, and gVJ is the nuclear statistic weight. Within the framework of the method of effective operators, the line strength is calculated using the eigenfunctions of an effective Hamiltonian. In this paper we use the eigenfunctions of the effective

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4x10-4

3x10-4

2x10-4

1x10-4

-30

-20

-10

0 m

10

20

30

Transition dipole moment squared in debye2

Fig. 6. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n1 þ 2n25  n15 (solid triangles are for Pee- and Ree-branches, and open triangles for the Qfe-branch; solid squares are for Pff- and Rff-branches, and open squares for the Qef-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

1x10-4 9x10-5 8x10-5 7x10-5 6x10-5 5x10-5 4x10-5 3x10-5 2x10-5 1x10-5 -40

-30

-20

-10

0

10

20

30

m

Fig. 7. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n1 þ ðn4 þ n5 Þ0þ  n14 (solid triangles are for P- and R-branches, and open triangles for the Q-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

Hamiltonian suggested by Perevalov et al. [1], which have been obtained by Lyulin et al. [2] in the result of a fit of the parameters of this Hamiltonian to the collected from the literature observed line positions lying in the 0–10000 cm1 wavenumber region. This Hamiltonian is based on the assumption of Abbouti Temsamani and Herman [23] about cluster structure of the vibrational energy levels of the acetylene molecule, due to the following approximate relations between harmonic frequencies o1  o3  5o4  5o5 , o2  3o4  3o5 .

(9) (10)

A cluster (or polyad) of vibrational states numbered by integer value P is formed by vibrational states, the vibrational quantum numbers of which satisfying the equation P ¼ 5V 1 þ 3V 2 þ 5V 3 þ V 4 þ V 5 .

(11)

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1x10-4 9x10-5 8x10-5 7x10-5 6x10-5 5x10-5 4x10-5 3x10-5 -40

-30

-20

-10

0

10

20

30

m

Transition dipole moment squared in debye2

Fig. 8. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n1 þ ðn4 þ n5 Þ0  n14 (solid triangles are for P- and R-branches, and open triangles for the Q-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

1x10-4

5x10-5

-30

-20

-10

0 m

10

20

30

Fig. 9. Variation of the transition dipole moment squared jRj2 , in D2 (1D ¼ 3.33546  1030 C m), vs. m, for the band n1 þ ðn4 þ n5 Þ2  n14 (solid triangles are for Pee- and Ree-branches, and open triangles for the Qfe-branch; solid squares are for Pff-and Rff-branches, and open squares for the Qef-branch). The solid lines have been calculated using the constants found in this work (see Table 12).

The operators of resonance interactions (interactions between vibrational states belonging to the same polyad) are included into an effective Hamiltonian in explicit form, and those of nonresonance interactions (interactions between vibrational states belonging to different polyads) are accounted for by effective Hamiltonian parameters. The vibration-rotational states of a linear molecule in its ground electronic state, within the framework of the effective operator approach, can be labeled by four labels: P, N, J, e, where N is the ranking index of eigenvalues in a (P, e, J) block. Eigenfunctions of effective Hamiltonian can be presented in the following way X J V 1 V 2 V 3 V 4 V 5 ‘4 ‘ 5 jV 1 V 2 V 3 V 4 V 5 ‘4 ‘5 JKi, Ceff C PN (12) PNJ ¼ 5V 1 þ3V 2 þ5V 3 þV 4 þV 5 ¼P ‘4 ‘5

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1 V 2 V 3 V 4 V 5 ‘ 4 ‘5 where jV 1 V 2 V 3 V 4 V 5 ‘4 ‘5 JKi is Wang-type basis function (see Ref. [3]) and J C V is expansion PN coefficient. Using these notations the line transition moment squared can be given as [3]    X X  J V 1 V 2 V 3 V 4 V 5 ‘4 ‘5 W P0 N 0 J 0 0 PNJ ¼ ð2J þ 1Þ C 5V þ3V þ5V þV þV 5 ¼P 5DV þ3DV þ5DV þDV þDV ¼DP PN 1 2 3 4 5 1 2 3 4  ‘ ‘ D‘ D‘ 4 5



J0

4

5

1 þDV 1 V 2 þDV 2 V 3 þDV 3 V 4 þDV 4 V 5 þDV 5 ‘ 4 þD‘ 4 ‘5 þD‘5 4 D‘5 CV M D‘ FDJDK ðJ; KÞ P0 N 0 0 DV

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   4 D‘ 5 f D‘ ðV ; ‘4 ; ‘5 Þ 1 þ d‘4 ;0 d‘5 ;0 þ d‘4 þD‘4 ;0 d‘5 þD‘5 ;0  2d‘4 ;0 d‘5 ;0 d‘4 þD‘4 ;0 d‘5 þD‘5 ;0 DV !2  X DV D‘ D‘  DV D‘4 D‘5 4 5  1þ ki V i þ F DJDK ðJ; K Þ  , ð13Þ  i 

where the functions FDJDK ðJ; KÞ for DK ¼ 0; 1 coincide with the Clebsh–Gordan coefficients  FDJDK ðJ; KÞ ¼ ð1DK JK J þ DJK þ DKÞ,

(14)

4 D‘5 4 D‘ 5 ðV ; ‘4 ; ‘5 Þ and the Herman–Wallis type function DV F D‘ the functions f D‘ DV DJDK ðJ; KÞ are given in Ref. [3]. Eq. (13) is presented here in more simple form compared to the respective Eq. (12) in Ref. [3] because the second part of the latter is not relevant to the present work. The parameters of the matrix elements of the effective dipole moment operator 4 D‘ 5 4 D‘ 5 M D‘ ¼ M D‘ ; DV DV

D‘4 D‘5 D‘4 D‘5 ¼ d DV ; d DV J J

D‘4 D‘5 D‘4 D‘5 kDV ¼ kDV ði ¼ 1; 2; 3; 4; 5Þ; i i

D‘4 D‘5 D‘4 D‘5 bDV ¼ bDV , J J

D‘4 D‘5 D‘4 D‘5 d DV ¼ d DV JQ JQ

4 D‘5 involved into Eq. (13) (see also expression for DV F D‘ DJDK ðJ; KÞ in Ref. [3]) describe simultaneously the line intensities of hot and cold bands, belonging to the same series of transitions determined by the value of DP. Within the framework of this semi-empirical approach, these parameters are fitted to the observed line intensities. Then, they can be used for the prediction of the line intensities of the hot bands, and for the prediction of the intensities of lines with high values of the quantum number J.

3.2. Line intensity fitting Using the approach described above, we have performed least-squares fitting of the line intensities of the bands belonging to two series of transitions: recently published DP ¼ 4 series [4] (3.8-mm region) and measured 1 V 2 V 3 V 4 V 5 ‘4 ‘5 in this paper DP ¼ 6 series (2.5-mm region). The values of the expansion coefficients J C V of the PN eigenfunctions have been obtained from the global fitting of the effective Hamiltonian parameters, to the observed line positions of the vibration–rotation transitions involving energy levels below 10000 cm1 [2]. The partition function QðTÞ is taken from Gamache et al. [21]. The value of isotopic abundance is C ¼ 1 because we used the observed line intensities recalculated for pure 12C2H2. The aim of the fitting procedure is to minimize the value of the dimensionless weighted standard deviation w, defined according to the usual formula ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , u N  obs uX S  S calc 2 i i w¼t ðN  nÞ, (15) d i i¼1 calc are, respectively, observed and calculated values of the intensity for the ith line, where Sobs i and S i obs di ¼ Si si =100%, si is the measurement error of the ith line in %, N is the number of fitted line intensities, and n is the number of adjusted parameters. In order to characterize the quality of a fit, it is sometimes more

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convenient to use the root-mean square (RMS) deviation defined according to the equation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 !2 , 1 u X N calc Sobs  S u@ i i RMS ¼ t N A  100. S obs i i¼1

(16)

The third statistical characteristic, which is used in this paper, is the value of the mean residual (MR) for a given band. The MR is defined according to the equation MR ¼

Nb 1 X S obs  Scalc i i  100, N b i¼1 S obs i

(17)

where Nb is the number of fitted line intensities for a given band. 3.3. The 3.8-mm region The spectrum in this region is formed by DP ¼ 4 series of transitions. The line intensities of two cold and three hot bands belonging to this series were published in our recent paper [4]. Using these published data we have fitted five vibrational and three vibration–rotation parameters of the effective dipole moment operator. The weighting of the line intensities has been performed according to their experimental uncertainties equal 5%. The fit showed very good consistency of the line intensity data. Because of this we decided to enlarge the weights for the line intensities using si ¼ 3%. In this last case the standard deviation of the fit w ¼ 0:46 has been achieved. This could mean that the precision of the measurements is better than 3%. The fitted values of the effective dipole moment parameters are presented in Table 13. The band statistics is presented in Table 14. In Fig. 10 the quality of the fit is demonstrated on the example of the resonance interacting n3n14 and n2+(n4+n5)0+n14 bands. Table 13 The set of effective dipole moment parameters describing the line intensities of acetylene in the 3.8-mm region Parametera

DV1

DV2

DV3

DV4

DV5

Dl4

Dl5

Valueb

Order

M k4 bJ M bJ M bJ M

0 0 0 1 1 0 0 0

1 1 1 0 0 0 0 0

0 0 0 0 0 1 1 0

0 0 0 0 0 1 1 3

1 1 1 1 1 0 0 1

0 0 0 0 0 1 1 1

1 1 1 1 1 0 0 1

0.2377(2) 19.1(4) 0.557(8) 1.121(1) 0.398(8) 1.0360(7) 0.361(6) 0.01621(2)

102 102 102 102 102 102 102 102

a

Parameters M are given in Debye (1D ¼ 3.33546  1030 C m), while the other parameters are dimensionless. Confidence intervals (1 SD, in unit of the last quoted digit) are given between parenthesis.

b

Table 14 Band statistics of the global fit in the 3.8-mm regiona Band

Region (cm1)

Jmax

N

MR(%)

RMS(%)

n2+n15 (3n4+n5)0+ n1-n15 n3-n14 n2+(n4+n5)0+ n14

2649.8–2752.4 2515.5–2609.0 2583.7–2694.1 2627.3–2737.2 2614.6–2713.7

25 21 24 24 23

61 33 55 57 39

0.02 0.03 0.01 0.03 0.10

1.44 1.78 0.83 0.87 1.90

a Jmax is the maximum value of the rotational quantum number in the file of the observed data for a given band, N is the number of the observed line intensities for a given band, MR (%) is the mean value of the residuals for a given band (Eq. (17)), RMS (%) is the root mean square of the residuals for a given band (Eq. (16)). See text for details.

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3.4. The 2.5-mm region As we have discussed above the spectrum in this region is formed by DP ¼ 6 series of transitions. The line intensities of four cold and five hot bands belonging to this region and measured in this paper were simultaneously fitted. The weighting of the line intensities has been performed according to their experimental uncertainties equal 5%. The standard deviation of the fit is 0.7. So the fit reached experimental accuracy. In this fit we used six vibrational and four vibration–rotation effective dipole moment parameters. They are presented in Table 15. The band statistics is presented in Table 16, where the mean residuals and RMS are given in the column denoted as ‘‘Complete’’. To test the predictive ability of the model we have excluded two last bands in Table 16 from the fit and in the the result of the new fit we have obtained the new set of effective dipole moment parameters from the line intensities of the rest seven bands. The line intensities of two excluded bands have then been predicted. The mean residiuals and RMS of this new fit together with the predictions for the two excluded bands are given in the column denoted as ‘‘Incomplete’’ in Table 16. The obtained sdandard deviation of this new fit is equal to 0.6. Figs. 11 and 12 give the comparison of the residuals between fitted and observed and between predicted and observed line intensities for these two bands. Both Table 16 and Figs. 11, 6

(Obs.-Calc.)/Obs.(%)

4

2

0

-2

-4 -30

-20

-10

0 m

10

20

30

Fig. 10. The residuals for the line intensities of the two resonance interacting bands obtained in the global fit in the region 3.8-mm: open triangles are for the n3  n14 band and solid squares are for the n2 þ ðn4 þ n5 Þ0þ  n14 band.

Table 15 The set of effective dipole moment parameters describing line intensities of acetylene in the 2.5-mm region Parameter M k4 bJ M bJ M bJ dJQ M M a

a

DV1

DV2

DV3

DV4

DV5

Dl4

Dl5

Valueb

Order

1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 1 1 1 1 0

0 0 0 1 1 0 0 0 0 1

0 0 0 1 1 0 0 0 2 1

1 1 1 0 0 3 3 3 1 2

0 0 0 1 1 0 0 0 2 1

1 1 1 0 0 1 1 1 1 2

0.4873(5) 1.8(2) 0.700(5) 0.4401(6) 0.857(5) 0.2938(4) 7.22(8) 0.147(5) 0.095(3) 0.19(1)

102 102 102 102 102 103 103 103 103 103

Parameters M are given in Debye (1D ¼ 3.33546  1030 C m), while the other parameters are dimensionless. Confidence intervals (1 SD, in unit of the last quoted digit) are given between parenthesis.

b

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Table 16 Band statistics of the global fit in the 2.5-mm regiona Band

n2+(2n4+n5)1II n3+n14 n1+n15 n2+3n15 n1+2n05-n15 n1+(n4+n5)0+n14 n1+(n4+n5)0n14 n1+2n25n15 n1+(n4+ n5)2n14

Rang, cm1

Jmax

3768.4–3967.4 3786.5–3979.3 3989.4–4162.6 4087.5–4208.5 3991.8–4133.6 3975.9–4121.1 3983.5–4133.4 4003.5–4151.9 4009.8–4137.8

43 42 39 31 32 35 36 33 28

Complete

Nb

75 69 89 66 61 61 64 102 78

Incomplete

MR(%)

RMS(%)

MR(%)

RMS(%)

0.05 0.25 1.07 0.11 0.61 0.16 0.34 0.91 0.01

3.16 3.49 2.71 3.29 3.13 3.08 2.56 4.24 3.33

0.05 0.25 0.27 0.13 0.20 0.16 0.32 1.74 4.06

3.16 3.50 2.50 3.29 3.11 3.01 2.57 4.52 5.39

a

Jmax is the maximum value of the rotational quantum number in the file of observed data for a given band, Nb is the number of observed line intensities for a given band, MR (%) is the mean value of the residuals for a given band (Eq. (17)), RMS (%) is the root mean square of the residuals for a given band (Eq. (16)). See text for details.

14 12

(Obs.-Calc.)/Obs.(%)

10 8 6 4 2 0 -2 -4 -6 -8 -30

-20

-10

0

10

20

30

m

Fig. 11. The residuals between observed and fitted line intensities (solid squares) and between observed and predicted line intensities (open circles) for the n1+(n4+n5)2(e)n14(e) band.

12 demonstrate good predictive ability of our model: the majority of the residuals between observed and predicted line intensities are within the experimental uncertainty. The fit which involves all the observed bands only shifts slightly all the residiuals of the band on the same value. 4. Conclusion In this paper, measured line intensities of perpendicular bands of the acetylene molecule, in the 2.5-mm region, have been presented. The vibrational transition dipole moment and Herman–Wallis parameters of each studied band have been determined. Involving these measured values together with those previously published in the 3.8-mm region, the simultaneous fit of all these line intensities has been performed within the framework of the method of effective operators. Two sets of the effective dipole moment parameters for DP ¼ 4 series of transitions (3.8mm region) and for DP ¼ 6 series of transitions (2.5-mm region) have been found. On the example of the line intensities of two hot bands n1+(n4+n5)2n14 and n1+2n25n15 the good predictive ability of the effective operators approach has been demonstrated.

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(Obs.-Calc.)/Obs.(%)

10

5

0

-5

-10

-30

-20

-10

0

10

20

30

m

Fig. 12. The residuals between observed and fitted line intensities (solid squares) and between observed and predicted line intensities (open circles) for the n1 þ 2n25 ðeÞ  n15 ðeÞ band.

Acknowledgements The authors thank Prof. A. Barbe and Mr. J.-J. Plateaux for their contribution to this research. This work was partly supported by CNRS-RFBR PICS Grant 05-05-22001. VIP and OML acknowledge the Program 2.10.1 of Russian Academy of Sciences ‘‘Optical Spectroscopy and Frequency Standards’’.

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