Room temperature line lists for CO\2 symmetric isotopologues with \textit{ab initio} computed intensities

Remote sensing experiments require high-accuracy, preferably sub-percent, line intensities and in response to this need we present computed room temperature line lists for six symmetric isotopologues of carbon dioxide: $^{13}$C$^{16}$O$_2$, $^{14}$C$^{16}$O$_2$, $^{12}$C$^{17}$O$_2$, $^{12}$C$^{18}$O$_2$, $^{13}$C$^{17}$O$_2$ and $^{13}$C$^{18}$O$_2$, covering the range 0-8000 \cm. Our calculation scheme is based on variational nuclear motion calculations and on a reliability analysis of the generated line intensities. Rotation-vibration wavefunctions and energy levels are computed using the DVR3D software suite and a high quality semi-empirical potential energy surface (PES), followed by computation of intensities using an \abinitio\ dipole moment surface (DMS). Four line lists are computed for each isotopologue to quantify sensitivity to minor distortions of the PES/DMS. Reliable lines are benchmarked against recent state-of-the-art measurements and against the HITRAN2012 database, supporting the claim that the majority of line intensities for strong bands are predicted with sub-percent accuracy. Accurate line positions are generated using an effective Hamiltonian. We recommend the use of these line lists for future remote sensing studies and their inclusion in databases.


Introduction
Remote sensing of carbon dioxide for atmospheric applications requires a detailed knowledge of the rotational-vibrational spectra of all its major isotopologues. It is commonly believed that such measurements should be supported by reference line intensities of 1% accuracy or better [1]. This is the main requirement for successful interpretation of data from the NASA Orbiting Carbon Observatory 2 (OCO-2) space mission, which is designed to monitor the concentration of carbon dioxide in Earth's atmosphere. As was shown in our previous papers considering the main 12 C 16 O 2 isotopologue (denoted here as 626), a fully ab initio dipole moment surface is capable of providing such a level of accuracy [2,3]. The next step is to address other isotopologues, which are ubiquitous in natural samples and may interfere with spectral lines of the main isotopologue as well as other species [4,5,6,7].
Concentration measurements of trace compounds also require very accurate line positions, intensities and line profiles of several isotopologues at the same time.
Experimental line-intensity measurements for isotopologues other than the main 626 one are much more challenging due to their low natural abundance, which ranges from 1% for 13 C 16 O 2 (636) down to 10 −10 % for the unstable 14 C 16 O 2 (646). Enriched samples feature the need for precise (a priori ) knowledge of isotopologue concentration. Therefore experimental accuracies of line intensities for less abundant isotopologues of carbon dioxide are in general lower than for the main isotopologue. A comprehensive source of spectroscopic data for carbon dioxide is the HITRAN database [8]. The 2012 release of this database contains line lists for the 13 C 16 O 2 (636), 12 C 17 O 2 (727), 12 C 18 O 2 (828) and 13 C 18 O 2 (838) isotopologues, all featuring spectral gaps because of lack of experimental data. Line intensities included in the database have uncertainties of 5-20%, thus well above current requirements.
The database comprises both semi-empirical (taken from the pre-release of the CDSD database [9,10]) and purely experimental entries. Multiple data sources are reflected by discontinuities in intensity patterns of some bands [3].
Several measurements have been made since the last edition of the database, to pursue the elusive naturally abundant molecules, although these studies also have uncertainties well above the desired 1% threshold. The most accurate recent experiments by Devi et al. [11], Jacquemart et al. [12,13,14,15], Karlovets et al. [5] and Durry et al. [16] address several bands of carbon dioxide isotopologues with unprecedented, though still unsatisfactory, intensity accuracies between 1% and 20%.
Among the few theoretical attempts to model high-resolution infrared spectra of carbon dioxide, the study by Huang et al. from the NASA Ames Research Center proved to be very accurate [17,18]. Line lists from Huang et al. provide both line positions and intensities covering the infrared and visible spectral region (J = 0−150) for room temperature (296 K) and 1000 K. Line positions, derived from a variational approach and based on semi-empirical potential energy surface (PES), are accurate to 0.03 -0.2 cm −1 . Although the Ames-1 line lists [17,18] show the best agreement with experiment among all variational calculations, semi-empirical approaches based on effective Hamiltonians can provide line positions with an accuracy at least one order of magnitude better [9]. On the other hand, effective Hamiltonian models strongly depend on the quality of the input data, thus the accuracy and completeness of this technique are limited by experiment.
A major advantage of the use of variational nuclear motion programs is that, within the limits of the Born-Oppenheimer approximation, line intensities can be computed with the same accuracy for all isotopolgues. In the present study the computation of line intensities was based on ab initio dipole moment surfaces (DMSs), and these were preliminarily tested by several authors [3,10,11]. Some minor inaccuracies and discontinuities were discovered; however, comparisons showed overall good agreement with experiment. A comprehensive literature review on both experimental and theoretical approaches to line positions and intensities is given in the recent work by Tashkun et al. [10]. In this study the authors indicate and support the need for a unified theoretical treatment of line positions and intensities; whereas the former are largely covered by experiments facilitated by effective Hamiltonian models [9], the latter still await high levels of accuracy, which we target in the present study. Due to lack of experimental data, a line list for the radioactive isotopologue 14 C 16 O 2 is not included in the HITRAN2012 and CDSD-296 databases. The Ames-1 line list is presently a unique source of high accuracy spectra for this species.
The unstable 14 C 16 O 2 isotopologue is of particular importance because of its usage in dating of biosamples and, more recently, in monitoring emissions, migrations and sinks of fossil fuel combustion products [19,20,21] as well as for the assessment of contamination from nuclear power plants [22].
Until recently, monitoring fossil fuel emission relied mostly on β-decay count measurements or mass spectrometry, both of which are high cost, invasive methods.
Despite its low natural atmospheric abundance, radiocarbon dioxide has been probed via optical spectroscopy methods [6,7,23,24]. Recent advances in absorption laser spectroscopy provided an unprecedented tool for detection of species containing radiocarbon of ratios 14 C/ 12 C down to parts per quadrillion. These measurements exploit saturated-absorption cavity ring down (SCAR) spectroscopy technique [25] for the strongest fundamentals of 14 CO 2 [7,24]. The knowledge of accurate line intensities for several isotopologues at the same time is therefore a necessity for eliminating the unwanted noise sourced in traces of different isotopic carbon dioxide representatives.
For instance the P (20) line of the 00011 -00001 band in 646, which is dedicated for radiocarbon measurements, above certain temperatures, heavily interferes with the Lorentzian tail of the P (19) line in the 05511 -05501 band of the 636 isotopologue [4]. This raises difficulties in retrieving unbiased concentrations of the radioactive isotopologue. Similar problems occurred in measurements based on the P (40) line of the ν 3 band of 14 CO 2 [26]. In both cases accurate values of line intensities are required. Otherwise, as shown in [26], calculation of the fraction of 14 C in measured samples that employed a line strength taken from a theoretical approach, led to over 35 % error in retrieved concentrations (as later confirmed by alternative experiments (AMS)). These observations were explained in terms of both inaccuracies of the line intensity and drawbacks of the spectroscopic fit model used, which fuels the need for reliable line intensity sources. Another successful technique further supporting this need was recently introduced by Genoud et al. [6], cavity ring-down spectroscopy with quantum cascade laser for monitoring of emissions from nuclear power plants. High quality line intensities for 12 C 16 O 2 , 13 C 16 O 2 and 16 O 12 C 18 O are also required for real-time detection methods based on quantum cascade lasers to monitor 13 C/ 12 C isotope ratios in identification of bio-geo-chemical origins of carbon dioxide emissions from the soil-air interface [27]. Spectra of isotopologues can be used for a variety of different tasks such as the recent suggestion that observations of absorptions by 13 C 16 O 2 in breath analysis provides a non-invasive means of diagnosing gastrointestinal cancers [28].
In this work we aim to provide highly accurate line intensities in the 0 -8000 cm −1 spectral region together with reliable semi-empirical line positions for six symmetric isotopologues of carbon dioxide; five naturally abundant: 636, 727, 737, 828, 838 and one radioactive (646). An important advantage of a first principles theoretical approach is wide spectral coverage, in contrast to limited laser tuning capability of some measurements. For this reason databases like HITRAN, containing ro-vibrational spectra of small molecules, often utilize various experimental data sources, thereby giving up consistency of entries. This is not the case for the ab initio approaches, which are believed to provide consistent accuracy of line intensities within a vibrational band [29].
Another issue related to post-processing of experimental data is the functional form of the Herman-Wallis factors. These, as arbitrarily chosen and empirically fitted, can become a source of unnatural biases for high J transitions [3]. The theoretical model implemented here should be inherently free from this problem, as the rotational contribution to line intensities is computed directly. The only underlying source of errors within the Born-Oppenheimer approximation is potential energy surface (PES) and dipole moment surface (DMS). The quality of both surfaces is assessed here by employing the line sensitivity analysis procedure introduced by Lodi and Tennyson [29], which requires computation of at least four line lists for each isotopologue. This allows us to detect resonances [3], that affect line intensities and significantly diminish the reliability of data provided.
The final results are given in the form of line lists with associated uncertainties, which are available in the supplementary materials. Uniformity of errors (except for the regions affected by resonance interactions) accompanied by the precise reproduction of observed line intensities at the sub-percent level for the majority of strong bands [3] suggests our results provide a viable update to the current, 2012 version of the HITRAN database.

Methodology
The Lodi-Tennyson methodology presented in detail elsewhere [3,29] is used to validate line lists on a purely theoretical basis. For each isotopologue four line lists were computed, based on set of two PESs and two DMSs. These surfaces come from two sources: the semi-empirical Ames-1 PES and DMS from Huang et al. [17], an ab initio PES and DMS (UCL DMS) computed by us, where the former one was subsequently fitted to observed J = 0 − 2 levels (called below: Fitted PES). Details of all surfaces were presented before [3].
These four high-quality surfaces are used in nuclear motion calculations to obtain rotational-vibrational energy levels, wavefunctions and line intensities.

Nuclear motion calculations
The strategy for solving the nuclear-motion problem employed here is analogous to the one presented in [3]. In the Born-Oppenheimer approxima-tion the PES and DMS are isotopologue independent. Therefore the only parameters that distinguish between different isotopologues are the nuclear masses entering the expression for the kinetic energy operator (KEO) of the nuclei. Our approach is based on an exact, within the Born-Oppenheimer approximation, KEO which is used in a two step procedure [30,31,32] [34]. In the first step of the computation (program DVR3DJZ) the Born-Oppenheimer ro-vibrational wavefunctions were expanded in Morse-like oscillator basis functions for stretching coordinates and Legendre polynomials for the bending one; symmetrised Raudau internal coordinates were used. The converged DVR basis set associated with Gauss-Legendre quadrature points contained 30 radial and 120 angular functions, respectively. In the second step (program ROTLEV3b) we employed a J−dependent basis set of symmetry-adapted symmetric-top rotational functions. The J ranges considered were chosen to match those present in the HITRAN2012 database (see Table 1). The same set of parameters was used to evaluate ro-vibrational energies computed from Ames-1 and fitted PESs. Calculation of line strengths used the DIPOLE program and require both DMS and ro-vibrational wavefunctions as input functions.
Transformation from line strengths to transition intensities included scaling by natural isotopic abundance and multiplication by the appropriate spin statistical weights. Partition functions at 296 K were taken from Huang et al. [17] and they agree to better than 0.1% with partition functions from the present computation based on Ames-1 PES (see Table 1). A naturalabundance weighted intensity cut-off was then set to 10 −30 cm/molecule. In the case of the radioactive isotopologue (646) we assumed unit abundance and increased the cut-off value to 10 −27 cm/molecule.

Estimation of the intensity uncertainties
The ab initio DMS is a primary source of inaccuracies in line intensities. The accuracy of the UCL DMS has been proven to be at sub-percent level for several strong bands (stronger than 10 −23 cm/molecule) below 8000 cm −1 [2,3]. A key feature of using an ab initio DMS with a variational nuclear motion calculation is that entire vibrational bands are reproduced with similar accuracy. The reliability of line intensities obtained theoretically is correlated with the quality of J-dependent ro-vibrational wavefunctions, hence with an underlying PES. Wavefunctions play an important role in capturing the interaction between different vibrational states. Such resonance interactions can lead to intensity stealing and, particularly for so-called dark states, huge changes in transition intensities.
Here we adopt a scheme from our previous works [3,29], where a method and unstable (ρ ≥ 4.0). These values are the same for all isotopologues. We believe that this descriptor gives a robust measure of sensitivity of line intensities to small PES changes, and hence reflects the reliability of a theoretically evaluated line intensity.

Line positions
As discussed above line positions for the recommended UCL-IAO line list are taken from the CDSD-296 database [10]. These line positions were calculated within the framework of the effective Hamiltonian approach for which the partly-reduced polyad model was used [36,37,38]. This model takes into account the resonance anharmonic, anharmonic+l -type and Coriolis interac- The good predictive ability of the resulting effective Hamiltonian parameters has been demonstrated many times [5,12,13,14,40,41]. The exceptions are several bands of the asymmetric isotopologues which are perturbed by the interpolyad resonance anharmonic interactions. Some of the examples are given in Refs. [5,40,41]. Usually these perturbed bands are very weak.

Pseudo-experimental refinements
Energy levels taken from the effective Hamiltonian (EH) are generally considered to be accurate to 0.002 cm −1 or better, which is usually more accurate than those given by PES based studies [10]. Multi-isotopologue fits allowed for completeness of EH line lists even for less abundant isotopologues [42]. From this reason our recommended line lists include line positions from the Effective Hamiltonian calculations provided by the CDSD-296 database [10]. However, the very limited experimental data for 14 C 16 O 2 did not allow us to extract EH parameters for this isotopologue. Therefore an alternative source of line positions is needed here. Thus, an attempt to correct DVR3D line positions with the effective Hamiltonian values was made. First, energy differences between corresponding EH and DVR energy levels for the main isotopologue were taken: Next, each energy level of a given less abundant isotopologue was refined by adding respective difference to the DVR-computed value: . These refined energy levels were then compared to EH values. Application of the above procedure to symmetric isotopologues of CO 2 resulted however in increased deviations between DVR and EH energy levels, hence should be considered here as unsuccessful. The reason for this is not entirely clear but the cited study for water [29] took considerable care over the corrections to the Born-Oppenheimer approximation while the results here were based on a PES simultaneously fitted to data from several isotopologues with no allowance for the beyond Born-Oppenheimer effects.

Results and Discussion
Below we present an overview of our line lists. The next subsection contains statistical analysis of the scatter factor for each isotopologue, which will be further used in detecting resonances in the following subsection. After that, we compare our present results to recent highly accurate measurements, which were not included in the latest release of the HITRAN database. Next, a detailed comparison with the HITRAN2012 and CDSD-296 databases is performed. Finally we discuss the radioactive 646 isotopologue in the environmental context, as one of possible applications of present results. Table 1 contains general information about our line lists.  [43]; e HITRAN2012 abundances were taken from Ref. [17]; f For 10 −27 cm/molecule intensity cut-off for the 646 isotopologues and 10 −30 cm/molecule after scaling for natural abundance for the other isotopologues; g UCL-IAO line list with 10 −33 cm/molecule intensity cut-off was used in the comparison.

Overview
Partition functions in this work were calculated using Ames-1 PES, and these are compared to HITRAN2012, Ames and CDSD-296 line lists. We can see from the Table 1 that for isotopologues: 636, 646 and 828 the agreement between Ames work and our values is at 0.002% level, as expected from runs based on the same PES. Partition functions calculated by us should however be treated as provisional for the 737 and the 838 isotopologue, as J-range employed here did not allow for full convergence of the partition function.
On average, the computed partition functions agree excellent with CDSD-296, and are systematically shifted by -0.3% with respect to HITRAN2012. This fact must be taken into account in comparisons aimed at sub-percent accuracy of line intensities. For line intensity calculations we used Ames partition functions from Huang et al. [17]. The HITRAN2012 database has significant spectral gaps for less abundant isotopologues, which are all covered by our line lists. Lines present in HITRAN2012 were completely matched to our lines using the energy level comparison technique.

Resonances
Detailed information on the energetic distribution of the scatter factor may be extracted from a 'transition stability map', which is a useful tool in searching for resonances, as exemplified in Figure 2, which presents an example of such a map for the 828 isotopologue. The advantage of this particular representation is that one gains a full overview of all energetic regions, where transition intensities appear to be sensitive to minor inaccuracies of the PES.
These lines are marked as red dots in Figure 2.    In Figure 4 colour coding shows the stability of the transition intensity.
A J-localized resonance is visible around m = +36, clearly correlating with both high instability of lines (marked by red points) and large deviations from HITRAN2012 line intensities.
This quasi-singularity in line intensity occurs due to Coriolis interaction with the strong 00011-00001 band, which equally perturbs P and R branches of the 11101-00001 band, and manifests itself by intensity borrowing, which in turn leads to the strengthening of the P-branch and to suppression of the R-branch. This observation confirms our previous predictions for existence of such perturbation in the main isotopologue [3]. For this reason, in the final recommended line list we replace line intensities perturbed by these Coriolis interactions with semi-empirical data from the CDSD-296 database.
A view of the 636 isotopologue in Figure 5 supports this thesis. Similar behaviour is observed for other isotopologues. The 636 case clearly shows that this type of interaction is branch-specific and J-specific as illustrated in Another example of the intrapolyad interaction is the pair: 23301 (perturber) and 12212 -02201 (perturbed band), for which we depict the intensities scheme in Figure 6.      [16] plotted against J quantum number for several databases. Sources considered are HITRAN2004 [47], HITRAN2008 [48], the 2008 release of CDSD [9] and the present work. The 1% deviation region is represented by green edge-blurred strip.

Isotopologue 727
In recent measurements performed on 17    Here, highly enriched sample allowed for more precise measurements than in the 727 isotopologue case. Figure   Above results for 636, 727 and 828 isotopologues are summarized in Table   2.

Comparison with HITRAN2012, Ames and CDSD-296
The I exp,(i) − 1 · 100%) and the total band strength in cm/molecule. The last column (marked UCL-IAO) contains the data from the present study, the total number of lines in the band, suggested accuracy for the band (in %) and the total band strength in cm/molecule. can be found in the official release of CDSD [10], which can be used to get more realistic information about the uncertainties of the line parameters.
Intensities from Toth et al.are supposed to be accurate to better than 2% (uncertainty code 7) or 5% (code 6). As discussed elsewhere [3], the stated uncertainty estimates of all current entries are insufficiently accurate for remote sensing applications. In addition to that, several bands feature unrealistic jumps in line intensities originating from switching data sources.
Therefore a unified approach giving line positions of spectroscopic quality combined with significantly more accurate transition intensities is needed.
In the present paper we aim in fulfilling these requirements.
In order to relate results from the present study to data given in HITRAN we compared line intensities for matched lines between the two line lists (see Table 1). As a primary measure of relative intensity deviation from HITRAN data we used the following formula: where I U CL stands for line intensity from UCL-IAO line list given in cm/molecule and I HIT is HITRAN2012 intensity.
This measure is adequate for small deviations but poorly illustrates highly discrepant intensities, due to its asymmetric functional form. For larger deviations, especially for line intensities weaker than HITRAN by more than 100%, the quasi-symmetry is noticeably broken, resulting in a biased picture.
In such cases, for example to show graphically a general overview, we decided to use a symmetrized measure to account for proper representation of large deviations: This measure, in turn, yields far from intuitive numbers near 0% deviation.

Isotopologue 636
The  blue filled triangles in the right panel of Figure 12) remain within the claimed 5% uncertainty, additionally exhibiting a very narrow spread.

Isotopologue 727
HITRAN2012 line list for the 727 isotopologue contains 5187 lines below 8000 cm −1 , all of which were taken from the effective Hamiltonian calculations by Tashkun and Perevalov [51]. We have already shown that inaccuracies of our model are largely reflected in systematic shifts of whole bands, rather than statistical scatter, which is assumed to remain almost constant as a function of J. Two bands in Figure 13 lie outside the tolerance given by the HITRAN2012 uncertainty code 3. These are: the 00031 -00001 band and the 30013 -00001 band (both indicated with arrows in Figure 13). The discrepancy for the former band has been explained in terms of rather poor reproduction of the 3ν 3 series of bands by our DMS and needs to be replaced in our recommended line list. The behavior of the latter band however is not clearly understood at this stage and requires further investigation. Our working hypothesis is that the −6% systematic shift applies to all bands, hence the 30013 -00001 band when shifted by +6%, should match the 20% tolerance region, which is also regarded as provisional. New measurements by Karlovets et al.  Figure 13: Relative intensities (cf. eq. (2)) plotted against HITRAN2012 line intensities for the 727 isotopologue. Green dashed horizontal line represents deviation from HITRAN2012 data equal to ±20%.

Isotopologue 828
HITRAN2012 line list for the 828 isotopologue contains 7071 lines below 8000 cm −1 . There are three sources of line intensities: 6280 lines taken from CDSD-296 [9] with ier equal to 3 and 4, 722 lines taken from a 1994 update to older variational calculations [52] with ier equal to 2, and finally 69 lines taken from measurements by Toth et al. [50] with ier assigned to 3. Figure 14 compares intensities from the present study to HITRAN2012 data.
Despite the low uncertainty index, line intensities originating from Rothman et al. [52] agree within ±20% with our results. Transitions around 2.06 µm measured by Toth et al. [50] are enclosed in 10% region reflecting the ier value for this set. Data points originating from CDSD-296 are divided into two sets with differing uncertainty index. The more accurate subset (marked with orange rotated crosses) is clearly squeezed along the relative deviation axis and exhibits almost no systematic shift. In contrast, the lower accuracy subset from CDSD spreads over a large region in relative deviation space.
This suggests that both sets were calculated with separate input parameters of different quality. The 30013 -00001 band (ier = 4) deviates around +2% from CDSD predictions, while the relatively strong 00031 -00001 band (ier = 4) lies 11% below the zero deviation line (visible in Figure 14). It should be noted that large deviations of the lower accuracy CDSD-296 data (ier = 3) occur for very weak lines, for each the respective experimental data to fit the effective dipole moment parameters are absent. In these cases the parameters of the principal isotopologue were used in CDSD-296. Relative intensities from the present study plotted against HITRAN2012 line intensities for the 828 isotopologue. Only ±50% region is depicted. Dashed grey and green lines correspond to 10% and 20% deviation, respectively. Blue crosses correspond to a subset of lines taken from Perevalov et al. [9] which has been assigned to ier = 4. Consequently, rotated orange crosses represent ier = 3 from the same reference. Red filled triangles refer to Rothman et al. [52], while purple filled squares stand for the small set of lines provided by Toth et al. [50].  2)) plotted against CDSD-296 line intensities for the 828 isotopologue. Only the ± 500% region is depicted. Figure 15 shows that all strong lines (> 10 −28 cm/molecule) follow funnel shape envelope, thereby reflecting the typical relation between intensity and accuracy of lines. However several weaker lines, which constitute whole bands, align in wide arc structures with large systematic shift. This is particularly visible for lowered accuracy lines from HITRAN2012 (blue crosses in Figure 15). These lines were directly incorporated from HITRAN2008.
The current release of the CDSD database improved on accuracy of these weak lines. A comparison between UCL and CDSD-296 intensities is given in Figure.   It is instructive to follow changes in relative intensity deviation among different isotopologues for the selected band. Two data sources were used for this band: a small set of low J lines from experiment by Toth et al. [50] for the 636 isotopologue (given uncertainty index 6) and CDSD-296 for rest of the lines (ier = 3). All lines compared above match the stipulated HITRAN uncertainty, that is lines with ier = 6 fit the 5% tolerance, and the rest of the lines are 20% or less away from HITRAN2012 values. Minor discontinuity related to change of source of data is seen for the 636 isotopologue. We conclude that the results illustrated in Figure 18 give a tentative validation of the consistency of our approach with the analysed database. Relatively good overall agreement between our line list and HITRAN2012, revealing only sporadic deviations that exceed the claimed HITRAN accuracy, but yet justified and facilitated with comparisons with recent and highly accurate measurements, allow us to draw a conclusion that replacing current HITRAN line intensities with our computed values would significantly increase the accuracy, reliability and consistency of the database.

Ames-1
In On balance we believe the second explanation is more likely.  Here we provide line intensities which are internally consistent and proved to agree within experimental uncertainty to state-of-the-art measurements. Table 4 also lists our prediction for the line intensity at T = 170 K, a temperature which is commonly used for intensity measurements for the P (20) line. The 3ν 3 family of bands discussed in ref. [3] and in the preceding sections of this article, has been proven to be on average 12% too weak in present calculations. For isotopologues other than the radioactive 14 C 16 O 2 intensities for these bands were replaced with CDSD-296 entries. For 14 C 16 O 2 these intensities were scaled by 1.12 to correct for the systematic differences known from other isotopologues. The scheme used for the line intensities is similar the one used for the main isotopologue. The 3ν 3 bands and unstable lines were taken from the effective Hamiltonian calculations, and appropriate uncertainty indices were assigned.

Recommended line lists
Intensities of stable lines belonging to bands stronger than 10 −23 cm/molecule (for unit abundance) were taken from UCL DMS calculations and assigned HITRAN uncertainty code 8 (i.e. accuracy of 1% or better). Stable lines belonging to parallel bands weaker than 10 −23 cm/molecule also come from UCL DMS computation and were given uncertainty code 7 (i.e. accuracy 1 -3 %). Intermediate lines and stable lines belonging to perpendicular bands weaker than 10 −23 cm/molecule feature HITRAN uncertainty code 6 (i.e. accuracy 3 -5%). All line positions and line intensities for which scatter factor was not assigned were taken from CDSD-296. This was the case for only 3700 weak lines in total for all isotopologues. Both types of line lists are given in the supplementary materials with appropriate explanation in text files. Abundances were taken from the HITRAN2012 database with intensity cut-off 10 −30 cm/molecule. For the radiocarbon isotopologue unit abundance was assumed and 10 −27 cm/molecule intensity cut-off cm/molecule.

Conclusion
In the present study we compute new line lists for six symmetric isotopologues of carbon dioxide: 13  This paper completes our analysis of the transition intensities of symmetric isotopolgues of CO 2 . We are currently analysing the transition intensities of the asymmetric isotopologues. This work raises some theoretical issues as the loss of symmetry has consequences for both the DVR3D nuclear motion calculations and the CDSD effective Hamiltonian studies. Results will be reported in the near future [54].