P Manceron

Groner

Laurent ; Soleil

Peter ; Groner

Agnès Perrin
email: agnes.perrin@lmd.polytechnique.fr

J M Flaud

F Kwabia-Tchana

L Manceron

Investigation of the ν 8 and ν 21 bands of propane CH 3 CH 2 CH 3 at 11.5 and 10.9 µm: evidence of large amplitude tunnelling effects

Keywords: Propane (CH3CH2CH3), two equivalent methyl rotors, High resolution Infrared spectroscopy, vibration-rotation resonances, large amplitude tunnelling High resolution Infrared spectroscopy, Propane (CH 3 CH 2 CH 3 ), two equivalent methyl rotors, vibration-rotation resonances, large amplitude tunnelling

    Investigation of the ν 8 and ν 21 bands of propane CH 3 CH 2 CH 3 at 11.5 and 10.9 µm: evidence of large amplitude tunnelling effects

Investigation of the ν 8 band (B-type band, C-C symmetric stretch at 870.348 cm -1 ), and ν 21 band (A type band, CH 3 rock at 921.382 cm -1 ) of propane (C 3 H 8 ) using high resolution FTS spectra recorded at low temperature on the SOLEIL synchrotron source. The rotational energy levels in the upper states are split because of interactions with the internal rotations of the methyl groups, leading to the identification of AA, EE, AE and EA torsional components. An effective "vibration -torsion-rotation" Hamiltonian model and its associated line intensity code were built in G 36 symmetry group.

a) Atmospheric and planetary importance

Propane (C 3 H 8 ) has been detected in the atmosphere of giant planets [1]. Also, in the Titan limb spectra recorded by the Cassini CIRS instrument, the presence of C 3 H 8 was detected through several infrared bands like the ν 26 (748cm -1 ), ν 8 (869 cm -1 ), ν 21 (922cm -1 ), ν 20 (1053cm -1) and ν 25 (1192 cm -1 ) bands [2]. At this occasion, it was noticed that the shape of the ν 21 band in the cold atmosphere Titan spectrum appears somewhat different from the room temperature (PNNL spectrum) [3], due to the presence of hot bands on the short-wavelength side of the fundamental band in the laboratory spectrum.

.In the Earth's atmosphere, propane (C 3 H 8 ) is produced by biomass burning and is an indicator of pollution in urban areas [4]. The available high resolution spectroscopic data on propane are, however, rather sparse, and, as a consequence, only the ν 26 and ν 18 bands, located at 748 cm -1 and 1376 cm -1 respectively, have been used to retrieve propane abundances using line lists generated during previous high resolution detailed spectroscopic studies [5,6].

b) Spectroscopic data

The main isotopologue ( 12 CH 3 12 CH 2 12 CH 3 ) of propane is an asymmetric top molecule with a C 2V symmetry equilibrium structure [7]. Among its 27 vibrational modes [START_REF] Shimanouchi | Tables of Molecular Vibrational Frequencies, Consolidated Volume I[END_REF], the two lowest energy ones are associated to the two large amplitude methyl torsional modes located at ~217(8) cm -1 (ν 14 , A 2 in the C 2V symmetry group) and ~265(8) cm -1 (ν 27 , B 2 ), respectively [START_REF] Shimanouchi | Tables of Molecular Vibrational Frequencies, Consolidated Volume I[END_REF]9], while the ν 9 band located at 369.22 cm -1 [10] corresponds to the low amplitude C-C-C (A 1 ) symmetric deformation. The barrier to internal rotation is estimated to be between 1139 and 1353 cm -1 depending on authors and methods [11,12,13]. Due to the presence of internal rotation, each rotational energy level is split into four components (AA, AE, EA and EE), with tunnelling splittings ranging from ~10 -4 cm -1 to ~2×10 -3 cm -1 for the ground and the first two torsional excited states, ν 14 and ν 27 .

The rotational transitions within the vibrational ground state [7,[14][15][16] and the first two torsional excited states (ν 14 and ν 27 ) [16] were measured using microwave and submillimeter techniques. In Ref. [16] the observed transition frequencies were satisfactorily reproduced using the SPFIT [17] and the ERHAM [18][19][20][21][22] theoretical approaches.

Cross sections of propane have been measured at low or medium resolution ( [3,[23][24][25][26], and references therein) or at high resolution [26][27][28]. Also several Raman spectroscopy [11,12] and references therein) and ab initio calculations ( [13] and references therein) have been performed.

As far as detailed line by line infrared investigations at high resolution are concerned, only a few bands among the 22 infrared active fundamental bands of propane have been analysed. This is because assigning the propane spectrum in the infrared region represents quite a challenge [5,6,10,29,30]. The spectra are indeed very dense because of the small values of the rotational constants (A= 0.9742562313 cm -1 , B= 0.2817271865 cm -1 and C = 0.2488055608 cm -1 ) [16]. Moreover, lines belonging to hot bands contribute significantly, complicating the assignment process even at low temperatures.

During these first high resolution infrared investigations which involve, starting from the low frequency range, the ν 9 band located at 369.22 cm-1 [10] and the ν 26 band at 748.531 [6,29], only "classical" vibration rotation resonances were observed (without torsional splittings). This was also more or less the case for the ν 19 , ν 18 , ν 5 , ν 17 , ν 24 and ν 4 bands, located at 1338.965, 1376.850, 1461.072, 1462.488 1471.875 and 1476.384 cm -1 respectively [5]. These bands involve indeed vibrational states with vibrational energies rather close to the torsional barrier heights. However, some series of transitions, belonging mainly to the ν 18 band, appeared to be split into weak components [5]. The existence of these splittings (about 0.001 to 0.005 cm -1 ) was explained by resonances coupling the 18 1 upper state energy levels with levels belonging to highly excited states involving the torsional modes ν 14 and/or ν 27 .

On the contrary, the overall set of transitions of the ν 21 A-type band and of the ν 8 Btype band located at 921.384 cm -1 and 970.348 cm -1 respectively are split into torsional transitions. In addition, during the investigation of the ν 21 band, some K a series (mainly K a =6 and 7) appeared to be perturbed by a dark state through a local C-type Coriolis resonance.

The main result of the present study is to explain the existence of these torsional components through the presence of anharmonic and Coriolis resonances coupling the 21 1 and 8 1 states with nearby highly excited torsional states involving ν 9 , ν 14 and ν 27 . c) Torsional splitting in the ( 9 1 , 14 1 , 27 1 ) excited states of propane The information can be derived from two sets of computations which combined ab initio calculations and/or low resolution Raman measurements or infrared data. The first one is a two-state computation involving the ν 14 and ν 27 torsional modes [11,12]. The second one [13] is a three-state computation involving the ν 9 low amplitude mode together with ν 14 and ν 27 . The outputs of these studies are predicted lists for the AA, EE, EA, and AE subcomponents band centres (E AA , E EE , E EA and E AE ) for a set of (v 9 =0,v 14 ,v 27 ) states covering the 0-1380 cm -1 energy range (Table 4 of Ref. [12]) and for a set of (v 9 ,v 14 ,v 27 ) states covering the 0-780 cm -1 energy range (Table 6 of Ref. [13]). In both calculations, the EA and AE torsional sub states have almost equal band centre values identified as E AE&EA :

E EA ≈E AE =E AE&EA Eq.(1)
Also for a molecule with a high torsional barrier, the following well known relation links the AA, EE and EA and AE band centres [31,32]:

( ) ( ) AA , EE AA EE AA AE & EA 2 E E 2 E E ∆ × = - × ≈ - Eq.( 2
)
where ( )

AA EE AA , EE E E - = ∆ Eq. ( 3
)
It is interesting to compare the results of the two computations for the 0 to 780 cm -1 energy range. For E AA , the values differ only by a few cm -1 (up to 3 cm -1 ). However, for the

∆

, the absolute values quoted by Villa et al. [13] are about four times larger than those of Engeln et al. [12].

Therefore, when focussing on the 850-950 cm -1 energy range, the only information that one can extract from the computations are (i) a crude estimate of the AA band centres for the (v 9 ,v 14 ,v 27 )=14 2 27 2 and (v 9 ,v 14 ,v 27 )= 14 1 27 3 dark vibrational states located around 872.8 and 929.1 cm -1 [13], respectively, (ii) the positive sign for the EE-AA torsional splittings ( = ∆ AA , EE 0.79 cm-1 and = ∆ AA , EE 0.44 cm-1 for 14 2 27 2 and 14 1 27 3 respectively). We also need information for the (v 9 ,v 14 ,v 27 )= 9 1 27 2 dark state which is presumed to be responsible for the existence of a C-type Coriolis resonance which perturbs the 21 1 energy levels for K a ≈6.

Since the ν 9 mode is not a torsional mode, we can assume that the

∆

≈ 0.005 cm -1 and 0.02 cm -1 in Ref. [12] and [13] respectively). This value is significantly smaller than for 14 2 27 2 and 14 1 27 3 . Therefore, the torsional splitting in the 9 1 27 2 dark state was not accounted for during this study.

The paper is organised in the following way. First, the experimental spectrum used during this investigation is described. Then, the symmetry properties are discussed. The

II Experimental details and wavenumbers calibration a) Experimental details

The high-resolution absorption spectrum of propane (810-970 cm -1 spectral range) was recorded using the AILES Beamline at SOLEIL, coupled to a Bruker IFS125HR Fourier transform spectrometer [33,[START_REF] Brubach | AIP Conference Proceedings[END_REF] and the SOLEIL-LISA cryogenic cell [START_REF] Kwabia Tchana | [END_REF]. The instrument was equipped with a KBr/Ge beamsplitter. The synchrotron source was operated with a 430 mA ring current and a liquid nitrogen-cooled HgCdTe (MCT) detector was used in conjunction with a 500-1300 cm -1 band pass optical filter to improve the S/N ratio. The spectrometer was evacuated to about 5×10 -3 Pa in order to minimize H 2 O and CO 2 absorptions. The spectrum was recorded with a nominal resolution of 0.0015 cm -1 . The temperature was regulated to 142 ± 2 K along the entire optical path, set to 45.14 m.

Propane was purchased from Sigma Aldrich and used without further purification. The chemical purity of propane was specified as 99.97%. The following procedure was used for measurements: first a background spectrum was recorded at the same temperature of 142 K while the cell was continuously evacuated; next the infrared gas cell was filled with 14.0 (3)

Pa of pure propane. The sample pressure in the cell was measured using a home-made cryogenic precision capacitive pressure gauge [36].

b) Wavenumbers calibration

The spectrum was calibrated with residual CO 2 lines observed in the spectrum with their wavenumbers taken from HITRAN [37]. The quantity ε = ((ν 3716 /ν this work ) -1) has been calculated using about 30 transitions between 650-680 cm -1 (CO 2 ). The mean value < ε > = -8.48×10 -7 has been found and the resulting precision is ± 0.000063 cm -1 (RMS) for well isolated lines. We then estimated the absolute accuracy of the measured propane line positions as the square root of the sum of squares of the accuracy of the line positions of CO 2 from [37] (better than 0.0001 cm -1 ) and two times the RMS of the wavenumber calibration, yielding 0.0002

III Theoretical considerations

The overall structure of propane [7] is presented on Fig. 2. Since torsional tunneling has to be considered, the G 36 inversion permutation group [38] which is quoted in Table 1, has to be used instead of the C 2V point group, which is suitable only for a rigid molecule [5]. The tables giving the reverse correlation between G 36 and the C 2v point group can be found in

Refs. [30,[START_REF] Bunker | Molecular Symmetry and Spectroscopy[END_REF]. ( )

K J K J 2 1 K J - γ + = γ with 1 ± = γ Eq.(4,a) and 0 K J 1 0 K J = = + = γ = for K=0.
Eq.(

It can be seen that only the first four (A 1 to A 4 ) representations of the G 36 group are to be used for the classification of any rotational wave function [J K a K c > or of any |v 1 ,…v 27 > vibrational state of propane.

For propane, the α 1 and α 2 angles ( with α 1 and α 2 ∈[-π,π]) [12] describe the torsions of the CH 3 -groups about C-C bonds with the convention that these angles are taken to be zero if the predetermined hydrogen of the top lies in C-C-C plane. The AA, AE, EA and EE torsional wave functions are expansions of tensorial products of the ( ) ( )

2 n 3 1 n 3 2 2 1 1 α α σ σ + + Ψ ⊗ Ψ
torsional basis wavefunctions, with:

( ) ( ) [ ] k k k n 3 i exp 2 1 k n 3 k k σ α π α σ + = + Ψ , with k=1,2 Eq.( 5
)
In this expression, 1 σ and 2 σ are the torsional phase factors, modulo 3, which can take the values σ =0, -1, +1. The AA, AE, EA and EE subcomponents are associated to the (σ 1 ,σ 2 )= (0,0), {(1,-1), (-1,1)}, {(1,1),(-1,-1)} and {(1,0), (0,1), (-1,0), (0,-1)} phase factors, respectively. 3 gives also the nuclear spin statistical weights (g Nuclear ) associated to the torsion-rotation levels in the ground vibrational state of propane.

IV Assignment process a) The ground state energy levels

The effects of the torsional motion in the ground state have negligible impact in the infrared region; so, in the present work, the ground state energy levels were computed using the ground state parameters of Drouin et al. [16] using an A-type Watson Hamiltonian in the I r representation [START_REF] Watson | Aspects of quartic and sextic centrifugal effects on rotational energy levels[END_REF]. For medium and high K a values, the [J, K a , J-K a ] and [J, K a , J-K a +1] levels become "degenerate" due to the particular values of the rotational constants (A~0.97426 cm -1 , B~0.28173 cm -1 , and C~0.24881cm -1 ). In the rest of this paper, the notation "K c =d" and "K c ≠d" stands for "degenerate" [J, K a , K c ] levels (with K c =J-K a and K c =J-K a +1), and "non-degenerate" levels respectively.

As will be seen later, the observed torsional structure changes dramatically depending on the existence or non-existence of a K c degeneracy for the rotational transitions involved in the cluster. Therefore, at the starting point of the present analysis, it was necessary to guessapproximatively -the form of the observable torsional clusters and the possible occurrence of a K c degeneracy. To do so we used the computed ground state energy levels [16] and defined the following separation limit:

F o r P e e r R e v i e w O n l y δ= |E( [J, K a , K c =J-K a ])-E([J, K a , K c =J-K a +1]| Eq.(7.a) )
δ≤ δ limit for "K c =d", δ> δ limit "K c ≠d" Eq.(7.b) with δ limit ≈0.001 cm -1 Eq.(7.c) Indeed, for ν 8 (resp. ν 21 ) transitions involving ground state energy levels with δ≤ δ limit , the effects of an A-type Coriolis resonance within the 8 1 (resp. 21 1 ) upper level are quite significant for EE and AE torsional subcomponents, affecting strongly the form of the cluster.

In such conditions, Fig. 3 gives the K c =d threshold limit for the ground vibrational state energy levels, according to K a and J values. This figure is restricted to the range of K a and J values (K a ≤ 18 and J ≤ 50) which could be reached during the assignment process. All the [J, K a =1 K c ] ground state levels are "non-degenerate", while for K a ≥ 10 all [J, K a , K c =d] levels are "degenerate" K c levels.

b) The ν ν ν ν 8 and ν ν ν ν 21 bands : structure of the torsion -rotation transitions

According to symmetry considerations in the C 2V point group, the ν 8 and ν 21 bands are B-and A-type bands with ( )

odd c K , odd a K = ∆ = ∆
and ( )

odd c K ; even a K = ∆ = ∆
selection rules, respectively. However, this is true only in the C 2V point group approach and for the AA subcomponents. Additional transitions are indeed symmetry-allowed for the other torsional components.

The assignment process proved to be difficult for both the ν 8 and ν 21 bands because we are dealing with a very dense and highly perturbed spectrum. In addition, each infrared vibration-rotation transition is split into several torsional subcomponents. Then, since the ground state combination differences cannot fully help in the identification of the torsional

J K a K c =J-K a ] and [J K a K c =J-K a +1]
levels for the EE or AE torsional substates. The consequence is that the structure of the torsional clusters in the observed spectrum looks quite differently from the expected pattern, depending on the existence (or non-existence) of a K c degeneracy.

• (Low K a values): For series involving "non degenerate" K c values, triplets of lines are generally observed, with a central line about twice as strong as the two other ones. Using the predictions of Table 4 which states that the relative intensities are in the ratio AA/EE/(AE+EA)= 36/64/36 or AA/EE/(AE+EA)= 28/64/28 for even or odd values of

      + " c K " a K
, respectively, it is easy to deduce that the central line is the EE torsional component, while the other two, of equal intensities, are the AA and the (AE+EA) components. Examples of this type of triplet patterns are observable on the left hand side of • (High K a values): For series involving "degenerate" K c values the K c degeneracy is lifted for the EE and AE subcomponents, but not for the AA and EA subcomponents. This is why six lines are then observed with a 64/64/64/32/16/16 intensity ratio. The two weakest components (relative intensity ≈ 16) are the "non-degenerate" (K c =J-K a ) and (K c =J-K a +1) AE transitions, while the EA "degenerate" (K' c =d) component is twice as strong (relative intensity ≈ 32). Finally, the three strongest lines, with similar intensities (intensity ratio 64/64/64) correspond to the (K' c =d) AA "degenerate" pair, together with the two split (K' c =J'-K' a ) and (K' c =J'-K' a +1) EE transitions. For such subcomponents, the torsional assignment (one AA, and two EE) was a tedious task, because no information can be provided by the intensity pattern. Figure 5 and the high frequency range of Figure 4 provide good examples of the "high K a " torsional structure.

• (Intermediate K a values) For J and K a values corresponding to border cases between "degenerate" and "non-degenerate" K c values, the cluster system is even more complex, and this is evidenced on Figures 6 and7. Indeed, C 2V non-allowed transitions (with ( )

even c K ; even a K = ∆ = ∆
for ν 21 and ( )

even c K ; odd a K = ∆ = ∆
for ν 8 are also observable for EE and AE in addition to the classical A and B-type transitions, respectively.

In conclusion, one can say that setting unambiguously a torsional assignment to a given [J' K' a K' c ]-[J K a K c ] torsional subcomponent was not an easy task. Once a few lines were firmly assigned, an initial fit was performed yielding preliminary values for the band centers and rotational and torsional constants and interacting parameters for the 21 1 and 8 1 vibrational states. These preliminary constants enabled the computation of a synthetic spectrum of line positions and relative intensities, which was then used to perform new assignments leading to an improvement of the upper state parameters. This process was pursued until it was not possible to perform additional assignments. The final results are given in Table 5 for the ν 21 and ν 8 bands.

V Hamiltonian matrix a) Introduction

Owing to the complexity of the vibration rotation resonances observed during the present analysis, it proved inadequate to use the codes which were developed in the microwave literature [18][19][20][21][22][START_REF] Ilyushin | [END_REF][42][43].

During our previous investigation of the ν 21 band [30] we used a rather "primitive" theoretical approach which consists in building up three different Watson-type Hamiltonians, each of them being associated to a given set of AA, EE or EA-AE torsional energy levels.

Clearly, this approach is not satisfactory, because it cannot account for the two types of clustering of the torsion-rotation energy levels and leads to erroneous torsional assignments.

Furthermore, the C-type Coriolis resonance, which perturbs several rotational series of the ν 21 band, was not considered at that time.

Therefore we use here a different approach in which a single "vibration -torsion-rotation" effective Hamiltonian is used for the whole set of vibration-rotation-torsion energy levels.

Using such an effective approach is justified because information concerning the high excited torsional states of propane is sparse and inaccurate. Therefore, it was decided to build a Hamiltonian matrix which is issued from our experience on C 2V molecules and is adapted to the G 36 symmetry group. This means that the model is able to account for any local anharmonic or Coriolis resonance and takes into consideration the torsional structure by adding to each rotational operator a periodic expansion which accounts for the (σ 1 ,σ 2 ) torsional quantum number of the considered torsional substate.

b) General form of the Hamiltonian matrices

The general form of the Hamiltonian matrices used for the {8 1 , 14 2 27 2 } and {21 1 , 14 1 27 3 , 9 1 27 2 } interacting states is given in Table 6. Table 7 gives the different type of rotational

c) Fourier transform expansion of the parameters

In order to account for the torsional motion, we perform, in Table 9, a Fourier expansion of the parameters involved in the Hamiltonian matrix. The Our approach is rather similar to the one adopted by Drouin et al [16] since our ρ parameter in Table 9 is in fact "ρ a ", the A-axis component of ρ. The present formalism is less sophisticated than the dual rotor effective Hamiltonian model (ERHAM) developed by P.Groner [18][19][20][21][22], which involves Wigner's type

) ( d ) J ( ' K K β matrices together with sinusoidal terms in 3 K avg πρ = χ
in the expansion of the parameters. However, the ERHAM model which was successfully used during the investigation of acetone [20][21] and dimethyl ether [19], cannot account for vibration rotation resonances, which are one of the major problems that we had to face to during this study.

As compared to its "torsion free" 0 v E counterpart, the vibrational energies of the AA, EE, EA, or AE torsional substates of a given v vibrational state are:

AA : v 4 v 3 v 2 v 1 0 v 0 , 0 , v
v 4 v 3 v 2 v 1 v 0 0 v 1 , 1 , v Tor Tor 2 / Tor 2 / Tor Tor 2 E E 2 + + - - × + = = σ σ AE : 2 / Tor Tor 2 / Tor 2 / Tor Tor 2 E E v 4 v 3 v 2 v 1 v 0 0 v 1 , 1 , v 2 - + - - × + = - = σ σ
Eqs [START_REF] Shimanouchi | Tables of Molecular Vibrational Frequencies, Consolidated Volume I[END_REF] With these definitions of v q Tors , the ν 8 and ν 21 least squares fit calculations lead for q>0 to v 0 q Tors ≠ values for the v=14 2 27 2 or v=14 1

VI Results of the energy level calculations

The experimental vibrational -torsional -rotational energy levels were introduced into two least squares fit calculations to derive a set of {8 1 , 14 2 27 2 } and {21 1 , 14 1 27 3 , 9 1 27 2 } vibrational energies, torsional, rotational and anharmonic or Coriolis type interacting parameters. These parameters are quoted, together with their associated uncertainties, in Table 10 and 11 for the 8 1 and 21 1 computations, respectively. During these computations we tried to adjust the ρ internal rotation parameter. However it proved that the ρ term is highly correlated with the v 0 Tors torsional shifts (v=14 2 27 2 or v= 14 1 27 3 ). Therefore it was decided to maintain ρ at the value determined by Drouin et al. [16]:

ρ=0.1511
Eq.( 9)

Let us notice that, because of the complexity of the vibration-rotation resonances and because of possible errors in the torsional labelling, the parameters gathered in Tables 10 and11 are only effective. For example the values obtained for the A rotational constants are rather large (0.993 ≤. A ≤. 1.14 cm -1 as compared to the ground state value A 0 =0.9742562313 cm-1 [16]).

The values of some centrifugal distortion constants (mainly ∆ K ) are also quite large.

Nevertheless, the values obtained for the band centers of the 14 2 27 2 and 14 1 27 3 dark states ν 14142727 = 872.5168(1) and ν 14272727 =932.8751(2) cm -1 differ little from the predictions given in Ref. [12] ( Pred ν 14142727 = 872.8 and Pred ν 14272727 =929.1 cm -1 ).

The statistical analysis on the results of the energy levels calculations are presented in -The imperfect quality of the model used to account for the resonances and for the torsional structure.

-Possible incorrect torsional or rotational assignments during the analysis process.

VII Line intensity calculations

The intensity of a line is given (in cm -1 /(molecule.cm -2 ) by [START_REF] Flaud | Water vapour line parameters from microwave to medium infrared[END_REF]:

B A A nucl 0 3 N ~R kT hcE exp kT hc exp 1 g ) T ( Z hc 3 4 8 k         -             ν - - πε ν π = ν
Eq. (10) where A and B are the lower and upper levels of the transition, respectively, ν

~=(E B -E A )/hc is the wavenumber of the transition, g nucl are the nuclear spin factors, given in Table 3 according to the 9 symmetry species in G 36 , and Z(T) is the total partition function at the temperature T.

Several computations of Z(T) were performed for propane in the literature [16,26,[START_REF] Fischer | [END_REF]. As a final result, these authors provide an empirical polynomial expansion of Z(T) as a function of the temperature:

i 4 , 0 i i T a ) T ( Z × = ∑ = Eq.( 11
)
One can notice that the Z(T) values proposed by Drouin et al. [16] or Sung et al. [26] and Fischer and Gamache [START_REF] Fischer | [END_REF] differ significantly, not only in their absolute values, but also in their temperature dependences. For the present study, we cannot use any of these expressions of Z(T) [26,[START_REF] Fischer | [END_REF]. Indeed, both computations were performed using incorrect values of the g nucl nuclear spin factors (a constant value g nucl =1028 for all levels, and the acetone nuclear spin weights, in Ref. [START_REF] Fischer | [END_REF] and in Refs. [16,26], respectively). As usual for molecules with low amplitude modes [46], performing a precise computation of Z(T) requires a good knowledge of the rotational, vibrational and torsional spectroscopic parameters, and this is clearly not the case for propane. In Refs. [16,26] the computation of Z(T) accounts for the fact that the rotational constants in the 14 1 and 27 1 states differ from those of the ground state. However, in order to get accurate Z(T) values, other sources of errors should be considered, like the rather unprecise knowledge of the fundamental torsional modes (ν 14 and ν 27 ), and of the overtones states involving ν 9 , ν 14 , and ν 27 (see for example Table 4 of Ref. [12] and Table 6 of Ref. [13]).

Knowing these limits, we decided to compute Z(T) in the harmonic approximation. More explicitly Z(T) was written as the product of its vibrational and rotational contribution.

Z(T)=Z vib (T) Z rot (T)

Eq.( 12)

The vibrational part Z vib (T) was computed in the harmonic oscillator approximation using the vibrational frequencies quoted in Ref. [START_REF] Shimanouchi | Tables of Molecular Vibrational Frequencies, Consolidated Volume I[END_REF] or, when existing, derived from high resolution investigations [5, 10, 6, 30, this work]. For the calculation of the rotational part Z rot (T), we used the ground state rotational constant of Ref. [16], together with the g nucl values of Table 3. The results of the present computations performed up to J=99 are presented in Table 12 for various temperatures. The T-polynomial dependence of Z(T) in the 70 K ≤T≤ 320K temperature range from these calculations is: Z(T) = 7.41945x10 6 --1.67279x10 5 ×T-1.33661x10 3 × T 2 -4.17648×T 3 +0.00527×T 4 Eq.( 13)

The last term of Eq.( 8), B A R , is the square of the matrix element of the transformed transition moment operator ' Z µ :

2 " c " a " ' Z ' c ' a ' ' B A K K J , 0 K K J , v R µ =
Eq. (14) In this expression, ' Z µ is the Z-component of the transformed dipole moment operator [START_REF] Flaud | Water vapour line parameters from microwave to medium infrared[END_REF], which can be expanded as usual as: The upper and ground state rovibrational wavefunctions are expanded as:

' v 0 ' Z v ' B ' v ' Z µ = µ
' ' ' ' ' ' ' ' γ = ∑ ∑ ∈ γ γ K J v C K K J , v B ' v , K ' ' K ' v ' c ' a ' ∑ γ = = γ " K " " K " 0 c a K J C K K J , 0 v ' '
Eqs. (16) In Eqs.( 16) the In the C 2v approach, only B-type or A-type transitions are to be observed for the ν 8 or ν 21 band, respectively. Although this is formally exact only for the AA subcomponents, the transition moment operator was written as:

X j A X ' j j ' v 0 ' Z ' v 0 " v µ = µ ∑ = , Eq.(17)
Where X stands for B or A for the ν 8 or ν 21 band, respectively, and the expansions of the X j A rotational operators are quoted in Ref. [START_REF] Flaud | Water vapour line parameters from microwave to medium infrared[END_REF]. In our case the expansions were restricted to the main terms, namely: and S 11µm (296K)= 2.81± 0.55 [24].

Eqs. (19) In the present work, we used for S 11µm (296K) a weighted mean value deduced from these rather consistent measurements: S 11µm (296K) = 2.853 ± 0.107 ×10 -19 cm -1 /(cm -2 .molecule) at 296K Eq.( 20)

In Eqs. ( 19) and ( 20), S 11µm (296K) is the total band intensity at 11 µm, which includes the contribution of hot bands. Using the method described in Ref. [47] it is possible to estimate the intensity of the associated (ν 8 and ν 21 ) cold bands associated to the 11 µm region by using the expression: The other information which is of importance is the intensity ratio of the ν 21 and ν 8 bands. Since a minimum of the infrared signature is observed around 892 cm -1 in the propane cross sections, it is possible to consider that below (resp. above) this 892 cm -1 limit, ν 8 and its associated hot bands (resp. ν 21 and its associated hot bands) provide the bulk of the contribution to the observed infrared signature. This is confirmed by our high resolution simulation (see Fig. 1). Also Giver et al. [25] provide separately band intensity values in the 817-892 and 892-1000 cm -1 spectral ranges. Using the various literature data, we estimate the intensity ratio of the cross sections above and below 892 cm -1 . One gets:

S 11µm (T) ≈Z Vib (T)× ( ) ( ) ( ) T
( ) [28] Eq.( 22)

) K 296 ( S ) K 296 ( S ) K 296 ( R
Averaging these values, one gets:

= = ) K 296 T ( R high low 2.15 ±0.12
Eq.( 23) Using all the information described previously, we can then determine the ν 8 and ν 21 line intensity parameters. For this, we calculated synthetic spectra and summed the ν 8 and ν 21 line intensities in the 817-892 and 892-1000 cm -1 spectral ranges. The Eq.( 24)

Let us notice that, for both parameters, only the absolute value can be obtained.

VIII Synthetic spectrum and validations: (a) Line intensity calculations:

Using the vibrational energies and the rotational, torsional and coupling constants given in Tables 10 and11 for the upper interacting states, and in Ref. [16] for the ground state, together with the B ' 1 8 0 µ and A ' 1 21 0 µ transition moment constants given Eqs. (24) we have generated at 142 K and 296K comprehensive lists of line positions and intensities for the ν 8 and ν 21 bands of propane. 3 As the energy level calculation was not fully satisfactory, the calculated upper state energy levels were replaced by the experimental ones whenever possible. The calculations were performed for a "pure" isotopic sample of propane using an intensity cutoff of 1.x10 -25 cm - 2 (1 Debye= 3.33564 ×10 -30 C× m) 3 These linelists can be obtained upon request from the authors. 4. The torsional identification of the weakest lines (EA for the medium one, and AE for the two weakest ones) was unambiguous. Note that the two AE subcomponents involve the same E 1 or E 2 submatrix for the upper levels. For the three strongest components, with equal intensities, only the calculations can provide the correct torsional assignment, since two of them are split into K c =J-K a and K c =J-K a +1 EE transitions, while the third one is the EA "degenerate" (K' c =d) transition.

Figures 6 and7 give a portion of the R branch of the ν 8 band at 881.0 cm -1 and of the P branch of the ν 21 band at 915.6 cm -1 . In these spectral regions the transitions involve "intermediate" K a values (see Figure 3 and Table 4). Therefore, for the EE subcomponents, Ctype transitions for the ν 8 band, (resp. "forbidden" transitions with ∆K a =even, ∆K c =even for the ν 21 band) are clearly observable in the spectrum, together with the "classical" B-type (resp. A-type) lines Figure 9 presents a portion of the ν 8 band in the 872 cm -1 spectral range. The agreement between observed and calculated spectra is generally satisfactory, except for the transitions involving the EE components of the [10 2 8] level of the bright v 8 =1 and the |10 1 10] level of the 14 2 27 2 dark state, which are affected by a local C-type resonance. This is one example where the computation is not fully satisfactory.

Figure 10 shows the central part of the Q branch of the ν 21 band. As compared to our previous investigation of the same band (see Fig. 5 of Ref. [30]) significant progress is achieved. Indeed, the previous Q branch investigation was restricted to K a values with K a ≤ 15 instead of K a ≤ 18 in the present case.

c) Validation with the existing cross sections and "JPL pseudo lines":

For this validation, we computed "artificial" low resolution cross sections at 296K and at 204K using our line list for the ν 8 and ν 21 bands of propane. These simulations, here after identified as "ThisWork_Cross-296K" and "ThisWork_Cross-204K" were generated for a propane sample diluted in one atmosphere (Pressure=1013 hPa) of nitrogen, (line width of ~0.1 cm -1 ) and for this calculation we used a Lorentzian line shape at the temperature of calculation in order to simulate approximatively the experimental conditions of the PNNL spectra.. Similarly the "JPL_Cross_204K" low resolution cross sections were generated using the JPL 296K pseudo line list and the pseudo "lower state energy" provided by the authors and the partition function given in Table B1 of Ref. [26]. Finally for "Wong_Cross_204K", we degraded at the PNNL resolution the experimental cross sections from Wong et al. [28].

All these cross sections are plotted in Figure 11.

The upper trace of this figure is devoted to T=296K. On average, as expected, "This_Work_Cross_296K" are weaker than the PNNL values by a factor of ≈ 2.76033, which is the Z vib (296K) vibrational partition function at 296K (see Table 12) . In the central part of the ν 21 band, three Q branch peaks are observable on the PNNL cross sections instead of one, at 921.8 cm -1 , in our model. ). We notice that at 204K the JPL [26] and Wong's [28] data are in perfect agreement, while, as expected, our cross sections are globally weaker. More precisely, the integrated band intensities in the 830-970 cm -1 spectral region are JPL S 204K =2.70,

The bottom part of

(142635) ×(78) ×(C1C2)* ( 14) × ( 25

) × (36) × (78) × (C1C2) (142536) × (C1C2) × (78) (23) × (56) × (78)* A 1 1 1 1 1 1 1 1 1 1 T b A 2 1 1 1 1 1 1 -1 -1 -1 T c J a A 3 1 1 -1 1 1 -1 1 1 1 J b A 4 1 1 -1 1 1 -1 -1 -1 -1 T a J c E 1 2 2 2 -1 -1 -1 0 0 0 E 2 2 2 -2 -1 -1 1 0 0 0 E 3 2 -1 0 2 -1 0 2 -1 0 E 4 2 -1 0 2 -1 0 -2 1 0 G 4 -2 0 -2 1 0 0 0 0
Caption: The permutation operations involve the eight hydrogen nuclei (identified by numbers 1 to 8 (see Fig. 2) and the carbon nuclei (C1 and C2) in the methyl groups. The * sign is for the inversion.

The last two columns are for the a, b and c molecular fixed components of the translational (T α ) and rotational (J α ) operators.

Γ (σ' , σ)* K a K c J even J odd A 1 AA (0 ,0) ν 1 -ν 9 ee E + E - A2 ν 22 -ν 27 eo E - E + A 3 ν 10 -ν 14 oo O + O - A 4 ν 15 -ν 21 oe O - O + E 1 AE (1 ,-1), (-1 ,1) E 2 E 3 EA (1 ,1), (-1 ,-1) E 4 G EE (1,0),(-1,0),( 0 ,1),(0 ,-1)
*(σ 1 , σ 2 ): torsional phase factor modulo 3, as defined in the text. Symmetry of the rotational energy levels and Wang's type wavefunctions in G 36 . Associated g spin nuclear statistical weights are given.

Torsional states Γ Rotational J even J odd g Spin AA (σ 1 , σ 2 )=(0,0)

A 1 .ee E+ E- 36 A 2
.eo E-E+ 28 Symmetry-allowed Watson's type, anharmonic, and Coriolis type operators in the v-diagonal or v' v off diagonal blocks, according to the AA, AE, EA and EE torsional substates.

A 3 .oo O+ O- 36 A 4 .oe O- O+ 28 AE (σ 1 , σ 2 )=(1,2) etc E 1 ee & eo E+ & E-E+ & E-16 E 2 oo & oe O+ & O-O+ & O-16 EA (σ 1 , σ 2 )=(1,1) etc E 3 ee & oo E+ & O+ E-& O- 20 E 4 eo & oe E-& O-E+ & O+ 12 EE (σ 1 , σ 2 )=(1,0) etc G All all
(a) Vibrational diagonal terms:

AA (σ 1 ,σ 2 )=(0,0) AE (σ 1 ,σ 2 )=(1,-1) EA (σ 1 ,σ 2 )=(1,1) EE (σ 1 ,σ 2 )=(1,0) W vv H v v W v v CorA H + v v W v v CorB H + v v v v v v W v v CorC CorB .. ... CorA H + + + + (b) Off diagonal vibrational terms :
. 14 1 27 3 21 1 . or 8).

14 2 27 2 8 1 v ' v Anh v ' v v ' v CorA Anh + v ' v v ' v CorB Anh + v ' v v ' v v ' v v ' v CorC CorB CorA Anh + + + 9 1 27 2 21 1 v ' v CorC v ' v v ' v CorB CorC + v ' v v ' v CorA CorC + v ' v v ' v v ' v v ' v CorC CorB ...
The torsional dependence of these operators is given in Table 9. Note that several of these parameters are presumed to be dependent on the considered torsional substate (see Table 9). However, for clarity, the (σ 1 ,σ 2 ) superscripts are written explicitly only when necessary. Overall structure of propane [7]. The numbering of the nuclei correspond to the convention adopted in Table 1. Range of "degenerate" ("K c = d") and "non degenerate" ("K c ≠ d") energy levels of propane in the ground vibrational state, with a boundary limit of δ limit = ~0.001 cm -1 for the energy separation of the [J, K a , K c = J-K a ] and [J, K a , K c = J-K a +1] levels, see text). CAPTION: For transitions involving low K a values, three components (AA, EE, and AE&EA) are observed with the intensity pattern described in Table 4. The structure of the torsional substructure is significantly more complex for transitions involving high K a values.

( ) [ ] ( ) ( ) 2 xy v v 2 1 2 v v 2 1 2 z v v 2 1 v v W v J C B J C B J C B A E H - + + + + - + = ( ) { } 2 2 xy v J 2 xy 2 z v K 2 2 v J 2 2 z v JK 4 z v K J 2δ J , J δ ∆ J ∆ J ∆ J J J - - - - - ( ) ( ) 3 2 v J 2 2 2 z v JK 2 4 z v KJ 6 z v K H J H J H J H J J J + + + + { } { } ( ) 2 2 2 y x v J 2 2 xy 2 z v KJ 2 xy 4 z v K J 2h J , J h J , J h J J + + + +… ( ) 4 4 4 2 xy 3 2 z 2 2 1 0 v ' v J J Anh J Anh J Anh Anh Anh Anh - + + × + × + × + + = J { } 3 z v ' v 4 z 2 v ' v 3 x y v ' v 2 z v ' v 1 v ' v J CorA J CorA J , iJ CorA J CorA CorA + + + = J { } { } 2 Z x v ' v 4 x 2 v ' v 3 z y v ' v 2 x v ' v 1 v ' v J , J CorB J CorB J , iJ CorB J CorB CorB + + + = J { } { } ( ) 3 3 5 2 z y 4 y 2 3 z x 2 y 1 v ' v J J CorC J , iJ CorC iJ CorC J , J CorC iJ CorC CorC + -- + + + + = J with { } BA AB B , A + =

  Hamiltonian and of the line intensity codes are then given. Finally the results of the assignments performed for the ν 8 band and ν 21 bands are presented and discussed.

7 cm - 1 .

 71 Figure 1 gives an overview of the experimental spectrum together with the modelling of the B-type ν 8 and A-type ν 21 bands.

  labels, we had to rely on the intensity pattern of the sub-components. The intensities of the torsional patterns depend indeed on the associated vibration rotation transitions. Normally, each vibration-rotation transition of ν 8 and ν 21 should appear split into three subcomponents AA, EE, (EA and AE), because the EA and AE torsional sub states have almost equal band centers (see Eqs. 1-3). However, except for the AA subcomponents, a Coriolis resonance within the v 8 =1 or v 21 =1 excited state may significantly affect the torsional structure. In principle, A-type, B-type, or C-type Coriolis resonances are symmetry allowed in the (EE and AE), (EE and EA), or EE subcomponents, respectively. Among these the possibilities quoted above, the A-type Coriolis resonance leads to the most significant effects "degenerate" sublevels by splitting apart the [

Figure 4 .

 4 Figure 4.

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  -type, anharmonic or Coriolis type) which, according to symmetry considerations, may be considered in the v-diagonal as well as in the v' v off diagonal blocks, depending on the considered torsional submatrix (AA, EE, AE et EA).Table8details the expansion of the Watson type operators (written in the I r representation and in the A-type reduction), and of the anharmonic, A-type, B-type and Ctype Coriolis operators.

  expressions quoted in Table9,b represent the torsional contribution to the matrix element of the effective Hamiltonian operators. These expressions are sinusoidal expansions in 3 K avg πρ = χ terms, where K avg = (K+K')/2 is the average value of the K and K' values of the two Wang-type basis functions involved in the considered matrix element and ρ is the internal rotation parameter.

  27 3 dark states which are significantly smaller (in absolute value) than their associated q=0 v 0 Tors corresponding values. Therefore the AA- Eqs. (1) and (2) are fulfilled.

  ://mc.manuscriptcentral.com/tandf/tmph Email: TMPH-peerreview@journals.tandf.co.belongs to the upper B' = {8 1 , 142 27 2 } (resp. B'={21 1 , 14 1 27 3 , 9 1 27 2 } vibrational polyads of interacting states and where |v=0> stands for the ground state.

C

  γ coefficients are obtained from the diagonalization of the Hamiltonian matrices (Tables6).

2 .

 2 Where ϕ α is the direction cosine between the Z-laboratory fixed axis and the α molecular axis are the transition moment parameters.Since the spectrum used for the present investigation was not optimized for measuring precisely intensities, we had to scale the intensities, we had to scale the intensity values measured in the literature. They are (in 10 -19 cm -1 /(cm - molecule) units) at 296K 1 : S 11µm (296K)= 2.82 ± 0.017[26], S 11µm (296K)= 2.98± 0.069[3],

21 ν

 21 , Z Vib (T) is the vibrational partition function at the temperature T are the intensities of the cold ν 8 and ν 21 bands, together with their associated resonating dark bands (2ν 14 +2ν 27 ; ν 9 +2ν 27 and ν 14 +3ν 27 for ν 8 and ν 21 , respectively).

  were determined so that Eqs. (20),(21), and (23) are fulfilled.The final values that we obtained for the

Figure 1

 1 Figure 1 presents an overview of the comparison between the observed spectrum and the calculation for the ν 8 and ν 21 bands. Also figures 4 to 10 give detailed views of various portions of the spectrum.

Figures 4 and 5

 5 Figures 4 and 5 present portions of the R branch of the ν 21 band around 931.3 cm -1 and 927.88 cm -1 , respectively. On the left hand side of Figure 4, the triplet patterns (AA/ EE/ AE&EA) are characteristic of lines involving low K a values. The right hand sides of Figures 4 and 5 present the structures which are observed for transitions involving high K a values. In this case, six torsional subcomponents are observable, and their relative intensities are given in Table 4. The torsional identification of the weakest

Figure 8 20 "

 820 Figure 8 gives a portion of the P branch of the ν 21 band at 915.06 cm -1 . The transitions involve low K a values. A classical "triplet" structure in (AA/ EE/ AE&EA) is observed for [11 2 10]-[12 2 11], with equal intensities for the AA and the (EA&AE) cluster. For the [11 39]-[12 3 10] transition, the AA and EE components behave normally, while the AE and EA torsional components do not coincide because a local resonance is affecting the EA subcomponent. Such an anomalous behaviour for a "low K a cluster" confirms, at posteriori, that the "non-degenerate" AA transitions are located in the low frequency range of the

21 (

 21 Fig. 11 compares computed cross sections at 204K. The Q branch of the ν 21 band at 921 cm -1 looks prominent on the three plots, while the additional PNNL 923.5 and 924.9 cm -1 peaks which have almost disappeared at 204K are clearly due to hot bands. A similar behavior is observed on the Cassini CIRS spectra of the Titan atmosphere see Fig. 5 of Ref. 2]

WongS

  204K =2.60, and ThisWork S 204K =1.63 (in 10 -19 cm -1 /(molecule.cm -2 ) units). This leads to an average band intensity ratio R(JPL/This-Work) or R(Wong/This-Work) of R~ 1.618(±0.034) and this is quite close to the value of the vibrational partition function at 204K (Z Vib (204K)= 1.645645), which is satisfactory.It is then clear that the hot bands contribution must be accounted for in order to model properly the 296K signature at 11 µm. A first priority should concern the hot bands involving the ν 14 and ν 27 modes which contribute for 36% and 27% respectively relative to their cold band counterpart to the total band intensity.ConclusionThe first high resolution investigation of the ν 8 band (A 1 , C-C stretch) located at 870.348 cm -1 , together with an extended analysis of the ν 21 band (A 4 , CH 3 rock, at 921.382 cm -1 ) of propane, (C 3 H 8 ), are presented. For both bands, the overall set of transitions is affected by torsional splittings, leading to the identification of AA, EE and AE and EA sub components. A Hamiltonian model was set up for the computation of line positions taking into account these torsional splittings. More precisely an effective "vibration -torsionrotation" Hamiltonian model was built which accounts both for anharmonic and Coriolis resonances coupling the 21 1 and 8 1 states with nearby highly excited states involving the two types of internal rotations of the methyl groups. In parallel, a line intensity code was set up for the computation of line intensities helping in the unambiguous assignment of the torsional components. The final result is a linelist of positions and intensities of the ν 8 and ν 21 bands which could be used for the detection of propane in the Earth and outer planets atmospheres.

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  type, B-type and C-type Coriolis operators (see Table

  expansion of the parameters involved in the Watson's type or anharmonic rotational Hamiltonians

Figure 1

 1 Figure 1Overview of the ν 8 and ν 21 bands of propane. Spectrum recorded with the high resolution FT spectrometer at SOLEIL (partial pressure of 14 Pa, temperature = 142 K and optical path length of 45.14 m). The distinctive shapes of the typical B-type (ν 8 ) and A-type bands (ν 21 ) are clearly visible. The experimental spectrum is compared to the calculation performed during this study.

21 Figure 2

 212 Figure 2

Figure 3 :

 3 Figure 3 :

Figure 4 :

 4 Figure 4: Portion of the R branch of the ν 21 band around 931.3 cm -1 .

Figure 5 : 25 Figure 6 :Figure 3 ) 26 Figure 7 27 Figure 8 :Figure 9 : 29 Figure 10 : 30 Figure 11 :

 52563267278929103011 Figure 5 : Portion of the R branch of the ν 21 band in the 926.88 cm -1 . CAPTION: For the [10, 5, K' c ] -[9, 5, K'' c = d] rotational transition six torsional subcomponents are observable since we are dealing with rather high K a values (K a = 5).

Table 2

 2

	lists the symmetry types of the 27 vibrational states of propane
	together with those of the rotational energy levels |Rot>= [J, K a K c ]. This table gives also in
	the I r representation (z=A, x=B, y=C) the symmetry type of the E+, O+, E-and O-(|J K=
	even or odd, γ=±1 > ) symmetrized Wang's type base functions, which, are:
	for K≠0

Table 2

 2

	gives the symmetry types of the four torsional wavefunctions (|Tors 0 >=AA,
	AE, EA and EE) in the ground vibrational state |V=0>. Using Tables 1 and 2 it is easy to

Table 5 .

 5 It is clear that it was not possible to reproduce the energy levels to within A very poor knowledge of the torsional structure for the states involving high ν 14 or ν 27 vibrational quanta. This is why the local resonances involving the 14 2 27 2 and 14 1 27 3 dark vibrational states are so difficult to identify and reproduce correctly.

	experimental accuracy.

-

Table List Table 1 :

 List1 Character table for the G 36 inversion permutation group of propane.

	G 36 E	(123)× (456)	(14)×(26)× (35)× (C1C2)*	(123) ×(465)

Table 2 :

 2 Symmetry species of the torsional wavefunctions (AA, AE, EA and EE) of the [J K a K c ] rotational energy levels in the ground vibrational state , of the 27 vibrational modes and of the E + , E -, O + and O - Wang's type wavefunctions.

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Table 4

 4 Nuclear weights associated to the torsional subcomponents for a [J' K' a K' c ]-[J K a K c ] vibration rotation transition (a) Low K a values ("non degenerate" K c ) and clustered AE and EA subcomponents.

	all	64

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Table 5 :

 5 Results of the analysis and of the energy levels calculations

	Page 31 of 67	Molecular Physics
		Results of the analysis
		ν 8	ν 21
	Number of lines	6047	5054
	Number of levels	1858 (v 8 bright )	2244 (v 21 bright )
		6 (2ν 14 +2ν 27 dark)	7 (ν 9 +2ν 27 dark)
		AA: 497	AA: 584
		EE: 745	EE: 781
		AE # : 207	AE # : 412
		EA # : 415	EA # : 474
	Maximum J	43	44
	Maximum K a	17	16

# For low K a values, the given EA or AE torsional labelling corresponds actually to clustered (EA & EA) levels (see text).

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Table 7

 7

Table 8

 8 Expansion of the Watson's type rotational operator (I r representation and A-type reduction), of the

	anharmonic (	v Anh , ) and of the A-type, B-type and C-type Coriolis ( ' v	v CorA , ' v	v CorB , and ' v
		v CorC ) operators. ' v

Table 10

 10 List of the Hamiltonian parameters determined during the energy level computations for the {8 1 , 14 2 27 2 } interacting states of propane.

Table 10 ,

 10 Part A: 8 1 or 142 27 2 vibrational -diagonal operators:

	Page 37 of 67				Molecular Physics
			(a)	Watson's type « torsion free » parameters
					0 ($)	8 1	14 2 27 2
		E V				870.39570(160) 872.5168(370)
		Band centers (£)			870.3136(1)	872.4896(1)
		A			0.9742562313 0.9742783(270) 1.141686(590)
		B			0.2817271865 0.28019447(670) 0.279266(110)
		C			0.2488055608 0.24757661(530) 0.2466803(970)
		∆ Κ ×10 6			5.324721011	4.4996(940)	509.75(850)
		∆ KJ ×10 7			-8.998168326	-10.355(410)	321.1(370)
		∆ J ×10 7			2.402258565	2.3561(340)	($)
		δ Κ ×10 7			1.035202827	($)	($)
		δ J ×10 8			4.659136555	4.274(300)	20.13(600)
		All higher order terms:	($)	($)	($)
		(b)	Higher order Watson's type rotational "torsion free" constants ($)
	H K ×10 10	1.147493844	h K ×10 12	-2.568443533	L K ×10 15	-2.625149429
	H KJ ×10 11	-4.071816910	h JK ×10 12	1.377953277	L KKJ ×10 16	4.703253742
	H JK ×10 13	5.193592962	h J ×10 13	1.197995448	L J ×10 19	-7.104915228
	H J ×10 13	2.755239426

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Table 11

 11 List of the Hamiltonan parameters determined during the energy level computations for the {21 1 , 14 1 27 3 , 9 1 27 1 } interacting states of propane.

Table 11 ,

 11 Part A: 21 1 , 14 1 27 3 or 91 27 2 vibrational -diagonal operators:

				Molecular Physics	Page 40 of 67
			(a)	Watson's type « torsion free » parameters
				21 1	14 1 27 3	9 1 27 2
	E V			921.5042(170)	932.0429(380)	909.4694(180)
	Band centers (£)		921.3756(400)	932.1715(400)	909.4694(180)
	A		0.9825827(110)	0.973887(450)	1.050674(630)
	B		0.28155148(998)	0.2814147(470)	0.2894370(610)
	C		0.24840146(710)	0.2479347(180)	0.2390320(760)
	∆ Κ ×10 6			9.374(190)	($)	383.79(210)
	∆ KJ ×10 7			-5.48(130)	($)	($)
	∆ J ×10 7			3.256(120)	($)	($)
	δ Κ ×10 7			($)	($)	($)
	δ J ×10 8			2.137(710)	($)	($)
	Higher order terms:	($)	($)	($)
	(b)	q-type torsional dependence of the Watson-type rotational parameters
				21 1	. 14 1 27 3
		Tor	v 0	0.0 (#)	0.3972(180)
		v 1 Abc ×10 4	-1.887(190)
		v 2 BC ×10 6	-9.38(220)

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Table 12 Partition

 12

	Page 43 of 67	Molecular Physics
		function for propane
	Temperature (K)	Z rot (T)	Z vib (T)	Z tot (T)
	75.	3.27410E+05 1.02296 3.34927E+05
	100.	5.03968E+05 1.07474 5.41636E+05
	142..	8.52718E+05 1.23641 1.05431E+06
	150.	9.25793E+05 1.27754 1.18274E+06
	225.	1.70119E+06 1.83681 3.12476E+06
	273.15	2.27615E+06 2.40767 5.48022E+06
	296.	2.56781E+06 2.76033 7.08799E+06
	300.	2.62008E+06 2.82871 7.41146E+06

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/(molecule.cm -2 ). The results of these calculations (maximum values of J and K a , frequency

Where the ( ) avg , q K S 2 1 σ σ are the sinusoidal terms and K avg =(K+K')/2 is the average value of the K rotation quantum number in the (f) .q-type torsional dependence of 14 2 Tor is fixed to zero for the 8 1 vibrational state (ν 8 is a "low amplitude" vibrational mode) (£) Computed position for the "J=0" energy level of the AA component of the "v" vibrational state.

The results are in cm -1 and the quoted uncertainties are one standard deviation. (b) q-type torsional dependence of the 9 1 27 2 21 1 anharmonic off diagonal operator Tor is fixed to zero for the 21 1 vibrational state (ν 21 is a "low amplitude" vibrational mode). Torsional splittings were not accounted for in the .9 1 27 2 dark state. (£) Computed position for the "J=0" energy level of the AA component of the "v" vibrational state.

The results are in cm -1 and the quoted uncertainties are one standard deviation.


Patrick Bouthemy
email: bouthemy@irisa.fr

Ronan Fablet

Motion Characterization from Temporal Cooccurrences of Local Motion-based Measures for Video Indexing

This paper describes an original approach for motion interpretation with a view to content-based video indexing. We exploit a statistical analysis of the temporal distribution of appropriate local motion-based measures to perform a global motion characterization. We consider motion features extracted from temporal cooccurrence matrices, and related to properties of homogeneity, acceleration or complexity. Results on various real video sequences are reported and provide a first validation of the approach.

Introduction

Multimedia databases are of growing importance in various application fields, such as television archives (movies, documentaries, news, . . . ), multimedia publishing, road traffic surveillance, medical imaging, remote sensing and meteorology (satellite images), . . . Then, efficient use of these databases requires to identify pertinent information related to their content to satisfy a given query or to access particular pieces of information. There is obviously a real need of indexing and retrieving multimedia documents by their content in an automatic way (at least partly). Several pioneering systems already exist for still images, [START_REF] Aigrain | Content-based representation and retrieval of visual media : A stateof-the-art review[END_REF][START_REF] Flicker | Query by image and video content : the QBIC system[END_REF], and a large research effort is currently undertaken to handle image and video databases, [START_REF] Aigrain | Content-based representation and retrieval of visual media : A stateof-the-art review[END_REF][START_REF] Gupta | On summarization of video[END_REF][START_REF] Idris | Review of image indexing techniques[END_REF][START_REF] Pickard | Vision texture for annotation[END_REF][START_REF] Zhang | An integrated system for content-based video retrieval and browsing[END_REF]. Nevertheless, due to the complexity of image interpretation and dynamic scene analysis, several important issues remain to be further investigated.

As far as video sequences are concerned, contentbased video indexing, browsing, editing, or retrieval, have motivated specific investigations focusing first on the structuration of the video in elementary shots, [START_REF] Aigrain | Content-based representation and retrieval of visual media : A stateof-the-art review[END_REF][START_REF] Bouthemy | Video partioning and camera motion characterization for content-based video indexing[END_REF][START_REF] Gupta | On summarization of video[END_REF][START_REF] Zhang | An integrated system for content-based video retrieval and browsing[END_REF], and concentrating more recently on image mosaicing [START_REF] Irani | Efficient representations of video sequences and their applications[END_REF], on image layering [START_REF] Ayer | Compact representations of videos through dominant and multiple motion estimation[END_REF], on object motion characterization in case of a static camera [START_REF] Courtney | Automatic video indexing via object motion analysis[END_REF], or on segmentation and tracking of moving elements [START_REF] Gelgon | Determining a structured spatio-temporal representation of video content for efficient visualization and indexing[END_REF]. Motion segmentation methods usually rely on 2D parametric motion models, and aim at localizing the different types of motions present in a scene. However, they turn out to be unadapted to certain classes of sequences, particularly in the case of unstructured motions of rivers, flames, foliages in the wind, or crowds, . . . , (see Fig. 1). Besides, it seems pertinent to provide a direct global characterization without any prior motion segmentation or without any complete motion estimation in terms of parametric models or optical flow fields. These remarks emphasize the need for the design of new low-level approaches in order to supply a direct global motion description, [START_REF] Nelson | Qualitative recognition of motion using temporal texture[END_REF][START_REF] Polana | Detecting activities[END_REF][START_REF] Szummer | Temporal texture modeling[END_REF].

We follow this point of view and we propose an original method for video indexing with respect to motion content. It uses local non-parametric motion-related information, and extracts, from temporal cooccurrence statistics, global motion features relative to motion complexity, coherence or acceleration. In Section 2, we outline the analogy between our approach and texture analysis. Section 3 describes the local motion-related information used. In Section 4, we introduce temporal cooccurrence matrices and the extracted global motion features. Section 5 contains results obtained on various real video sequences and concluding remarks.

Problem statement

Our approach consists in analyzing, within each shot previously extracted from the processed video sequence, the whole spatio-temporal motion distribution, as spatial grey level distribution in texture analysis. In particular, we aim at adapting in that context cooccurrence measurements which supply a texture characterization in terms of homogeneity, contrast or coarseness [START_REF] Haralick | Textural features for image classification[END_REF].

Preliminary work in that direction, developed by Polana and Nelson, [START_REF] Nelson | Qualitative recognition of motion using temporal texture[END_REF][START_REF] Polana | Detecting activities[END_REF], introduces the notion of temporal texture, related to fluid motions. Indeed, motions of rivers, foliages, flames, or crowds, . . . , can be regarded as temporal textures. Maps of local motion-related measures along the image sequence, required as input of cooccurrence measurements, could be provided by dense optical flow fields. However, first, it is really time consuming, and second, the quality of the estimated displacement fields cannot be ensured in the case of temporal content corresponding to such complex dynamic scenes. Therefore, we prefer to consider local motion-related information, easily computed from the spatio-temporal gradients of the intensity. Contrary to [START_REF] Nelson | Qualitative recognition of motion using temporal texture[END_REF], where the normal velocity is considered, we make use of a more reliable information as explained in the next section.

Local motion-related measures

By assuming intensity constancy along 2D motion trajectories, the image motion constraint relating the 2D apparent motion and the spatio-temporal derivatives of the intensity function can be expressed by :

¢¡ ¤£ ¦¥ ¨ § © ¡ £ ¥ !¡ £ ¥ #" %$ (1)
where is the 2D motion vector in the image, ¡ ¤£ ¦¥ the intensity function at point £ , © &" '¡ ( !) 0 1 32 4¥ the intensity spatial gradient, and ¡ ¤£ ¦¥ the intensity partial temporal derivative. Then, we can infer the normal velocity 5 46 :

5 46 ¡ £ ¥ #" 7 8 © ¡ ¤£ ¥ 8 (2)
This quantity 5 46 can in fact be null whatever the mo- tion magnitude, if the motion direction is perpendicular to the spatial intensity gradient. 5 46 is also very sensi- tive to noise attached to the computation of the intensity derivatives. However, if the spatial intensity gradient is sufficiently distributed in terms of direction in the vicinity of point £ , an appropriately weighted average of 5 46 in a given neighbourhood forms a more relevant motionrelated quantity :

5 @9 1A CB ¡ £ ¥ D" FE B HG 4I QP ¤R !S 8 © & ¡ UT V¥ 8 W § YX 5 6 ¡ UT V¥ `X a cb 4d e¡ gf W 0 E B hG 4I ¢P iR 3S 8 © ¡ UT V¥ 8 W ¥ (3)
where p ¡ ¤£ ¦¥ is a q sr tq window centered on £ . f W is a predetermined constant, related to the noise level in uniform areas, which prevents from dividing by zero or by a very low value. In [START_REF] Odobez | Separation of moving regions from background in an image sequence acquired with a mobile camera[END_REF], this motion-related measure 5 @9 hA CB has been successfully used to process sequences compensated by the estimated dominant motion with a view to detecting moving objects in a scene, and confidence bounds, depending on the intensity gradient distribution within p ¡ ¤£ ¦¥ , have been derived to assess its reliability.

Thus, 5 9 hA uB provides us with a local motion measure, easily computed and reliably exploitable. The loss of the information relative to motion direction is not a real shortcoming, since we are interested in interpreting the general type of dynamic situations observed in a given video shot. Nevertheless, in practice, it turns out to be irrelevant due to the spreading out of motion quantities, as shown in Figure 2. The sequence Merry-go-round is a static camera shot with, in the foreground, a merry-go-round which undergoes a rotation, and, in the background, walking persons and a bus just leaving the square. Therefore, we introduce bounds which define an interval where measures are regarded as pertinent. Since the motion quantities are positive, 0 is taken as the lower bound. Moreover, in motion estimation, it is generally considered that a single resolution analysis is un-able to correctly estimate displacements of large magnitude. It appears relevant to introduce a limit beyond which measures are no more regarded as usable. In the experiments,practice, sampling within v $ 0 @ on ` levels proves accurate, as shown in Figure 2.

Extraction of global motion features

Motion cooccurrences

Polana and Nelson have combined spatial cooccurrence distribution with normal flow fields to classify processed examples in pure motions (rotational, divergent) or in temporal textures (river, foliage), [START_REF] Nelson | Qualitative recognition of motion using temporal texture[END_REF]. However, temporal evolution cannot be handled in that way , since studied interactions are purely spatial, and only stationary motions can be characterized. Moreover, considering spatial cooccurrences is highly time-consuming, since matrices relative to several configurations of spatial interactions have to be computed. That is the reason why we have looked for transferring cooccurrence to the temporal domain.

The temporal cooccurrence for the pair of quantified motion quantities ¡ g H0 C Y¥ at the temporal distance is de- fined as follows :

H ¡ g H0 C Y¥ #" ed 3f ¡ gg h0 HT V¥ ji sk Hl hm 4n T Y¡ gg 4¥ D" o H0 m 4n T Y¡ UT V¥ #" p ¦q X k X

where m 4n T holds for the quantified version of 5 @9 hA uB , and k " f ¡ gg h0 HT ¥ at the same spatial position in the image grid l 4r ts 0 1g ui image ¡ s ¥ and T vi image ¡ s 7 ¥ `q . These temporal cooccurrences are evaluated over all the images of the given shot of the video sequence (typically, about 20 images for the examples reported below).

Global motion features

From cooccurrence matrices, we can now extract global motion features similar as those defined in [START_REF] Haralick | Textural features for image classification[END_REF] : The average feature indicates the importance of the observed motion, whereas the variance and the Dirac features express the degree of spreading out of the motion distribution. The contrast feature is related to the average acceleration. The ASM feature quantifies the temporal coherence. It varies within P } j S g 0 ! ; it is equal to 1, if motion is close to uniformity in space and time. On the contrary, if the motion coherence falls, it tends to P } j S .

This set of global motion features is computed over all the image grid. This could also be achieved either on predefined blocks or on extracted regions resulting from a spatial segmentation. This ensures feasible cooperations with other methods for video content analysis.

Results and concluding remarks

This approach has been validated by experiments with several kinds of real video sequences. We report here results obtained on four video sequences representative of different classes of dynamic situations. The first sequence, called Fire (Fig. 1), is temporal texture with a poorly structured motion in space and time. The second processed example is the sequence Merry-go-round, shown in Figure 2, which involves quite an important motion activity. In the third sequence Mobi (Fig. 3a), several rigid motions are combined ; along with the camera panning, the train and the calendar undergo a translation respectively from right to left, and towards the top of the scene, whereas the ball rolls towards the left. The fourth sequence is a static shot of the Concorde Place in Paris (Fig. 3b), with a weak motion activity resulting from the presence of cars around the Obelisk. The results reported in Figure 4 show that the computed motion features can discriminate between the different motion configurations and quantify their degree of homogeneity, spatial and temporal uniformity, as well a) b)

4

¢ e ¢ 3 of values computed for the shot zoom as complexity. The characterization of secondary motions in presence of camera motion seems also possible. In the case of sequence Mobi, the camera is panning the scene. Nevertheless, our global motion characterization method accesses directly to the content of the filmed scene. Moreover, other experiments prove that this analysis is quite independent of subsampling in space or time.

Results given in Figure 5 corresponds to another example involving a camera zooming on a curtain. They show that the four features relative to coherence and homogeneity, i.e., average, variance, Dirac, and ASM features, are almost independent of the chosen temporal distance . Consequently, we can compute these four motion features for only one cooccurrence parameter value, which greatly saves calculation time. Concerning the contrast feature, its evolution with respect to yields a characterization of the motion acceleration. Indeed, in the video shot zoom, the acceleration is positive and the contrast feature increases with .

We have described in this paper an original and efficient method of global motion characterization for content-based video indexing. It relies on a secondorder statistical analysis of temporal distributions of relevant, local, and quantified motion-related information. It exploits global motion features extracted from temporal cooccurrence matrices. This approach allows us to deal with various kinds of motion configurations, and to describe properties of the image dynamic content as complexity, uniformity and homogeneity. It does not require parametric motion models nor the computation of optical flow fields. Obtained results on a reasonable range of real scenes demonstrate that these global motion features can provide us with a set of pertinent and discriminating indexes with a view to video indexing by the dynamic content. In future work, a first step will be to determine optimal sets of global features corresponding to specific video databases, and to design a complete classification scheme. Moreover, we hope to benefit from the flexibility of our method by introducing a multi-scale approach in order to refine the analysis of the motion structure.

Figure 1 .

 1 Figure 1. Examples of temporal textures : a) foliage b) fire (by courtesy of MIT).Maps of local motion-related measures along the image sequence, required as input of cooccurrence measurements, could be provided by dense optical flow fields. However, first, it is really time consuming, and second, the quality of the estimated displacement fields cannot be ensured in the case of temporal content corresponding to such complex dynamic scenes. Therefore, we prefer to consider local motion-related information, easily computed from the spatio-temporal gradients of the intensity. Contrary to[START_REF] Nelson | Qualitative recognition of motion using temporal texture[END_REF], where the normal velocity is considered, we make use of a more reliable information as explained in the next section.

4. 1 .Figure 2 .

 12 Figure 2. Motion-related information for the video shot Merry-go-round : (a) first image of the sequence, (b)-(c) maps of motion quantities 5 9 1A CB with a linear quantization (b) and with the introduction of a-priori bounds (c)

  w x x x x y x x x x z Average : { " E P }| (~ 1S ¡ ( h0 C t¥ Variance : W " E P }| (~ 1S ¡ ( 7 { ¥ W ¡ g H0 C Y¥ Dirac : " { W l W Angular Second Moment : { # " E P }| (~ 1S ¨ H ¡ g H0 u t¥ W Contrast : k m V s " E P }| (~ 1S ¡ g 7 t¥ W H ¡ g H0 C Y¥

Figure 3 .

 3 Figure 3. Video shots: a) Mobi, b) Concorde.

Figure 4 .

 4 Figure 4. Global motion features : from left to right and from top to bottom, { , W , , ASM, and Cont, computed with " , for four sequences, Fire, Merry-go-round, Mobi and Concorde (from left to right within each sub-figure).

Figure 5 .

 5 Figure 5. Influence of the temporal distance on the global motion features. (a) One image of the shot zoom, (b) Tableofvalues computed for the shot zoom as complexity. The characterization of secondary motions in presence of camera motion seems also possible. In the case of sequence Mobi, the camera is panning the scene. Nevertheless, our global motion characterization method accesses directly to the content of the filmed scene. Moreover, other experiments prove that this analysis is quite independent of subsampling in space or time.Results given in Figure5corresponds to another example involving a camera zooming on a curtain. They show that the four features relative to coherence and homogeneity, i.e., average, variance, Dirac, and ASM features, are almost independent of the chosen tempo-