Theoretical prediction on the R-branch lines for the first overtone transitions in the ground electronic state of 12C16O

An analytical formula for the diatomic R-branch emission lines that was recently tested as a universal expression has been further modified based on the difference algebraic converging method. The tiny experimental line errors that may lead to amplified errors in the determination of high J lines were taken into account in the formula. Applications are presented for the R-branch emission spectra of (2-0), (3-1), (6-4), and (7-5) overtone bands of the ground electronic state of 12C16O. The rotational constants and band origins that are consistent with those reported previously are determined through the analytical formula for predicting frequency for emission lines up to J = 110. The results are shown to not only compare favorably with available lower J lines, but also generate reasonable higher J lines for the overtone bands, which agree with data from the HITRAN database and other works.An analytical formula for the diatomic R-branch emission lines that was recently tested as a universal expression has been further modified based on the difference algebraic converging method. The tiny experimental line errors that may lead to amplified errors in the determination of high J lines were taken into account in the formula. Applications are presented for the R-branch emission spectra of (2-0), (3-1), (6-4), and (7-5) overtone bands of the ground electronic state of 12C16O. The rotational constants and band origins that are consistent with those reported previously are determined through the analytical formula for predicting frequency for emission lines up to J = 110. The results are shown to not only compare favorably with available lower J lines, but also generate reasonable higher J lines for the overtone bands, which agree with data from the HITRAN database and other works.


I. INTRODUCTION
Accurate spectroscopic parameters for carbon monoxide (CO) are of great need in studies of astrochemistry, 1 planetary atmospheres, 2-7 atmospheric carbon cycle, 8,9 and thermodynamic properties. 10 There has been a growing interest in the first overtone transitions in the ground electronic state of CO. [11][12][13][14][15][16][17][18][19][20][21][22][23] The spectroscopic parameters of carbon monoxide, such as transition frequencies and rotational constants, are extensively studied with sufficient accuracy, where methods from both experiment and theory are well developed.  For example, in early 1965, Rank et al. published heated absorption tube measurements of rotational lines of the 2-0, 3-1, and 4-2 overtone bands of 12 C 16 O to an accuracy of ±0.0032 cm −1 when compared with the calculated data. 11 The overtone Δυ = 2 sequence bands are recorded up to υ ′ = 7 in the X 1 ∑ + -X 1 ∑ + transition of 12 C 16 O, in which the R-and P-branches were observed by Mantz and Maillard in 1974 with a high-resolution Fouriertransform Spectrometer (FTS). 12 Later in 1983, the heterodyne frequency measurements of the 2-0 band in X 1 ∑ + of CO were performed by Pollock et al., 13 which give accurate line positions for R-and P-branches and improved centrifugal distortion constants. In addition, extensive line lists for rovibrational transitions of the first overtone bands up to υ ′ = 22 have been recorded by Goorvitch 14 for astrophysical applications. Recently, the "hot" overtone 2-0 band was measured for rovibrational analyses over a wide spectral coverage by Zou, 15 Sung, 16 Mishra,17 Malathy Devi, 18 Hashemi, 19 and Esteki. 20 On the other hand, the HITRAN (HIgh-resolution TRANsmission) 21,22 and HITEMP (HIgh-TEMPerature) 23,24 molecular spectroscopic databases provide line lists of high-resolution spectral parameters for molecules of atmospheric science.

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Spectroscopic constants for different transitions in the observed states can be determined by fitting the rotational or rovibrational lines using the corresponding term energy expressions. Thus, it is believed that extrapolation of the experimental lines to the higher rotational quantum numbers in these cases can also be achieved using spectroscopic constants that have sufficient accuracy. Based on this assumption, previous work has been directed toward predicting the high-lying transitional emission spectral lines using the difference converging method (DCM) for diatomic molecules. [27][28][29][30][31] The results from these works yielded not only deeper insights into the delicate relationship between the rotational constants and the transitional lines, but also generated the formulae that can give high-lying transitional lines with precision. Despite these remarkable achievements, previous research was limited in two key ways: (a) In order to give accurate lines up to high J, power series expansions in J (J + 1) for rotational energy part of the Herzberg expansion were set to be unchangeable in the DCM treatment of our past work [27][28][29][30][31] and (b) the uncertainty of each experimental line is neglected, which may mainly affect the predictions of the weaker high J transitions.
The present study will introduce an algebraic technique in the DCM procedure to evaluate the contribution of the rotational constants {Bυ, Dυ, Hυ, . . .}, which are used to obtain high-lying transitional lines toward improved precision assisted by variational treatment during the calculation process. Our work will focus on several Δυ = 2 bands of the X 1 ∑ + ground electronic state of 12 C 16 O and spectroscopic information concerning this state could also be derived from these bands. This paper is organized as follows. The improved method will be described briefly in Sec. II. In Sec. III, the theoretical applications to some overtone bands in the ground electronic state of 12 C 16 O are given and discussed. Section IV summarizes this study.

A. Method for R-branch transition lines
For a given transition band of the CO system in the X 1 ∑ + state, which would be considered Hund's case (b), the R-branch transition lines and molecular constants can be expressed by using the wellknown formula, 32 is the origin of the υ ′ -υ ′′ band, and {B, D, H, L, . . .} are the rotational constants.

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Next, a further modification is possible to simplify Eq. (8) giving Equation (9) can be written in a matrix form as in which the expansion coefficient matrix δ, the molecular constants matrix χ, and the matrix k of the transition lines are given by Note that, in the matrix k in Eq. (11), the ν J l ′ could be any one of the m known experimental lines (νJ) m other than νJ l , l = 1, 2, 3. Thus, the exact experimental lines used in every algebraic procedure of Eq. (10) will be {imax + 3}(imax = even).
One may choose a subset of (imax + 3) transition lines that have sufficient spectral accuracy out of the m(>imax + 3) known experimental ones to solve Eq. (10). Then, is determined by the following physical converging requirements: Max(π J+1 11 π 1,J 1 ′ + 1 , . . . , π J+1 11 π 1,J imax ′ + 1 ) → sufficiently small, (14) where the total accuracy for a transition band is given by the root mean square (rms) error ΔERMS listed in Eq. (13), in which m stands for the number of known experimental R-branch lines for a given band. Equation (14) ensures that the computational errors in Eqs. (9) and (10) are as small as possible. Then, the converged molecular constants (ν 0 , B υ ′ , B υ ′′ , D υ ′ , D υ ′′ , . . . , χi max ) will be used to predict the unknown R-transition lines for a given band at higher J. The typical workflow can be seen in Fig. 1.
According to the derivation above, a multi-difference method [27][28][29][30][31] combined with an algebraic technique is found to  13) and (14) ensure that the calculated transition lines have sufficient spectral accuracy. Therefore, one may name this theoretical method as the difference algebraic converging method for transition lines (DACMt) and the converged transition lines as ν DACMt J,cal .

B. Variational treatment for the R-branch lines
It has to be noted that tiny experimental line errors have a non-negligible impact in predicting the high-lying R-branch lines. Consequently, the uncertainty given to each observation should be taken into consideration as part of the process, and this can be done by varying each observation in the form of displacement from its initial value νJ,expt by an increment δνJ, which has a numerical value within the uncertainty or error limit. The "true" experimental lines νJ shown in Eq. (11) then become νJ(=νJ,expt + δνJ). Of course, δνJ could be equal to zero in the limit of infinite accuracy.
For example, the "true" value of an experimental transition can be expressed as νJ = νJ,expt ± ΔνJ, in which ΔνJ is the error limit of the reported measurement νJ,expt. This corresponds to a spectral range of or a Mathematical expansion For each n, i can be 1 or −1. Then adjustments to each transition can be done in DACMt calculation, where the calculated variational transition lines are named ν VDACMt J,cal . In Ref. 12, the difference between the experimental and calculated R-branch lines is at the 10 −3 cm −1 or 10 −4 cm −1 level, which is in good agreement with the experimental uncertainty. Thus, in this work, we set ΔνJ = 0.01 cm −1 as a tentative value for every R-branch line used in the DACMt calculation. Then, the final value of each Δν ′ J is variationally determined by applying the converging standards in Eqs. (13) and (14).

III. APPLICATIONS AND DISCUSSIONS
The DACMt of calculating the high-lying R-branch transitions by solving Eq. (10) is applied to the (2-0), (3-1), , and  bands of the ground electronic state of CO. The DACMt calculated R-branch transition lines, which cover the range of 3935-4361 cm −1 with J values up to 110, are compared to the values from HITRAN, 21,22 DCM, 31 and available measurements. As can be seen from Table I, the rotational quantum numbers and the corresponding variational transition lines used in the DACMt process are listed for these overtone bands. The differences between our calculated band origins and experimental and theoretical values exhibit small absolute errors, and the rotational constants Bυ, Dυ, and Hυ (0 ≤ υ ≤ 7) are in good agreement with past works as can be seen in Tables II and III, respectively.   TABLE I. The rotational quantum numbers J k and the corresponding variational transition lines ν J k for calculation of (2-0), (3-1), , and (7-5) overtone bands of the X 1 Σ + → X 1 Σ + transition of CO (in cm −1 ).

Band
(J k , νJ k ) (2-0) 9 a (3-1) 9 (6-4) 9 (7-5) 9 (J        Table A1 in the Appendix of the supplementary material (SM) for more transition lines], for the 2-0 band, there are eight relevant sets of data from previous calculations and measurements for comparison. It can be seen that, for the (2-0) overtone band, the best R-branch transition lines ν VDACMt J,cal (9) that are given to 8 decimal places up to J = 110 could be calculated using nine experimental transition lines (in bold-type) from the work by Mantz and Maillard, 12 which has given satisfactory results with the error ΔERMS of 1.945 × 10 −3 cm −1 . We have compared our transition lines ν VDACMt J,cal (9) to the HITRAN database 21,22 and available measurements. For example, the benchmark values from Malathy Devi et al. 18 have R-branch lines up to J = 30 that are listed in the third column. These lines are included as they have the most precise line positions to 6 decimal places for comparison of the low J lines. The line positions recorded by the atmospheric chemistry experiment-Fourier transform spectrometer (ACE-FTS) 6 are collected in the fourth column. The transition lines from HITRAN08 21 recorded by atmospheric trace molecule spectrometer-Fourier transform spectrometer (ATMOS-FTS), 5 which were calculated by the PGOPHER program 33 that is a general program for simulation molecular spectra, are listed in the fifth column. These two versions of HITRAN lines are presented here for comparison for high J lines to validate the accuracy of the predicted transition lines beyond the observed ones. 12 As seen in Table IV, the transition lines ν VDACMt J,cal (9) are found to be consistent with those measured by Malathy Devi et al., 18 Mishra et al., 17 Mantz and Maillard, 12 Hashemi et al., 19 Pollock et al., 13 and Zou and Varanasi 15 for the low J lines. However, the calculated ν VDACMt J,cal (9) lines are slightly higher than the values from HITRAN for some higher rotational quantum numbers with J values from 103 up to 110, which gives the difference errors at the 10 −2 cm −1 level (see Table A1 of the supplementary material). Moreover, plots of the differences between the ν VDACMt J,cal

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Additionally, to test the theoretical treatment of DACMt, we have shown variational (VDACMt) and non-variational (DACMt) treatments to yield results for the 2-0 overtone band. This allows us to compare the transition lines obtained from these two channels with the line values from the HITRAN database and some other works. However, along with a test for the variational and non-variational treatments in the calculation itself, there are also tests for the theoretical treatments of ν VDACMt J,cal (m)|m ≠ 9 calculation and DCM treatment of our recent work. 31 Comparisons of the transition lines among these work are shown in Fig. 3 (these lines from Table A2 in

B. (3-1) band
It is also interesting to see the calculated R-branch transition lines for the (3-1) overtone band that are based on the measurement lines reported by Mantz and Maillard,12 which are collected in Tables A3 and A4 of the supplementary material. The best R-branch transition lines ν VDACMt J,cal (9) that give the lowest error ΔERMS of 1.52 × 10 −3 cm −1 are calculated using the corresponding molecular constants (ν 0 , Bυ, Dυ, Hυ) listed in Tables II and III. Just as shown in Fig. 4, comparing ν VDACMt J,cal (9) with the lines of Mantz and Maillard, 12 Hase et al., 6 HITRAN08 database, 21 and Mishra et al., 17 we gain the conclusion that good agreement exists between them. Figure 5 illustrates the general trends in line differences between the DACMt, DCM treatments, and the values of HITRAN08 21 for the (3-1) overtone band. As can be seen in the inset of Fig. 5 (lines   FIG. 3. Differences between the R-branch transition lines of DACMt, DCMt, and the values from HITRAN08 for the (2-0) band.   (11)-HITRAN08] increase, which can be explained using the inappropriate expansions of rotational energy form for these three cases. In addition, the results of ν VDACMt J,cal (5), which are in poor agreement with the values of HITRAN08, 21 should be abandoned. However, comparing with the results of ν DACMt J,cal (9), the ν VDACMt J,cal (9) treatment can be a better candidate to improve the accuracy of the predicted R-branch lines by utilizing the variational treatment (see the Fig. 5 inset).

C. (6-4) band
We also performed the DACMt method for the (6-4) transition band. This is illustrated in Fig. 6, where the differences between R-branch transition lines of ν VDACMt J,cal (9) and some other works are plotted against the rotational quantum number J for this transition  band. The calculated precision of the (6-4) transition band is approximately 1.70 × 10 −3 cm −1 . It is clear that the ν VDACMt J,cal (9) lines, and the lines of Mantz and Maillard, 12 Hase et al., 6 and HITRAN08 21 compare very well, but it is worth noting that the blended lines R 48,49,50 of Mishra et al., 17 used for comparison of ν VDACMt J,cal (9) calculation are at −0.0123 cm −1 , −0.0377 cm −1 , and 0.0142 cm −1 in the line differences, slightly more than the values comparing the work of Mantz and Maillard, 12 Hase et al., 6 and HITRAN08. 21 Similarly, as can be seen in Fig. 7, the lines of ν VDACMt J,cal (9) could enable a reasonable prediction of the behavior of high-lying transitional lines in the VDACMt approach, thus indicating a representation of its validity, while an obvious deviation can be seen between the ν DACMt J,cal (9) values and these HITRAN08 results 21 ], where the inconsistency is visible for most lines, given some differences at the 10 −2 cm −1 level. Another illustration of comparison is given in Fig. 9, which displays the line differences between the DACMt, DCM theoretical calculations, and the values of HITRAN08. 21 As it has been used above, the transition lines ν VDACMt J,cal (9) will be a good interpretation FIG. 9. Differences between the R-branch transition lines of DACMt, DCMt, and the values from HITRAN08 for the  band.

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scitation.org/journal/adv of the spectroscopic information for this band, other than the lines of ν VDACMt J,cal (m)|m = 5, 7, 11, ν DACMt J,cal (9), and ν DCMt J,cal (11). Such a disagreement clearly expresses that among these approaches, difference in treatment for the experimental lines or polynomial truncation with respect to the [J(J + 1)] n indicates the failure of the prediction for high J R-branch transition lines for the  band.
It is worth to note that the matter of reassessing the predictions of the R-transition lines is not a simple one, and may arise from the accuracy of the experimental ones selected. As such, this is to say, the better representation of the data from the experiment could help us improve the accuracy of the DACMt prediction for higher J lines.

IV. SUMMARY
The difference algebraic converging method (DACM) originates from a close cooperation between the theoretical method and experimental measurements. Based on some experimental lines, algebraic Eq. (10) and analytical formula Eq. (9) are used to predict R-branch transition spectral lines with variational treatment for (2-0), (3-1), , and (7-5) overtone bands in the X 1 Σ + → X 1 Σ + transition of CO. With J values up to 110, our results ν VDACMt J,cal are in good agreement with the line values from HITRAN and some other experiments, and also are expected to give reliable predictions. In addition, an improved set of molecular constants is derived for rovibrational analyses of the R-transition lines for these first overtone bands. In the case of each band of CO, there is a typically well-seen connection between transition lines and the molecular constants. Such deep insights shall undoubtedly enhance our understanding of how the diversities of molecular constants influence the transition lines, and of course, improve the development of a better method. Therefore, the DACMt suggested in this work for the calculation of the R-transition lines provides a promising method for diatomic molecules. In addition, the improved spectroscopic parameters of CO will be important to the astrophysical research and astronomical community.