Eugenia N Starikova

Alain Barbe

Marie-Renée De Backer

Vladimir G Tyuterev
email: vladimir.tyuterev@univ-reims.fr

S A Tashkun

S Kassi

A Campargue

M.-R De Backer-Barilly

Isotopic shifts in vibration levels of ozone due to homogeneous substitution: Band centres of 18O3 derived from analysis of CW-CRDS spectra in the 5900-7000 cm-1 range

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Despite considerable effort from the scientific community devoted to the investigation of ozone properties and spectra, there remain many issues and puzzling problems which are far from being completely solved, like, for example, the interpretation of anomalous isotopic enrichment effect in the ozone formation [1][2][3], the calculation of the dissociation and recombination rates [4][5], a detailed description of non-local thermodynamic equilibrium processes in the upper atmosphere [6], or the search of optimal channels for multi-photon laser excitation of ozone. A solution for many of these problems requires detailed information on high-energy rovibrational states of the ozone isotopic species. Accurate measurements and calculations of these states in the energy range approaching the dissociation limit are a particular challenging issue.

The high-resolution spectroscopy of ozone has a long history. Up to 5800 cm -1 the infrared spectra have been mostly obtained by FTS (Fourier Transform Spectroscopy). The range of fundamental and first overtone and combination bands of the principal isotopologue, 16 O 3 , has been covered in series of studies by Flaud, Camy-Peyret et al. [7][8][START_REF] Flaud | Atlas of ozone line parameters from microwave to medium infrared[END_REF] resulting in an atlas of line parameters up to 3000 cm -1 [START_REF] Flaud | Atlas of ozone line parameters from microwave to medium infrared[END_REF]. Barbe et al. [START_REF] Barbe | [END_REF][11][12][13] have extended the range of observations assigning more than 30 bands of 16 O 3 up to 5800 cm -1 . Ozone spectroscopy studies available up to 2003 have been reviewed by Steinfeld et al. [14], Flaud and Bacis [15], Mikhailenko et al. [16], and Rinsland et al. [17].

FTS spectra of ozone enriched in 18 O and 17 O were analysed in Refs. [15,[18][19][20][21][22][23][24][25][26]. The most extended spectral regions were investigated by Chichery et al. [21,22] for the homogeneous isotopic variant 18 O 3 and by De Backer, Barbe et al. [23][24][25][26] for the mixed variants 16 O 18 O 16 O, 18 O 16 O 18 O, 16 O 16 O 18 O and 16 O 18 O 18 O. All previously available studies of isotopically substituted ozone spectra were performed below 4900 cm -1 and involved bands having at maximum a 5

i i ∆ ≡ ∆ = ∑ v v
variation of the vibration quantum numbers. A large part of the related results are available in the S&MPO (Spectroscopy & Molecular Properties of Ozone) information system [27], which is accessible via the websites ozone.univ-reims.fr and ozone.iao.ru.

There are two major problems in deriving information about the vibration levels of ozone approaching the dissociation limit (situated near 8500 cm -1 [15]) from high-resolution absorption spectra. One is experimental: band strengths are quickly decreasing with the vibrational excitation, and a higher sensitivity than that provided by FTS is necessary: up to eight orders of magnitude are noted between the band strength of the ν 3 fundamental and that of the bands near 7000 cm -1 corresponding to 8 ∆ = v . The other problem is related to an increasing complexity of analyses due to many accidental vibration-rotation resonance perturbations and missing information on "dark" states.

A breakthrough in the extension of experimental measurements to higher energy has been achieved with the development of very sensitive laser techniques [28][29][30][31][32][33][34]. Wentz et al. [28] measured more than 3500 lines in the 16 O 3 spectrum recorded with a tunable single mode diode laser between 6430 and 6670 cm -1 but only about 400 of them could be assigned. A further progress has been achieved with the CW-Cavity Ring Down Spectrometer (CW-CRDS) developed at Grenoble University [34] in the wide 5850-7000 cm -1 region [29][30][31][32][33][34], where more than 7500 lines of 16 O 3 have been recently recorded and assigned in collaboration with the Reims group. These results have been recently reviewed by Campargue et al. [34].

On the theoretical side, considerable efforts have been devoted to the determination of the potential energy surface (PES) [5, 35-41 and references therein], for the electronic ground state of ozone. Extended ab initio calculations have been performed by Siebert and Schinke et al. [36,37]. Babikov et al. [38] and Grebenshchikov et al. [39] used these calculations to study the metastable states and the van der Waals states, respectively. Related dynamical studies have been summarized and reviewed in Ref. [5]. Up to date, the most accurate empirical PES for the spectroscopic calculations at the open configuration have been determined by Tyuterev et al. [40,41] from the fit of FTS experimental vibration-rotation data of 16 O 3 below 5800 cm -1 .

This PES has been used, for a theoretical interpretation of rates of formation of ozone isotopologues [3], for the assignment of CW-CRDS spectra of 16 O 3 [29][30][31][32][33][34] and for the extrapolation towards highly excited vibrational states [35].

In this Letter, we report the first results of experimental and theoretical studies of the isotopic effect in high-resolution spectra of the ozone molecule due to the 16 O 3 → 18 O 3 substitution in the 5900 -7000 cm -1 wavenumber range. Ro-vibrational levels corresponding to the high energy edge of this range are at about 82% of the dissociation limit. Contrary to the low energy range, the isotopic shifts of the band centres appear to be irregular even though the change in masses is small and homogeneous. This reflects the complexity of anharmonic interactions in the ozone molecule. Due to the shifts of about 340 cm -1 towards lower wavenumbers for the band centres with respect to the major isotopologue, we were able to record three 18 O 3 new bands which correspond to the largest vibrational excitation ever detected in ozone spectra at high resolution. One of the goals of this study was to use these new observations to check to what extent the PES of Refs. [40,41] determined from 16 O 3 data, is valid for accurate predictions of the 18 O 3 bands.

In Section 2, theoretical calculations of energy levels and wave functions relevant to the observed bands are briefly described; Section 3 is devoted to the experimental set up and recorded spectra; Section 4 to the spectral analysis and Section 5 to the discussion of the obtained results.

Band centres predicted from PES

At low vibrational excitation, the assignment of ozone bands has been traditionally based on Dunham-type expansions involving Darling-Dennison resonance terms which couple stretching vibrations due to the proximity of the ω 1 and ω 3 harmonic frequencies [15,16]. This simple approach is no more valid at higher energies because of the increasing density of states and of the overlapping of stretching polyads defined by the condition const + =

1 3
v v [11-13, 16, 21-34]. The overlapping of the polyads results in numerous unusual resonances corresponding to large differences in vibrational quantum numbers up to 10 ∆ = v .

In previous analysis of the CW-CRDS spectra of 16 O 3 [29][30][31][32][33][34], our assignments relied on extrapolations from the empirical PES of Refs. [40,41]. For the 21 band centres of 16 O 3 reported in the 5850-7000 cm -1 range, the deviations of the predictions from observations vary from 0.1 to 2.4 cm -1 with a rms error of about 1 cm -1 . For the vibrational assignments of the present 18 O 3 spectra, we used two versions of the same PES, one adapted to the Morse-cosine kinetic model [40], another adapted to the exact kinetic energy operator [41]. The corresponding calculations using both versions of the PES result in energies for J = 0 and J = 1 levels which are in a good agreement at least up to 7000 cm -1 (see Table 1): the rms deviation between two calculations is 0.9 cm -1 .

As far as techniques of calculations are concerned, two complementary and completely independent methods were applied. The first method uses variational calculations involving numerical integration in a large basis set of primitive wave functions defined in internal bond length and bond angle or Radau coordinates, as described in Refs. [40,41]. In order to achieve a convergence of calculations in the considered energy region, it was necessary to extend the size of the basis by at least one order of magnitude compared to similar calculations for water molecule (see Ref. [41] for the corresponding discussion). The second method is purely algebraic and uses successive high-order Contact Transformations (CT) of the complete vibration-rotation Hamiltonian into series of effective Hamiltonians defined for groups of strongly interacting vibrational states. In our analysis, the MOL_CT suite of computational codes [42] was used to allow for a systematical building of effective Hamiltonians in normal coordinates [42,43]. These calculations account for various resonance interactions specific to the ozone molecule including inter-polyad couplings. This method has the advantage to directly provide with the expansion of eigen-functions

| V N Γ Ψ > of the final transformed
Hamiltonian in the normal mode basis:

0 | | V V N N C Γ Γ Ψ >= > ∑ 1 2 3 1 2 3 v v v 1 2 3 v v v v v v (1)
Here Γ is the symmetry type of the state and N V is the rank number of the vibration state according to the increasing energy for a given Γ. The normal mode basis

0 0 0 0 | | | | > = > > > 1 2 3 1 2 3 v v v v v v
corresponds to vibrations which are decoupled near the minimum of the potential well. Eq. ( 1) stands for vibration wave functions. Similar expansions are also obtained for vibration-rotation wave functions involving asymmetric top rotational basis |JK a K c > as a direct product in the right hand side of Eq.( 1). This offers a rigorous procedure for normal mode spectroscopic assignments of ro-vibrational states from a given PES.

The mixing coefficients are defined as squares of the expansion coefficients: 2 ( )

V N n P C Γ = 1 2 3 v v v with 1 n n P = ∑ (2)
Here n = 1 corresponds to the largest contribution in Eq. ( 1) and the terms with n = 2, 3, … correspond to the decreasing order of the mixing coefficients. The convergence of MOL_CT algebraic calculations with increasing orders of CT was checked against variational calculations with the same PES. For the 16 O 3 and the 18 O 3 vibration levels, the rms deviation between these two calculations up to 7000 cm -1 are less than one wavenumber for the 8 th order of contact transformations. For this reason, only CT calculations ("calc_2") corresponding to the PES of [41] are provided in Table 1.

In this work, we use two types of vibration state identification. The first one is the (Γ, N V ) global assignment containing the symmetry species Γ (=A 1 or B 1 ) and the ranking number N V only. If a PES is sufficiently accurate i.e. (obs.-calc.) errors are much smaller than the energy separation, this global assignment is unambiguous. Further comparisons with observations clearly show that this is our case: there is no ambiguity in the global (Γ, N V ) assignment in the considered spectral range. Between 5900 and 7000 cm -1 , both PES [40,41] result in 28 calculated (B 1 ←A 1 ) A-type bands due to the dipole moment component parallel to the "linearization" z-axis and in 33 calculated (A 1 ← A 1 ) B-type bands due to the dipole moment component perpendicular to the "linearization" z-axis. Symmetry labels correspond to the standard choice of axes for the open configuration of the ozone molecule as described in [START_REF] Flaud | Atlas of ozone line parameters from microwave to medium infrared[END_REF]15]. The second type of assignment is the traditional normal mode assignment relying on 1 2 3 v v v labels corresponding to the major normal mode contribution in the expansion (1).

According to the nuclear spin statistics only ro-vibrational levels of A 1 and A 2 symmetry types are allowed in the case of 16 O 3 and 18 O 3 [START_REF] Flaud | Atlas of ozone line parameters from microwave to medium infrared[END_REF]15]. This means that the lowest allowed

levels correspond to 1 {( )( )( ) [ 0 0 0]} even a c A JK K = 1 2 3 v v v for A 1 type vibrations and to 2 {( )( )( ) [ 101]} odd a c A JK K = 1 2 3 v v v
| n W = > 1 2 3 v v v
for n =1, 2, 3.

Experiment

For this work, the CW-CRDS spectra of 18 O 3 were recorded with the same CW-CRDS spectrometer as used in our previous studies of the main ozone isotopologue [29][30][31][32][33][34] and of other species of atmospheric importance such as H 2 O [45, 46] and 12 CO 2 [47,48]. The detailed description of the fibered set up can be found in Refs. [29,44,48]. Briefly, the full 5900-7000 cm -1 range was covered with a set of about 50 Distributed Feed-Back (DFB) laser diodes. The tuning range of each DFB laser is about 7 nm (~ 30 cm -1 ) for a temperature variation from -10 to 65°C. The typical sensitivity (noise equivalent absorption α min ~2-5×10 -10 cm -1 ), the wide spectral coverage and the four to five decades linear dynamic range make this CW-CRDS spectrometer an ideal tool for high sensitivity absorption spectroscopy in the important atmospheric window of transparency around 1.5 µm. The typical ring down times are on the order of 60 µs. A few hundreds ring down events are averaged for each spectral data point, and the duration of a temperature scan is about 70 minutes for each DFB laser. The pressure, measured by a capacitance gauge, as well as the ring down cell temperature, was monitored during the spectrum acquisition. Note that the DFB line width is much smaller than the Doppler broadening (about 10×10 -3 cm -1 (FWHM) at 6000 cm -1 ), so that the observed resolution is mostly Doppler limited since the pressure broadening is about 5×10 -3 cm -1

(FWHM) at a typical pressure of 20 Torr.

The 140 cm long CRDS cell fitted by the super mirrors was filled with a typical pressure of 30 Torr of 18 O enriched oxygen. The quasi complete conversion of oxygen in ozone (P = 20 Torr) was obtained after a few minutes with a silent electric discharge (12 kV, 400 Hz). The 18 O enrichment of the molecular oxygen used for the ozone synthesis was 95 %, leading to an 86 % partial pressure of the 18 18 O transitions were identified on the basis of the HITRAN database [49] and of our recent CW-CRDS investigation of the CO 2 spectrum [48].

The wavenumber calibration of the spectrum is based on the wavelength values measured by a wavemeter (Burleigh WA1640) during the DFB frequency scan. When necessary, it is further refined by simply stretching the frequency scale (with an origin at 0) in agreement with highly accurate positions of reference lines due to impurities. H 2 18 O and H 2 16 O line positions as provided by the HITRAN database [49] were used as reference lines. The obtained absolute wavenumber calibration is estimated accurate within 2×10 -3 cm -1 .

The entire spectrum range of our recordings with band assignments and impurity absorption features is shown in Fig 1.

Band centres determined from the rovibrational analysis of the CRDS spectra

Among triatomic atmospheric molecules, the ozone molecule is a relatively heavy one with dense rotational structure. In the case of an absorption band at room temperature, depending on the band type, the most intense lines correspond to J values in the 15-35 range.

As seen in Table 1, most of the bands falling in our spectral region correspond to 7 or 8 vibration quanta of the excited state. These bands are extremely weak and a complete set of transitions from low-populated levels with small J values could not be detected for some of them, even with the achieved CW-CRDS sensitivity. This makes difficult an experimental determination of the band centres.

The centre of a rovibrational band is defined as the J → 0 extrapolation of the upper state energies. This J → 0 limit could be accurately determined by following line series in P, Q

or R branches in a sufficiently large range of J, K a values. This implied the full rovibrational assignment of the recorded spectrum. As illustrated in our previous analysis of the CW-CRDS spectrum of 16 O 3 case [29][30][31][32][33][34], the assignment and modelling of the thousands of 18 O 3 lines observed between 5900 and 7000 cm -1 represents a considerable task. The very large number of possible rovibrational resonance interactions, in particular with dark states, made an accurate modelling in the frame of the effective Hamiltonian, very difficult. The obtained spectra, displayed in Fig. 1, were divided in five spectral intervals corresponding to interacting bands systems which could be treated independently: 5930-6080, 6170-6280, 6320-6400, 6490-6650 and 6745-6840 cm -1 . Conceptually, the assignment procedure had many features in common with those encountered for 16 O 3 spectra [29][30][31][32][33][34]. However as a consequence of the irregular behaviour of the isotopic shifts (see below), the resonance situation is in fact quite different.

The detailed presentation of these analyses and of the derivation of the rotational, centrifugal distortion and resonance coupling parameters is beyond the scope of this letter and will be published separately. As an illustration of the achieved results, we include in Table 2 the total number of transitions assigned for the different bands. Here, we limit our discussion to the derivation of the band centre with the best possible accuracy.

The simultaneous modelling of the bands belonging to coupled band systems made it possible to identify the levels perturbed by resonance interactions. The line series remaining after the exclusion of these strongly perturbed levels were fitted using an isolated band model i.e. by using the standard Watson rotational Hamiltonian. The band centre was then obtained through the J → 0 extrapolation. For most of the bands a sufficient accuracy of the band centre was achieved using line series truncated at J and K a maximum values of 20 and 4, respectively.

In the case of the ν 1 +3ν 2 +4ν 3 band, rovibrational lines up to J = 30 have been included into the fit because the number of lines observed for J ≤ 20 was too limited to achieve a sufficient accuracy on the band centre determination. The obtained band centres are listed in Table 2 with their respective accuracy taken as the rms deviation between the truncated (see above) sets of observed and calculated line positions for each considered band. The uncertainty for all observed retrieved band centres is smaller than 3×10 -2 cm -1 which is largely sufficient in comparison with the isotopic shifts which are observed to vary between 296 and 372 cm -1 in our range.

Once the experimental value of the band centre was obtained, the global (Γ, N V ) vibrational assignment was rather straightforward relying on the good agreement between predicted and experimentally determined band centres (Table 2). (The rms value of the deviations is 1.1 cm -1 ). The situation is more complicated for the normal mode assignment. As stated in previous works about 16 O 3 [29][30][31][32][33][34]40], a normal mode labelling becomes ambiguous for some high vibration states above half the dissociation energy. The main reason is a strong mixing of the basis wavefunctions due to various anharmonic interactions. Table 1 shows that for many bands no normal mode contribution is truly dominant. In addition to the global (Γ, N V ) assignment, we provide in Table 1 the normal mode vibrational quantum numbers (v1,v2,v3) corresponding to the major contributions in Eq. ( 1). Since for small values of the J rotational quantum numbers, the normal mode mixing coefficients remain nearly unchanged, we use the decomposition given in Table 1 for the normal mode assignment of the observed bands.

Labelling the vibrational bands with the dominant (v1,v2,v3) state follows the spectroscopic tradition of ozone studies, but one should keep in mind that these "nominative" notations may be ambiguous at high energy range: for instance, the two bands at 6392 and 6643 cm -1 correspond to upper states having the same principal normal mode contribution (see Table 1).

However the knowledge of the normal mode mixture is of great help for a better understanding of possible resonance interactions.

Note that the local mode labeling [50] does not provide a satisfactory alternative in a sufficiently wide frequency range for the ozone vibrations either [40], because strong interactions with the bending mode break down the pure stretching model for the normal-tolocal mode transitions.

Discussion: isotopic effect on band centres and assignments

Fig. 2 shows a comparison of the simulated spectra using assigned rovibrational lines for 18 O 3 (upper panel) and 16 O 3 (lower panel) in the region of the 5ν 1 +ν 3 band as obtained in this work and in Ref. [33], respectively. On average, the 18 O 3 bands lie at about 340 cm -1 below the corresponding 16 O 3 bands, but it is clear that individual isotopic shifts are not regular. For instance, the overlapping between the 5ν 1 +ν 3 and 2ν 2 +5ν 3 bands is quite different for the two isotopologues.

Irregular variations of isotopic shifts due to isotopic substitutions have been observed

for other molecules, however contrary to these previous studies the 16 O 3 → 18 O 3 substitution is homogeneous that makes this situation rather unique. Indeed, usually an isotopic substitutions change the forms of mass dependent normal vibrations probing then different directions onto the molecular PES. In our case the comparison of L matrices which link internal and normal coordinates makes it obvious that they are exactly proportional. The ratio of matrix elements L nm (666) / L nm (888) = 1.0300 is the same in all significant digits. Consequently the forms of normal mode vibrations are identical for 16 O 3 and 18 O 3 . Also the relative mass changes are very small. In these conditions, if one assumes that ozone normal modes form regular stretching Darling-Dennison (DD) polyad sequences (due to the ω 1 ≈ ω 3 resonance), these polyad structures have to be very similar for both isotopic species and consequently the isotopic shift would be regular as well. This is not what we observe in our energy range.

Isotopic substitutions modify importantly the effect of resonance interactions leading, in some cases, to different vibrational mixing and then different vibrational labelling as illustrated by the highest frequency bands of full calculations and with experiment for both isotopologues: the error < 1 cm -1 . This is consistent with the normal mode decomposition of Table 1, which does not contain extrapolyad contributions for the 45B state. But the situation is quite different for other two bands of Fig 2 . For the right hand side 46B←1A band, the DD polyad model gave the same normal mode assignment ν 1 + ν 2 + 5ν 3 for both isotopologues contrary to full calculations (Tables 1,2 and [ ]), the error being -20 cm -1 for 16 O 3 and -10 cm -1 for 18 O 3 . For the left hand side 44B←1A band, the DD polyad model gave the error with the opposite sign: +17 cm -1 for 16 O 3 and +8 cm -1 for 18 O 3 . For both bands the test result is consistent with the normal mode decomposition of wave functions (Table 1 and [33]), because the extrapolyad contributions are larger for 16 O 3 than for 18 O 3 . This means that accidental exptrapolyad normal more couplings contribute to the isotopic shift in a rather irregular manner.

coupling terms 2 | | 2 eff H ± > ∓ 1 3 < v v , 4 | | 4 eff H ± > ∓ 1 3 < v v ,
Fig. 3 represents the energy dependence of the isotopic shift in two ways:

i) in the upper panel, we plotted the difference of the 16 O 3 and 18 O 3 band centres corresponding to the same (Γ, N V ) global assignment. All observed 18 O 3 bands given in Table 2 were included in this comparison. The filled symbols stand for the partner bands of the 16 O 3 isotopologue which were directly observed in Refs. [29][30][31][32][33][34], the grey-filled symbols correspond to the 16 O 3 values which were determined via "dark" state resonance perturbations as described in Refs. [29][30][31][32][33][34], open symbols correspond to the bands which have not yet been observed in 16 O 3 spectra. In the latter case, we used calculated positions for 16 O 3 which proved to be sufficiently accurate at the scale of the Figure .   ii) in the lower panel of Fig. 3, the comparison of the band centres is based on the normal mode assignments which are traditionally employed in ozone spectroscopy: we associated the bands of 16 O 3 and 18 O 3 for which the upper states correspond to the most similar normal mode mixtures. As stated before, this procedure leads to ambiguities in some cases because the resonance situations change quite substantially from one isotopologue to the other. This is clearly observed in Fig. 4, where the first expansion coefficients for the partner bands of 16 O 3 and 18 O 3 are plotted. Certain correlations are noted between the irregular variation of P 1 (Fig. 4) and that of the isotopic shifts (Fig 3).

In conclusion, fourteen new band centres of 18 O 3 were determined from high-sensitive CRDS observations. The rms deviation of theoretical predictions from observed band centres is 1.1 cm -1 confirming the validity of the empirical PES of Refs. [40,41] for this isotopologue. In particular, the three highest ozone bands observed so far at high resolution with upper states (Γ, N V ) = (A 1 , 82), (B 1 , 64), (B 1 , 66), which were beyond the accessible range for 16 O 3 , are also in a good agreement with calculations.

Previous studies of 18 O 3 spectra [15,19,21,22] below 5000 cm -1 suggested that isotopic shifts varied smoothly according to the simple relation for harmonic vibration frequencies, * * / / m m ω ω ∼ , allowing for a direct transfer of the vibrational labelling to the partner bands. This is no more valid in our spectral range reaching 82% of the ozone dissociation energy. The variation of the isotopic shifts still depends on normal mode quantum numbers ( , )

v , v v but appears to be more irregular as a consequence of drastic changes in the resonance interactions pattern and in the normal mode composition of the considered states.

As discussed above, the extrapolyad interactions play an important role in such behaviour. This work gives thus an experimental evidence for the strong mixing of the normal modes vibrations for some ozone states in the 6000-7000 cm -1 range. The coupling of normal modes should also have a considerable impact on the probabilities for dipole excitation channels in this high energy range.

Very few molecules allow studying effects of homogeneous isotopic substitutions, and to our knowledge for none of them a sufficiently large set of experimental data on high-energy

vibrations was yet available. A further step of this study will be to determine a full set of accurate rotational constants, centrifugal distortion, dipole moment transition and resonance coupling parameters as well as "dark" state perturbations for all bands involved in order to produce a synthetic spectrum. In addition, we plan to study the isotopic effects on the spectrum of the non-homogeneously substituted species (

Fig 2 . 3 . 3 v

 233 Note that upper states of these two bands have the same global assignment 46B but the normal mode assignments belong to different stretching In terms of normal modes these are mixed states (Tables1 and [33]) with the extra-polyad contributions different for16 O 3 and 18 O 3 . Extra-polyad interactions responsible for this re-assignment depend on accidental coincidences of polyad patterns corresponding to different 2 v values. This suggests that such interactions localised in narrow energy intervals and distributed in an irregular way are very sensitive to mass changes even though the latter are small and homogeneous. In order to verify this suggestion we have proceeded with the following test: band centres were calculated via the vibrational extrapolation by the polyad effective Hamiltonian model following increasing values of P = + 1 v . This simplified DD model contains all intra-polyad

  …accounting for the Darling-Dennisson ω 1 ≈ ω 3 resonance but excludes all extra-polyad couplings terms. The computed band centre for the 45B←1A band (in the middle of Fig 2) was in excellent agreement with our

Figure 1 :

 1 Figure 1:Overview of the CW-CRDS spectrum of18 O 3 between 5925 and 6850 cm -1 . The 18 O 3 sample pressure was about 18 Torr. The displayed spectrum was obtained by concatenating 39 individual spectra recorded successively with different DFB laser sources (the corresponding ranges are shown with different colors / grey levels). The CW-CRDS spectrum is observed above the baseline fixed by the reflectivity of the used super mirrors. The A-type parallel bands are indicated and assigned with the usual normal mode label. Note the rapid decrease of their band strength with the energy. The stronger absorption bands due to impurities (CO 2 and H 2 O) are also indicated.

Figure 2 :

 2 Figure 2:Comparison of the simulated spectra using assigned ro-vibrational lines for18 O 3 (upper panel) and16 O 3 (lower panel) in the region of the 5ν 1 +ν 3 bands. The global and normal mode band assignments are shown. The vertical axe represents the absorption coefficient in arbitrary units at the same scale for both isotopologues.

Figure 3 :

 3 Figure 3: Isotopic shift due to the 16 O 3 → 18 O 3 homogeneous isotopic substitution for the centers of the 18 O 3 bands observed by CW-CRDS between 5900 and 6850 cm -1 . Upper panel (diamonds): wavenumber difference between the band centres having the same global assignment; Lower panel (circles): wavenumber difference between the centers of bands associated with the most similar compositions of the normal mode upper states. See text of the Section 5 explaining the distinctions between filled, grey and open symbols.

Figure 4 .

 4 Figure 4. Principal normal mode mixing coefficients for upper vibration states involved in Fig 3. Upper panel: 18 O 3 states observed in this work. Lower panel: corresponding 16 O 3 states (lower panel of Figure 3 ). The vertical axe represents the mixing coefficients P 1 for the normal mode expansion of the wavefunctions, Eqs. (1,2). Triangles correspond to ψ for the [JK a K c ]=[0 0 0] level of A 1 vibration states. Stars correspond to ψ for the lowest possible [JK a K c ]=[1 0 1] level of B 1 vibration states.

  Figure 3

  for B 1 type vibrations. Predictions for these lowest allowed levels corresponding to the bands observed in this work are given in Table1together with their three major mixing coefficients P n associated with normal mode contributions 0

  O 3 homogeneous isotopologue. Consequently, the other minor isotopic species -mainly16 O 18 O 18 O and 18 O 16 O 18 O -contributed to the absorption spectra through many weak absorption lines superimposed on the 18 O 3 spectrum. The spectrum analysis was further complicated by the presence of absorption lines of several impurities which were generally 18 O enriched: H 2 18 O (and H 2 16 O), 12 C 18 O, 16 O 12 C 18 O or 12 C 18 O 2 . The H 2 18 O, 16 O 12 C 18 O and 12 C

  16 O18 O16 O, 18 O 16 O 18 O, 16 O 16 O 18 O, and 16 O 18 O 18 O) for which even stronger changes are expected.

Table 2 Bands of 18 O 3 analysed in the CW-CRDS spectrum between 5900 and 7000 cm -1 and comparison with the calculations from the PES

 2 In case of a large normal mode mixing of the upper states, several contributions are given in column 2 according to Table1. Column 3 represents the global unambiguous assignment (see text), dσ is the uncertainty estimation for the experimental band centre determination. The bands at 6392 and 6643 cm -1 have the same principal normal mode contributions (Table1) for the upper states.

	Observed bands: spectroscopic notations	Normal mode assignment of the upper vibration state	Global band assignment	Number of assigned ro-vib. lines	Band centres σ dσ (cm -1 ) (10 -3 cm -1 ) σ(exp.)	Calc_1 (cm -1 )	obs-calc (cm -1 )
				B-type bands
	4ν 1 + 3ν 2 2ν 1 + ν 2 + 4ν 3 ν 2 + 6ν 3 ν 1 + 3ν 2 + 4ν 3	(430) (214) (016) (134/224)	65A ← 1A 66A ← 1A 71A ← 1A 82A ← 1A	32 184 91 339	6011.836 6047.101 6245.039 6592.661	9 14 11 14	6012.31 6047.95 6245.81 6591.83	-0.5 -0.8 -0.7 0.9
				A-type bands
	2ν 2 + 5ν 3 5ν 1 + ν 3 ν 1 + ν 2 + 5ν 3 2ν 1 + 5ν 3 2ν 1 + 3ν 2 + 3ν 3 (I) 3ν 2 + 5ν 3 5ν 1 + ν 2 + ν 3 2ν 1 + 3ν 2 + 3ν 3 (II) ν 1 + 2ν 2 + 5ν 3 4ν 1 + 3ν 2 + ν 3	(025) (501) (115/313/223) (205/403) (233/143) (035) (511) (233/323) (125/413/215) (431/323)	44B ← 1A 45B ← 1A 46B ← 1A 51B ← 1A 54B ← 1A 58B ← 1A 59B ← 1A 60B ← 1A 64B ← 1A 66B ← 1A	507 567 599 659 344 574 254 355 385 165	5984.439 6013.048 6072.132 6270.604 6392.214 6556.788 6611.039 6642.896 6796.463 6825.512	6 8 6 7 9 2 34 21 6 6	5984.58 6013.12 6072.49 6272.27 6393.72 6557.49 6610.48 6644.23 6795.31 6827.74	-0.1 -0.1 -0.4 -1.7 -1.5 -0.7 0.6 -1.3 1.2 -2.2

Acknowledgements

We acknowledge the support from a collaborative program between CNRS-France and RFBR-Russia (PICS grant No 05-05-22001). We are grateful to S.Mikhailenko for stimulating discussions. The support from IDRIS computer centre of CNRS France and Champagne-Ardennes regional computer centre for global calculations is also acknowledged.

Table 1

Theoretical prediction of the lowest rovibrational levels of excited states corresponding to the observed bands of 18 O 3 and their global and normal mode assignment N V : global ranking number for A 1 and B 1 vibrational states. Calc_1: variational predictions from the PES V M of Ref. [40] in internal coordinates ( r 1 , r 2 ,θ ); Calc_2: non-empirical effective Hamiltonian predictions derived from the PES of Ref. [41] in normal coordinates q 1 , q 2 , q 3 using 8-th order Contact Transformations [42]; J K a K c : rotational quantum numbers; Columns P 1 , P 2 , P 3 indicate the first three mixing coefficients (in %) according to Eqs. (1,2); Columns W 1 , W 2 , W 3 indicate the corresponding vibration normal mode quantum numbers ( v 1 v 2 v 3 ). The subscript "0" of ( v 1 v 2 v 3 ) 0 means that these contributions correspond to the harmonic normal mode basis.