ROVIBRATIONAL LINE LISTS FOR NINE ISOTOPOLOGUES OF THE CO MOLECULE IN THE X¹Σ⁺ GROUND ELECTRONIC STATE

Transitions of the 4–0 band were investigated in Grenoble with a newly developed cw-CRDS spectrometer using a fibered External Cavity Diode Laser (ECDL: Toptica DL pro, 1200 nm) as the light source. The total spectral coverage of this spectrometer is about 7900–8320 cm⁻¹. High P(J) 4–0 lines are located in the high-energy part of this interval. They were measured one by one on the basis of their predicted line positions.

The set up is very similar to the CRDS spectrometer based on distributed feed back (DFB) which is used routinely in the 5850–7920 cm⁻¹ region (Kassi & Campargue 2012; Tran et al. 2013). The 1.40 m long CRDS cell is fitted with high reflectivity mirrors leading to ring down times of about 200 μs. The CRDS cell was filled with carbon monoxide (Alphagaz 99% stated purity) at a pressure of 10.0 Torr. The pressure and the ring down cell temperature were monitored during the spectrum acquisition. The typical mode-hop free tuning range of this ECDL is about 0.8 cm⁻¹. The laser frequency was tuned over a 1.6 cm⁻¹ wide region around the line center by changing the grating angle together with the laser current. Three consecutive and partially overlapping spectra were typically recorded for each line. About 10 ring down events were averaged for each spectral data point separated by 8 × 10⁻⁴ cm⁻¹. The noise equivalent absorption evaluated as the rms of the baseline fluctuation is better than α_min ∼ 1 × 10⁻¹¹ cm⁻¹, as illustrated in Figure 1. An important advantage of this ECDL compared to the DFB diode lasers used in Kassi & Campargue (2012) and Tran et al. (2013) lies in its smaller frequency jitter (typically 100 kHz compared to 2 MHz), which limits the noise due to the conversion of the frequency jitter in intensity fluctuations on the rising and falling portions of a line profile.

Zoom In Zoom Out Reset image size

The wavenumber of the light emitted by the laser diode was measured by a commercial Fizeau type wavemeter (HighFinesse WSU7-IR, 5 MHz resolution, 20 MHz accuracy over 10 hr) that allows laser frequency to be determined at a typical 100 Hz refresh rate. In order to further refine the absolute calibration, a very low pressure scan was recorded previous to this experiment, showing H₂O lines (present as impurity) for which accurate line positions were taken from the HITRAN2008 database (Rothman et al. 2009). We estimate the accuracy of the absolute calibration of the wavenumber scale of our spectra at 1 × 10⁻³ cm⁻¹.

In the line profile fitting, the ECDL line width contribution (FWHM) was neglected compared to the Doppler broadening (∼0.1 MHz and ∼0.57 GHz FWHM, respectively). In a first step, a Voigt function of the wavenumber was adopted for the line shape. The half width at half maximum (HWHM) of the Gaussian component was fixed to its theoretical value. The local baseline (assumed to be a linear function of the wavenumber), the line center, the integrated absorbance, and the HWHM of the Lorentzian component were fitted. Figure 1 shows the best reproduction of the P(21) line recorded at 10.0 Torr obtained with a Voigt profile. The W shape residual is a typical signature of the Dicke narrowing effect. Much reduced residuals were obtained using a Galatry line profile. Note that weak absorption lines due to water and carbon dioxide had to be taken into account in order to reproduce the spectrum.

In this way, the center and intensity of 10 P(J) 4–0 lines with J values between 19 and 34 were determined (Tables 1 and 2). They include the weakest CO lines that were ever measured quantitatively by absorption spectroscopy, such as the P(34) line with an intensity of about 1.9 × 10⁻³⁰ cm⁻¹/(molecule × cm⁻²). (Note that these units are often given by their reduced form, cm molecule⁻¹, but the longer version emphasizes the sense of wavenumber per column density.)

During the spectra analysis, we found that a few additional CO lines were accidentally present in the spectral interval around the targeted 4–0 ¹²C¹⁶O lines (see Tables 1 and 2). They include the P(6) and P(9) lines of the 5–1 hot band of ¹²C¹⁶O and one 4–0 line for each of the ¹³C¹⁶O, ¹²C¹⁸O, and ¹²C¹⁷O isotopologues present in natural abundance in the sample.

Tables 1 and 2 also list the line positions and intensities from the HITRAN database, and the semi-empirical and ab initio results of the present study for both the 4–0 and 5–1 bands. The observed line intensities have been converted to the intensities at 296 K using Equation (1) in the paper by Rothman et al. (2010). Discussion of these results in comparison with other sources of data is offered in Section 5.

The transitions of the 6–0 band of CO in the range of 12347–12497 cm⁻¹ were recorded using a continuous wave cavity ring-down spectrometer (cw-CRDS). The experimental setup has been presented by Cheng et al. (2012). In brief, the 1 m long ring-down cavity is composed of two mirrors with a reflectivity of 99.995%. The spectral calibration is based on the frequencies of the passive modes of a Fabry–Perot interferometer (FPI) made of ultra-low-expansion glass. The 10 cm long FPI is placed in a vacuum chamber and thermo-stabilized. The absolute frequencies have been calibrated with an accuracy of about 2 MHz using the atomic transitions of ⁸⁷Rb at 780 nm and 795 nm. The spectra were recorded at room temperature (296 ± 1 K), using sample pressures in the range of 0.2–0.8 atm. The gas pressure was measured by a capacitance gauge with a stated accuracy of 0.5%. The CO spectra of each line recorded at different pressures were fitted using a Voigt profile by fixing the Gaussian width at the calculated Doppler width. The positions, intensities, and Lorentzian widths were derived from the fitting. Here only the intensities of the R(5), R(7), R(9), and R(10) lines are presented in Table 3. The full analysis of this band will be presented in a separate study (Y. Tan et al. 2014, in preparation). Discussion of these results in comparison with other sources of data is offered in Section 5.

Conventionally, the semi-empirical DMF is constructed based on the rotationless matrix elements R_v-0(m = 0), where v = 0, 1, 2, 3,...,v_max, and m = −J for P-branch lines and J + 1 for the R-branch lines. The rotationless matrix elements are determined by fitting the rotational matrix elements within a single vibrational band with a second-order polynomial function, which is known as the Herman–Wallis fit (Chackerian & Tipping 1983; Tipping & Ogilvie 1982). In our previous study (Li et al. 2011), we showed that it is possible to obtain the DMF by directly fitting all available rovibrational matrix elements with appropriate weightings of the experimental uncertainties. This method has been successfully applied to hydrogen halides, HF, HCl, HBr, and HI (Li et al. 2011, 2013a, 2013b). The direct-fit method is superior to the conventional method in the following ways: (1) it is now possible to obtain a good fit of a vibrational band using the high-precision measurements of just a few or even a single rotational line within that band, while a good Herman–Wallis fit needs substantially more lines covering both the P and R branches; (2) the overall accuracy and reliability of the fit has been improved by including more measurements from different studies and different bands with appropriate weightings; (3) better accounting for the rotational dependence is obtained than when only rotationless matrix elements are used.

The details of the direct-fit method have been explained in our previous study of the DMF of the HCl molecule (Li et al. 2011). A brief explanation is given as follows. First, the rovibrational matrix elements, $M_{v^{\prime \prime},J^{\prime \prime} }^{v^{\prime},J^{\prime} } {\rm }$, are retrieved from the best available experimental measurements of rovibrational line intensities, using the equations in Šimečková et al. (2006). Here, double and single primes correspond to the lower and upper states, respectively. Second, the DMF (μ(r)) is usually approximated as a power series expansion (Bernath 2008)

$\begin{equation} \mu \left(r \right) = \mathop \sum \limits_{i = 0}^n M_i x^i, \end{equation} \tag{ 1 }$

where M_i are the coefficients of the polynomial expansion and x = (r − r_e)/r_e denotes the dimensionless Cartesian displacement with r_e as the equilibrium internuclear distance. Using this expansion, $M_{v^{\prime \prime},J^{\prime \prime} }^{v^{\prime},J^{\prime} }$ can be rewritten as

$\begin{equation} M_{v'',J''}^{v',J'} = \langle v'J'{\rm |}\mu \left(r \right){\rm |}v''J''\rangle = \mathop \sum \limits_{i = 0}^n M_i \langle v'J'{\rm |}x^i {\rm |}v''J''{\rm }\rangle. \end{equation} \tag{ 2 }$

At this stage, the determination of the M_i coefficients becomes a problem of solving the over-determined systems of equations, with the expectation values, 〈v'J'|xⁱ|v''J〉, being calculated using the potential energy function. A weighted least-squares fit computer code by LeRoy (2004) has been accordingly modified to determine the M_i coefficients.

In the present study, the best spectroscopically determined potential energy functions of CO of Coxon & Hajigeorgiou (2004) have been employed to calculate the expectation values. We have also identified the best available line intensity measurements in the literature. In general, we used the measurements from Meerts et al. (1977) for the permanent dipole, Birk et al. (1996) for the pure rotational band, Zou & Varanasi (2002) for the fundamental band, Malathy Devi et al. (2012b) for the 2–0 band, Jacquemart et al. (2001) for the 3–0 band, our CRDS experiments in the 4–0 band described in Section 2, Chung et al. (2005) for the 5–0 band, and Y. Tan et al. (2014, in preparation) for the 6–0 band (also briefly described here in Section 2). A few individual rovibrational dipole moments for the 3–0 and 4–1 bands derived from intensities that were measured with lower uncertainty have also been added from the results of Wójtewicz et al. (2013). For the fundamental band, the results from Varanasi & Sarangi (1975) were not chosen because they have fitted the line intensities with a Voigt line-shape profile which could not resolve the line mixing properly. For the 3–0 band, there were two other different measurements that were carried out at a parallel time by Sung & Varanasi (2004a) and Chackerian et al. (2001). Interestingly, recent accurate measurements of high-J lines by Wójtewicz et al. (2013) could be fitted very well together with Jacquemart et al. (2001), while it was not in accord with the Sung & Varanasi (2004b) and Chackerian et al. (2001) data. Therefore the latter two sources were not included in the fit, although the differences were only of the order of 2%. A sequence of signs of dipole moment matrix elements (−+−+−−+) for the 0–0 through 6–0 bands was adopted to imply the C⁻O⁺ polarity as used in previous studies (Chackerian & Tipping 1983; Chackerian et al. 1984). It is interesting to note that the dipole moment matrix element flips its sign from R(16) − to R(17) + transitions of the 5–0 band as shown in Figure 2.

Zoom In Zoom Out Reset image size

Ultimately, a grand fit with appropriate weightings (taking into account experimental uncertainties) has been performed, and the M_i coefficients determined from the present study and the other studies are listed in Table 4. For the reader's reference, the input rovibrational matrix elements and the corresponding expectation values can be found in Table S1. First of all, we can see from Table 4 that two extra coefficients, M₅ and M₆, have been determined owing to the availability of intensity measurements of the 5–0 and 6–0 bands. Second, the M₀, M₁, M₂, and M₃ coefficients from the present study agree well with the ones from Chackerian & Tipping (1983). Nevertheless, the present power series expansion of the DMF is the most accurate, up-to-date one and should be superior to the ones determined by Chackerian & Tipping (1983) as a consequence of the new accurate experimental data. In addition, the present study should serve well as a reference for ab initio studies near equilibrium internuclear distance. Finally, we can see that the M₂, M₃, and M₄ coefficients from Kiriyama & Rao (2000) do not agree with either Chackerian & Tipping (1983) or the present study, especially for the M₂ coefficient, which is approximately an order of magnitude smaller than the other two studies. The likely reason for this is that Kiriyama & Rao (2000) failed to identify the best experimental measurements of line intensities and used only pre-1980s experimental data in their fitting.

Experimental measurements for the intensities of high overtone bands are difficult due to the fact that these transitions are usually very weak. Ab initio calculation can provide a global DMF as a complement to the semi-empirical DMF. Therefore, it is desirable to compare and combine the two methods, i.e., the semi-empirical method described in Section 3.1 with the ab initio method, in order to cover the entire range of meaningful internuclear distances. In the present study, the best DMF of CO is presented as a piecewise function consisting of two parts: μ₁ equals the semi-empirical calculation near equilibrium internuclear distance covering up to v = 6, while μ₂ equals the ab initio DMF up to r = 4 a₀, where a₀ is the Bohr radius.

Langhoff & Bauschlicher (1995) have studied the DMF for the ground state of CO as a function of one and n-particle treatment. Their best calculated DMF using the finite-field approach has been found to be far superior to the ones calculated as expectation values. For evaluation purpose, we have employed their theoretical DMF to calculate the rovibrational line intensities of CO. The work has been carried out using the LEVEL program (LeRoy 2007) by combining the DMF with a PEF from Coxon & Hajigeorgiou (2004). A cubic spline function has been used to interpolate the dipole moment points from LB95. It has been found that the calculated line intensities for the 0–0 to 3–0 bands are in excellent agreement with the ones derived from the semi-empirical DMF of the present study (differences ⩽ 2%). However, unexpected large discrepancies have been found for the 4–0 and 5–0 bands, by approximately 20% and 80%, respectively, which are much larger than the reported experimental uncertainties used for fitting. Because of the excellent agreement for the 0–0 through 3–0 bands, we were perplexed about the discrepancies for the 4–0 and 5–0 bands until we re-calculated the DMF using the same approach as LB95 but with a finer grid (with a bond length interval of 0.02 a₀ instead of 0.1 a₀). When the new DMF was used to calculate the line intensities, the results for the 4–0 and 5–0 bands were in much better agreement with our semi-empirical values. The differences between the calculated and semi-empirical values at the band origin are now 2% and 10% for the 4–0 and 5–0 bands, respectively. Meanwhile, the new calculated line intensities for the 0–0 to 3–0 bands are almost identical to the previous calculated ones using the DMF of LB95. This gives us a hint that the interpolation method may not be sufficiently accurate at certain internuclear distances where the DMF behaves irregularly. Prior study also suggests that the intensity calculation is sensitive to the small change of the DMF at certain bond lengths (Li et al. 2013b). Therefore, we conclude that a finer grid of DMF should be a safer choice for quantitative studies in order to reproduce accurate line intensities.

LB95 carried out a calibration study to determine the sensitivity of the dipole moment to improvements in both the one- and n-particle basis sets. To do this, they selected two internuclear distances: 2.1 a₀, which is near r_e, and 3.7 a₀, which is near the maximum of the DMF. Although their best dipole moment agrees within 0.001 Debye with the one deduced from the semi-empirical DMF at 2.1 a₀, the difference at 3.7 a₀ is as large as 0.04 Debye. In addition, due to the limitation of computer power, they were not able to finish a number of more sophisticated calculations, leaving some gaps in their calibration table. In this work, we have extended their calibration study with more reference points at different internuclear distances and extended the size of the CASSCF/ACPF calculations further to (200/533) and (200/644) active space with the aug-cc-pVXZ basis set, where X = T, Q, 5 (abbreviated as AVXZ hereafter). For the new calculation, we have selected points at r = 2.1, 2.23, 2.30, 2.36, 2.41, 2.46, 3.70 a₀, which are approximately r_e and the classic turning points for v = 0–4 and v = 40, respectively. Our newly calculated dipole moments at various internuclear distances using the finite-field approach, together with the corresponding values from LB95 are listed in Table 5. We can see from the table, first of all, that the results at 2.1 and 3.7 a₀ from LB95 are almost identical to our newly calculated values. Second, poorer agreement has been found between experiment and calculation when the active space has been increased from (300/322) to (200/644). The difference between the values calculated using active space (300/322) and (200/533) is very small (<0.00005 atomic units (A.U.)). It has been found that the line intensities calculated using (300/322) agree better with experiments than the ones using (200/533) active space. Finally, systematic improvement has been seen when increasing the basis set from QZ to 5Z, with a difference in dipole moment between 0.0001 and 0.004 A.U., as shown in Figure 3. Considering the above three findings, the DMF has finally been calculated using the active space (300/322) and a basis set at 5Z level.

Zoom In Zoom Out Reset image size

For the finite-field approach, a small artificial perturbation is introduced by adding an external electric field. To investigate the effect of the field strength, we have calculated the dipole moment at three different field strength values of ±0.0001, ±0.003, ±0.005 A.U. at the internuclear distance r_e = 2.13222 a₀ using the ACPF(300/322) method plus a 5Z basis set. The calculated dipole moments are −0.04794, −0.04804, and −0.04812 A.U. at the three field strengths, respectively. Therefore, for the CO molecule, the choice of field strength does not affect the results very much (Δ ⩽ 0.0001 A.U.). In the present study, a field of 0.005 A.U. has been chosen for the rest of the calculations.

Another useful measure of the quality of the calculation is the rotationless matrix elements, R_v-0(0), which are proportional to the square root of the line intensities. LB95 have compared their ab initio R_v-0(0), where v = 0, 1, 2, 3, and 4, with the semi-empirical ones reported by Chackerian & Tipping (1983). They have found good agreement between the two studies, except for R_4–0(0) where the difference is approximately 12%. In Table 6, we have carried out a similar comparison by adding the R_v-0(0) values calculated using both the semi-empirical and ab initio DMFs of the present study. As can be seen from this table, the theoretical R_1–0 and R_2–0 of LB95 agree better with the present semi-empirical study than the previous semi-empirical study of Chackerian & Tipping (1983). This suggests that for the 1–0 and 2–0 bands, the theoretical DMF of LB95 is actually of better quality than the semiempirical DMF of Chackerian & Tipping (1983).

For R_0–0(0), R_1–0(0), R_2–0(0), and R_3–0(0), the values calculated using the present ab initio DMF are almost identical to those from LB95. However, for R_4–0(0), R_5–0(0), and R_6–0(0), the values from the present ab initio DMF are in much better agreement with the present semi-empirical values than those from LB95. With that, we note that the extreme disagreement between LB95 and our fitted and ab initio values for R_6–0(0) (and to a lesser extent R_5–0(0)) is not only an indication that ab initio calculation is better but largely due to the fact that one needs to calculate DMF points at a finer grid, especially at internuclear distances corresponding to higher overtones. The R_6–0(0) value that we calculated from LB95 had proven to be extremely sensitive to the interpolation methods used and varied by orders of magnitude. We picked this particular interpolation to present here as it yielded the best results everywhere except for the R_6–0(0) value.

Chackerian et al. (1994) raised their concern about the quality of the semi-empirical DMF of Chackerian & Tipping (1983) and the ab initio DMF of LB95 because of the large differences between the intensities calculated using these two DMFs. With the availability of the new CRDS measurement of the 6–0 band (Y. Tan et al. 2014, in preparation, see Table 3), we decided to test both ab initio DMFs by comparing the calculated line intensities using these two DMFs. In Table 3, the line intensities calculated using the present ab initio DMF are approximately 30%–38% higher than measurements of Y. Tan et al. (2014, in preparation), those calculated using LB95 by about three orders of magnitude, while the one calculated from the DMF of Chackerian & Tipping (1983) overestimated the line intensities by about fivefold (the latter not shown). It can thus be concluded that the DMF of Chackerian & Tipping (1983) is less accurate at larger internuclear distance.

We have compiled a complete database of rovibrational transitions of nine isotopologues of carbon monoxide including radioactive ones. The extent of the database is given in Table 8. We originally calculated substantially larger data sets with transitions with Δv up to 41. However, using validation tools developed by Medvedev (2012), it was found that the intensities calculated for lines with Δv ⩾ 12 are unreliable. At the moment, it is unclear if this is due to any unforeseen problem with ab initio calculation at these internuclear distances or the numerical approach implemented by LEVEL. This will have to be determined in a separate study, probing several molecular systems. Nevertheless, the database with Δv ⩽ 11 is complete for any practical application. The intensities go down to 10⁻²⁰⁰ cm molecule⁻¹ to accommodate even the most challenging high-temperature applications. For terrestrial and planetary atmosphere applications, it is informative to compare the calculated line positions and line intensities of the present study with experimental measurements as well as the ones in the HITRAN and HITEMP databases. Table 8 compares the present CO line lists with the HITRAN and HITEMP databases, and the Goorvitch data sets. We can see that the present study provides more comprehensive v, J, and Δv coverage, plus data for three additional radioactive isotopologues. To visualize the difference, the present line list (black square), HITEMP2010 (blue circle), and HITRAN2012 (red triangle) are plotted in Figure 4. The lower panel of this figure covers a spectral range of 0–70000 cm⁻¹, with the line intensities plotted on logarithm scale. The upper panel is a blow up that covers all the Δv = 0–6 bands.

Zoom In Zoom Out Reset image size

The current version of the HITRAN database for CO consists of 4606 lines belonging to six isotopologues covering a spectral range of 3.46–8464.88 cm⁻¹. In general, six vibrational bands 1–0, 2–0, 2–1, 3–0, 3–1, and 4–1 are included for the first six isotopologues; two pure rotational bands 0–0 and 1–1 are included for the first five isotopologues; and the 2–2, 3–3, and 4–0 bands are included only for the principal isotopologue. The line positions of the current HITRAN CO line list are largely based on the Dunham coefficients of Farrenq et al. (1991), but with the 0–0 band of ¹²C¹⁶O, ¹³C¹⁶O, and ¹²C¹⁸O, and the 1–1, 2–2, and 3–3 bands of ¹²C¹⁶O supplemented with the values from the CDMS database (Müller et al. 2005) and the 0–0 band of ¹²C¹⁷O and ¹³C¹⁸O, and the 1–1 band of ¹³C¹⁶O, ¹²C¹⁸O, ¹²C¹⁷O, and ¹³C¹⁸O supplemented with data from Varberg & Evenson (1992). Figure 5 shows the comparison of line positions of ¹²C¹⁶O between the HITRAN 2012 database and the present study which uses the PEF of Coxon & Hajigeorgiou (2004). For the pure rotational bands, 0–0, 1–1, 2–2, and 3–3, excellent agreement has been found between the two data sets for lines with J'' < 50. However, for the 0–0 band with J'' > 50, the HITRAN database starts to quickly diverge from the present study. For the 1–0 to 4–0 vibrational bands, the HITRAN database agrees very well with the present study near band origins but diverges quadratically for high-J lines, reaching differences of 0.0011, 0.0024, 0.0039, and 0.0063 cm⁻¹ at J'' = 49, respectively, for the four vibrational bands. Similar comparison is also shown for the 2–1, 3–1, and 4–1 hot bands in Figure 5, with line position differences of 0.0013, 0.0028, 0.0042 cm⁻¹ at J'' = 49, respectively. In general, the line positions from the present study are superior to the HITRAN database through the inclusion of new high-precision data for various CO isotopologues that were not contained in Farrenq et al. (1991) and through better predictability of the parameterized potential energy function for high-J lines.

Zoom In Zoom Out Reset image size

However, a possible minor problem may still exist in the available line position data. As one of the illustrations, one can take a look at the comparisons of different data sets and measurements in the 4–0 and 5–1 bands carried out in this work. It can be seen that for the 4–0 band, the line positions in HITRAN agree well with the CRDS measurements. However, the agreement for calculated values using the potential energy function of Coxon & Hajigeorgiou (2004) is slightly worse, and moreover exhibits a definite rotational dependence reaching a maximum difference of 0.0034 cm⁻¹ for the P(34) line. In 2010, Tashkun et al. (2010) carried out a Ritz analysis of ¹²C¹⁶O rovibrational energy levels that were involved in multiple (and often multiply) measured transitions. We generated transition frequencies from these energy levels and it appears that they also agree better with experimental ones than our values calculated based on the PEF of Coxon & Hajigeorgiou (2004). Nevertheless, the differences are marginal even at high-J values. For the 5–1 band, our line positions and those generated from the energy levels by Tashkun et al. (2010) are similar to each other and agree better with experimental values more than those for the 4–0 band. More high-precision experiments are desirable to improve existing line position analyses. On the other hand, the current line positions from any of the existing sources are of acceptable quality for most present-day applications.

The line intensity of the current HITRAN line list for CO was originally derived from the DMF of Chackerian & Tipping (1983), and was later replaced with the Goorvitch data sets (Goorvitch 1994; Goorvitch & Chackerian 1994a, 1994b). Recent (post 2000) updates to HITRAN for line intensity include the replacements for the pure rotational bands, 0–0, 1–1, 2–2, and 3–3, with the CDMS database (Müller et al. 2005), the 2–0 band with Malathy Devi et al. (2012a, 2012b), and the 3–0 band with Sung & Varanasi (2004a). Figure 6 shows the comparison of line intensities for the 0–0 to 6–0 bands of ¹²C¹⁶O between the experimental measurements, the HITRAN 2012 database, the ones calculated using the present ab initio DMF, and the ones calculated from the present semi-empirical DMF.

Zoom In Zoom Out Reset image size

According to this figure, for the 0–0 band, the present semi-empirical intensities (referred to as PS(fitted)) agree well with the four sets of measurements of Birk et al. (1996). Similar agreement has also been found between HITRAN and Birk et al. (1996), which is expected since the same data from Birk et al. (1996) were adopted in HITRAN. The line intensities for the 0–0 band derived from the ab initio DMFs of both the present study (referred to as PS(ab initio)) and LB95 differ from the semi-empirical values by approximately −1% and +3%, respectively.

For the 1–0 band, the PS(fitted) intensities agree well with the measurements of Zou & Varanasi (2002) within the reported uncertainty except for a few high-J, R-branch lines. In addition, the PS (ab initio) intensities, the LB95 intensities, and the HITRAN intensities differ systematically from the PS(fitted) intensities by approximately 0.7%, 0.4%, and −2.1%, respectively. The difference of 2.1% for the 1–0 band between the PS (fitted) and HITRAN2012 essentially arises from different DMFs employed for calculating the line intensities. The semi-empirical DMF used to generate the Goorvitch data set was based on the old measurements of Varanasi & Sarangi (1975) for the 1–0 band, while the present study adopted the more recent measurements of Zou & Varanasi (2002). Zou & Varanasi (2002) compared their measurements with HITRAN2000 (Rothman et al. 2003) and plotted the results in Figure 3 in their paper. However, we are not sure why they have found excellent agreement (<1% difference) between the two, while our new comparison clearly shows an average difference of approximately 2% between their experiment and HITRAN.

For the 2–0 band, the PS (fitted) intensities reproduce very well the accurate measurements of Malathy Devi et al. (2012b) within approximately 0.1% from the P(15) to R(25) lines, but the difference becomes larger for high-J lines, reaching 0.5% and −0.2% for the P(30) and R(30) lines, respectively. Although the difference is still within the experimental uncertainty for R-branch lines, it is slightly beyond the uncertainty of high-J, P-branch lines. In addition, the quadratic J-dependence of high-J lines shown in Figure 6 might be introduced due to the multi-spectrum fit approach used by Malathy Devi et al. (2012b), which fits intensity information to the Herman–Wallis coefficients rather than individual intensities, or the limitation of the power series formulism of the DMF when an accuracy of 0.2% is required. Both ab initio intensities (PS and LB95) agree remarkably well with the semi-empirical intensities (PS (fitted)) within 0.6%.

For the 3–0 band, the PS (fitted) intensities reproduce the precise CRDS measurements of the P(29) and P(28) transitions of Wójtewicz et al. (2013) within the reported uncertainties, whereas it appears that for the measurements of Jacquemart et al. (2001) the experimental uncertainties are slightly underestimated. For comparison purposes, the measurements of Sung & Varanasi (2004a) are also plotted, although these data were not included in the fit. The measurements of Sung & Varanasi (2004a) are approximately 1% higher than HITRAN, which is odd considering the latter should be derived from the measurements of Sung & Varanasi (2004a) according to the HITRAN2004 paper (Rothman et al. 2005). Furthermore, the two ab initio intensities (PS(ab initio) and LB95) are approximately 1.2% and 1.5% higher than the present semi-empirical values (PS (fitted)), respectively.

For the 4–0 band, our new CRDS measurements described in Section 2.1 and Table 2 were used in the fit. One can see that these experimental values have excellent precision based on how well they agree with the fitted values. Although one can argue that this is not surprising since these are the values used in the fit, most of the other experimental measurements used in the fit in other bands do not exhibit the same precision. The difference between the PS(fitted) intensities and the measurements of Chackerian & Valero (1976) is larger than the reported uncertainties. The HITRAN line intensities are about 7% lower than the PS(fitted) intensities at the band origin with a different J-dependence. The PS(ab initio) intensities are approximately 2% higher than the PS(fitted) intensities at the band origin. In comparison, the other ab initio intensities (LB95) are 18% higher than the present study with a very different J-dependence. In Table 2, one can also see the comparison of measured, semi-empirical and ab initio intensities of two weak lines in the 5–1 hot band and observe a fairly good agreement (considering weakness of these lines). This comparison is quite informative as the 5–1 intensities were not used in the fit and therefore it gives a good confirmation of the predictive abilities of our method.

Comparison of the measured intensities of the lines from three less abundant isotopologues provides rather interesting food for thought. It is clear from Table 2 that the agreement of experimental values to the ones calculated in this work is different for different isotopologues. Intensities agree very well for ¹²C¹⁸O, while the experimental values are noticeably lower for ¹³C¹⁶O and higher for ¹²C¹⁷O. There are three reasons that could contribute to these discrepancies. First, HITRAN abundances are somewhat arbitrarily chosen numbers and may be different from the abundances of minor isotopologues in the experimental sample. Second, the minor isotopologue lines are weaker and therefore the experimental uncertainty is higher. Finally, there is a probable break down of the Born–Oppenheimer approximation which results in deviations when using the dipole moment function derived from ¹²C¹⁶O experiments or ab initio calculations that were carried out on the principal isotopologue. We compared isotopologue intensities measured in the second overtone by Wójtewicz et al. (2013) and observed a very similar behavior that the ¹²C¹⁸O intensities agreed very well with our values, but the experimental values are lower for ¹³C¹⁶O and higher for ¹²C¹⁷O, although the discrepancies are more subtle than in the 4–0 band. Nevertheless, our new values for these three minor isotopologues agree with the experimental ones much better than HITRAN2012. To make things more confusing, comparison of our results with the 2–0 band results of Malathy Devi et al. (2012a) show excellent agreement for ¹³C¹⁶O and a systematic 2% offset for ¹²C¹⁸O. More measurements for minor isotopologues in different bands are needed to construct fitted dipole moment functions for individual isotopologues. Ab initio calculations carried out directly for each isotopologue are needed as well.

For the 5–0 band, the line intensity measurements of Chung et al. (2005) are rather coarse since high pressure had to be used to observe weak lines of that band. Although the authors did not provide any estimation of the experimental uncertainty, we estimated it as 25% to 44% based on rotational scatter, according to the DMF fitting. The PS (ab initio) intensities are approximately 10% higher than the PS(fitted) intensities, while the other ab initio intensities (LB95) are 80% lower.

For the 6–0 band, the recent accurate CRDS measurements (Y. Tan et al. 2014, in preparation) reproduced well by the present study within the estimated uncertainty of 1.5%, while the PS (ab initio) intensities are approximately 26%–37% higher than the PS (fitted) intensities for the P(15) to R(15) lines. The other ab initio intensities (LB95) are approximately three orders of magnitude larger than the semi-empirical values and are not shown in Figure 6. We note, however, that at the corresponding internuclear separation, the calculation of intensities strongly depends on the interpolation used for LB95 points and the values really vary orders of magnitude. The bottom line is that LB95 lacks the density of points calculated at this part of DMF.

In general, the present semi-empirical line intensities (PS(fitted)) agree well with the best available experimental values. An overview is shown in Figure 7 using a three-dimensional plot to compare all the eleven bands of ¹²C¹⁶O in the HITRAN2012 database with the semi-empirical values of the present study. One can easily observe the vibration and rotational dependencies of the intensity ratios on this plot. The Δv values are color coded while v'' values are given on the x axis. For instance, the green circles (Δv = 2) along v'' = 1 value represent the 3–1 band. The projection of each plot is also shown on the xz plane to aid the reader. It can be seen from this figure that for the 2–0 band, the present semi-empirical line intensities agree well with HITRAN2012 for J < 30, but diverge quickly from HITRAN2012 after J > 30, reaching approximately 4% at J'' = 70. It shows that for the high-J lines, the PS(fitted) intensities are different from the ones predicted using a polynomial extrapolation (the Herman–Wallis fit) of the experimental measurements of Malathy Devi et al. (2012b). Similarly for the 3–0 band, the PS(fitted) intensities diverge from HITRAN2012 after J > 15, with the biggest difference of 16% at J'' = 60. It has been known for a long time that the Herman–Wallis polynomial extrapolations diverge after the highest observed lines. In comparison, the DMF should provide better predictability for unmeasured high-J lines by taking into account data from other vibrational bands, and therefore accessing a broader range of internuclear distances.

Zoom In Zoom Out Reset image size

Moreover, for the 0–0 band, HITRAN2012 agrees very well with the present semi-empirical values, but starts to diverge slowly after J = 70. For the 1–1 band, a discontinuity of line intensity is observed at J = 30 for HITRAN2012. This is because the line intensities in HITRAN for J < 30 lines have been replaced with the CDMS data (Müller et al. 2005), while for J > 30 lines, previous values calculated using an old DMF have been retained. Relatively large discrepancies of approximately 7% have been found for the 3–3 pure rotational band, while a large difference of 10%–16% is shown for the 4–1 hot band.

The current HITEMP database for the CO molecule includes 113,631 transitions belonging to six isotopologues, ¹²C¹⁶O, ¹²C¹⁷O, ¹²C¹⁸O, ¹³C¹⁶O, ¹³C¹⁷O, and ¹³C¹⁸O, and covers the same spectral region as HITRAN, 3.46–8464.88 cm⁻¹, but with v_MAX = 23, J_MAX = 149, and Δv ⩽ 4. The HITEMP 2010 database of CO uses data from Goorvitch (1994) and Goorvitch & Chackerian (1994a, 1994b) for intensity, with the transition wavenumbers calculated from an RKR potential using the Dunham coefficients of Farrenq et al. (1991). The line positions in the HITEMP database are the same as in HITRAN but with an extended rovibrational ensemble. Figure 8 shows the comparison of line positions between the present study (based on the PEF of Coxon & Hajigeorgiou 2004) and the HITEMP database in a 3D plot. In general, good agreement has been found for low v, J lines, but starts to diverge quickly for high v, J lines, with maximum differences approximately 2, 4, 6 cm⁻¹ for Δv = 1, 2, 3 bands at J = 149, respectively. The line positions of the rotational bands, 0–0, 1–1, 2–2, and 3–3, which are marked in red, are the same as the ones in the HITRAN database, which is based on the CDMS data (refer to Section 1 for details).

Zoom In Zoom Out Reset image size

Figure 9 shows the comparison of line intensities between the HITEMP database and the present semi-empirical study for J ⩽ 149, Δv ⩽ 4, v ⩽ 4 bands. For Δv = 0 bands (red), 0–0, 1–1, 2–2, and 3–3, the line intensities in HITEMP2010 are identical to HITRAN2012. Therefore the comparison with the present study is the same as shown in Figure 7. For Δv = 1 bands (yellow), the present semi-empirical study agrees very well with HITEMP2010, even for high-J lines. For Δv = 2 bands (green), the difference between the two data sets increases for high-J lines, reaching a maximum difference of 20% at J = 149 for the 2–0 band. For Δv = 3 bands (blue), a large difference has been found between HITEMP2010 and the present semi-empirical study, especially for high-J lines. For example, the line intensities of the 4–1 band from HITEMP2010 are higher than the present semi-empirical study by 11% and 70% at the band origin and J = 149, respectively. However, the HITEMP2010 line intensities of low-J lines of the 3–0 band agree reasonably well with the present study because they are based on the measurements of Sung & Varanasi (2004a).

Zoom In Zoom Out Reset image size

In this section, we will discuss the assembly of the complete set of air-, CO-, H₂-, and CO₂-broadened parameters for all the CO lines presented here. The line-shape parameters of the CO lines have been studied in dozens of publications where authors probe different spectral regions, different broadeners and different line-shape profiles. It is important to note that all of the parameters derived here assume Voigt line shape. If one seeks accuracies of the retrievals to be below the one to two percent level, this line shape is not adequate. In fact, in HITRAN2012, we have already employed the speed-dependent Voigt profile for the first overtone band. Nevertheless, we believe that for the majority of applications, the Voigt line shape will be satisfactory.

6.1.1. Air-broadened Widths

6.1.2. Self-broadened Widths

6.1.3. CO₂-induced Widths

6.1.4. H₂-induced Widths

There are quite a few measurements of the H₂-induced broadening and shift parameters available in the literature. It is interesting to compare three studies that have carried out measurements in the 2–0 band with a wide coverage of the rotational lines (Malathy Devi et al. 2004; Régalia-Jarlot et al. 2005; Sung & Varanasi 2004b). The H₂-induced broadening parameters for the 2–0 band from these three studies have been depicted in Figure 10. The results of Sung & Varanasi (2004b) and Malathy Devi et al. (2004) are in good agreement with each other, unlike the results from Régalia-Jarlot et al. (2005). Finally, the data from Malathy Devi et al. (2004) have been chosen because of the smaller uncertainties of their measurements. For the high-J values whose H₂-induced widths are not measured, the available measured value for the highest-J line (J = 24) was adopted. These values were used for all other bands, neglecting vibrational dependence. This assumption seems reasonable based on the finding of Sung & Varanasi (2004b) who measured broadening in the fundamental and first and second overtone bands and found the widths to be very similar. Also, if one compares the values from the first overtone with ones measured or calculated for the pure rotational band (Dick et al. 2009; Faure et al. 2013), excellent agreement is found. All broadening values are well within 5%, whereas temperature-dependent coefficients are all within 10%. We believe this is more than satisfactory for planetary applications.

Zoom In Zoom Out Reset image size

6.2.1. Methodology for Calculating the Semi-empirical Line Shifts

Hartmann & Boulet (2000, hereafter HB00) have successfully adopted the classic path method proposed by Neilsen & Gordon (1973) to calculate the width and shifts for HF molecules in an argon bath. Their results are in satisfactory agreement with both recent experimental measurements and the fully quantum-mechanical calculations. Most conveniently, in the Appendix, they have given a pertinent formalism that simplifies the line shift calculation. According to their theory, the line shift consists of an isotropic part and an anisotropic part which are introduced from the isotropic and anisotropic parts of the potential, respectively. The isotropic part can be further separated into symmetric and antisymmetric parts, as described below:

$\begin{equation} \delta (R(J)) = \delta \left(R_{{\rm iso}}^a (J)\right) + \delta \left(R_{{\rm iso}}^s (J)\right) + \delta (R_{{\rm aniso}} (J)) \end{equation} \tag{ 3a }$

$\begin{equation} \delta (P(J + 1)) = \delta \left(P_{{\rm iso}}^a (J + 1)\right) + \delta \left(P_{{\rm iso}}^s (J + 1)\right) + \delta (P_{{\rm aniso}} (J + 1)) \end{equation} \tag{ 3b }$

with the relation:

$\begin{equation} \delta \left(R_{{\rm iso}}^a (J)\right) = - \delta \left(P_{{\rm iso}}^s (J + 1)\right) \end{equation} \tag{ 4a }$

$\begin{equation} \delta \left(R_{{\rm iso}}^s (J)\right) = \delta \left(P_{{\rm iso}}^s (J + 1)\right) \end{equation} \tag{ 4b }$

$\begin{equation} \delta (R_{{\rm aniso}} (J)) = - \delta (P_{{\rm aniso}} (J + 1)), \end{equation} \tag{ 4c }$

where δ(R(J)) and δ(P(J+1)) denote the line shifts for two corresponding lines R(J) and P(J+1), respectively.

Combining Equations (3) and (4), we can obtain the following relations:

$\begin{equation} \delta \left(R_{{\rm iso}}^a (J)\right) = \delta \left(P_{{\rm iso}}^s (J + 1)\right) = 0.5 \times (\delta (R(J) + \delta (P(J + 1)) \end{equation} \tag{ 5a }$

$\begin{eqnarray} \delta \left(R_{{\rm iso}}^a (J)\right) + \delta (R_{{\rm aniso}} (J)) &=& {-} \left(\delta \left(P_{{\rm iso}}^a (J + 1)\right) + \delta (P_{{\rm aniso}} (J + 1))\right) \nonumber \\ &=& 0.5 \times (\delta (R(J) - \delta (P(J + 1)). \end{eqnarray} \tag{ 5b }$

It has been proven in HB00 that the symmetric part (Equation (5a)) is proportional to the difference (v'−v''), while the asymmetric part (Equation (5b)) is identical for all vibrational bands. Therefore if we choose the measurements from the first overtone (2–0) band as reference data, we could generalize the equation for both P and R branches of other rovibrational bands as follows:

$\begin{eqnarray} \delta _{v' - v''} (m) &=& \left[ {(v' - v'')/(v'_{ref} - v'' _ {ref})} \right] \nonumber \\ &&\times \delta _{(2 - 0){\rm symmetric}} (m) + \delta _{(2 - 0){\rm asymmetric}} (m), \end{eqnarray} \tag{ 6 }$

where $\delta _{v^{\prime} - v^{\prime \prime} }$ represents the line shift for the (v' − v'') band; $(v^{\prime}_{ref} - v^{\prime \prime} _{ref})$ is (2–0) = 2 in this case, δ_{(2 − 0)symmetric} (m) = 0.5 × (δ_{2 − 0}(m) + δ_{2 − 0}(− m)); δ_{(2 − 0)symmetric} (m) = 0.5 × (δ_{2 − 0}(m) − δ_{2 − 0}(− m)); and m = −J for P branch and J + 1 for R branch transitions. Equation (6) has been adopted in our calculation of line-shift parameters for the bands where no experimental measurements of shifts are available.

6.2.2. Air-induced Shifts

The air-induced line shifts have been measured experimentally for the 1–0, 2–0, and 3–0 bands (Zou & Varanasi 2002; Malathy Devi et al. 2012b; Sung & Varanasi 2005). Among these works, the measurements of Malathy Devi et al. (2012b) for the 2–0 band have achieved an unprecedented accuracy (uncertainty <1%) using a multispectral fit procedure. To test the theoretical treatment of HB00, we took the accurate measurement of δ_air for the 2–0 overtone band to predict the values of δ for the 1–0 and 3–0 bands using Equation (6). The calculated results were then compared with the measurements of Zou & Varanasi (2002) and Sung & Varanasi (2005) for the 1–0 and 3–0 bands, respectively. Figure 11 shows the comparison between calculated and experimental δ_air for the 1–0 band. In this figure, the calculated values show a very similar pattern to the experimental ones. However, the calculated values are systematically higher than the measured values by approximately 0.0007 cm⁻¹ atm⁻¹ for most of the transitions. Zou & Varanasi (2002) have also calculated the air-induced shifts using the O₂- and N₂- induced shifts measured by Bouanich et al. (1996) assuming δ_air = 0.79δ_N2 + 0.21δ_O2. Their calculated results are systematically higher than their measurements by approximately 0.001 cm⁻¹ atm⁻¹. Since Zou & Varanasi (2002) have also reported the δ_air values for the 2–0 band, we compared their results with the accurate measurements of Malathy Devi et al. (2012b). It was then found that the results of Zou & Varanasi (2002) are systematically lower than those of Malathy Devi et al. (2012b) by approximately 0.0003 cm⁻¹ atm⁻¹. Another test was performed by calculating δ_air for the 1–0 band using the δ_air measurements of Zou & Varanasi (2002). The calculated results shown in Figure 11 are very similar to the values calculated using the data of Malathy Devi et al. (2012b). It would be interesting to see a new independent study of the 1–0 band to understand if the discrepancy is due to an oversimplified model or to issues in the actual experiment.

Zoom In Zoom Out Reset image size

Figure 12 shows the comparison between the calculated and experimental δ_air values of Sung & Varanasi (2005) for the 3–0 band. In this figure, the calculated values using Equation (6) agree reasonably well with the experimental values within experimental uncertainties, apart from the R(0) and R(1) lines where the experimental values are higher than the calculated values by approximately 0.0015 cm⁻¹ atm⁻¹. Moreover, the asymmetric features of δ_air between P and R branches are correctly interpreted by our calculation. It thus can be concluded that the HB00 theory should provide reasonable predictions for line-shift parameters for higher overtone bands where no experimental data are available.

Zoom In Zoom Out Reset image size

6.2.3. CO₂-induced Shifts

6.2.4. H₂-induced shifts