# Laboratory Measurement of CH<sup>17</sup>_3OH Transitions in the Frequency Range from 216 to 264 GHz for Astronomical Application

Akemi Tamanai, Takahiro Oyama, Yoshimasa Watanabe, Takeshi Sakai, Riouhei Nakatani, Shaoshan Zeng, Isabelle Kleiner, Nami Sakai

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## ► To cite this version:

Akemi Tamanai, Takahiro Oyama, Yoshimasa Watanabe, Takeshi Sakai, Riouhei Nakatani, et al. Laboratory Measurement of CH<sup>17</sup>_3OH Transitions in the Frequency Range from 216 to 264 GHz for Astronomical Application. The Astrophysical Journal, 2025, 980 (1), pp.110. 10.3847/1538-4357/ad9b16. [hal-05344382](https://hal.science/hal-05344382v1)

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# Laboratory Measurement of CH<sub>3</sub><sup>17</sup>OH Transitions in the Frequency Range from 216 to 264 GHz for Astronomical Application

Akemi Tamanai<fnref n="1" />, Takahiro Oyama<fnref n="1" /> ID, Yoshimasa Watanabe<fnref n="1" />,<fnref n="2" /> ID, Takeshi Sakai<fnref n="3" /> ID, Riouhei Nakatani<fnref n="4" />,<fnref n="1" /> ID, Shaoshan Zeng<fnref n="1" /> ID, Isabelle Kleiner<fnref n="5" /> ID, and Nami Sakai<fnref n="1" /> ID

Received 2024 June 14; revised 2024 November 29; accepted 2024 December 3; published 2025 February 6

## Abstract

Methanol is one of the most abundant complex organic molecules in interstellar environments. Molecular lines of its rare isotopologues ¹²CH₃¹⁷OH and ¹²CH₃¹⁸OH therefore play a crucial role in examining the column density of ¹²CH₃¹⁶OH, which serves as a reference for organic molecular chemistry in interstellar clouds. In this study, we have recorded the spectroscopic emission spectrum of ¹²CH₃¹⁷OH in the frequency range between 216 and 264 GHz by making use of an emission-type millimeter and submillimeter spectrometer. We have specifically paid attention to the *Q*-branch transitions, which are the strongest line series in this frequency region. Among the stable oxygen isotopes, ¹⁶O, ¹⁷O, and ¹⁸O of methanol, only ¹²CH₃¹⁷OH obviously shows line profiles having double and/or triple peaks in low-*J* transitions, due to the nuclear quadrupole interaction. The newly obtained ¹²CH₃¹⁷OH data will play an important role in facilitating a deeper understanding of the organic chemistry related to star and planet formation. The ¹²CH₃¹⁷OH line data allow us to trace and constrain the isotopic ratio ¹⁷O/¹⁸O in methanol, which is efficient to investigate the galactic-scale evolution of elements. In addition, we also assigned some transitions of ¹³CH₃¹⁷OH in the recorded spectrum.

*Unified Astronomy Thesaurus concepts: Molecular spectroscopy (2095); Line positions (2085); Experimental data (2371); Astrochemistry (75)*

## 1. Introduction

Methanol is known as one of the most abundant and well-studied organic molecules in interstellar clouds. Because of its high abundance, rare isotopologues have also been observed toward various types of astronomical objects. ¹³CH₃OH has been detected toward Sagittarius B2 (Sgr B2) with the 11 m radio telescope of the National Radio Astronomy Observatory (C. A. Gottlieb et al. 1979). Similarly, the transition lines of oxygen isotopologue methanol, ¹²CH₃¹⁸OH, were confirmed in Sgr B2 with the 70 m antenna of NASA's Canberra Deep Space Communication Complex (F. F. Gardner et al. 1989). Observed rare isotopologue lines are frequently used to examine the optical depth of lines obtained from major isotopologues. For instance, the abundances of ¹²CH₃¹⁶OH could roughly be estimated from those of ¹²CH₃¹⁸OH by assuming the isotopic ratio of ¹⁶O/¹⁸O to be 1/560 (e.g., R. Lucas 1998).

Recent highly sensitive observations allow us to trace isotopic ratios spatially and/or to investigate the formation pathway of more complex molecules (e.g., N. Sakai et al. 2007; T. Soma et al. 2015; J. K. Jorgensen et al. 2018). Since isotope fractionation effects in molecules are known to be significant especially for hydrogen and carbon isotopologues, formation mechanisms of molecules in interstellar clouds have been investigated mainly by observing those isotopologues (e.g., P. Caselli & C. Ceccarelli 2012; N. Sakai et al. 2013). However, oxygen isotopic ratios have not been given that much

attention in comparison to hydrogen and carbon isotopic ratios, due to observational difficulties of the lines detected from extremely low abundance isotopologues of molecules, except for CO. As for methanol, only several astronomical observations of ¹²CH₃¹⁸OH have indeed been conducted (K. P. Johns et al. 1998; J. K. Jorgensen et al. 2018) since the first detection in Sgr B2 (F. F. Gardner et al. 1989). For ¹²CH₃¹⁷OH, there have been no reports of clear detection in any interstellar clouds to date (see comments in the last part of Section 4.6).

Nondetection of ¹²CH₃¹⁷OH lines is caused not only by its low abundance but also by insufficient spectroscopic information in millimeter- and submillimeter-wave regions. Only a limited number of transitions have been measured in the microwave domain, including *b*-type *Q*-branch and *a*-type *R*-branch transitions. Molecular constants were determined within this limit by L.-H. Xu (1994) and Y. Hoshino et al. (1991). This is because they utilized a commercial sample of ¹²CH₃¹⁸OH containing only 1% of ¹²CH₃¹⁷OH in their investigation. Although rotation-torsion transitions were assigned in the far-infrared frequency range between 1948.6 and 35975 GHz (G. Moruzzi et al. 2011), transition assignments in the millimeter- or submillimeter-wave regions have not been carried out yet. Due to the internal rotation of the methyl group, spectral modeling of methanol is less straightforward than that of semirigid molecules. The torsion and the overall rotation are strongly coupled, and thus it is very difficult to consider them separately. The quadrupole hyperfine structure arising as a result of the nuclear spin of ¹⁷O (I = 5/2) makes analysis more intricate. In consequence, it is essential to determine more precise transition frequencies of ¹²CH₃¹⁷OH experimentally.

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Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹ RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan; nami.sakai@riken.jp</footnote>

<footnote n="2">² Materials Science and Engineering, College of Engineering, Shibaura Institute of Technology, 3-7-5 Toyosu Koto-ku, Tokyo 135-8548, Japan</footnote>

<footnote n="3">³ Graduate School of Informatics and Engineering, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan</footnote>

<footnote n="4">⁴ NASA Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA</footnote>

<footnote n="5">⁵ Laboratoire Interuniversitaire des Systèmes Atmosphériques (LISA), Université Paris Cité and Univ Paris Est Créteil, CNRS, LISA, F-75013 Paris, France</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 1. A schematic diagram of the experimental apparatus SUMIRE.</figcaption></figure>

Oxygen is the third most abundant element in the Universe, as well as the most abundant element on Earth. It is hence important to trace its isotopic ratios along star-forming regions for further understanding of formation mechanisms of various bio-related molecules and/or the origin of the ratios found in the solar system (e.g., H. Yurimoto & K. Kuramoto 2004). However, the abundance ratio of $^{18}$O/$^{17}$O in the local interstellar clouds is difficult to derive because of the low abundance even for the simple and abundant interstellar molecules like CO as mentioned previously. Observations of C$^{18}$O and C$^{17}$O line emission toward several dark clouds revealed the $^{18}$O/$^{17}$O ratio to be 2.8 ± 0.2, which is lower than the solar system value ($^{18}$O/$^{17}$O = 5.5) by a factor of ~2 (A. A. Penzias 1981; R. W. Wilson et al. 1981). For the ρ Oph molecular cloud, a value of 4.11 ± 0.14 has been reported (J. G. A. Wouterloot et al. 2005), closer to that of the solar system. More observations are essential for obtaining better estimates and understanding of the $^{18}$O/$^{17}$O ratio. Such observational data should include a larger sample of molecular clouds and other molecular species with the $^{17}$O and $^{18}$O isotopes in order to have a more accurate picture. It is not straightforward to determine the $^{18}$O/$^{17}$O ratio by observing C$^{18}$O and C$^{17}$O line emission, due to the exceedingly high abundance of CO in molecular clouds. The optical depth of spectral lines from the CO isotopologues could still be high. Consequently, observation of other molecular isotopologues is crucial to determine the highly accurate $^{18}$O/$^{17}$O ratio.

Considering the importance of observing $^{17}$O species of methanol, spectroscopic measurements of this molecule in the millimeter to submillimeter region should be carried out. In this paper, we report the laboratory-based molecular emission spectroscopy for $^{12}$CH$_3$$^{17}$OH by use of the Spectrometer Using superconductor MIXer REceiver (SUMIRE; Y. Watanabe et al. 2021) in the frequency range between 216 and 264 GHz and present the assignments for the main spectral lines (Q-branch transitions) and the molecular constants obtained from our measurements.

## 2. Experiment

### 2.1. Measurement Procedure and Sample

The laboratory-based molecular emission spectrum measurements by means of SUMIRE have been carried out in the frequency range between 216 and 264 GHz. SUMIRE is the self-manufactured millimeter and submillimeter spectrometer apparatus coupled to an Atacama Large Millimeter/submillimeter Array (ALMA)-type cartridge receiver using a Superconductor-Insulator-Superconductor (SIS) mixer (A. R. Kerr et al. 2014). Figure 1 shows a schematic diagram of the SUMIRE setup. We adopted a gas cell made of a borosilicate glass with a diameter of 100 mm and a length of 1997 mm for the measurements. A liquid sample is placed in a customized branched cannula with a cock. Initially, the gas cell is evacuated to a pressure of $2 \times 10^{-4}$ Pa. When the cock is opened, the vaporized sample enters the gas cell until its pressure reaches 0.5 Pa.

The radio waves emitted from the gas molecules inside of the gas cell are concentrated by the ultrahigh molecular weight polyethylene (UHMWPE) lenses attached to both ends of the cell and heading to the ALMA-type cartridge receiver. As shown in Figure 1, a chopper covered by an absorber at room temperature is located between the one end of the gas cell having an UHMWPE lens and the liquid nitrogen container. When the gas cell is evacuated, we can measure the background emissions at liquid nitrogen temperature (77 K) and at room temperature (296 K) by removing and inserting the chopper, respectively. When we remove the chopper and fill the glass cell with sample gas at room temperature, we can obtain emission lines from the sample gas against the liquid nitrogen (77 K). The background (blackbody) emissions measured in two different temperatures are used to scale the intensities of the molecular lines. This is the so-called Hot-Cold method (Y-factor method) with a hot temperature load ($T_{\text{hot}}$: room temperature, 296 K) and a cold one ($T_{\text{bg}}$: liquid nitrogen temperature, 77 K) (more details in Y. Watanabe et al. 2021; Y. Ohno et al. 2022).
The SIS mixer downconverts the high frequency (~230 GHz) to a signal in the range 4–8 GHz by mixing the local oscillator signal, whose frequency is called the local frequency. After amplification of the converted signal under low-temperature conditions, it again downconverts to frequencies of 0.2–2.2 GHz that can be input to the autocorrelation spectrometer. Since the receiver is a two-sideband (2SB) system, the signal is separated into upper sideband (USB) and lower sideband (LSB) by shifting the phase using two 90° hybrids and outputting them. The measurements are carried out while switching the local frequency from 224 to 256 GHz by 4 GHz. Each local frequency has four basebands, LSBO, LSBI, USBI, and USBO (O: Outer; I: Inner), which correspond to -8 to -6 GHz offset from the local frequency in LSB, -6 to -4 GHz in LSB, +4 to +6 GHz in USB, and +6 to +8 GHz in USB, respectively.

A commercially available liquid ¹²CH₃¹⁷OH sample (methanol ¹⁷O; Sigma-Aldrich, Merck KGaA) is used for the measurements. The conditions of this sample are chemical purity (CP) of 98% (i.e., 98% is methanol) and isotopic purity (IP) ≥80% (i.e., ≥80% of oxygen atom is ¹⁷O). For comparative verification, spectral lines of ¹²CH₃¹⁶OH and ¹²CH₃¹⁸OH are also measured using the same method. The ¹²CH₃¹⁶OH sample fabricated by the Fujifilm Wako Pure Chemical Corporation (CP: 99.8+%; natural isotopic ratios) and the ¹²CH₃¹⁸OH one produced by Merck KGaA (CP: 98%; IP: 95% ¹⁸O) are employed.

## 2.2. Analytical Method

All the spectral lines measured with each local frequency are submitted to a Gaussian fitting in order to determine the transition frequency and the line width and peak intensity (more details in T. Oyama et al. 2023). For that, we have developed a code, which is named the Automatically Quality-Updating Application for Line Investigation by the Onground-Universe Spectroscopy (AQUALIOUS), which performs the Gaussian fitting automatically for all lines. This code is specialized for the data obtained from SUMIRE. In our experiment, background and sample measurements were repeatedly performed at least 10 times under a temperature of 295.8±1.5 K and a gas pressure of 0.51±0.03 Pa. The signal-to-noise ratio can be improved by increasing the cumulative number of measurements. For that, AQUALIOUS has a function that enables us to average the 10 measured line spectral data sets at first. Subsequent baseline corrections, Gaussian fitting, and spurious elimination lead to an improved averaged spectrum and an increased accuracy for the quantities retrieved in the next step such as the temperature and the column density of the sample gas in the cell.

The column density and the temperature of ¹²CH₃¹⁷OH have been derived by using the rotation diagram method under the assumption of the local thermodynamic equilibrium (LTE) approximation. The experimentally measured intensity ΔT_m can be described as

$$ \Delta T_m = (T - T_{bg})(1 - \exp(-\tau_{\nu})), \quad (1) $$

where T is the temperature of the molecular gas, Tbg is the liquid nitrogen temperature of 77 K, and τν is the optical depth

of the transition with a frequency of ν expressed as

$$ \tau_{\nu} = \frac{8\pi^3 S \mu^2 N}{3h\Delta\nu U(T)} \left\{ \exp\left(\frac{h\nu}{kT}\right) - 1 \right\} \exp\left(-\frac{E_u}{kT}\right), \quad (2) $$

where Sμ² is the intrinsic line strength of each transition, ν is the frequency of the transition line, N is the column density, h is the Planck constant, U(T) is the partition function at the temperature of T, Δν is the line width, k is the Boltzmann constant, and E_u is the upper-state energy (see N. Sakai et al. 2008). The partition function is written as follows (T. Oyama et al. 2023):

$$ U(T) = U_r(T) U_v(T) U_t(T), \quad (3) $$

$$ U_r(T) = \sqrt{\frac{\pi (kT)^3}{h^3 ABC}}, \quad (4) $$

$$ U_v(T) = \prod_{i=1}^{11} \frac{1}{1 - e^{-\omega_i/kT}}, \quad (5) $$

and

$$ U_t(T) = \sum_{v_t=0}^{4} e^{-E_{vt}/kT}, \quad (6) $$

where A, B, and C are rotational constants; ωᵢ is the fundamental vibration frequency for the ith mode (i = 1–11) excluding the torsional mode; and Eᵣᵢ is the energy of the torsional substate of vᵣ, which are caused by the internal rotation of the methyl group. The vibrational frequency and energies of the torsion-rotation levels are fixed to the values for ¹²CH₃¹⁶OH. When we assume an optically thin condition for the detected spectral lines, the formula can be simplified as

$$ \ln \left( \frac{3kI}{8\pi^3 S \mu^2 \nu} \right) = \ln \left\{ \frac{N}{U(T)} \left( 1 - \frac{77 K}{T} \right) \right\} - \frac{E_u}{kT} \quad (7) $$

(more details are given in Y. Watanabe et al. 2021). Here I represents the integrated intensity of the spectral line. In Section 4.2, the column density of each isotopologue is calculated using this equation.

## 3. Theoretical Model

For assigning and fitting the line positions, we used the so-called rho-axis method (RAM) described theoretically by J. T. Hougen et al. (1994) for internal rotors similar to methanol. This method takes its name from the choice of the axis system to calculate the energy levels and transitions. As described by C. C. Lin & J. D. Swalen (1959) in their review article, to speed up the numerical computations, D. G. Burkhard & D. M. Dennison (1951) proposed to use a different axis system than the principal axis system. Their objective was to eliminate two of the rotation-torsion coupling terms in the Hamiltonian, the -2FρₓpγPₓ and -2FρᵧpγPᵧ terms, where F is the internal rotation constant, pγ is the torsional angular momentum, P is the rotational angular momentum operator, and ρ is the coupling parameter. The resultant Hamiltonian operator then contains only the -2FρₓpγPₓ term, which gives rise to ΔK = 0 matrix elements. In the new axis system, the new z-axis is parallel to the ρ vector, since by definition we then have ρₓ = ρᵧ = 0. The method was thus named the “rho-axis method” (J. T. Hougen et al. 1994). This formalism has
been implemented in the BELGI code and applied with success to a number of internal rotor molecules with intermediate to low barriers, observed in the interstellar medium (ISM; I. Kleiner 2019), i.e., acetaldehyde, CH₃CHO (I. Kleiner et al. 1996), acetamide CH₃CONH₂ (J. M. Hollis et al. 2006), acetic acid CH₃COOH (V. Ilyushin et al. 2008), and various isotopologues of methyl formate HCOOCH₃ (M. Carvajal et al. 2007; 2009a; 2009b, 2010; B. Tercero & M. Carvajal 2012; I. Haykal et al. 2014; C. Favre et al. 2014; C. Duan et al. 2015). It was also applied to methanol ¹²CH₃¹⁶OH by L.-H. Xu et al. (2008) and its isotopologues ¹³CH₃¹⁶OH (L.-H. Xu et al. 2014), ¹²CH₃¹⁶OD (M. S. Walsh et al. 2000), and ¹²CD₃¹⁶OH (M. S. Walsh et al. 1998), as well as methyl mercaptan (CH₃SH; L.-H. Xu et al. 2012). The same RAM formalism is also implemented in the RAM36 code developed by V. V. Ilyushin et al. (2010), written initially for internal rotors with a sixfold symmetry torsional potential like toluene, but which can also deal with threefold symmetry torsional potential. It was applied to a number of astrophysical molecules and in particular to ¹²CD₃¹⁶OH (V. V. Ilyushin et al. 2022), ¹²CD₃¹⁶OD (V. V. Ilyushin 2023), and ¹²CH₃¹⁶OD (V. V. Ilyushin et al. 2024).

In the RAM “global” approach the A-symmetry and E-symmetry states are taken into account simultaneously in the fit. The torsional states $v_t = 0, 1, 2, ...$ (up to a truncation limit of $v_t = 8$) and the interactions within those torsion-rotation energy levels are also included in the rotation-torsion Hamiltonian matrix elements (I. Kleiner 2010). The various rotational, internal rotation, and coupling terms between internal and global rotation, as well as their physical meaning, in the Hamiltonian for the fit of the ¹²CH₃¹⁷OH have been previously described for the normal methanol species (L.-H. Xu et al. 2008) and will not be repeated here.

## 4. Results and Discussion

Experimentally measured molecular lines of ¹²CH₃¹⁷OH in the frequency range between 216 and 264 GHz are shown in Figure 2, together with the molecular lines of ¹²CH₃¹⁶OH and ¹²CH₃¹⁸OH obtained from SUMIRE as a comparison. In the following subsections, we describe line assignments of ¹²CH₃¹⁷OH and determination of the molecular constants. For fitting the line frequencies, we used the BELGI code (J. T. Hougen et al. 1994; L.-H. Xu et al. 2008) The values determined for the molecular constants are tentative since the measurements are carried out within the limited frequency range in this study; however, astronomical investigation, i.e., searching for ¹²CH₃¹⁷OH and derivation of its column density toward astronomical sources, could be conducted as discussed in Section 4.6. For that reason, after spectroscopic measurements are carried out over a wider frequency range, the molecular constants should be updated with line data over a wider frequency range (see comments in the last part of Section 4.6). In order to verify the experimentally measured ¹²CH₃¹⁷OH data and its transition assignments, we have also performed the spectroscopic measurements of ¹²CH₃¹⁶OH and ¹²CH₃¹⁸OH. Those spectral lines are analyzed based on their line parameters and molecular constants previously reported in the literature (L.-H. Xu et al. 2008; J. Fisher et al. 2007).

### 4.1. Line Assignments of ¹²CH₃¹⁷OH

In total, 528 lines have been identified as spectral lines emitted from the sample gas, with their intensities at least 3 times higher

than the rms noise level. Among them, 113 lines are assigned as ¹²CH₃¹⁶OH, which is included in the sample gas as mentioned in the previous subsection. The characteristic M-shape pattern of Q-branch transitions can be identified around 252 GHz with an intensity of 2–3 K. Here the assignment is confirmed by the Cologne Database for Molecular Spectroscopy (CDMS; Müller, H. S. P. et al. 2005). In the >100 GHz frequency region, molecular transitions of ¹²CH₃¹⁷OH have not yet been reported in either experiments or theoretical calculations. For that reason, the assignment of ¹²CH₃¹⁷OH was initially carried out based on experimental outcomes obtained from the ¹²CH₃¹⁶OH (Y. Watanabe et al. 2021; T. Oyama et al. 2023). We have preferentially assigned the strongest line series, which is composed of the Q-branch transitions in this frequency range showing the M shape seen in all the panels of Figure 2. These plots show that as the mass number of the oxygen isotope decreases, the frequency corresponding to the center of the M shape in the spectral pattern tends to decrease from 262 to 252 GHz. Despite being several times weaker than the strongest Q-branch lines forming the M-shaped feature, spectral lines of ¹²CH₃¹⁸OH are detected in astronomical sources. Even with its lower abundance, strong ¹²CH₃¹⁷OH lines, such as those in the Q-branch, might also still be detectable (see discussion in Section 4.6).

An important physical distinction among the three isotopologues is a nuclear spin quantum number. Neither ¹²CH₃¹⁶OH nor ¹²CH₃¹⁸OH displays quadrupole coupling hyperfine structure, whereas this coupling is displayed in ¹²CH₃¹⁷OH owing to the nuclear spin of 5/2 of the ¹⁷O atom. As shown in Figure 3, only ¹²CH₃¹⁷OH exhibits the clear double and/or triple peaks in low-J transitions, due to the nuclear quadrupole interaction, unlike in ¹²CH₃¹⁶OH and ¹²CH₃¹⁸OH. Detailed analysis on these splitting peaks will be the subject of future studies. In order to reproduce those hyperfine splittings with our theoretical model, it is necessary to perform the spectroscopic measurements in lower frequency ranges since only a few splittings were observed in the frequency range of the present measurements. Table 1 reports the 71 newly assigned transitions of ¹²CH₃¹⁷OH and the 23 transitions from previous work (Y. Hoshino et al. 1991). This table lists experimentally measured line frequency (Exp. Freq.) in MHz, theoretically calculated line frequency with the optimized molecular constants listed in Table 2 (Calc. Freq.: see Section 4.3 for details) in MHz, the experimental line intensity in K, the line width (FWHM) in MHz, the upper-state energy ($E_u$) in cm⁻¹, and the calculated line strength ($S\mu^2$) in D². The frequency error is limited by the hyperfine splitting. It is typically a few dozen to a hundred of kHz.

### 4.2. Rotation Diagram of ¹²CH₃¹⁷OH

The rotation diagram of ¹²CH₃¹⁷OH is shown in Figure 4. Sixty transitions out of 71 were selected in Table 1, as overlapping transitions were excluded. Additionally, as mentioned in the previous section, the ¹²CH₃¹⁷OH sample is not 100% isotopically pure and thus also contains lines belonging to ¹²CH₃¹⁶OH; 113 lines of this species are detected in the spectrum. Indeed, the characteristic M-shape that originated from ¹²CH₃¹⁶OH is seen around 251.91 GHz, with an intensity peak of approximately 2 K in the ¹²CH₃¹⁷OH measurement (Figure 2, middle panel). For verifying the column density of ¹²CH₃¹⁷OH derived from the rotation diagram, it is essential to derive the column density of ¹²CH₃¹⁶OH in the ¹²CH₃¹⁷OH sample as well. For ¹²CH₃¹⁷OH, the partition function $U(T_{gas})$ is calculated to be 9363.632 at the temperature of 295.8 ± 0.5 K,
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Experimentally measured spectra of <sup>12</sup>CH<sub>3</sub><sup>16</sup>OH (top panel), <sup>12</sup>CH<sub>3</sub><sup>17</sup>OH (middle panel), and <sup>12</sup>CH<sub>3</sub><sup>18</sup>OH (bottom panel) in the frequency range between 216 and 264 GHz. All spectra are measured by means of SUMIRE.</figcaption></figure>

which is the experimental condition of the measurement. This
value of the partition function is calculated using the same
method as in T. Oyama et al. (2023). The rotation term is

calculated using the value of the rotational constants *A*, *B*, and
*C* obtained from the fit with the BELGI code (L.-H. Xu et al.
2008) for methanol. The vibrational contribution is estimated
<figure><img src="image_6.png" /><figcaption>Figure 3. Experimentally obtained *Q*-branch transitions of $^{12}$CH$_{3}^{16}$OH (left panel), $^{12}$CH$_{3}^{17}$OH (middle panel), and $^{12}$CH$_{3}^{18}$OH (right panel).</figcaption></figure>

from the fundamental frequencies of $^{12}$CH$_{3}^{16}$OH (A. Serrallach & R. Meyer 1974) and the torsional contribution from its torsional energies (J. M. Bowman et al. 2007). Because of these assumptions, the partition function could vary by about a few to 10%. The partition function value of $^{12}$CH$_{3}^{18}$OH was also derived using the same method at 300 K and is roughly 3.9% larger than the value of 38749 reported in CDMS. Note that to match the definition of the present partition function with that of CDMS, we employ a statistical weight of 4 for $^{12}$CH$_{3}^{16}$OH and $^{12}$CH$_{3}^{18}$OH.

By the same token, we take the $U(295.8^{+1.5}_{-0.6})$ of $^{12}$CH$_{3}^{16}$OH to be 36591.944. The derived column densities of $^{12}$CH$_{3}^{16}$OH and $^{12}$CH$_{3}^{17}$OH are $(2.91 \pm 0.04) \times 10^{15} \text{ cm}^{-2}$ and $(2.44 \pm 0.03) \times 10^{16} \text{ cm}^{-2}$, respectively. Therefore, the total column density of the measured methanol ("CH$_{3}^{17}$OH" sample; Sigma-Aldrich, Merck KGaA) is roughly calculated to be $(2.73^{+0.03}_{-0.04}) \times 10^{16} \text{ cm}^{-2}$. As we describe in Section 4.5, some strong transitions by $^{13}$CH$_{3}^{17}$OH are detected in our data. The intensities are about 1/97 (1.0 $\pm$ 0.1%) of the $^{12}$CH$_{3}^{17}$OH, which is consistent with the natural $^{12}$C/$^{13}$C ratio of 1/92 (1.1%). We therefore assume that the contribution to the column density of $^{13}$CH$_{3}^{17}$OH and $^{13}$CH$_{3}^{16}$OH was 1/92 that of the $^{12}$C species. Consequently, we evaluate the total column density of methanol in our data to be $(2.76 \pm 0.21) \times 10^{16} \text{ cm}^{-2}$. The error is estimated by the root squared sum of the errors originating from the intensity calibration of the system (5.7%; see T. Oyama et al. 2023), from the partition function (5%), and from the fitting error in the LTE calculations ($^{+1.4}_{-1.5}$%). The temperatures of $^{12}$CH$_{3}^{16}$OH and $^{12}$CH$_{3}^{17}$OH derived from the rotation diagram are 297.6 $\pm$ 2.6 K and 291.5 $\pm$ 1.6 K, respectively. Note that when we fix the rotational temperature in the LTE fit to be room temperature (295.8$^{+1.5}_{-0.6}$ K), the total methanol column density is found to be $(2.68 \pm 0.21) \times 10^{16} \text{ cm}^{-2}$, which is not much different from the value above. There are few percent differences between the temperature values obtained from the rotation diagram of $^{12}$CH$_{3}^{16}$OH and $^{12}$CH$_{3}^{17}$OH and the actual experimental temperature value. The discrepancy between the experimental temperature value and the calculated one might be due to the uncertainty on the line intensity calibration and/or on the line strength calculation.

The total column density of the $^{12}$CH$_{3}^{17}$OH sample derived from the rotation diagram is verified by comparison with the value obtained from the ideal gas law, which is $(2.40^{+0.14}_{-0.11}) \times 10^{16} \text{ cm}^{-2}$ at the actual temperature in the experiment, 295.8$^{+1.5}_{-0.6}$ K. This value of the column density takes into account the CP of 98% and the molecule adsorption in the inner wall of the gas cell ($2.0 \pm 0.4$%; more details in T. Oyama et al. 2023). The pressure uncertainty includes several contributions: 2% due to round-off

errors, 0.8% from absolute scale error, and a 15% error due to zero-point offset, this last contribution being the largest. This induces a 16% error on the column density, leading to a value of $(2.40^{+0.39}_{-0.38}) \times 10^{16} \text{ cm}^{-2}$, which agrees within uncertainty with the column density derived from the measured spectrum.

### 4.3. Determination of Molecular Constants of $^{12}$CH$_{3}^{17}$OH

The experimentally measured spectrum of $^{12}$CH$_{3}^{17}$OH displays a line pattern similar to that of $^{12}$CH$_{3}^{16}$OH as shown in Figure 2, which is consistent with the one reported in our previous work (Y. Watanabe et al. 2021). A preliminary assignment of the strongest *Q*-branch and the relatively strong *R*-branch in the frequency ranges of ALMA Band 6 have been carried out by comparison with these spectra (Table 1). Rotational transitions displaying well-resolved hyperfine splittings are omitted from the table. A detailed analysis of the nuclear quadrupole interactions needs further experimental measurements covering a wider frequency range. Based on the assignment, the molecular constants have been determined by using the BELGI program (L.-H. Xu et al. 2008). The weights of the least-squares analysis are taken to be equal to the inverse of the experimental uncertainty squared. Twenty-three previously measured transitions in the range 29–95 GHz (Y. Hoshino et al. 1991) are included in our analysis, with an experimental uncertainty of 150 kHz. The uncertainty of the transitions measured in this work is 30 kHz for isolated lines and 70 kHz or 100 kHz for blended lines. The fitting procedure began considering low-*J* transitions and by first floating only the rotational parameters *A*, *B*, *C*, and *D*<sub>ab</sub>; the height of the barrier hindering the internal rotation (*V*<sub>3</sub>); and *ρ*, the rotation–torsion coupling parameter. Transitions characterized by a higher *J* value were then gradually included in the fit, allowing us to vary a total of 25 parameters, including fourth- and sixth-order parameters. The internal rotation constant *F* was constrained to the value determined for $^{12}$CH$_{3}^{16}$OH (L.-H. Xu et al. 2008). The optimized molecular constants are listed in Table 2.

Ninety-four lines were fitted with a unitless standard deviation of 1.037. However, we emphasize that, due to the restricted number of transitions in the excited states $v_t = 1$ and 2, the observed–calculated values for some of those line frequencies are high. The determined constants of $^{12}$CH$_{3}^{17}$OH are close to those of $^{12}$CH$_{3}^{16}$OH also listed in Table 2. Note that we do not include the whole set of 119 parameters for $^{12}$CH$_{3}^{17}$OH, although it is used in L.-H. Xu et al. (2008) for $^{12}$CH$_{3}^{16}$OH (see their Table 2). We only include those corresponding to our nonzero parameters in the $^{12}$CH$_{3}^{17}$OH fit. Due to the very limited spectral range and thus the limited range of *J* and *K* values investigated in our study, the
<table><caption>Table 1
Experimentally Measured and Predicted Transition Frequencies of $^{12}$CH$_{3}$$^{17}$OH</caption><thead><tr><th>v<sub>t</sub>'</th><th>J'</th><th>K'<sub>a</sub></th><th>K'<sub>c</sub></th><th>P<sup>ta</sup></th><th>v<sub>t</sub>''</th><th>J''</th><th>K''<sub>a</sub></th><th>K''<sub>c</sub></th><th>P''<sup>a</sup></th><th>Exp. Freq.<sup>b</sup> (MHz)</th><th>Calc. Freq.<sup>c</sup> (MHz)</th><th>Exp. - Calc.<sup>d</sup> (MHz)</th><th>Exp. I<sup>e</sup><sub>peak</sub> (K)</th><th>Exp. ΔI<sup>f</sup> (MHz)</th><th>E<sup>g</sup><sub>upper</sub> (cm<sup>-1</sup>)</th><th>Calc. Sμ<sup>2h</sup> (D<sup>2</sup>)</th><th>Exp. Sμ<sup>2k</sup> (D<sup>2</sup>)</th></tr></thead><tbody><tr><td colspan="18">A state</td></tr><tr><td>0</td><td>5</td><td>0</td><td>5</td><td>+</td><td>→</td><td>0</td><td>4</td><td>0</td><td>4</td><td>+</td><td>236506.555(70)</td><td>236506.433</td><td>0.122</td><td>9.255(19)</td><td>0.692(2)</td><td>23.670</td><td>4.045</td><td>4.55(80)</td></tr><tr><td>0</td><td>5</td><td>1</td><td>5</td><td>+</td><td>→</td><td>0</td><td>4</td><td>1</td><td>4</td><td>+</td><td>234538.716(70)</td><td>234538.742</td><td>-0.026</td><td>8.322(17)</td><td>0.731(2)</td><td>33.534</td><td>3.889</td><td>4.40(78)<sup>l</sup></td></tr><tr><td>0</td><td>5</td><td>1</td><td>4</td><td>-</td><td>→</td><td>0</td><td>4</td><td>1</td><td>3</td><td>-</td><td>238548.918(30)</td><td>238548.922</td><td>-0.004</td><td>10.389(11)</td><td>0.626(1)</td><td>33.936</td><td>3.890</td><td>4.60(81)</td></tr><tr><td>0</td><td>5</td><td>2</td><td>3</td><td>+</td><td>→</td><td>0</td><td>4</td><td>2</td><td>2</td><td>+</td><td>236595.725(30)</td><td>236595.718</td><td>0.007</td><td>6.408(18)</td><td>0.784(2)</td><td>49.834</td><td>3.428</td><td>3.82(68)</td></tr><tr><td>0</td><td>5</td><td>2</td><td>4</td><td>-</td><td>→</td><td>0</td><td>4</td><td>2</td><td>3</td><td>-</td><td>236553.557(70)</td><td>236553.510</td><td>0.047</td><td>9.256(16)</td><td>0.945(2)</td><td>49.832</td><td>3.428</td><td>3.82(68)<sup>m</sup></td></tr><tr><td>0</td><td>5</td><td>3</td><td>3</td><td>+</td><td>→</td><td>0</td><td>4</td><td>3</td><td>2</td><td>+</td><td>236546.113(100)</td><td>236545.948</td><td>0.165</td><td>6.143(14)</td><td>1.242(3)</td><td>58.430</td><td>2.579</td><td>2.90(51)<sup>n</sup></td></tr><tr><td>0</td><td>5</td><td>3</td><td>2</td><td>-</td><td>→</td><td>0</td><td>4</td><td>3</td><td>1</td><td>-</td><td>236546.113(100)</td><td>236545.290</td><td>-0.177</td><td>6.143(14)</td><td>1.242(3)</td><td>58.430</td><td>2.579</td><td>2.90(51)<sup>n</sup></td></tr><tr><td>0</td><td>5</td><td>3</td><td>3</td><td>+</td><td>→</td><td>0</td><td>5</td><td>2</td><td>4</td><td>-</td><td>257760.457(70)<sup>l</sup></td><td>257760.102</td><td>0.355</td><td>5.141(46)</td><td>0.698(8)</td><td>58.430</td><td>4.121</td><td>...o</td></tr><tr><td>0</td><td>6</td><td>3</td><td>3</td><td>-</td><td>→</td><td>0</td><td>6</td><td>2</td><td>4</td><td>+</td><td>257619.661(70)</td><td>257619.645</td><td>0.016</td><td>7.715(18)</td><td>1.052(3)</td><td>67.898</td><td>5.220</td><td>5.48(97)</td></tr><tr><td>0</td><td>6</td><td>3</td><td>4</td><td>+</td><td>→</td><td>0</td><td>6</td><td>2</td><td>5</td><td>-</td><td>257766.060(70)<sup>l</sup></td><td>257765.986</td><td>0.074</td><td>7.748(17)</td><td>1.260(4)</td><td>67.898</td><td>5.216</td><td>...o</td></tr><tr><td>0</td><td>7</td><td>3</td><td>4</td><td>-</td><td>→</td><td>0</td><td>7</td><td>2</td><td>5</td><td>+</td><td>257530.975(30)</td><td>257530.971</td><td>0.004</td><td>9.503(20)</td><td>0.895(2)</td><td>78.944</td><td>6.274</td><td>6.3(11)</td></tr><tr><td>0</td><td>7</td><td>3</td><td>5</td><td>+</td><td>→</td><td>0</td><td>7</td><td>2</td><td>6</td><td>+</td><td>257793.322(70)</td><td>257793.321</td><td>0.001</td><td>9.964(19)</td><td>0.958(2)</td><td>78.944</td><td>6.267</td><td>7.0(12)</td></tr><tr><td>0</td><td>8</td><td>3</td><td>5</td><td>-</td><td>→</td><td>0</td><td>8</td><td>2</td><td>6</td><td>+</td><td>257416.109(30)</td><td>257416.111</td><td>-0.002</td><td>10.901(21)</td><td>0.804(2)</td><td>91.567</td><td>7.302</td><td>7.0(12)</td></tr><tr><td>0</td><td>8</td><td>3</td><td>6</td><td>+</td><td>→</td><td>0</td><td>8</td><td>2</td><td>7</td><td>-</td><td>257851.308(70)</td><td>257851.310</td><td>-0.002</td><td>11.815(20)</td><td>0.867(2)</td><td>91.567</td><td>7.288</td><td>8.2(15)<sup>l</sup></td></tr><tr><td>0</td><td>9</td><td>3</td><td>6</td><td>-</td><td>→</td><td>0</td><td>9</td><td>2</td><td>7</td><td>+</td><td>257270.089(30)</td><td>257270.088</td><td>0.001</td><td>13.252(22)</td><td>0.749(1)</td><td>105.77</td><td>8.312</td><td>8.6(15)</td></tr><tr><td>0</td><td>9</td><td>3</td><td>7</td><td>+</td><td>→</td><td>0</td><td>9</td><td>2</td><td>8</td><td>-</td><td>257950.301(30)</td><td>257950.304</td><td>-0.003</td><td>13.202(21)</td><td>0.811(1)</td><td>105.77</td><td>8.287</td><td>9.2(16)</td></tr><tr><td>0</td><td>10</td><td>3</td><td>7</td><td>-</td><td>→</td><td>0</td><td>10</td><td>2</td><td>8</td><td>+</td><td>257087.598(30)</td><td>257087.600</td><td>-0.002</td><td>15.226(22)</td><td>0.721(1)</td><td>121.54</td><td>9.311</td><td>10.5(19)</td></tr><tr><td>0</td><td>10</td><td>3</td><td>8</td><td>+</td><td>→</td><td>0</td><td>10</td><td>2</td><td>9</td><td>-</td><td>258101.726(30)</td><td>258101.729</td><td>-0.003</td><td>14.362(24)</td><td>0.783(2)</td><td>121.54</td><td>9.270</td><td>10.5(19)</td></tr><tr><td>0</td><td>11</td><td>3</td><td>8</td><td>-</td><td>→</td><td>0</td><td>11</td><td>2</td><td>9</td><td>+</td><td>256863.143(30)</td><td>256863.145</td><td>-0.002</td><td>15.473(22)</td><td>0.697(1)</td><td>138.90</td><td>10.304</td><td>11.2(20)</td></tr><tr><td>0</td><td>11</td><td>3</td><td>9</td><td>+</td><td>→</td><td>0</td><td>11</td><td>2</td><td>10</td><td>-</td><td>258318.011(30)</td><td>258318.014</td><td>-0.003</td><td>14.951(22)</td><td>0.765(1)</td><td>138.90</td><td>10.238</td><td>11.6(21)</td></tr><tr><td>0</td><td>12</td><td>3</td><td>9</td><td>-</td><td>→</td><td>0</td><td>12</td><td>2</td><td>10</td><td>+</td><td>256591.171(30)</td><td>256591.170</td><td>0.001</td><td>15.536(23)</td><td>0.683(1)</td><td>157.83</td><td>11.293</td><td>12.1(21)</td></tr><tr><td>0</td><td>12</td><td>3</td><td>10</td><td>+</td><td>→</td><td>0</td><td>12</td><td>2</td><td>11</td><td>-</td><td>258612.507(30)</td><td>258612.506</td><td>0.001</td><td>15.026(22)</td><td>0.753(1)</td><td>157.83</td><td>11.193</td><td>12.5(22)</td></tr><tr><td>0</td><td>13</td><td>3</td><td>10</td><td>-</td><td>→</td><td>0</td><td>13</td><td>2</td><td>11</td><td>+</td><td>256266.256(30)</td><td>256266.255</td><td>0.001</td><td>15.774(23)</td><td>0.674(1)</td><td>178.34</td><td>12.283</td><td>13.4(24)</td></tr><tr><td>0</td><td>13</td><td>3</td><td>11</td><td>+</td><td>→</td><td>0</td><td>13</td><td>2</td><td>12</td><td>-</td><td>258999.387(70)</td><td>258999.387</td><td>0.000</td><td>14.917(19)</td><td>0.727(1)</td><td>178.33</td><td>12.135</td><td>13.1(23)</td></tr><tr><td>0</td><td>14</td><td>3</td><td>11</td><td>-</td><td>→</td><td>0</td><td>14</td><td>2</td><td>12</td><td>+</td><td>255883.317(30)</td><td>255883.316</td><td>0.001</td><td>15.100(16)</td><td>0.668(1)</td><td>200.42</td><td>13.276</td><td>14.2(25)</td></tr><tr><td>0</td><td>14</td><td>3</td><td>12</td><td>+</td><td>→</td><td>0</td><td>14</td><td>2</td><td>13</td><td>-</td><td>259493.574(30)</td><td>259493.570</td><td>0.004</td><td>13.686(22)</td><td>0.745(1)</td><td>200.41</td><td>13.064</td><td>13.7(24)</td></tr><tr><td>0</td><td>15</td><td>3</td><td>12</td><td>-</td><td>→</td><td>0</td><td>15</td><td>2</td><td>13</td><td>+</td><td>255437.853(30)</td><td>255437.852</td><td>0.001</td><td>14.378(16)</td><td>0.662(1)</td><td>224.08</td><td>14.274</td><td>15.0(27)</td></tr><tr><td>0</td><td>15</td><td>3</td><td>13</td><td>+</td><td>→</td><td>0</td><td>15</td><td>2</td><td>14</td><td>-</td><td>260110.613(30)</td><td>260110.609</td><td>0.004</td><td>13.202(15)</td><td>0.741(1)</td><td>224.07</td><td>13.979</td><td>14.6(26)</td></tr><tr><td>0</td><td>16</td><td>3</td><td>13</td><td>-</td><td>→</td><td>0</td><td>16</td><td>2</td><td>14</td><td>+</td><td>254926.222(30)</td><td>254926.221</td><td>0.001</td><td>13.962(16)</td><td>0.659(1)</td><td>249.31</td><td>15.282</td><td>16.4(29)</td></tr><tr><td>0</td><td>16</td><td>3</td><td>14</td><td>+</td><td>→</td><td>0</td><td>16</td><td>2</td><td>15</td><td>-</td><td>260866.585(30)</td><td>260866.582</td><td>0.003</td><td>12.580(15)</td><td>0.743(1)</td><td>249.30</td><td>14.879</td><td>15.7(28)</td></tr><tr><td>0</td><td>17</td><td>3</td><td>14</td><td>-</td><td>→</td><td>0</td><td>17</td><td>2</td><td>15</td><td>+</td><td>254345.949(30)</td><td>254345.949</td><td>0.000</td><td>13.005(16)</td><td>0.658(1)</td><td>276.12</td><td>16.302</td><td>17.6(31)</td></tr><tr><td>0</td><td>17</td><td>3</td><td>15</td><td>+</td><td>→</td><td>0</td><td>17</td><td>2</td><td>16</td><td>-</td><td>261777.989(30)</td><td>261777.987</td><td>0.002</td><td>11.677(15)</td><td>0.753(1)</td><td>276.10</td><td>15.762</td><td>16.6(29)</td></tr><tr><td>0</td><td>18</td><td>3</td><td>15</td><td>-</td><td>→</td><td>0</td><td>18</td><td>2</td><td>16</td><td>+</td><td>253696.078(30)</td><td>253696.078</td><td>0.000</td><td>11.881(14)</td><td>0.651(1)</td><td>304.50</td><td>17.338</td><td>18.1(32)</td></tr><tr><td>0</td><td>18</td><td>3</td><td>16</td><td>+</td><td>→</td><td>0</td><td>18</td><td>2</td><td>17</td><td>-</td><td>262861.617(30)</td><td>262861.623</td><td>-0.006</td><td>10.515(27)</td><td>0.761(2)</td><td>304.47</td><td>16.628</td><td>17.2(30)</td></tr><tr><td>0</td><td>19</td><td>3</td><td>16</td><td>-</td><td>→</td><td>0</td><td>19</td><td>2</td><td>17</td><td>+</td><td>252977.540(30)</td><td>252977.543</td><td>-0.003</td><td>10.672(14)</td><td>0.649(1)</td><td>334.46</td><td>18.392</td><td>18.7(33)</td></tr><tr><td>0</td><td>20</td><td>3</td><td>17</td><td>-</td><td>→</td><td>0</td><td>20</td><td>2</td><td>18</td><td>+</td><td>252193.577(30)</td><td>252193.579</td><td>-0.002</td><td>9.858(14)</td><td>0.646(1)</td><td>365.99</td><td>19.470</td><td>20.2(36)</td></tr><tr><td>0</td><td>21</td><td>3</td><td>18</td><td>-</td><td>→</td><td>0</td><td>21</td><td>2</td><td>19</td><td>+</td><td>251350.144(30)</td><td>251350.147</td><td>-0.003</td><td>8.634(15)</td><td>0.644(1)</td><td>399.10</td><td>20.574</td><td>20.8(37)</td></tr><tr><td>0</td><td>22</td><td>3</td><td>19</td><td>-</td><td>→</td><td>0</td><td>22</td><td>2</td><td>20</td><td>+</td><td>250456.376(30)</td><td>250456.377</td><td>-0.001</td><td>7.432(15)</td><td>0.645(2)</td><td>433.78</td><td>21.710</td><td>21.6(38)</td></tr><tr><td>0</td><td>23</td><td>3</td><td>20</td><td>-</td><td>→</td><td>0</td><td>23</td><td>2</td><td>21</td><td>+</td><td>249525.000(30)</td><td>249525.002</td><td>-0.002</td><td>6.836(12)</td><td>0.638(1)</td><td>470.04</td><td>22.881</td><td>23.4(41)</td></tr><tr><td>0</td><td>24</td><td>3</td><td>21</td><td>-</td><td>→</td><td>0</td><td>24</td><td>2</td><td>22</td><td>+</td><td>248572.791(30)</td><td>248572.791</td><td>0.000</td><td>5.977(12)</td><td>0.637(1)</td><td>507.87</td><td>24.091</td><td>24.5(43)</td></tr><tr><td>0</td><td>25</td><td>3</td><td>22</td><td>-</td><td>→</td><td>0</td><td>25</td><td>2</td><td>23</td><td>+</td><td>247620.933(30)</td><td>247620.931</td><td>0.002</td><td>5.260(12)</td><td>0.637(2)</td><td>547.27</td><td>25.346</td><td>26.3(47)</td></tr><tr><td>0</td><td>26</td><td>3</td><td>23</td><td>-</td><td>→</td><td>0</td><td>26</td><td>2</td><td>24</td><td>+</td><td>246695.376(30)</td><td>246695.374</td><td>0.002</td><td>4.287(12)</td><td>0.635(2)</td><td>588.25</td><td>26.649</td><td>26.4(47)</td></tr><tr><td>0</td><td>27</td><td>3</td><td>24</td><td>-</td><td>→</td><td>0</td><td>27</td><td>2</td><td>25</td><td>+</td><td>245827.101(30)</td><td>245827.095</td><td>0.006</td><td>3.890(11)</td><td>0.623(2)</td><td>630.80</td><td>28.003</td><td>29.3(52)</td></tr><tr><td>0</td><td>28</td><td>3</td><td>25</td><td>-</td><td>→</td><td>0</td><td>28</td><td>2</td><td>26</td><td>+</td><td>245052.256(30)</td><td>245052.252</td><td>0.004</td><td>3.261(11)</td><td>0.616(2)</td><td>674.92</td><td>29.413</td><td>30.3(54)</td></tr><tr><td>0</td><td>29</td><td>3</td><td>26</td><td>-</td><td>→</td><td>0</td><td>29</td><td>2</td><td>27</td><td>+</td><td>244412.228(30)</td><td>244412.224</td><td>0.004</td><td>2.729(11)</td><td>0.613(3)</td><td>720.62</td><td>30.879</td><td>30.9(55)</td></tr><tr><td>0</td><td>30</td><td>3</td><td>27</td><td>-</td><td>→</td><td>0</td><td>30</td><td>2</td><td>28</td><td>+</td><td>243953.502(30)</td><td>243953.501</td><td>0.001</td><td>2.244(13)</td><td>0.613(4)</td><td>767.89</td><td>32.402</td><td>32.5(58)</td></tr></tbody></table>
<table><caption>Table 1
(Continued)</caption><thead><tr><th>v<sub>t</sub>'</th><th>J'</th><th>K'<sub>a</sub></th><th>K'<sub>c</sub></th><th>P<sup>ra</sup></th><th>v<sub>t</sub>''</th><th>J''</th><th>K''<sub>a</sub></th><th>K''<sub>c</sub></th><th>P''<sup>ra</sup></th><th>Exp. Freq.<sup>b</sup><br>(MHz)</th><th>Calc. Freq.<sup>c</sup><br>(MHz)</th><th>Exp. - Calc.<sup>d</sup><br>(MHz)</th><th>Exp. I<sup>e</sup><sub>peak</sub><br>(K)</th><th>Exp. ΔI<sup>f</sup><br>(MHz)</th><th>E<sup>g</sup><sub>upper</sub><br>(cm<sup>-1</sup>)</th><th>Calc. Sμ<sup>2h</sup><br>(D<sup>2</sup>)</th><th>Exp. Sμ<sup>2k</sup><br>(D<sup>2</sup>)</th></tr></thead><tbody><tr><td>0</td><td>31</td><td>3</td><td>28</td><td>- →</td><td>0</td><td>31</td><td>2</td><td>29</td><td>+</td><td>243727.406(30)</td><td>243727.411</td><td>-0.005</td><td>1.838(13)</td><td>0.604(5)</td><td>816.74</td><td>33.983</td><td>33.4(59)</td></tr><tr><td>0</td><td>32</td><td>3</td><td>29</td><td>- →</td><td>0</td><td>32</td><td>2</td><td>30</td><td>+</td><td>243789.660(30)</td><td>243789.671</td><td>-0.011</td><td>1.484(13)</td><td>0.605(6)</td><td>867.16</td><td>35.619</td><td>33.9(60)</td></tr><tr><td>0</td><td>33</td><td>3</td><td>30</td><td>- →</td><td>0</td><td>33</td><td>2</td><td>31</td><td>+</td><td>244199.755(30)</td><td>244199.758</td><td>-0.003</td><td>1.245(11)</td><td>0.621(6)</td><td>919.15</td><td>37.306</td><td>37.8(67)</td></tr><tr><td>0</td><td>34</td><td>3</td><td>31</td><td>- →</td><td>0</td><td>34</td><td>2</td><td>32</td><td>+</td><td>245020.122(30)</td><td>245020.123</td><td>-0.001</td><td>1.012(11)</td><td>0.628(8)</td><td>972.72</td><td>39.037</td><td>39.5(70)</td></tr><tr><td>0</td><td>35</td><td>3</td><td>32</td><td>- →</td><td>0</td><td>35</td><td>2</td><td>33</td><td>+</td><td>246315.247(30)</td><td>246315.246</td><td>0.001</td><td>0.785(12)</td><td>0.630(11)</td><td>1027.9</td><td>40.804</td><td>39.6(71)</td></tr><tr><td>0</td><td>36</td><td>3</td><td>33</td><td>- →</td><td>0</td><td>36</td><td>2</td><td>34</td><td>+</td><td>248150.603(30)</td><td>248150.589</td><td>0.014</td><td>0.596(11)</td><td>0.656(15)</td><td>1084.6</td><td>42.596</td><td>41.0(74)</td></tr><tr><td>0</td><td>37</td><td>3</td><td>34</td><td>- →</td><td>0</td><td>37</td><td>2</td><td>35</td><td>+</td><td>250591.477(30)</td><td>250591.483</td><td>-0.006</td><td>0.440(14)</td><td>0.758(27)</td><td>1142.9</td><td>44.400</td><td>48.1(88)</td></tr><tr><td>0</td><td>38</td><td>3</td><td>35</td><td>- →</td><td>0</td><td>38</td><td>2</td><td>36</td><td>+</td><td>253701.991(70)</td><td>253701.998</td><td>-0.007</td><td>0.372(13)</td><td>0.759(30)</td><td>1202.8</td><td>46.202</td><td>50.9(92)<sup>l</sup></td></tr><tr><td>0</td><td>39</td><td>3</td><td>36</td><td>- →</td><td>0</td><td>39</td><td>2</td><td>37</td><td>+</td><td>257543.662(70)<sup>i</sup></td><td>257543.861</td><td>-0.199</td><td>0.259(21)</td><td>0.749(72)</td><td>1264.2</td><td>47.986</td><td>48.0(88)</td></tr><tr><td>0</td><td>40</td><td>3</td><td>37</td><td>- →</td><td>0</td><td>40</td><td>2</td><td>38</td><td>+</td><td>262174.777(70)<sup>i</sup></td><td>262175.466</td><td>-0.689</td><td>0.184(12)</td><td>0.926(72)</td><td>1327.2</td><td>49.734</td><td>52.4(101)<sup>l</sup></td></tr><tr><td>1</td><td>5</td><td>1</td><td>5</td><td>+</td><td>1</td><td>4</td><td>1</td><td>4</td><td>+</td><td>235691.898(100)</td><td>235691.944</td><td>-0.046</td><td>3.352(12)</td><td>0.672(3)</td><td>250.86</td><td>3.886</td><td>4.61(82)<sup>l</sup></td></tr><tr><td>1</td><td>5</td><td>1</td><td>4</td><td>- →</td><td>1</td><td>4</td><td>1</td><td>3</td><td>-</td><td>236163.163(100)</td><td>236163.200</td><td>-0.037</td><td>3.308(19)</td><td>0.662(4)</td><td>250.91</td><td>3.886</td><td>4.58(81)<sup>l</sup></td></tr><tr><td>2</td><td>5</td><td>0</td><td>5</td><td>+</td><td>2</td><td>4</td><td>0</td><td>4</td><td>+</td><td>235670.692(70)</td><td>235670.777</td><td>-0.085</td><td>1.940(12)</td><td>0.652(5)</td><td>377.64</td><td>4.039</td><td>4.79(85)<sup>l</sup></td></tr><tr><td>2</td><td>5</td><td>1</td><td>5</td><td>+</td><td>2</td><td>4</td><td>1</td><td>4</td><td>+</td><td>235201.640(70)<sup>i</sup></td><td>235201.418</td><td>0.222</td><td>1.035(12)</td><td>0.660(9)</td><td>500.61</td><td>3.877</td><td>4.69(83)</td></tr><tr><td>2</td><td>5</td><td>1</td><td>4</td><td>- →</td><td>2</td><td>4</td><td>1</td><td>3</td><td>-</td><td>236084.127(70)</td><td>236084.125</td><td>0.002</td><td>1.026(19)</td><td>0.660(14)</td><td>500.70</td><td>3.877</td><td>4.68(83)</td></tr><tr><td>2</td><td>5</td><td>2</td><td>3</td><td>+</td><td>2</td><td>4</td><td>2</td><td>2</td><td>+</td><td>235499.020(70)</td><td>235499.004</td><td>0.016</td><td>0.842(11)</td><td>0.748(11)</td><td>636.62</td><td>3.387</td><td>4.25(76)<sup>n</sup></td></tr><tr><td>2</td><td>5</td><td>2</td><td>4</td><td>- →</td><td>2</td><td>4</td><td>2</td><td>3</td><td>-</td><td>235499.020(70)</td><td>235499.031</td><td>-0.011</td><td>0.842(11)</td><td>0.748(11)</td><td>636.62</td><td>3.387</td><td>4.25(76)<sup>n</sup></td></tr><tr><td colspan="16">E state</td></tr><tr><td>0</td><td>5</td><td>-2</td><td>4</td><td>→</td><td>0</td><td>4</td><td>-2</td><td>3</td><td></td><td>236609.944(70)</td><td>236609.861</td><td>0.083</td><td>6.622(18)</td><td>0.780(2)</td><td>41.722</td><td>3.403</td><td>3.81(67)</td></tr><tr><td>0</td><td>5</td><td>-1</td><td>5</td><td>→</td><td>0</td><td>4</td><td>-1</td><td>4</td><td></td><td>236483.336(70)</td><td>236483.332</td><td>0.004</td><td>8.725(19)</td><td>0.699(2)</td><td>27.577</td><td>3.887</td><td>4.37(77)</td></tr><tr><td>0</td><td>5</td><td>0</td><td>5</td><td>→</td><td>0</td><td>4</td><td>0</td><td>4</td><td></td><td>236425.951(100)</td><td>236426.222</td><td>-0.271</td><td>8.874(19)</td><td>0.695(2)</td><td>32.725</td><td>4.041</td><td>4.58(81)</td></tr><tr><td>0</td><td>5</td><td>1</td><td>4</td><td>→</td><td>0</td><td>4</td><td>1</td><td>3</td><td></td><td>236584.285(70)</td><td>236584.222</td><td>0.063</td><td>8.723(19)</td><td>0.689(2)</td><td>38.248</td><td>4.021</td><td>4.53(80)</td></tr><tr><td>0</td><td>5</td><td>2</td><td>3</td><td>→</td><td>0</td><td>4</td><td>2</td><td>2</td><td></td><td>236615.569(30)</td><td>236615.575</td><td>-0.006</td><td>6.609(18)</td><td>0.781(2)</td><td>39.240</td><td>3.345</td><td>3.76(67)</td></tr><tr><td>0</td><td>5</td><td>3</td><td>2</td><td>→</td><td>0</td><td>4</td><td>3</td><td>1</td><td></td><td>236553.557(100)</td><td>236553.786</td><td>-0.229</td><td>9.256(16)</td><td>0.945(2)</td><td>57.042</td><td>2.590</td><td>2.89(51)<sup>m</sup></td></tr><tr><td>1</td><td>5</td><td>-2</td><td>3</td><td>→</td><td>1</td><td>4</td><td>-2</td><td>2</td><td></td><td>235915.077(30)</td><td>235915.060</td><td>0.010</td><td>2.298(11)</td><td>0.777(4)</td><td>278.76</td><td>3.402</td><td>4.13(73)</td></tr><tr><td>1</td><td>5</td><td>-1</td><td>4</td><td>→</td><td>1</td><td>4</td><td>-1</td><td>3</td><td></td><td>235964.081(100)</td><td>235964.370</td><td>-0.296</td><td>2.536(12)</td><td>0.667(4)</td><td>312.52</td><td>3.875</td><td>4.72(84)</td></tr><tr><td>1</td><td>5</td><td>0</td><td>5</td><td>→</td><td>1</td><td>4</td><td>0</td><td>4</td><td></td><td>235932.112(70)</td><td>235932.160</td><td>-0.049</td><td>3.773(12)</td><td>0.682(2)</td><td>233.45</td><td>4.044</td><td>4.83(85)<sup>l</sup></td></tr><tr><td>1</td><td>5</td><td>1</td><td>5</td><td>→</td><td>1</td><td>4</td><td>1</td><td>4</td><td></td><td>235929.796(100)</td><td>235929.430</td><td>0.358</td><td>3.723(12)</td><td>0.685(2)</td><td>227.27</td><td>3.877</td><td>4.64(82)<sup>l</sup></td></tr><tr><td>1</td><td>5</td><td>2</td><td>4</td><td>→</td><td>1</td><td>4</td><td>2</td><td>3</td><td></td><td>235935.427(30)<sup>i</sup></td><td>235935.340</td><td>0.078</td><td>2.129(11)</td><td>0.757(5)</td><td>303.54</td><td>3.407</td><td>4.18(74)</td></tr><tr><td>2</td><td>5</td><td>-2</td><td>4</td><td>→</td><td>2</td><td>4</td><td>-2</td><td>3</td><td></td><td>235668.310(70)</td><td>235668.310</td><td>-0.002</td><td>0.937(11)</td><td>0.776(10)</td><td>474.89</td><td>3.393</td><td>4.41(78)<sup>l</sup></td></tr><tr><td>2</td><td>5</td><td>-1</td><td>5</td><td>→</td><td>2</td><td>4</td><td>-1</td><td>4</td><td></td><td>235690.916(30)<sup>i</sup></td><td>235691.910</td><td>-0.995</td><td>1.830(12)</td><td>0.640(6)</td><td>394.69</td><td>3.876</td><td>4.81(85)<sup>l</sup></td></tr><tr><td>2</td><td>5</td><td>0</td><td>5</td><td>→</td><td>2</td><td>4</td><td>0</td><td>4</td><td></td><td>235605.123(100)</td><td>235604.850</td><td>0.264</td><td>0.884(12)</td><td>0.661(10)</td><td>536.55</td><td>4.037</td><td>5.00(89)</td></tr><tr><td>2</td><td>5</td><td>1</td><td>4</td><td>→</td><td>2</td><td>4</td><td>1</td><td>3</td><td></td><td>235556.138(30)<sup>i</sup></td><td>235555.870</td><td>0.263</td><td>0.721(12)</td><td>0.636(12)</td><td>582.05</td><td>3.874</td><td>4.68(83)<sup>l</sup></td></tr><tr><td>2</td><td>5</td><td>2</td><td>3</td><td>→</td><td>2</td><td>4</td><td>2</td><td>2</td><td></td><td>235685.747(30)</td><td>235685.750</td><td>-0.012</td><td>1.146(11)</td><td>0.741(8)</td><td>433.07</td><td>3.388</td><td>4.18(74)</td></tr><tr><td colspan="16">Previously published microwave transitions from the literature<sup>l</sup></td></tr><tr><td>0</td><td>1</td><td>0</td><td>1</td><td>+</td><td>0</td><td>0</td><td>0</td><td>0</td><td>+</td><td>47314.64(15)</td><td>47314.708</td><td>-0.068</td><td>...</td><td>...</td><td>1.578</td><td>0.809</td><td>...</td></tr><tr><td>0</td><td>2</td><td>1</td><td>2</td><td>+</td><td>0</td><td>1</td><td>1</td><td>1</td><td>+</td><td>93830.50(15)</td><td>93830.577</td><td>-0.077</td><td>...</td><td>...</td><td>14.757</td><td>1.216</td><td>...</td></tr><tr><td>0</td><td>2</td><td>1</td><td>1</td><td>- →</td><td>0</td><td>1</td><td>1</td><td>0</td><td>-</td><td>95435.10(15)</td><td>95435.245</td><td>-0.145</td><td>...</td><td>...</td><td>14.837</td><td>1.216</td><td>...</td></tr><tr><td>0</td><td>2</td><td>-1</td><td>2</td><td>→</td><td>0</td><td>1</td><td>-1</td><td>1</td><td></td><td>94624.30(15)</td><td>94624.018</td><td>0.282</td><td>...</td><td>...</td><td>8.643</td><td>1.215</td><td>...</td></tr><tr><td>0</td><td>1</td><td>0</td><td>1</td><td>→</td><td>0</td><td>0</td><td>0</td><td>0</td><td></td><td>47318.99(15)</td><td>47319.093</td><td>-0.103</td><td>...</td><td>...</td><td>10.638</td><td>0.809</td><td>...</td></tr><tr><td>0</td><td>2</td><td>0</td><td>2</td><td>→</td><td>0</td><td>1</td><td>0</td><td>1</td><td></td><td>94629.90(15)</td><td>94629.712</td><td>0.188</td><td>...</td><td>...</td><td>13.794</td><td>1.617</td><td>...</td></tr><tr><td>0</td><td>2</td><td>1</td><td>1</td><td>→</td><td>0</td><td>1</td><td>1</td><td>0</td><td></td><td>94637.20(15)</td><td>94637.350</td><td>-0.150</td><td>...</td><td>...</td><td>19.308</td><td>1.258</td><td>...</td></tr><tr><td>0</td><td>2</td><td>2</td><td>0</td><td>→</td><td>0</td><td>2</td><td>1</td><td>1</td><td></td><td>29690.76(15)</td><td>29690.766</td><td>-0.006</td><td>...</td><td>...</td><td>20.298</td><td>1.586</td><td>...</td></tr><tr><td>0</td><td>4</td><td>2</td><td>2</td><td>→</td><td>0</td><td>4</td><td>1</td><td>3</td><td></td><td>29702.73(15)</td><td>29702.595</td><td>0.135</td><td>...</td><td>...</td><td>31.347</td><td>3.901</td><td>...</td></tr><tr><td>0</td><td>5</td><td>2</td><td>3</td><td>→</td><td>0</td><td>5</td><td>1</td><td>4</td><td></td><td>29734.06(15)</td><td>29733.948</td><td>0.112</td><td>...</td><td>...</td><td>39.240</td><td>4.985</td><td>...</td></tr><tr><td>0</td><td>6</td><td>2</td><td>4</td><td>→</td><td>0</td><td>6</td><td>1</td><td>5</td><td></td><td>29796.12(15)</td><td>29796.243</td><td>-0.123</td><td>...</td><td>...</td><td>48.712</td><td>6.070</td><td>...</td></tr><tr><td>0</td><td>7</td><td>2</td><td>5</td><td>→</td><td>0</td><td>7</td><td>1</td><td>6</td><td></td><td>29901.82(15)</td><td>29901.905</td><td>-0.085</td><td>...</td><td>...</td><td>59.763</td><td>7.168</td><td>...</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th>v<sub>t</sub>'</th><th>J'</th><th>K'<sub>a</sub></th><th>K'<sub>c</sub></th><th>P<sup>ra</sup></th><th>v<sub>t</sub>''</th><th>J''</th><th>K''<sub>a</sub></th><th>K''<sub>c</sub></th><th>P''<sup>ra</sup></th><th>Exp. Freq.<sup>b</sup><br>(MHz)</th><th>Calc. Freq.<sup>c</sup><br>(MHz)</th><th>Exp. - Calc.<sup>d</sup><br>(MHz)</th><th>Exp. I<sup>e</sup><sub>peak</sub><br>(K)</th><th>Exp. Δv<sup>f</sup><br>(MHz)</th><th>E<sup>g</sup><sub>upper</sub><br>(cm<sup>-1</sup>)</th><th>Calc. Sμ<sup>2h</sup><br>(D<sup>2</sup>)</th><th>Exp. Sμ<sup>2k</sup><br>(D<sup>2</sup>)</th></tr></thead><tbody><tr><td>0</td><td>8</td><td>2</td><td>6</td><td>→</td><td>0</td><td>8</td><td>1</td><td>7</td><td>30063.92(15)</td><td>30064.094</td><td>-0.174</td><td>...</td><td>...</td><td>72.394</td><td>8.285</td><td>...</td></tr><tr><td>0</td><td>9</td><td>2</td><td>7</td><td>→</td><td>0</td><td>9</td><td>1</td><td>8</td><td>30295.58(15)</td><td>30295.633</td><td>-0.053</td><td>...</td><td>...</td><td>86.605</td><td>9.422</td><td>...</td></tr><tr><td>0</td><td>10</td><td>2</td><td>8</td><td>→</td><td>0</td><td>10</td><td>1</td><td>9</td><td>30607.49(15)</td><td>30607.546</td><td>-0.056</td><td>...</td><td>...</td><td>102.40</td><td>10.573</td><td>...</td></tr><tr><td>0</td><td>11</td><td>2</td><td>9</td><td>→</td><td>0</td><td>11</td><td>1</td><td>10</td><td>31007.29(15)</td><td>31007.217</td><td>0.073</td><td>...</td><td>...</td><td>119.77</td><td>11.729</td><td>...</td></tr><tr><td>0</td><td>12</td><td>2</td><td>10</td><td>→</td><td>0</td><td>12</td><td>1</td><td>11</td><td>31496.32(15)</td><td>31496.261</td><td>0.059</td><td>...</td><td>...</td><td>138.72</td><td>12.868</td><td>...</td></tr><tr><td>0</td><td>13</td><td>2</td><td>11</td><td>→</td><td>0</td><td>13</td><td>1</td><td>12</td><td>32068.47(15)</td><td>32068.301</td><td>0.169</td><td>...</td><td>...</td><td>159.25</td><td>13.964</td><td>...</td></tr><tr><td>0</td><td>14</td><td>2</td><td>12</td><td>→</td><td>0</td><td>14</td><td>1</td><td>13</td><td>32707.20(15)</td><td>32706.988</td><td>0.212</td><td>...</td><td>...</td><td>181.36</td><td>14.979</td><td>...</td></tr><tr><td>0</td><td>15</td><td>2</td><td>13</td><td>→</td><td>0</td><td>15</td><td>1</td><td>14</td><td>33384.69(15)</td><td>33384.670</td><td>0.020</td><td>...</td><td>...</td><td>205.05</td><td>15.867</td><td>...</td></tr><tr><td>0</td><td>16</td><td>2</td><td>14</td><td>→</td><td>0</td><td>16</td><td>1</td><td>15</td><td>34062.14(15)</td><td>34062.243</td><td>-0.103</td><td>...</td><td>...</td><td>230.31</td><td>16.578</td><td>...</td></tr><tr><td>0</td><td>17</td><td>2</td><td>15</td><td>→</td><td>0</td><td>17</td><td>1</td><td>16</td><td>34690.31(15)</td><td>34690.597</td><td>-0.287</td><td>...</td><td>...</td><td>257.15</td><td>17.062</td><td>...</td></tr><tr><td>0</td><td>20</td><td>2</td><td>18</td><td>→</td><td>0</td><td>20</td><td>1</td><td>19</td><td>35717.89(15)</td><td>35717.782</td><td>0.108</td><td>...</td><td>...</td><td>347.07</td><td>16.847</td><td>...</td></tr></tbody></table>

**Notes.**

a Upper- and lower-state quantum numbers $v'_t$ and $v''_t$ are the torsional quantum number, and $J'$, $K'_t$ and $K''_t$ and $J''$, $K''_t$ and $K''_t$ are the traditional asymmetric top rotational quantum numbers for the upper and lower states, respectively. For the A species, P is the so-called quasi-parity quantum number + or - . For the E species we use a signed value for $K_a$ (see E. Herbst & J. K. Messer 1984).

b Experimental frequency from present work in MHz. The numbers in parentheses represent the experimental uncertainties in units of the last significant digits.

c Calculated frequencies in MHz obtained using the BELGI program with the molecular constants in Table 2.

d Experimental - calculated values in MHz.

e Peak intensity of the experimentally measured line in K.

f FWHM of the experimentally measured line in MHz.

g Upper-state energies are values after taking out the zero-point energy (i.e., energy of the $J = K = 0$ level for the A species = 129.1131 cm$^{-1}$).

h Line strengths calculated by the BELGI program in D$^2$, using the values $\mu_a = 0.899$ D and $\mu_b = -1.44$ D reported for $^{12}\text{CH}_3\text{OH}$ (L.-H. Xu et al. 2008).

i Blended or distorted lines; weights were set to zero in the fits.

j Y. Hoshino et al. (1991).

k Derived from the experimentally measured integrated intensity of the line by assuming a partition function of 9363.632 at 295.8 K and a scaling factor of 1.053 (See text in Section 4.2). The errors are evaluated by using the root squared sum of errors originating from the scaling factor (5.7%), the partition function of (5%), the column density (16%), and the line-integrated intensity for each line (for details, see text).

l Calculated using the peak area derived by the Gaussian fit instead of the integrated intensity because part of the peaks are overlapped or the baseline is perturbed. The areas for each Gaussian function are also corrected by multiplying by a factor of 1.05 to include the expanded skirt.

m Transitions of $5_{24} - 4_{23}A^{-}v_{t} = 0$ and $5_{32} - 4_{31}E v_{t} = 0$ accidentally overlapped each other. The value of Exp. $S\mu^2$ is, therefore, calculated using the ratio of the intensities of two spectral lines estimated by the BELGI program.

n Unresolved asymmetry doublet. The value of Exp. $S\mu^2$ is calculated by assuming that the intensities of the two spectral lines are the same.

o Overlap of multiple lines.
<table><caption>**Table 2**
Molecular Constants of $^{12}\text{CH}_3^{17}\text{OH}$ and $^{13}\text{CH}_3^{17}\text{OH}$ in the RAM Axis System (cm$^{-1}$)</caption><thead><tr><th>Parameter</th><th>Operator<sup>a</sup></th><th><sup>12</sup>CH<sub>3</sub><sup>17</sup>OH</th><th><sup>13</sup>CH<sub>3</sub><sup>17</sup>OH</th><th><sup>12</sup>CH<sub>3</sub><sup>16</sup>OH<sup>b</sup></th></tr></thead><tbody><tr><td><i>F</i></td><td><i>P</i><sub>γ</sub><sup>2</sup></td><td>27.64684641<sup>c</sup></td><td>27.64684641<sup>c</sup></td><td>27.64684641(28)</td></tr><tr><td><i>V</i><sub>3</sub></td><td>(1/2)(1 − cos 3γ)</td><td>374.55(14)</td><td>375.75(15)</td><td>373.554746(12)</td></tr><tr><td>ρ<sup>d</sup></td><td><i>P</i><sub>γ</sub><i>P</i><sub>a</sub></td><td>0.80880(17)</td><td>0.81002(40)</td><td>0.8102062230(37)</td></tr><tr><td><i>A</i></td><td><i>P</i><sub>a</sub><sup>2</sup></td><td>4.1636(85)</td><td>4.2586(12)</td><td>4.2537233(71)</td></tr><tr><td><i>B</i></td><td><i>P</i><sub>b</sub><sup>2</sup></td><td>0.805100(13)</td><td>0.78445(11)</td><td>0.8236523(70)</td></tr><tr><td><i>C</i></td><td><i>P</i><sub>c</sub><sup>2</sup></td><td>0.775660(13)</td><td>0.75706(12)</td><td>0.7925575(71)</td></tr><tr><td><i>D</i><sub>ab</sub></td><td><i>P</i><sub>a</sub><i>P</i><sub>b</sub> + <i>P</i><sub>b</sub><i>P</i><sub>a</sub></td><td>−0.00694(11)</td><td>−0.01104(71)</td><td>−0.0038095(38)</td></tr><tr><td><i>V</i><sub>6</sub></td><td>(1/2)(1 − cos 6γ)</td><td>1.07(14)</td><td>...</td><td>−1.319650(85)</td></tr><tr><td><i>G</i><sub>v</sub></td><td><i>P</i><sub>γ</sub><sup>2</sup><i>P</i><sub>c</sub><sup>2</sup></td><td>−0.000114405(62)</td><td>−0.000128(17)</td><td>−0.0001373(31)</td></tr><tr><td><i>c</i><sub>1</sub></td><td>2<i>P</i><sub>γ</sub><sup>2</sup>(<i>P</i><sub>b</sub><sup>2</sup> − <i>P</i><sub>c</sub><sup>2</sup>)</td><td>−0.00003898(92)</td><td>...</td><td>−0.0001955(97)</td></tr><tr><td><i>F</i><sub>v</sub></td><td>(1 − cos 3γ)<i>P</i><sub>c</sub><sup>2</sup></td><td>−0.00233210(42)</td><td>−0.00248(19)</td><td>−0.0024324(69)</td></tr><tr><td><i>k</i><sub>5</sub></td><td>(1 − cos 3γ)<i>P</i><sub>a</sub><sup>2</sup></td><td>0.226(21)</td><td>...</td><td>0.01117844(23)</td></tr><tr><td><i>d</i><sub>ab</sub></td><td>(1 − cos 3γ)(<i>P</i><sub>a</sub><i>P</i><sub>b</sub> + <i>P</i><sub>b</sub><i>P</i><sub>a</sub>)</td><td>0.01194(19)</td><td>0.0174(18)</td><td>0.00907791(65)</td></tr><tr><td><i>c</i><sub>2</sub></td><td>(1 − cos 3γ)(<i>P</i><sub>b</sub><sup>2</sup> − <i>P</i><sub>c</sub><sup>2</sup>)</td><td>0.0000835(57)</td><td>0.000494(19)</td><td>−0.00008698(21)</td></tr><tr><td><i>L</i><sub>v</sub></td><td><i>P</i><sub>γ</sub><i>P</i><sub>a</sub><i>P</i><sub>c</sub><sup>2</sup></td><td>−0.000183045(50)</td><td>−0.000206(27)</td><td>−0.0002305(54)</td></tr><tr><td><i>k</i><sub>1</sub></td><td><i>P</i><sub>a</sub><i>P</i><sub>c</sub><sup>3</sup></td><td>0.0135(30)</td><td>...</td><td>−0.034254(13)</td></tr><tr><td>δ<sub>ab</sub></td><td><i>P</i><sub>γ</sub>(<i>P</i><sub>a</sub><sup>2</sup><i>P</i><sub>b</sub> + <i>P</i><sub>b</sub><i>P</i><sub>a</sub><sup>2</sup>)</td><td>−0.000383(14)</td><td>...</td><td>0.004496(33)</td></tr><tr><td><i>c</i><sub>4</sub></td><td><i>P</i><sub>γ</sub>{<i>P</i><sub>a</sub>, (<i>P</i><sub>b</sub><sup>2</sup> − <i>P</i><sub>c</sub><sup>2</sup>)}</td><td>−0.0000522(60)</td><td>...</td><td>−0.0007047(94)</td></tr><tr><td>Δ<sub>J</sub></td><td>−<i>P</i><sup>4</sup></td><td>0.000001642(14)</td><td>0.000001642<sup>c</sup></td><td>0.000001688465(31)</td></tr><tr><td>Δ<sub>JK</sub></td><td>−<i>P</i><sup>2</sup><i>P</i><sub>a</sub><sup>2</sup></td><td>0.000082793(56)</td><td>0.000091(11)</td><td>0.0000920(25)</td></tr><tr><td>Δ<sub>K</sub></td><td>−<i>P</i><sub>a</sub><sup>4</sup></td><td>−0.0105(25)</td><td>...</td><td>0.008524(10)</td></tr><tr><td>δ<sub>J</sub></td><td>−2<i>P</i><sup>2</sup>(<i>P</i><sub>b</sub><sup>2</sup> − <i>P</i><sub>c</sub><sup>2</sup>)</td><td>0.0000005155(23)</td><td>0.0000000496(11)</td><td>0.000000059414(33)</td></tr><tr><td>δ<sub>K</sub></td><td>−{<i>P</i><sub>a</sub><sup>2</sup>, (<i>P</i><sub>b</sub><sup>2</sup> − <i>P</i><sub>c</sub><sup>2</sup>)}</td><td>0.0000176(45)</td><td>...</td><td>0.000057361(89)</td></tr><tr><td><i>M</i><sub>v</sub></td><td><i>P</i><sub>γ</sub><sup>4</sup><i>P</i><sub>c</sub><sup>2</sup></td><td>0.0000000181(13)</td><td>...</td><td>0.00000009215(14)</td></tr><tr><td><i>c</i><sub>3</sub></td><td>2<i>P</i><sub>γ</sub><sup>4</sup>(<i>P</i><sub>b</sub><sup>2</sup> − <i>P</i><sub>c</sub><sup>2</sup>)</td><td>0.000000298(33)</td><td>...</td><td>0.0000000214(18)</td></tr><tr><td><i>k</i><sub>3</sub><i>P</i><sub>b</sub><sup>f</sup></td><td><i>P</i><sub>γ</sub><sup>3</sup><i>P</i><sub>b</sub></td><td>0.000821(26)</td><td>...</td><td>...</td></tr><tr><td>Num. of fitted lines</td><td>...</td><td>94</td><td>36</td><td>24722</td></tr><tr><td>σ<sub>fit</sub> (unitless)</td><td>...</td><td>1.04</td><td>0.67</td><td>1.45</td></tr><tr><td><i>J</i><sub>Max</sub></td><td>...</td><td>40</td><td>23</td><td>30</td></tr></tbody></table>

uncertainties on the fitted parameters are larger than for the normal species.

### 4.4. Line Intensities of $^{12}\text{CH}_3^{17}\text{OH}$

Since SUMIRE is an intensity-calibrated spectrometer, we also recorded the line strength of $^{12}\text{CH}_3^{17}\text{OH}$ transitions. In the LTE condition, $S\mu^2$ is written as follows (T. Oyama et al. 2023):

$$S\mu^2 = \frac{3\tau_0 hU(T)}{8\pi^3 N \left[ \exp\left(\frac{h\nu}{kT}\right) - 1 \right]} \exp\left(\frac{E_u}{kT}\right), \quad (8)$$

where

$$\tau_0 = 2\sqrt{\pi}\sigma \left( 1 - \sqrt{1 - \frac{If}{\sqrt{\pi}\sigma\Delta T}} \right). \quad (9)$$

In Equation (9),

$$\sigma = \frac{\Delta\nu}{2\sqrt{2\ln 2}}, \quad (10)$$

and

$$\Delta T = \frac{h\nu}{k} \left\{ \frac{1}{\exp\left(\frac{h\nu}{kT}\right) - 1} - \frac{1}{\exp\left(\frac{h\nu}{kT_{\text{bg}}}\right) - 1} \right\}, \quad (11)$$

where *f* and $\Delta T$ represent the scaling factor for the intensity calibration and the difference of the brightness temperature between a hot load with temperature *T* and a cold load with temperature *T*<sub>bg</sub>, respectively. Therefore, the $S\mu^2$ value of each transition line can be determined by substituting $E_u$ and $U(T)$ and the experimentally available parameters ($\nu$, $\Delta\nu$, *I*, *N*, *T*, and *T*<sub>bg</sub>). Note that Equation (8) is in principle the same as

<footnote><sup>a</sup> Note. {A,B} = AB + BA. The product of the parameter and operator from a given row yields the term actually used in the torsion-rotation Hamiltonian of the fitting program, except for *F*, *ρ*, and *A*, which occur in the Hamiltonian in the form $F(P_γ + ρP_a^2) + AP_a^2$.</footnote>

<footnote><sup>b</sup> Part of the 119 torsion-rotation parameters from L.-H. Xu et al. (2008). See their Table 2 for the complete set of parameters used in $^{12}\text{CH}_3^{16}\text{OH}$.</footnote>

<footnote><sup>c</sup> Constrained to the value of $^{12}\text{CH}_3^{16}\text{OH}$.</footnote>

<footnote><sup>d</sup> *ρ* is dimensionless.</footnote>

<footnote><sup>e</sup> Constrained to the values of $^{12}\text{CH}_3^{17}\text{OH}$; see text.</footnote>

<footnote><sup>f</sup> After unsuccessful attempts to fit our data set with the usual *k*<sub>3</sub> parameter that multiplies the $P_γ^3P_a$ term, it turns out that the fit of the $k_3P_b$ parameter, a parameter that was not used before and that multiplies the $P_γ^3P_b$ term, was decreasing the standard deviation of our fit.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Rotation diagram of $^{12}$CH$_{3}$$^{17}$OH at a temperature of 295.8 $^{+1.5}_{-0.6}$ K. Note that errors from the Gaussian fitting are smaller than the symbol size. The weights of all points are set to 1 for the least-squares analysis.</figcaption></figure>

Equation (2), except for the treatment of the optical depth described by $\tau_{\nu}$ and $\tau_{0}$. Although the effect of the optical depth is negligible for deriving the column density of the gas, it slightly affects the $S_{\mu}^{2}$ value of each transition, especially for strong transitions. We therefore applied the method to calculate $S_{\mu}^{2}$ by considering the effect of the optical depth up to its first- and second-order terms:

$$ \Delta T_{\text{m}} = \Delta T(1 - \exp(-\tau_{\nu})) \sim \Delta T\left(\tau - \frac{1}{2}\tau^{2}\right). \quad (12) $$

Details of the derivation of Equation (9) are described in T. Oyama et al. (2023). In Table 2, we list the $S_{\mu}^{2}$ value derived from this method (Exp. $S_{\mu}^{2}$), along with the value obtained by the BELGI program (Calc. $S_{\mu}^{2}$). Here we have used the experimentally derived column density, the one obtained from the gas pressure, to derive the Exp. $S_{\mu}^{2}$ and not the LTE calculation value, as it uses the $S_{\mu}^{2}$ value from the BELGI program. Note that the experimentally determined $S_{\mu}^{2}$ potentially has an offset < 17% that originates from the zero-point offset of the pressure gauge and the error of the partition function (See Section 4.2). This offset error can cause the entire plot to shift up or down, but it does not alter the relative positions between the plots. As shown in Figure 5, the $S_{\mu}^{2}$ values derived from the BELGI program agree very well with the $S_{\mu}^{2}$ directly derived from the measurement. For the b-type Q-branch series, the calculated $S_{\mu}^{2}$ values tend to become higher than the experimental ones when the lower-state energy ($E_{l}$) increases, which corresponds to transitions with higher J and K quantum numbers. However, this offset is still within the estimated systematic uncertainty in the experiment (indicated by a gray area in Figure 5), so no definitive conclusion can be drawn at this stage. Instead, the calculated $S_{\mu}^{2}$ of the a-type R-branch series (up to J = 4) tends to become lower than the experimental $S_{\mu}^{2}$ when $E_{l}$ increases. Note that the calculated values for $S_{\mu}^{2}$ from the BELGI code are based on the dipole moment of the normal species and ignore rotational and torsional distortion effects.

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Comparison of the $^{12}$CH$_{3}$$^{17}$OH line strength between the experimentally determined values and ones calculated by the BELGI code along with the lower-state energy. Overlapping transitions are excluded in the figure. The gray area indicates the experimental systematic uncertainty (see text).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Comparison with measured lines and estimated ones for $^{13}$CH$_{3}$$^{17}$OH.</figcaption></figure>

### 4.5. $^{13}$CH$_{3}$$^{17}$OH

According to the column densities of methanol isotopologues derived from the measurement, 88% ± 7% of the total methanol originates from $^{12}$CH$_{3}$$^{17}$OH. This confirms that the samples used in the experiment have a purity ≥80% for $^{17}$O. Since the natural abundance of $^{13}$C with respect to $^{12}$C is 1/92, the amount of $^{13}$CH$_{3}$$^{17}$OH in the sample should be on the order of 1%. In the spectrum recorded in this study, the intensities of the strong lines from the Q- and R-branches are estimated to be in the range 0.1–0.18 K. These lines could be observed, and Figure 6 shows a Q-branch consisting of transitions with the expected intensity. We therefore have succeeded in assigning some transitions from $^{13}$CH$_{3}$$^{17}$OH. The assigned lines of $^{13}$CH$_{3}$$^{17}$OH are listed in Table A1.

In total, 36 lines are assigned, mainly in the a-type R-branch and b-type Q-branch, which have characteristic line patterns. Since the measured frequency range is limited to be 48 GHz, only five transitions are assigned for the E state. We fit the lines
<figure><img src="image_10.png" /><figcaption>Figure 7. Spectra of $^{12}$CH$_{3}$$^{17}$OH emission estimated under the LTE approximation with the temperatures of (a) 200 K, (b) 100 K, and (c) 50 K. The frequency range is from 250.8 to 258.8 GHz. The intensities of the transition lines are estimated by using calculated $S\mu^2$ in Table 1.</figcaption></figure>

by using the BELGI program as in the case of $^{12}$CH$_{3}$$^{17}$OH. The unitless standard deviation of the fit for those 36 lines is 0.67.

Theoretically, rotational constants, R, could be estimated for $^{13}$CH$_{3}$$^{17}$OH as follows:

$$R_{\text{est}}(^{13}\text{CH}_{3}^{17}\text{OH}) = R_{\text{exp}}(^{12}\text{CH}_{3}^{17}\text{OH}) \times R_{\text{calc}}(^{13}\text{CH}_{3}^{17}\text{OH})/R_{\text{calc}}(^{12}\text{CH}_{3}^{17}\text{OH}), \quad (13)$$

where $R_{\text{est}}$ and $R_{\text{exp}}$ represent estimated and experimentally derived constants, respectively. DFT calculations were carried out to derive $R_{\text{calc}}$, calculated rotational constants, at the CAM-B3LYP level of theory (T. Yanai et al. 2004) with the cc-pVTZ basis set (T. H. Dunning 1989; R. A. Kendall et al. 1992) using the Gaussian 16 package (M. J. Frisch et al. 2016). From Equation (13), $R_{\text{est}} = A, B$, and $C$ are derived to be 4.1634, 0.78460, and 0.75657 cm$^{-1}$, respectively. Although the number of assigned transitions is limited, the molecular constants of $^{13}$CH$_{3}$$^{17}$OH are also derived by using the measured line parameters with the BELGI program and floating only 13 parameters to achieve the unitless standard deviation of 0.67 for 36 lines. After transforming the values of $A, B$, and $C$ of Table 2 that are in the RAM system into their values in the PAM system (I. Kleiner 2010), we found the values for $A_{\text{PAM}}, B_{\text{PAM}},$ and $C_{\text{PAM}}$ for $^{13}$CH$_{3}$$^{17}$OH to be 4.2586(12), 0.78441(11), and 0.75706(12) cm$^{-1}$, respectively. These values show good agreement with the

theoretical estimates from the DFT calculation. Note that we have also conducted a trial fit where the nonfloated parameters in Table 2 were fixed to values obtained for $^{12}$CH$_{3}$$^{17}$OH instead of zero. The $A, B$, and $C$ values derived in this trial are not significantly different from those of the first fitting, demonstrating consistency in the derived constants. Therefore, we have adopted the method, setting the nonfloated parameters to be zero, in this paper. For $^{13}$CH$_{3}$$^{17}$OH spectroscopy, there is no report on experimental study. Theoretically, molecular constants were estimated and reported by Y.-B. Duan et al. (2000). Considering the limited number of measured transitions and the uncertainties inherent in purely theoretical calculations, the $A, B$, and $C$ values from their study are in good agreement with our present results.

## 4.6. Astronomical Implications

Here we estimate the intensities of $^{12}$CH$_{3}$$^{17}$OH emission in astronomical observations invoking the LTE approximation. We employed the $^{12}$CH$_{3}$$^{17}$OH column density of $N = 5.6 \times 10^{15}$ cm$^{-2}$, which is estimated from the column density of $^{12}$CH$_{3}$$^{16}$OH in the hot corino source IRAS 16293–2422 B reported by J. K. Jorgensen et al. (2018) with the canonical isotope ratios of $^{16}$O/$^{18}$O = 560 and $^{18}$O/$^{17}$O = 3.2 in the local ISM (T. L. Wilson & R. Rood 1994). The dispersion of line profile is calculated from the FWHM of 1.0 km s$^{-1}$ corresponding to the typical line width of molecular line emission observed in IRAS 16293–2422 B (J. K. Jorgensen et al. 2018).
Figure 7 shows the spectra of $^{12}$CH$_{3}$$^{17}$OH from 250.8 to 258.8 GHz assuming temperatures of 200, 100, and 50 K, which are typical temperatures of molecular gas observed in the hot cores and hot corinos. This frequency range covers the strongest emission lines of Q-branch series observed by SUMIRE as demonstrated in Figure 2. The emission lines from 257 to 258 GHz are found to be particularly strong at all temperatures because the upper-state energies, $E_u$, of these transition lines are from 98 to 175 K, which are close to the assumed temperatures. With the expected line intensities of more than 5 K, these lines can be easily detected by ALMA with 1 hr observation. Since many transition lines with different $E_u$ are observed in this frequency range, the column density of $^{12}$CH$_{3}$$^{17}$OH can be estimated by the rotation diagram analysis with the aid of the molecular constants determined by SUMIRE. Because the frequency range covered in this study corresponds to ALMA Band 6, which is the most observed frequency band, transitions from $^{12}$CH$_{3}$$^{17}$OH might be detected toward various sources even in the archival data. In addition, a part of the same Q-branch series of $^{12}$CH$_{3}$$^{18}$OH at a frequency of 263 GHz (Figure 2) can be observed simultaneously in the same sideband of the ALMA Band 6 receiver because the IF bandwidth of the receiver is 5.5 GHz. Therefore, the isotopologue ratios of $^{12}$CH$_{3}$$^{18}$OH/$^{12}$CH$_{3}$$^{17}$OH are expected to be determined by the observation in the target sources.

Our experimental results demonstrate that the M-shape centers of the Q-branch spectral line pattern of the two isotopologues, $^{12}$CH$_{3}$$^{17}$OH (257.70 GHz) and $^{12}$CH$_{3}$$^{18}$OH (262.99 GHz), are subtly shifted from that for $^{12}$CH$_{3}$$^{16}$OH (251.91 GHz) by 6–11 GHz. Normally, considering that the frequency of spectral lines shifts toward the lower direction as the molecule becomes heavier, the observed shift in this measurement is in the opposite direction, indicating a characteristic of methanol molecules obviously. This subtle shift may also provide a clue to detect $^{12}$CH$_{3}$$^{17}$OH in observations toward various astronomical sources. Since $^{12}$CH$_{3}$$^{17}$OH is less abundant than other methanol isotopologues, there is a possibility that it would be blended with the strong emission lines from other, more abundant species including methanol isotopologues and be difficult to identify their lines. However, there is a frequency range where the spectral intensity of $^{12}$CH$_{3}$$^{17}$OH is relatively high between 254 and 260 GHz. The strong emission lines from 257 to 258 GHz can especially be expected with assumed typical temperatures of molecular gas, 200, 100, and 50 K, observed in the hot cores and hot corinos. Consequently, astronomical detection of $^{12}$CH$_{3}$$^{17}$OH will be potentially easier in these frequency regions.

In Galactic scale, the $^{18}$O and $^{17}$O abundance is important for nucleosynthesis. It is theoretically predicted that the synthesis of $^{17}$O is a consequence of hydrogen burning in stars via both cold and hot CNO cycles (e.g., M. Wiescher et al. 2010), and $^{16}$O and $^{18}$O have their origin in helium burning (e.g., E. E. Salpeter 1952; D. Clayton 2003). In other words, $^{17}$O is predominantly ejected by intermediate-mass stars through a longer production timescale, while $^{18}$O is primarily synthesized in massive stars. Recently, the $^{18}$O/$^{17}$O isotopic ratio in CO molecules has been reported as a function of galactocentric distance (J. S. Zhang et al. 2020). They represent a gradual positive gradient of the $^{18}$O/$^{17}$O ratio, which is consistent with the above hypothesis. However, the analysis has been done by using only one transition of C$^{18}$O and one transition of C$^{17}$O.

Confirmation of this trend by observing multiple transition lines with different energy states but with similar critical density is awaited.

Tracing the isotope ratio of oxygen in molecules is important so as to better understand the origin of the various values obtained in our solar system (e.g., H. Yurimoto & K. Kuramoto 2004; A. N. Krot et al. 2020). In interstellar clouds, for example, isotope-selective photodissociation of CO could be effective, which results in lower abundance of C$^{18}$O and C$^{17}$O in the central part of the cloud (e.g., J. Bally & W. D. Langer 1982; A. E. Glassgold & P. J. Huggins 1985; E. F. van Dishoeck & J. H. Black 1988). On the other hand, $^{16}$O atoms in the gas phase could be less abundant compared to the outer edge of the cloud since destruction of C$^{16}$O is insufficient in the central part of the cloud. This would cause less abundant $^{16}$O in H$_2$O ice compared to $^{16}$O in water ice at the outer cloud. Such isotope-selective photodissociation can make a variety of $^{16}$O/$^{18}$O or $^{16}$O/$^{18}$O ratios in protoplanetary disks depending on the mixing efficiency of gas and dust between the outer edge and inner part of the disk, which should be confirmed by observations of $^{17}$O and $^{18}$O isotopologues across different stages of star formation. As methanol is the most abundant saturated organic molecule in star-forming regions and a key precursor to more complex organics, further observational studies are required to understand its role in chemical evolution.

Note that after the submission of this paper a paper about a spectroscopic investigation of $^{12}$CH$_{3}$$^{17}$OH and its tentative detection toward the Sgr B2(N) region came out (Muller et al. 2024, accepted in A&A). While they focused on spectroscopic analysis of $^{12}$CH$_{3}$$^{17}$OH by using a wide measurement range (38–1095 GHz), which allowed them to assign $0 \le J \le 45$, $K_a \le 16$, and mainly the $v_t = 0$ and 1 torsional states, the present study focuses mostly on the measured line parameters including intensities in the ALMA Band 6 frequency band, since those parameters are also important for astronomical studies. Despite the limited number of measured transitions, the derived molecular constants of $^{12}$CH$_{3}$$^{17}$OH in this study agree well with their results. For $^{13}$CH$_{3}$$^{17}$OH, the present study is the first spectroscopic study.

## 5. Conclusions

In this paper, laboratory-based millimeter and submillimeter spectroscopic emission measurements of the methanol isotopologue, $^{12}$CH$_{3}$$^{17}$OH and $^{13}$CH$_{3}$$^{17}$OH, are reported. The measurements are performed with our apparatus, SUMIRE, in the frequency range between 216 and 264 GHz, paying special attention to the Q-branch transitions since their strong lines are useful for astronomical observations. Consequently, 71 rotational transition lines of $^{12}$CH$_{3}$$^{17}$OH are assigned, and their frequency and intensity are determined. The intrinsic line strength ($S\mu^2$) of each transition is also determined experimentally. Molecular constants have been derived based on our experimental outcomes by using the BELGI program. The accuracy of the line frequency is typically a few kHz or less. The $S\mu^2$ is determined within ~16% accuracy. In the measured spectra, we find that $^{12}$CH$_{3}$$^{17}$OH clearly exhibits double and/or triple peaks in low-$J$ transitions, unlike the other isotopologues $^{12}$CH$_{3}$$^{16}$OH and $^{12}$CH$_{3}$$^{18}$OH, because of the nuclear quadrupole interaction.
Investigation of methanol composed of stable oxygen isotopes, <sup>16</sup>O, <sup>17</sup>O, and <sup>18</sup>O, gives us a clue to determine more precise ratios among the isotopes for a better understanding of the galactic-scale chemical evolution of elements. Furthermore, highly accurate spectra of various substances including isotopologues and isotopomers obtained under well-defined experimental conditions are essential for identifying the "unidentified molecular lines" in observed spectra as many lines as possible. This requires the development and improvement of millimeter and submillimeter spectroscopic instruments that enable us to perform laboratory-based measurements over a wide frequency range. The spectral lines of <sup>13</sup>CH<sub>3</sub><sup>17</sup>OH may also be detected in hot cores, where the existence of extremely abundant methanol is expected, which allows us to verify optical depths of the <sup>12</sup>CH<sub>3</sub><sup>17</sup>OH lines.

## Acknowledgments

We greatly acknowledge Takashi Kojima and the NAOJ staff for maintaining the SIS receiver mounted on SUMIRE. We appreciate Li-Hong Xu and Ron Lees for providing the BELGI code for methanol. We also thank the anonymous reviewers for their constructive comments and suggestions. We

are thankful to Nadia Murillo Mejias, Kaori Kobayashi, and Satoshi Yamamoto for insightful comments and support. This project has been supported by MEXT/JSPS Grants-in-Aid for Scientific Research (KAKENHI) grant Nos. JP20H05845 and JP20H05844, as well as a pioneering project in RIKEN (Evolution of Matter in the Universe). The financial support of the Soroptimist Japan Foundation for our experimental project is highly appreciated. An analytical part of this research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (grant No. 80NM0018D0004). Part of this work is supported by the Program National "Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP cofunded by CEA and CNES.

# Appendix
## Line List of <sup>13</sup>CH<sub>3</sub><sup>17</sup>OH

For <sup>13</sup>CH<sub>3</sub><sup>17</sup>OH, we have listed their quantum numbers, observed and calculated rest frequencies, differences between the observed and calculated rest frequencies, peek intensities, FWHM line width, upper-state energies, and Sµ<sup>2</sup> values calculated by the BELGI code in Table A1.
<table><caption>Table A1
Experimentally Measured and Predicted Transition Frequencies of $^{13}$CH$_{3}$$^{17}$OH</caption><thead><tr><th>v<sub>t</sub>'</th><th>J'</th><th>K'<sub>a</sub></th><th>K'<sub>c</sub></th><th>P<sup>ra</sup></th><th>v<sub>t</sub>''</th><th>J''</th><th>K''<sub>a</sub></th><th>K''<sub>c</sub></th><th>P''<sup>ra</sup></th><th>Exp. Freq.<sup>b</sup><br>(MHz)</th><th>Calc. Freq.<sup>c</sup><br>(MHz)</th><th>Exp. - Calc.<sup>d</sup><br>(MHz)</th><th>Exp. f'<sub>peak</sub><br>(K)</th><th>Exp. Δt<sup>f</sup><br>(MHz)</th><th>E<sup>g</sup><sub>upper</sub><br>(cm<sup>-1</sup>)</th><th>S<sub>f</sub><sup>2h</sup><br>(D<sup>2</sup>)</th></tr></thead><tbody><tr><td colspan="17">A state</td></tr><tr><td>0</td><td>5</td><td>0</td><td>5</td><td>+</td><td>→</td><td>0</td><td>4</td><td>0</td><td>4</td><td>+</td><td>230600.794(100)</td><td>230600.812</td><td>-0.018</td><td>0.098(11)</td><td>0.737(95)</td><td>23.079</td><td>4.045</td></tr><tr><td>0</td><td>5</td><td>1</td><td>5</td><td>+</td><td>→</td><td>0</td><td>4</td><td>1</td><td>4</td><td>+</td><td>228728.919(50)</td><td>228728.918</td><td>0.001</td><td>0.088(9)</td><td>0.593(72)</td><td>32.966</td><td>3.889</td></tr><tr><td>0</td><td>5</td><td>1</td><td>4</td><td>-</td><td>→</td><td>0</td><td>4</td><td>1</td><td>3</td><td>-</td><td>232542.246(100)</td><td>232542.246</td><td>0.000</td><td>0.088(10)</td><td>0.739(103)</td><td>33.348</td><td>3.890</td></tr><tr><td>0</td><td>5</td><td>2</td><td>3</td><td>+</td><td>→</td><td>0</td><td>4</td><td>2</td><td>2</td><td>+</td><td>230682.978(100)</td><td>230682.977</td><td>0.001</td><td>0.069(9)</td><td>0.725(115)</td><td>49.337</td><td>3.430</td></tr><tr><td>0</td><td>5</td><td>2</td><td>4</td><td>-</td><td>→</td><td>0</td><td>4</td><td>2</td><td>3</td><td>-</td><td>230644.851(130)</td><td>230644.895</td><td>-0.044</td><td>0.082(11)</td><td>0.778(124)</td><td>49.335</td><td>3.430</td></tr><tr><td>0</td><td>6</td><td>3</td><td>3</td><td>-</td><td>→</td><td>0</td><td>6</td><td>2</td><td>4</td><td>+</td><td>260975.325(130)</td><td>260975.370</td><td>-0.045</td><td>0.083(11)</td><td>0.978(150)</td><td>67.276</td><td>5.216</td></tr><tr><td>0</td><td>7</td><td>3</td><td>4</td><td>-</td><td>→</td><td>0</td><td>7</td><td>2</td><td>5</td><td>+</td><td>260891.824(30)</td><td>260891.790</td><td>0.034</td><td>0.126(12)</td><td>0.766(86)</td><td>78.046</td><td>6.269</td></tr><tr><td>0</td><td>7</td><td>3</td><td>5</td><td>+</td><td>→</td><td>0</td><td>7</td><td>2</td><td>6</td><td>-</td><td>261128.635(130)</td><td>261128.686</td><td>-0.051</td><td>0.113(11)</td><td>0.957(114)</td><td>78.046</td><td>6.263</td></tr><tr><td>0</td><td>8</td><td>3</td><td>5</td><td>-</td><td>→</td><td>0</td><td>8</td><td>2</td><td>6</td><td>+</td><td>260783.868(40)</td><td>260783.859</td><td>0.009</td><td>0.131(11)</td><td>0.652(65)</td><td>90.353</td><td>7.296</td></tr><tr><td>0</td><td>8</td><td>3</td><td>6</td><td>+</td><td>→</td><td>0</td><td>8</td><td>2</td><td>7</td><td>-</td><td>261176.940(150)</td><td>261176.943</td><td>-0.003</td><td>0.109(12)</td><td>1.007(138)</td><td>90.353</td><td>7.283</td></tr><tr><td>0</td><td>9</td><td>3</td><td>6</td><td>-</td><td>→</td><td>0</td><td>9</td><td>2</td><td>7</td><td>+</td><td>260646.955(30)</td><td>260646.977</td><td>-0.022</td><td>0.168(11)</td><td>0.711(54)</td><td>104.20</td><td>8.304</td></tr><tr><td>0</td><td>9</td><td>3</td><td>7</td><td>+</td><td>→</td><td>0</td><td>9</td><td>2</td><td>8</td><td>-</td><td>261261.554(30)</td><td>261261.562</td><td>-0.008</td><td>0.148(12)</td><td>0.755(76)</td><td>104.20</td><td>8.282</td></tr><tr><td>0</td><td>10</td><td>3</td><td>7</td><td>-</td><td>→</td><td>0</td><td>10</td><td>2</td><td>8</td><td>+</td><td>260476.196(70)</td><td>260476.216</td><td>-0.020</td><td>0.171(12)</td><td>0.556(44)</td><td>119.58</td><td>9.302</td></tr><tr><td>0</td><td>10</td><td>3</td><td>8</td><td>+</td><td>→</td><td>0</td><td>10</td><td>2</td><td>9</td><td>-</td><td>261392.785(30)</td><td>261392.832</td><td>-0.047</td><td>0.105(13)</td><td>0.785(120)</td><td>119.58</td><td>9.265</td></tr><tr><td>0</td><td>11</td><td>3</td><td>8</td><td>-</td><td>→</td><td>0</td><td>11</td><td>2</td><td>9</td><td>+</td><td>260266.445(70)</td><td>260266.432</td><td>0.013</td><td>0.179(11)</td><td>0.583(44)</td><td>136.50</td><td>10.292</td></tr><tr><td>0</td><td>11</td><td>3</td><td>9</td><td>+</td><td>→</td><td>0</td><td>11</td><td>2</td><td>10</td><td>-</td><td>261581.989(30)</td><td>261581.957</td><td>0.032</td><td>0.168(14)</td><td>0.783(75)</td><td>136.50</td><td>10.233</td></tr><tr><td>0</td><td>12</td><td>3</td><td>9</td><td>-</td><td>→</td><td>0</td><td>12</td><td>2</td><td>10</td><td>+</td><td>260012.348(70)</td><td>260012.381</td><td>-0.033</td><td>0.189(12)</td><td>0.597(45)</td><td>154.96</td><td>11.279</td></tr><tr><td>0</td><td>12</td><td>3</td><td>10</td><td>+</td><td>→</td><td>0</td><td>12</td><td>2</td><td>11</td><td>-</td><td>261840.996(30)</td><td>261840.985</td><td>0.011</td><td>0.150(13)</td><td>0.757(76)</td><td>154.96</td><td>11.189</td></tr><tr><td>0</td><td>13</td><td>3</td><td>10</td><td>-</td><td>→</td><td>0</td><td>13</td><td>2</td><td>11</td><td>+</td><td>259708.863(40)</td><td>259708.880</td><td>-0.017</td><td>0.162(10)</td><td>0.642(48)</td><td>174.95</td><td>12.265</td></tr><tr><td>0</td><td>13</td><td>3</td><td>11</td><td>+</td><td>→</td><td>0</td><td>13</td><td>2</td><td>12</td><td>-</td><td>262182.737(30)</td><td>262182.732</td><td>0.005</td><td>0.153(12)</td><td>0.729(68)</td><td>174.95</td><td>12.133</td></tr><tr><td>0</td><td>14</td><td>3</td><td>11</td><td>-</td><td>→</td><td>0</td><td>14</td><td>2</td><td>12</td><td>+</td><td>259350.975(30)</td><td>259350.970</td><td>0.005</td><td>0.176(11)</td><td>0.740(56)</td><td>196.48</td><td>13.253</td></tr><tr><td>0</td><td>14</td><td>3</td><td>12</td><td>+</td><td>→</td><td>0</td><td>14</td><td>2</td><td>13</td><td>-</td><td>262620.714(70)</td><td>262620.703</td><td>0.011</td><td>0.169(15)</td><td>0.613(63)</td><td>196.48</td><td>13.064</td></tr><tr><td>0</td><td>15</td><td>3</td><td>12</td><td>-</td><td>→</td><td>0</td><td>15</td><td>2</td><td>13</td><td>+</td><td>258934.176(50)</td><td>258934.129</td><td>0.047</td><td>0.171(9)</td><td>0.613(40)</td><td>219.55</td><td>14.246</td></tr><tr><td>0</td><td>15</td><td>3</td><td>13</td><td>+</td><td>→</td><td>0</td><td>15</td><td>2</td><td>14</td><td>-</td><td>263168.847(130)</td><td>263169.001</td><td>-0.154</td><td>0.170(130)</td><td>0.970(92)</td><td>219.54</td><td>13.982</td></tr><tr><td>0</td><td>16</td><td>3</td><td>13</td><td>-</td><td>→</td><td>0</td><td>16</td><td>2</td><td>14</td><td>+</td><td>258454.503(30)</td><td>258454.495</td><td>0.008</td><td>0.172(11)</td><td>0.684(52)</td><td>244.15</td><td>15.246</td></tr><tr><td>0</td><td>16</td><td>3</td><td>14</td><td>+</td><td>→</td><td>0</td><td>16</td><td>2</td><td>15</td><td>-</td><td>263842.262(100)<sup>i</sup></td><td>263842.241</td><td>0.021</td><td>0.170(15)</td><td>0.546(58)</td><td>244.14</td><td>14.886</td></tr><tr><td>0</td><td>17</td><td>3</td><td>14</td><td>-</td><td>→</td><td>0</td><td>17</td><td>2</td><td>15</td><td>+</td><td>257909.124(30)</td><td>257909.130</td><td>-0.006</td><td>0.149(10)</td><td>0.759(63)</td><td>270.29</td><td>16.257</td></tr><tr><td>0</td><td>18</td><td>3</td><td>15</td><td>-</td><td>→</td><td>0</td><td>18</td><td>2</td><td>16</td><td>+</td><td>257296.306(30)</td><td>257296.311</td><td>-0.005</td><td>0.133(13)</td><td>0.654(75)</td><td>297.96</td><td>17.282</td></tr><tr><td>0</td><td>19</td><td>3</td><td>16</td><td>-</td><td>→</td><td>0</td><td>19</td><td>2</td><td>17</td><td>+</td><td>256615.862(70)</td><td>256615.846</td><td>0.016</td><td>0.136(11)</td><td>0.574(55)</td><td>327.17</td><td>18.323</td></tr><tr><td>0</td><td>20</td><td>3</td><td>17</td><td>-</td><td>→</td><td>0</td><td>20</td><td>2</td><td>18</td><td>+</td><td>255869.304(30)<sup>i</sup></td><td>255869.423</td><td>-0.119</td><td>0.113(9)</td><td>0.701(65)</td><td>357.91</td><td>19.385</td></tr><tr><td>0</td><td>21</td><td>3</td><td>18</td><td>-</td><td>→</td><td>0</td><td>21</td><td>2</td><td>19</td><td>+</td><td>255060.926(70)</td><td>255060.971</td><td>-0.045</td><td>0.111(11)</td><td>0.573(66)</td><td>390.18</td><td>20.470</td></tr><tr><td>0</td><td>22</td><td>3</td><td>19</td><td>-</td><td>→</td><td>0</td><td>22</td><td>2</td><td>20</td><td>+</td><td>254197.045(30)</td><td>254197.044</td><td>0.001</td><td>0.100(13)</td><td>0.690(108)</td><td>423.99</td><td>21.583</td></tr><tr><td>0</td><td>23</td><td>3</td><td>20</td><td>-</td><td>→</td><td>0</td><td>23</td><td>2</td><td>21</td><td>+</td><td>253287.295(160)</td><td>253287.208</td><td>0.087</td><td>0.084(10)</td><td>0.982(147)</td><td>459.34</td><td>22.727</td></tr><tr><td colspan="17">E state</td></tr><tr><td>0</td><td>5</td><td>-2</td><td>4</td><td>→</td><td>0</td><td>4</td><td>-2</td><td>3</td><td></td><td></td><td>230693.619(100)</td><td>230693.614</td><td>0.005</td><td>0.093(8)</td><td>0.722(72)</td><td>41.199</td><td>3.404</td></tr><tr><td>0</td><td>5</td><td>-1</td><td>5</td><td>→</td><td>0</td><td>4</td><td>-1</td><td>4</td><td></td><td></td><td>230579.483(200)</td><td>230579.476</td><td>0.007</td><td>0.088(9)</td><td>0.857(102)</td><td>26.992</td><td>3.888</td></tr><tr><td>0</td><td>5</td><td>0</td><td>5</td><td>→</td><td>0</td><td>4</td><td>0</td><td>4</td><td></td><td></td><td>230531.251(100)</td><td>230531.273</td><td>-0.022</td><td>0.097(8)</td><td>0.696(76)</td><td>32.104</td><td>4.040</td></tr><tr><td>0</td><td>5</td><td>1</td><td>4</td><td>→</td><td>0</td><td>4</td><td>1</td><td>3</td><td></td><td></td><td>230672.179(100)</td><td>230672.151</td><td>0.028</td><td>0.095(9)</td><td>0.704(82)</td><td>37.640</td><td>4.018</td></tr><tr><td>0</td><td>5</td><td>2</td><td>3</td><td>→</td><td>0</td><td>4</td><td>2</td><td>2</td><td></td><td></td><td>230699.470(160)</td><td>230699.408</td><td>0.062</td><td>0.079(9)</td><td>0.799(104)</td><td>38.698</td><td>3.349</td></tr></tbody></table>

### Notes.

a Upper- and lower-state quantum numbers, $v'_t$ and $v''_t$, are the torsional quantum number; $J'$, $K'_a$ and $K'_c$ and $J''$, $K''_a$ and $K''_c$ are the traditional asymmetric top rotational quantum numbers for the upper and lower states, respectively. For the A species, $P$ is the so-called quasi-parity quantum number + or - . For the E species we use a signed value for $K_a$ (see E. Herbst & J. K. Messer 1984).

b Experimental frequency from present fit in MHz. The numbers in parentheses represent the experimental uncertainties in units of the last significant digits.

c Calculated frequencies in MHz by the BELGI program with the molecular constants in Table 2.

d Experimental - calculated values in MHz.

e Peak intensity of the experimentally measured line in K.

f FWHM of the experimentally measured line in MHz.

g Upper-state energies are values after taking out the zero-point energy (i.e., energy of the $J = K = 0$ level for the A species = 128.9228 cm$^{-1}$).

h Line strengths calculated by the BELGI program in D$^2$, using the values $\mu_a = 0.899$ D and $\mu_b = -1.44$ D reported for $^{12}$CH$_3$$^{16}$OH (L.-H. Xu et al. 2008).

i Blended or distorted lines; weights were set to zero in the fits.
ORCID iDs

Takahiro Oyama https://orcid.org/0000-0002-2301-7951

Yoshimasa Watanabe https://orcid.org/0000-0002-9668-3592

Takeshi Sakai https://orcid.org/0000-0003-4521-7492

Riouhei Nakatani https://orcid.org/0000-0002-1803-0203

Shaoshan Zeng https://orcid.org/0000-0003-3721-374X

Isabelle Kleiner https://orcid.org/0000-0002-8715-9764

Nami Sakai https://orcid.org/0000-0002-3297-4497

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