# First high resolution infrared spectra of 1-13C-Propane analyses of the ν26 (B2) C-Type, ν8 (A1) and ν9 (A1) B-Type bands

Stephen Daunt, Robert Grzywacz, Colin Western, Walter Lafferty, Jean-Marie Flaud, Brant Billinghurst

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► To cite this version:

Stephen Daunt, Robert Grzywacz, Colin Western, Walter Lafferty, Jean-Marie Flaud, et al. First high resolution infrared spectra of 1-13C-Propane analyses of the ν26 (B2) C-Type, ν8 (A1) and ν9 (A1) B-Type bands. Journal of Molecular Spectroscopy, 2023, 391, pp.111717. 10.1016/j.jms.2022.111717. hal-04568703

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# First High Resolution Infrared Spectra of
## 1-$^{13}$C-Propane

Analyses of the *v*₂₆ (B₂) C-Type,
*v*₈ (A₁) and *v*₉ (A₁) B-Type Bands

Stephen J. Daunt* and Robert Grzywacz

Department of Physics & Astronomy, The University of Tennessee, Knoxville, TN

Colin M. Western†

School of Chemistry, University of Bristol, Bristol, UK

Walter J. Lafferty†‡

Optical Technology Division, NIST, Gaithersburg, MD

Jean-Marie Flaud

Universités Paris Est Créteil et Paris Diderot, LISA, Créteil, France

Brant E. Billinghurst

Canadian Light Source, University of Saskatchewan, Saskatoon, SK, Canada

†Deceased September 21, 2021

†† Deceased December 27, 2018

Corresponding authors: sdaunt@utk.edu
## Highlights

* First high-resolution infrared spectra of 1-¹³C-Propane molecule

* First experimentally determined centrifugal distortion constants for this isotopologue of propane

* Simultaneous fitting of the upper states for the $v_8$, $v_9$, $2v_9 - v_8$, $2v_9$ and $v_{26}$ bands including resonances and “forbidden” transitions

* Produced simulated spectra and line lists for these bands that reproduce the observed data and that can be used to identify this species in future atmospheric and astronomical studies

## ABSTRACT

This paper presents the first high resolution ($\Delta v = 0.00096 \text{ cm}^{-1}$) IR investigation of 1-¹³C - propane. Spectra of the $v_9$ (A₁) CCC skeletal bending mode near 366.40365 $\text{cm}^{-1}$ (B-type band) and the $v_{26}$ (B₂) methylene (CH₂) wagging mode near 748.470 $\text{cm}^{-1}$ (C-type band) were recorded at the Canadian Light Source (CLS) synchrotron. The spectra were assigned both traditionally and with the aid of the PGOPHER program [1]. As only limited MW data are available for this molecule, we had to use the present data to determine a new set of ground state constants that included centrifugal distortion terms. Upper state constants for both bands have been determined that provide a good simulation of the spectra. The analysis also included the strong A-Type Coriolis resonance between the $v_{26}$ and $2v_9$ states that causes strong perturbation allowed transitions to appear in the spectrum. Hot band lines of the $2v_9-v_9$ band were also assigned and included in our analysis of the bending region. In addition, the $v_8$ (A₁) CC stretching band at 862.076 $\text{cm}^{-1}$ (B-type band) which somewhat overlapped $v_{26}$ was included in the analysis as well. The hope is that this data and the derived molecular constants will be useful in identifying isotopic propane lines in Titan and other astrophysical objects.

**Keywords:**

High resolution IR

Propane isotopologues

Vibration-rotational spectra

Ground state rotational constants

CLS synchrotron spectra

PGOPHER

Vibration-rotation resonances
**INTRODUCTION**

This is the second paper of an ongoing project to obtain and analyze high resolution IR and Far-IR spectra of singly substituted ¹³C and D isotopologues of propane. The first propane paper [2] in this series dealt with the IR spectra of the v₉ and v₂₆ regions of the species 2-¹³C-Propane. That study also involved the 2v₉ state's interaction with the v₂₆ state as well as an analysis of the 2v₉-v₉ hot band. In this present paper our spectra have once again been run on the Bruker IFS 125 HR at the Far-IR beamline of the Canadian Light Source (CLS) synchrotron [Fig. 1]. A comparison of observed values for the normal and 1-¹³C-Propane are listed in Table 1.

<figure><img src="image_3.png" /><figcaption>**Fig. 1** Spectrum of 1-¹³C-Propane taken at the CLS Far-IR beamline with the Bruker FTS 125HR.
Resolution=0.00096 cm⁻¹.</figcaption></figure>
<table><caption>**Table 1.** Origins (in cm⁻¹) of selected vibrational bands of Propane</caption><thead><tr><td>Band</td><td>Normal Propane</td><td>1-<sup>13</sup>C-Propane<sup>a</sup></td><td>Shift</td></tr></thead><tbody><tr><td>v<sub>9</sub> (A<sub>1</sub>)</td><td>369.222 808(25)<sup>b</sup></td><td>366.403 646 0(95)</td><td>-2.819 162</td></tr><tr><td>v<sub>26</sub> (B<sub>2</sub>)</td><td>748.530 882(80)<sup>c</sup></td><td>748.470 279 4(81)</td><td>-1.916 731</td></tr><tr><td>v<sub>8</sub> (A<sub>1</sub>)</td><td>870.395 75(160)<sup>d</sup></td><td>862.075 848(15)</td><td>-8.319 902</td></tr><tr><td>v<sub>21</sub> (B<sub>1</sub>)</td><td>921.375 6(400)<sup>d</sup></td><td>920.85</td><td>-0.53</td></tr><tr><td>v<sub>20</sub> (B<sub>1</sub>)</td><td>1054.2<sup>e</sup></td><td>1048.07</td><td>-14.7</td></tr><tr><td>v<sub>7</sub> (A<sub>1</sub>)</td><td>1157<sup>e</sup></td><td>1152</td><td>-5</td></tr><tr><td>v<sub>25</sub> (B<sub>2</sub>)</td><td>1192<sup>e</sup></td><td>1187.1</td><td>-4.9</td></tr></tbody></table>

Bands corresponding to those already detected on Titan by Voyager and Cassini for the normal species of propane are one of our main aims. The other region of concentration is the lowest vibrational (non-torsional) mode near 366 cm⁻¹ that is in the THz region and appears unperturbed. This band has allowed us to obtain accurate rotational constants through 6th order for the different species. With these we hope to predict rotational lines that can be used in radio and IR astronomy searches for these isotopic species in the ISM and planetary atmospheres.

Many studies of normal propane have been reported over the years. The recent articles by Perrin, et al. [3 (Table 1)-4] summarize the several high-resolution studies that have been done which supported NASA missions (Voyager, Galileo & Cassini) to the outer Solar System. There are also studies reported from other satellite, aircraft and ground based telescopic observations where propane data are useful. The interstellar medium where new solar systems exist or are forming is also a place to expect to find propane since other organic molecules have been detected by these same observing methodologies [5]. Our earlier work [6-7] on normal C₃H₈ has been used to help with understanding the atmospheres of Jupiter, Saturn and Titan [8-11].

A thorough review of previous propane work was given in the introduction section of the first paper of this series [2] and will not be recounted here. That review covered spectroscopic work relating to the astronomical, environmental and geophysical applications.

Now we need to start to study the isotopic versions of propane that will be detectable with modern spectroscopic tools used in environmental and astronomical problems. Only a few of these isotopic species of propane have ever been studied and that work was done over 50 years ago [12-15]. Henry L. McMurry and his colleagues continued to work on propane over many decades mainly on the deuterated propanes and propane force fields. They were also the first to directly observe the torsional fundamentals of C₃H₈ via neutron diffraction studies [16].

The numbering notation for the normal modes and the inertial axes/vibrational dipole moment orientations are the same as that used in our first high resolution propane paper [17, see Fig. 1] even though technically the 1-<sup>13</sup>C-Propane molecule has C₅ rigid point group. This is so we can be consistent across all the isotopologues being studied in our work. It also keeps things consistent with the notation and descriptions of the molecular motions given in the *ab initio* study of Villa, et al. [18] and with the force field

<footnote>a This work, b Ref [7a], c Ref [7b], d Ref [3], e From the PNNL database as available from HITRAN [21]</footnote>
calculations/symmetry coordinates listed in Gough, et al. [19, Table II] for several of the isotopologues of
propane.

We note here that normal propane and 2-¹³C-Propane both belong to the C₂v rigid point group so v₉ has A₁ symmetry and v₂₆ has B₂ symmetry. That notation is kept in this paper although in reality the 1-¹³C-Propane modes belong to irreducible representations A' and A'' of the C₅ point group, respectively. PGOPHER has a thorough online discussion about handling the complexities of choosing the proper axes, Watson representations and transition dipole directions for C₂v type molecules. [See the section titled: "Axis systems and symmetries for asymmetric tops".]

Here the CCC molecular plane is the xz-plane with y perpendicular to it [17, see Fig. 1]. The 27 normal modes then follow these rules for this orientation: v₁- v₉ (A₁→A'→μₓ→B-type bands), v₁₀-v₁₄ (A₂→A''→IR inactive), v₁₅- v₂₁ (B₁→A'→μᵧ→A-type bands) and v₂₂-v₂₇→μᵧ→C-type bands) following the descent in symmetry.

Only six pure rotational microwave transitions for a few isotopic species were recorded by Lide in 1960 [20], including the species of interest here. These yielded the main rotational constants A₀, B₀, and C₀ but none of the centrifugal distortion constants for the singly substituted isotopologues. A recent set of *ab initio* calculations has been published trying to predict the constants for these isotopic species of propane [18]. The astronomical community prompted the authors of reference 18 and they stressed the scarcity of experimental data for propane isotopologues in that paper.

The use of ground state combination differences for the bands studied in this paper now fills in the blank
column for the 1-¹³C species seen in Table 1 of the *ab initio* article by Villa, et al. [18].

In this paper we make use of the high resolution of the CLS Far-IR beamline to record spectra of the v<sub>26</sub> and v<sub>9</sub> regions. The v<sub>26</sub> band is the corresponding feature detected in Titan and Saturn for normal propane. The v<sub>9</sub> band provides an unperturbed band to determine accurate ground state constants. Its accompanying hot band 2v<sub>9</sub>-v<sub>9</sub> provides constants for the 2v<sub>9</sub> state that perturbs transitions in the v<sub>26</sub> band. All the data in this work was analyzed and fit using the PGOPHER program. As ground state information in the literature has been extremely limited a new ground state analysis was required. We used an extensive set of ground state combination differences generated from our assigned spectra. A detailed analysis of both regions is presented here, sufficient to reproduce all the assigned bands to an accuracy of about 0.0002 cm<sup>-1</sup>, of a quality to give a significant improvement to spectroscopic databases.
# EXPERIMENTAL

The sample of 1-¹³C-Propane was purchased from Cambridge Research Labs and used without further purification. The gas was quoted to be of 99% ¹³C purity and no obvious spectral impurities were detected in our experiments. The sample came in a 75 mL stainless steel Whitey straight/male cylinder. The sample was connected via stainless steel tubing and a needle valve to the filling system of the CLS Far-IR FTS 2m base length White cell. Due to the limited beamtime available for these experiments the spectra were taken only at room temperature.

## CLS Far-IR Synchrotron Spectra

All the spectra discussed in this paper were recorded with the Bruker 125HR FTS at the Canadian Light Source Synchrotron Far-IR beamline during Cycle 24 (Sept. 7-12, 2016). The data analyzed here were all recorded at a resolution of 0.00096 cm⁻¹. At that time period the instrument was using a liquid Helium cooled Ge:Cu detector. The samples were contained in a 2m White type cell set for an optical path length of 72 m. The White cell can be cooled by a recycling refrigerated liquid coolant system. Due to the limited beamtime allocated for that cycle these experiments were only run at room temperature. Cooling the White cell and realignment can use up to a complete 8-hour schedule shift. Recording in two different wavelength regions also requires beam splitter and cell window changeovers. Returning the cell to room temperature, repumping down, realignment and refilling the absorption cell also takes up even more precious scheduled beamtime.

Spectra in the v₂₆ region were therefore taken only at room temperature (290 K). As for the 2-¹³C species we would have liked to have recorded the spectral regions at both room and cooled temperatures (~200 K) to assist in identifying hot band transitions. That would have required many more beamtime shifts then we had available to us. We also only had a small amount of gas available due to its price. This meant we used all our available beamtime trying to record as many scans as possible to get acceptable S/N ratios in the averaged spectra. Of all the propane isotopologues we have studied this species spectra have the lowest S/N ratios. This can be seen in the figures below when compared to our other studies [2, plus refs. 55-59 therein].

The v₂₆ spectra were recorded with a pressure of 0.083 Torr in the White cell. The beam splitter material was KBr. An entrance aperture of 1.15 mm was used. The analyzed spectrum was computed from an averaged total of 166 scans [Fig. 2]. The spectrum was calibrated using the v₂ CO₂ residual lines in the spectrometer and wavenumber values taken from the HITRAN Online database [21].
<figure><img src="image_4.png" /><figcaption>**Fig. 2** The $v_{26}$ band of 1-$^{13}$C-Propane with sharp CO$_2$ lines to the left. Tic marks above are lines in the fit. Blue tics for $v_{26}$, red for $2v_9$ lines. Observed spectrum (maroon) above the PGOPHER calculated (blue/red). PGOPHER vertical scale is for Relative Intensity.</figcaption></figure>

The $v_9$ spectra [Fig. 3] were recorded with a pressure of 1.0403 Torr in the White cell at room temperature. A 6 µm Mylar multilayer beam splitter was used in this region along with an entrance aperture of 1.5 mm. The spectrum used here consisted of an average of 505 scans. Water lines in the room temperature spectra were used to calibrate the spectrum using values from HITRAN Online. These strong water lines were later truncated to make the observed 1-$^{13}$C-Propane lines be of comparable intensities to the PGOPHER simulated lines.

<figure><img src="image_5.png" /><figcaption>**Fig. 3** The $v_9$ band of 1-$^{13}$C-Propane. Observed spectrum is in green. PGOPHER simulation of $v_9$ is in black. Red is for the $2v_9-v_9$ hot band. Tic marks at top show lines included in the fitting. Vertical scale of PGOPHER simulation is for Relative Intensity.</figcaption></figure>
We first used the Bruker OPUS software to process to measure the line positions. The spectra were then plotted out in small sections and taped together to create a continuous roll. Estimates of line spacings expected were calculated from the inertial constants of Lide [20]. Then using mechanical drawing dividers, we tracked series of lines back to their estimated sub-band origins. Later we used PGOPHER and the Lide constants plus changes ($α^x$, x = A, B, C) in the inertial constants estimated from our earlier work on normal propane [5] to begin to simulate the spectra. We also used PGOPHER to create Loomis-Wood plots to add evidence to support the assignments and to track each series back to its respective sub-band origin.

Finally, we then began to use PGOPHER to further measure and assign the lines in each band extending the assignments to ever higher J and $K_a$ values. We began our analysis of each region separately while continuing to refine the constants.

## Analysis of the v$_{26}$ Region

We began our analyses working on the v$_{26}$ band as in Ref. [21] since it was most pertinent to our interest of obtaining astrophysically relevant information for planetary atmospheres such as Titan. Our earlier study of the same band for both normal and 2-$^{13}$C-Propane propane gave us a familiarity with the basic structure of this C-type band and experience in assigning such lines.

This analysis was also started before the automated assignment tools [22] in PGOPHER were available, so the traditional approach to assignment was used based on picking out related sub-branch lines. Once the roll of the spectral plot was constructed, drafting dividers and estimated values of A, B, C were used to track the $^rR_{Ka}$(J) and $^{pP}_{Ka}$(J) series lines to their assumed $^rQ_{Ka}$ and $^{pQ}_{Ka}$ band heads. Those sub-band heads were assigned readily by comparing them to what was seen in the normal and 2-$^{13}$C-propane spectra as well as by their approximate distances from the front of the central Q-branch head near 748.256 cm$^{-1}$. The correctness of these $^r{pQ}_{K}$ assignments could then be verified by the “missing lines” in a particular $K_a$ series of J lines as they were tracked back to their respective sub-band origins near the front of each sub-band Q-branch head.

Preliminary assignments were then penciled onto the spectrum chart for the strongest $^{pP}$ and $^rR$ lines. We also made trial assignments for a few of the $^rQ_K$(J) and $^{pQ}_K$(J) lines if they were well resolved and the first line in that particular sub-band Q-branch was clearly resolved.

Once this was done, some PGOPHER trial simulations were made with the MW ground state A", B", C" values [20] and estimated upper state A', B', C' constants. The estimated changes of $α^A$ = (A"-A'), $α^B$, $α^C$ were taken from the earlier $C_3H_8$ [6] and 2-$^{13}$C- Propane work [2] to start. All centrifugal distortion constants were set to zero since no experimental information was available in the literature.

This gave us a rough approximation of the band appearance and we were then able to make incremental changes in the A', B', C' constants until both the $^r{pQ}_K$ band heads and the stronger $^rR$ and $^{pP}$ line patterns became closer to being in the right areas. As the simulation appearance got close to the observed spectrum PGOPHER was then used to measure selected lines and those assignments were transferred to the chart. This created an assigned line list within PGOPHER. After a few hundred lines with a reasonable distribution of J and K values were included in PGOPHER we started the fitting process. As observed spectra and simulation grew much closer, the measurement line list was extended. Once most $^rR$ and $^{pP}$ lines looked in reasonable agreement we added in the tentative Q-branch assignments.

Eventually as this went further it became apparent from the residual plots and the Q-branch lines that centrifugal distortion constants were needed. At first only $Δ_J$, $Δ_{JK}$ and $Δ_K$ values were allowed to vary. As the
observed and simulated spectra began to match even more closely, we decided that the assignments were
now good enough to try to do ground state combination fitting.

Eventually, we were able to get a fourth order set of ground state constants with an average residual of about 2 x 10⁻⁴ cm⁻¹. Then the ground state constants were fixed, and the assigned lines were fit to obtain upper state constants. These results were reported at the 2017 ISMS and are available on the ISMS archive website [2, ref. 57] or via ResearchGate.

The quality of the present analysis is illustrated in the following figures below for selected regions of the v₂₆ band. The central Q-branch region is seen in Figure 4 where the complex underlying structure of multiple overlapping band heads for the low K_a = 0-5 sub-branches is well reproduced. Similar central Q-branch structures were found in both the normal and the 2-¹³C-Propane v₂₆ bands.

<figure><img src="image_6.png" /></figure>

**Figure 4.** Central Q-branch region for 1-¹³C-Propane. Observed spectrum in maroon. PGOPHER calculated spectrum in blue. Tic markers above observed data are the lines included in the fit.

Figure 5 shows the expanded PQ₁₁ to PQ₉ branch region and the strong PP<sub>K</sub>(J) lines occurring throughout that area.

<figure><img src="image_7.png" /></figure>

**Fig. 5.** Observed (maroon) and PGOPHER calculation (blue) for the PQ₁₁- PQ₉ region of v₂₆ of 1-¹³C-Propane
In Figure 6 the clumping together of $^1R_K(J)$ lines can be seen and are well reproduced in the present analysis.

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** The $^1R_K(J)$ line cluster features on the high wavenumber side of $v_{26}$ of 1-$^{13}$C-Propane. Observed data in maroon and the PGOPHER calculation in blue. Tic markers are for lines included in the fit.</figcaption></figure>

This type of pattern is caused in bands when the subband origin spacing [~2(A-B<sub>bar</sub>)] is approximately an integer multiple of the P and R line spacing (~2B<sub>bar</sub>). In the case here that is equal to about the number three. In a particular $^1R$ cluster of lines the first line on the high wavenumber side might be $^1R_K(J=K)$ with the next lower line being $^1R_{(K-1)}(J+3)$, then $^1R_{(K-2)}(J+6)$, and so on until the intensity dies out. A similar pattern occurs with the $^Pp$ line clusters on the lower wavenumber side of the band.

Eventually, we could also see that a problem first noticed in our earlier work on $v_{26}$ of normal propane and also in the 2-$^{13}$C-Propane $v_{26}$ band was also present in this analogous band of 1-$^{13}$C-Propane. Lines with values of $K_a$ greater than 15 or so started to have ever increasing residuals. There were also "extra" lines appearing among the $^1R$ and $^Pp$ lines out in the wings of $v_{26}$ due to perturbation allowed transitions of $2v_9$ [Fig. 4]. This required a more complicated model discussed in the Combined Fit section below.

<figure><img src="image_9.png" /><figcaption>**Fig. 7** High wavenumber R-side of the $v_{26}$ band of 1-$^{13}$C-Propane. Maroon (upper trace) is the observed CLS spectrum. Below that is the PGOPHER calculated spectrum with $v_{26}$ lines in blue and the perturbation allowed $2v_9$ lines in red. Calculated vertical scale is normalized intensity.</figcaption></figure>
Analysis of the v<sub>9</sub> Region

We began our analysis on the v<sub>9</sub> band as we had done for the analogous region of 2-<sup>13</sup>C-Propane [2]. The v<sub>9</sub> band has the classic asymmetric rotor B-type band appearance with strong P and R branches and a central gap (Figs. 3 and 8), just as in normal propane.

An initial simulation was set up as for the v<sub>26</sub> band above but changing to a B-type band calculation and using
the changes in rotational constants from v<sub>9</sub> of normal propane [7a]. The first useful features for assignment
were the clear rQ<sub>0</sub>(J) lines from 367.0 – 368.1 cm<sup>-1</sup> (Fig. 8). This then allowed the <sup>p</sup>Q<sub>K</sub> and rQ<sub>K</sub> band heads to be
easily assigned, and the fit extended to the entire band, walking up in J and K. For v<sub>9</sub> we used PGOPHER to plot
out some Loomis-Wood diagrams to check some of the branches since this band was denser in line
distribution. There were also many secondary band heads caused by lines in some Q branches turning back on
themselves especially at the lower K<sub>a</sub> values. The v<sub>9</sub> band proved to be unperturbed, as in normal propane and
2-<sup>13</sup>C-Propane. The fit could be extended to the entire band and yielded values for upper and lower state
rotational constants and centrifugal distortion constants to 4<sup>th</sup> order.

<figure><img src="image_10.png" /><figcaption>**Fig. 8** Band center region for v<sub>9</sub> of 1-<sup>13</sup>C-Propane. Noted are band heads for some of the lower K<sub>a</sub> values and some J values of lines in the rQ<sub>0</sub>(J) subband. Black lines belong to the v<sub>9</sub> band and red lines are for the hot band lines of 2v<sub>9</sub>-v<sub>9</sub>. Vertical scale for PGOPHER simulation is Relative Intensity.</figcaption></figure>
As in the 2-¹³C-Propane species spectra the PQK(J) branches of the 2v₉-v₉ band of the 1-¹³C species were readily seen between the PQK branches of the cold v₉ band (Fig. 9).

<figure><img src="image_11.png" /></figure>

**Fig. 9.** Close up of part of the PQK region of v₉ for 1-¹³C-Propane. Cold band (v₉) in black and hot band (2v₉-v₇) (HB) in red. From the left are PQ₇ (v₉), PQ₈ (HB), PQ₆ (v₉) and PQ₇ (HB) branches. Tic marks above are for lines included in the fits.

Also, 'R lines in clusters that mirrored the cold band R<sub>K</sub>(J) patterns on the higher wavenumber side of v₉ could be seen (Fig. 10).

<figure><img src="image_12.png" /></figure>

**Fig. 10.** Closer look at the 'R side of the v₉ band of 1-¹³C-Propane. Prominent black lines are clusters of 'R lines of the v₉ band and the red lines are 'R clusters of the 2v₉-v₇ hot band. Tic marks above the observed spectrum (maroon) are for the lines included in the fits.
We used similar starting assumptions to those made in the 2-¹³C paper [2] for the 1-¹³C band hot band. An approximate simulation for the hot band was calculated using a similar origin shift and alphas that were ~2x the cold band values. From this we were then able to make tentative assignments and begin to fit the hot band parameters. The specific assignment to v₉ = 1 as the lower state was confirmed via combination differences. Eventually it was found that as Kₐ values neared higher values (>17) lines shifted significantly This was being caused by the Coriolis interaction with the v₂₆ state as seen earlier for the normal and 2-¹³C species. This is discussed in the section on combining the states.
# Analysis of the v<sub>8</sub> Region

While assigning the higher wavenumber side of the v<sub>26</sub> band it was noticed that there seemed to be lines appearing in the spectrum that were from neither v<sub>26</sub> nor 2v<sub>9</sub>. As the data was taken at room temperature they could have been from hot bands. The other possibility was that they might be from the nearby v<sub>8</sub> (A<sub>1</sub>) band centered near 862 cm<sup>-1</sup>. This is a B-type band whose lines were then later assigned using the recently added PGOPHER nearest lines facility [22]. The band was first fit as an isolated band then later added to the overall data set and fit as part of the combined analysis discussed below.

Lines with K<sub>a</sub>" = 20 and above appear perturbed and although definitively assigned are not included in the final fit. The perturbation could possibly be like that observed in normal propane by Perrin, et al. [4] where v<sub>8</sub> interacts with 2v<sub>14</sub> + 2v<sub>27</sub> torsional states. CMW attempted to model a perturbing state P similar to what was done in the 2-<sup>13</sup>C-Propane paper. In that study some of the perturber level lines were observed and could be fit but none could be found in the current CLS spectra of 1-<sup>13</sup>C-Propane.

No torsional splittings were seen and no perturbation allowed lines could be found in the v<sub>8</sub> band. That is very unlike the situation for the normal propane species [3,4] and on re-examining our 2-<sup>13</sup>C-Propane data we could find splittings there. That v<sub>8</sub> band was not assigned nor included in our 2-<sup>13</sup>C-Propane analysis [2] as we did not see its lines overlapping with that v<sub>26</sub> band.

There are still no experimental data specific to the torsional states of 1-<sup>13</sup>C-Propane. *Ab initio*, MW and Raman torsional studies have only been done for the normal species. Thus, no attempt has been made here to try to model those complex torsional interactions with these dark states at this time. Such modelling was also far beyond the main purpose of this project to determine GS constants and model accurately the appearance of bands with possible astrophysical/atmospheric applications.

The quality of this current v<sub>8</sub> band analysis is illustrated in Figs [11-12].

<figure><img src="image_13.png" /><figcaption>**Fig. 11** The v<sub>8</sub> band of 1-<sup>13</sup>C-Propane. Observed data in maroon and PGOPHER simulation in green. Tic markers at top are for lines included in the fit. Green tics are for v<sub>8</sub>, blue for v<sub>26</sub> and red for 2v<sub>9</sub>. Vertical scale in the PGOPHER simulation is normalized intensity. The band to the right in the observed data is the v<sub>21</sub>(B<sub>1</sub>)/A-type band. The P-branch overlaps the higher R-side of v<sub>8</sub>. The sharp peak near 920 cm<sup>-1</sup> is the v<sub>21</sub> central Q-branch.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 12** The $^{{}^{13}C}$PQ$_6$ region of $v_8$ for 1-$^{{}^{13}C}$Propane showing the quality of the assignments and the fit. Maroon is the observed data and green is PGOPHER simulated spectrum. Tic marks are lines in the fit. The vertical scale in the PGOPHER simulation is for normalized intensity. Please note that no torsional splittings were observed for any lines of this band in our data. In the $v_8$ spectrum of normal propane torsional splittings were observed [4].</figcaption></figure>
**<u>Combined Fit to All States</u>**

To model the interaction between $v_{26}$ and $2v_9$ we added an a-type Coriolis resonance, as was done for the $v_{26}$ band of normal propane [7b] and $2\text{-}^{13}\text{C-propane}$ [2]. This was done using the general purpose perturbation mechanism built into PGOPHER [1]. The first order Coriolis mixing term, $\langle v_{26}=1 | \hat{J}_a | v_9=2 \rangle$, clearly improved the fits for both $v_{26}$ and $2v_9$ and a straight-forward trial and error process accompanied by re-fitting could then be used to assign all the remaining lines from $v_{26}$ and $2v_9$ in our spectra. The fitting process at this stage used was a simultaneous fit to all states, which predicted some perturbation-allowed $2v_9 \leftarrow 0$ transitions. (It was clear from the initial analysis that this overtone transition is otherwise too weak to see under the conditions we used.) The perturbation-allowed transitions are strong for many lines with $K_a'$ between 18 and 26 [Fig. 7].

In the end we included all of lines for $v_9$, $2v_9-v_9$, $v_{26}$, $2v_9$ and $v_8$ bands in the analysis. We first used a GSCD fit for the $v_9$ fundamental to obtain constants for the ground state (GS) $v=0$. We have previously found in almost all our studies of the propane isotopologues that the $v_9=1$ states appear unperturbed and yield excellent GS constants. We then held those GS values plus the $v_9=1$ state values unchanged while allowing the upper states of all the other bands to vary.

The only possible exceptions observed have been for $v_9$ bands of the $-D_3$, $-D_6$ and $-D_8$ species. These all have calculated torsional modes whose first overtones would fall very near to their $v_9=1$ states.

There was an attempt to find a perturbing state P responsible for some of the observed localized shifts in some sub-bands of $v_{26}$. Some connecting terms were included in the combined fitting for this hypothetical state. Unlike in $2\text{-}^{13}\text{C}$ we could not find any observed lines due to this possible perturbing state. Being unsure of the significance of any numbers derived we do not list them here. States causing these shifts are again most likely due to some unidentified dark combination states of the two torsional modes. This was also our explanation for the high $K_a > 20$ sub-band lines that are shifted in $v_8$.

In conclusion, we have attained the original goals set for this project on $1\text{-}^{13}\text{C-Propane: (1) determine accurate GS constants and (2) effective upper state constants that model and accurately simulate the vast majority of observed spectral lines for these bands. Examination of the various figures displayed in this paper seem to demonstrate that second goal has been successful.}$

The results of the fittings are listed in Tables 2 and 3. There were well over 20,000 transitions assigned in this study. Some distributions of residuals are seen in Figs 13 and 14.
<table><caption>**Table 2.** Rotational constants for the ground vibrational state of 1-$^{13}$C-Propane.</caption><thead><tr><th></th><th>This work</th><th>Microwave<sup>a</sup></th><th>Ab initio<sup>b</sup></th></tr></thead><tbody><tr><td>A</td><td>0.970 407 45(26)</td><td>0.970 408 4(33)</td><td>0.969 580</td></tr><tr><td>B</td><td>0.274 479 76(17)</td><td>0.274 483 3(33)</td><td>0.274 612</td></tr><tr><td>C</td><td>0.242 892 75(17)</td><td>0.242 893 0(33)</td><td>0.242 869</td></tr><tr><td>Δ<sub>K</sub></td><td>5.312 88(45) x 10<sup>-6</sup></td><td></td><td>5.517 x 10<sup>-6</sup></td></tr><tr><td>Δ<sub>JK</sub></td><td>-8.861 38(373) x 10<sup>-7</sup></td><td></td><td>-10.04 x 10<sup>-7</sup></td></tr><tr><td>Δ<sub>J</sub></td><td>2.303 35(135) x 10<sup>-7</sup></td><td></td><td>2.399 x 10<sup>-7</sup></td></tr><tr><td>δ<sub>K</sub></td><td>1.078 1(88) x 10<sup>-7</sup></td><td></td><td>1.348 x 10<sup>-7</sup></td></tr><tr><td>δ<sub>J</sub></td><td>4.364 5(15) x 10<sup>-8</sup></td><td></td><td>4.669 x 10<sup>-8</sup></td></tr><tr><td>Φ<sub>K</sub></td><td>1.098 7(25) x 10<sup>-10</sup></td><td></td><td></td></tr><tr><td>Φ<sub>KJ</sub></td><td>-4.006 4(312) x 10<sup>-11</sup></td><td></td><td></td></tr><tr><td>Φ<sub>JK</sub></td><td>1.486(157) x 10<sup>-12</sup></td><td></td><td></td></tr><tr><td>Φ<sub>J</sub></td><td>1.57(34) x 10<sup>-13</sup></td><td></td><td></td></tr><tr><td>Φ<sub>K</sub></td><td>0</td><td></td><td></td></tr><tr><td>Φ<sub>JK</sub></td><td>0</td><td></td><td></td></tr><tr><td>Φ<sub>J</sub></td><td>0</td><td></td><td></td></tr><tr><td>σ</td><td>0.000 14</td><td>3.3 x 10<sup>-6</sup></td><td></td></tr><tr><td>n<sub>obs</sub></td><td>5443</td><td>6</td><td></td></tr></tbody></table>

<sup>a</sup> Reference [20]; only 6 K<sub>a</sub> = 1 ← 0 lines were observed. <sup>b</sup> Reference [18]

<table><caption>**Table 3.** Rotational constants for the v<sub>9</sub>, 2v<sub>9</sub>, v<sub>26</sub> and v<sub>8</sub> vibrational states of 1-$^{13}$C-Propane.</caption><thead><tr><th>Constants / cm<sup>-1</sup></th><th>v<sub>9</sub> = 1</th><th>v<sub>9</sub> = 2</th><th>v<sub>26</sub> = 1</th><th>v<sub>8</sub> = 1<sup>a</sup></th></tr></thead><tbody><tr><td>v<sub>0</sub></td><td>366.403 646 0(95)</td><td>734.619 138(21)</td><td>748.470 305 1(93)</td><td>862.075 701(16)</td></tr><tr><td>A</td><td>0.981 092 29(14)</td><td>0.991 857 46(25)</td><td>0.970 667 45(11)</td><td>0.971 143 48(66)</td></tr><tr><td>B</td><td>0.274 287 952(37)</td><td>0.274 090 75(13)</td><td>0.273 671 407(44)</td><td>0.273 046 41(13)</td></tr><tr><td>C</td><td>0.242 263 162(41)</td><td>0.241 636 97(12)</td><td>0.242 204 118(35)</td><td>0.241 713 93(14)</td></tr><tr><td>Δ<sub>K</sub></td><td>6.020 74(44) x 10<sup>-6</sup></td><td>6.781 63(109) x 10<sup>-6</sup></td><td>5.345 35(26) x 10<sup>-6</sup></td><td>5.529 56(926) x 10<sup>-6</sup></td></tr><tr><td>Δ<sub>JK</sub></td><td>-8.523 7(13) x 10<sup>-7</sup></td><td>-8.133 92(863) x 10<sup>-7</sup></td><td>-8.714 5(13) x 10<sup>-7</sup></td><td>-8.073 7(60) x 10<sup>-7</sup></td></tr><tr><td>Δ<sub>J</sub></td><td>2.252 40(20) x 10<sup>-7</sup></td><td>2.203 1(12) x 10<sup>-7</sup></td><td>2.286 06(15) x 10<sup>-7</sup></td><td>2.261 585(524) x 10<sup>-7</sup></td></tr><tr><td>δ<sub>K</sub></td><td>2.049 5(44) x 10<sup>-7</sup></td><td>2.73(12) x 10<sup>-7</sup></td><td>1.270(10) x 10<sup>-7</sup></td><td>1.030(48) x 10<sup>-7</sup></td></tr><tr><td>δ<sub>J</sub></td><td>4.302 88(69) x 10<sup>-8</sup></td><td>4.207 8(60) x 10<sup>-8</sup></td><td>4.404 16(94) x 10<sup>-8</sup></td><td>4.263 4(76) x 10<sup>-8</sup></td></tr><tr><td>Φ<sub>K</sub></td><td>1.894 6(36) x 10<sup>-10</sup></td><td>2.345 1(83) x 10<sup>-10</sup></td><td>1.107 5(17) x 10<sup>-10</sup></td><td>3.513(457) x 10<sup>-10</sup></td></tr><tr><td>Φ<sub>KJ</sub></td><td>-4.066(17) x 10<sup>-11</sup></td><td>-3.452(44) x 10<sup>-11</sup></td><td>-3.985(11) x 10<sup>-11</sup></td><td>2.620 0(729) x 10<sup>-11</sup></td></tr><tr><td>Φ<sub>JK</sub></td><td>1.699(36) x 10<sup>-12</sup></td><td>0</td><td>0</td><td>-1.044(214) x 10<sup>-11</sup></td></tr><tr><td>Φ<sub>J</sub></td><td>2.690(145) x 10<sup>-13</sup></td><td>0</td><td>0</td><td>-8.798(321) x 10<sup>-13</sup></td></tr><tr><td>Φ<sub>K</sub></td><td>0</td><td>0</td><td>0</td><td>-1.203(662) x 10<sup>-10</sup></td></tr><tr><td>Φ<sub>JK</sub></td><td>0</td><td>0</td><td>0</td><td>-1.55(23) x 10<sup>-11</sup></td></tr><tr><td>Φ<sub>J</sub></td><td>0</td><td>0</td><td>0</td><td>-1.46(23)x 10<sup>-13</sup></td></tr><tr><td>L<sub>K</sub></td><td>0</td><td>0</td><td>0</td><td>5.747(694) x 10<sup>-13</sup></td></tr><tr><td>&lt;v26=1|J&#770;<sub>a</sub>|v9=2&gt;</td><td></td><td>0.041 631 8(14)</td><td></td><td></td></tr><tr><td>&lt;v26=1|J&#770;<sub>a</sub>J&#770;<sup>2</sup>|v9=2&gt;</td><td></td><td>1.000 50(92) x 10<sup>-6</sup></td><td></td><td></td></tr><tr><td>σ</td><td></td><td>0.000 233</td><td></td><td></td></tr><tr><td>N</td><td></td><td>13 803<sup>b</sup></td><td></td><td></td></tr></tbody></table>

a. This state was restricted to K<sub>a</sub> of 20 or so due to perturbation from torsional states not yet clearly identified but must involve one or more quanta of the two torsional modes as for Propane-H<sub>8</sub> [3,4].
b. There were 22,101 assigned lines fitted. After analyzing $v_9$ and the GS separately and after the use of the Merge lines facility in PGOPHER for very close lying lines these were reduced to this number of “lines”. Within the various .pgo files used: $v_9$, $v_9$ hotband, $v_{26}$ with $2v_9$ lines and $v_8$ there were 6,059, 1917, 9,426 and 4,699 assigned lines fitted, respectively.

<figure><img src="image_15.png" /></figure>

**Fig. 13** Residuals plot vs wavenumber for the combined states fitting. Legend: black for $v_9$ and red for $2v_9-v_9$ transitions in the lower wavenumber region; blue for $v_{26}$, red for $2v_9$, green for $v_8$ transitions in the higher wavenumber region. Red symmetry symbols are obscured by the darker black or blue ones. Vertical scale is Obs-Calc value of the residuals in cm$^{-1}$.

<figure><img src="image_16.png" /></figure>

**Fig. 14** Distribution of the combined state residuals as a function of $K_a$. Note that there are assigned transitions with high $K_a$ values. Residuals are plotted on a vertical scale of +0.0012 to -0.0012 with an average of 0.00022 cm$^{-1}$. Color codes as in Figure 8 for the assigned and fitted transitions.
# Acknowledgements

This study was being worked on by Colin and SJD until Colin informed me via email in late June of 2021 that he was in hospital and seriously ill. He said he might be there awhile. We were trying to get ready for ISMS 2021 and had put this manuscript on a temporary hold. After that SJD communicated with Colin via his sister Isabel Western Morris and he said initially he was hopeful he would get better and soon get back to work. Little did we know that Colin had only three more months left to him.

In 2015 SJD first met Colin at the ISMS meeting and then spent time with him learning about PGOPHER at the CLS Workshop organized by BB held immediately after that ISMS meeting. Later BB said that Colin wanted to work on the propane isotopologues project with us since he wished to include torsional items into PGOPHER. That began a close interaction with Colin that lasted for more than five years as we worked on the spectra of various propane isotopologues and he added the various improvements to PGOPHER. Colin was actually going to give a lecture on his incorporation of torsion into PGOPHER at the 2021 ISMS meeting. Sadly this was not to be.

SJD also wants to note that this isotopologues project was only made possible originally by the late Jon Hougen. He organized this collaboration setting up our first meetings at NIST with WFL and JMF in 2014. He also arranged the initial collaboration with BB at 2014 ISMS. Jon kept encouraging SJD at subsequent meetings, via phone calls/emails and nurtured this project until his untimely passing.

Dr. Bob McKellar (NRCC) made the initial suggestion at ISMS 2014 to work on the isotopologues when SJD was told by Olivier Pirali (SOLEIL-Ailes) that JMF and WFL had recently finished a re-analysis of the $v_{26}$ band of normal propane that Don Jennings (NASA/GSFC) and SJD had begun years earlier [6].

SJD thanks the University of Tennessee Knoxville for its support through continued access to library and computer facilities after his retiring from teaching and advising duties. He also thanks the UTK Department of Physics & Astronomy, especially former Department Head Professor Hanno Weitering, for his support of this project.

The authors would also like to thank Dr. Louli Gordon at the Harvard-Smithsonian Center for Astrophysics for his help with the HITRAN listings needed for checking our calibrations.

The spectra described in this paper were obtained at the Canadian Light Source, which is supported by the Canada Foundation for Innovation, Natural Sciences and Engineering Research Council of Canada, the University of Saskatchewan, the Government of Saskatchewan, Western Economic Diversification Canada, the National Research Council Canada, and the Canadian Institutes of Health Research.

## sDeclaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## CRediT authorship contribution statement

**Stephen J. Daunt:** Conceptualization, Project administration, Writing - original draft, Investigation, Resources, Formal analysis, Supervision, Visualization, Validation. **Robert Grzywacz:** Conceptualization. **Colin M. Western:** Writing - original draft, Investigation, Supervision, Methodology, Formal analysis, Software Visualization, Validation, Data curation. **Walter J. Lafferty:** Conceptualization, Methodology. **Jean-Marie Flaud:** Conceptualization, Methodology. **Brant E. Billinghurst:** Investigation, Resources, Software, Supervision.
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