# Spectroscopic Shifts in Deuterated Methanol Induced by Variation of $m_{\rm
e}/m_{\rm p}$

J. S. Vorotyntseva1,2 S. A. Levshakov1 M. G. Kozlov2,3 1 Ioffe Institute,
Polytekhnicheskaya Str. 26, 194021 St. Petersburg, Russia
2 St. Petersburg Electrotechnical University “LETI”, Prof. Popov Str. 5,
197376 St. Petersburg, Russia
3 Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, Gatchina,
Leningrad District, 188300, Russia

###### Abstract

Numerical calculations of the sensitivity coefficients, $Q_{\mu}$, of
microwave molecular transitions in the ground torsion-rotation state of
deuterated methanol (CH3OD, CD3OH, and CD3OD) to small variations in the
fundamental physical constant $\mu=m_{\rm e}/m_{\rm p}$ – the electron-to-
proton mass ratio – are reported. Theoretical motivation for changes in $\mu$
comes from a variety of models beyond the Standard Model of particle physics
which are invoked to explain the nature of dark matter and dark energy that
dominate the Universe. The calculated values of $Q_{\mu}$ range from $-300$ to
+73 and, thus, make deuterated methanol promising for searches for small
space-time changes in $\mu$. It is also shown that among the calculated
sensitivity coefficients $Q_{\mu}$ using different Hamiltonians in the present
and previous works, there are several pronounced outliers of unclear nature.

##  I Introduction

Since the pioneering work of [1] the $\mu$-test has been widely used in
experimental studies of theoretical models beyond the Standard Model of
particle physics which predict small variations of fundamental physical
constants and, in particular, the electron-to-proton mass ratio, $\mu=m_{\rm
e}/m_{\rm p}$, in space and time. Theories that connect varying constants to
dark matter consider the dependence of $\mu$ on the local density of dark
matter which differs across the Milky Way ([2]). The coupling of physical
constants to either dark matter, or to gravitational fields is related to the
chameleon scenario (e.g., [3], [4]) and it can be probed in the Milky Way, or
in the nearby galaxies by astrophysical observations ([5], [6]).

Regular quantitative measurements of fractional changes in $\mu$ between its
extraterrestrial ($\mu_{\scriptscriptstyle\rm obs}$) and terrestrial
($\mu_{\scriptscriptstyle\rm lab}$) values,
${\Delta\mu}/{\mu}=(\mu_{\scriptscriptstyle\rm obs}-\mu_{\scriptscriptstyle\rm
lab})/\mu_{\scriptscriptstyle\rm lab}$, became possible after the
identification of molecular hydrogen lines in the absorption-line spectra of
distant quasars ([7]) and calculation of sensitivity coefficients, $Q_{\mu}$,
for individual molecular transitions of the Lyman and Werner bands of H2
([8]).

It should be noted that at the first stage these studies were carried out
mainly in the optical range on the H2 lines for which the sensitivity
coefficients are not very high, $|Q_{\mu}|\sim 10^{-2}$, and the spectral
resolution (channel width) does not exceed 1 km s-1. The tightest upper limits
on the variability of $\mu$ on the cosmological time scale were obtained in
the framework of the VLT/UVES Large Programme for testing fundamental physics,
$\Delta\mu/\mu$ $<10^{-5}$ ([9]), and at the highest redshift $z=4.22$ towards
the quasar J1443+2724, $\Delta\mu/\mu$ $<8\times 10^{-6}$ ([10]) 111All upper
limits on $\Delta\mu/\mu$ throughout the paper are given at the confidence
level of $1\sigma$..

The second stage of development of the $\mu$-tests took place in the microwave
radio range after [11] showed the enhanced sensitivity of the inversion
tunneling transition at 23 GHz in ammonia molecule NH3, $Q_{\mu}=+4.46$.
Comparing the positions of the ammonia inversion transition with pure rotation
transitions in CO, HCO+, and HCN, for which $Q_{\mu}=1$, they obtained a
constraint on the changes in $\mu$ at redshift $z=0.6847$ towards the quasar
B0218+357, $\Delta\mu/\mu$ $<2\times 10^{-6}$.

The accuracy of $\Delta\mu/\mu$ estimates depends linearly on the accuracy of
position measurements of spectral lines and is inversely proportional to the
difference in the sensitivity coefficients of a pair of lines used in the
analysis ([12]):

$\frac{\Delta\mu}{\mu}=\frac{V_{j}-V_{i}}{c(Q_{\mu,i}-Q_{\mu,j})},$ (1)

where $V_{j}$ and $V_{i}$ are the local standard of rest radial velocities,
$V_{\rm LSR}$, of a pair of molecular transitions having sensitivity
coefficients $Q_{\mu,j}$ and $Q_{\mu,i}$, and $c$ is the speed of light.

The next step in improving $\Delta\mu/\mu$ estimates was the search for
molecules with large differences in the sensitivity coefficients of one
molecule to avoid spectroscopic shifts (the so-called Doppler noise) caused by
segregation effects in the spatial distribution of molecules within molecular
clouds when $\Delta\mu/\mu$ is calculated by comparing the positions of the
lines of different molecules. The methanol molecule CH3OH turned out to be
most suitable for this purpose. The parent methanol and its isotopologues are
among the most abundant molecules in interstellar space. The sensitivity
coefficients for CH3OH lie in the range $-53\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;Q_{\mu}\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;+43$ ([12], [13])
which made it possible to obtain restrictions on cosmological variations of
$\mu$ at the level of $5\times 10^{-8}$ at $z=0.88582$ towards the quasar
PKS1830-211 ([14]).

The same order of magnitude constraints on the spatial $\mu$-variation in the
Milky Way disk over the range of the galactocentric distances $4\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;R\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;12$ kpc are based
on the methanol thermal emission lines ([15]) and methanol masers observations
([16], [17]): $\Delta\mu/\mu$ $<(2-3)\times 10^{-8}$.

It should be emphasized that over the past five decades, improvements in
$\Delta\mu/\mu$ estimates have always followed improvements in the quality of
astronomical equipment and corresponded to the limiting parameters of
observations in optical and radio ranges (see, e.g., Fig. 1 in [18]). Such
extreme measurements may introduce unaccounted systematic errors into the
final results. Therefore, to obtain robust results, it is necessary to use
various instruments and expand the list of molecular transitions with high
sensitivity to changes in $\mu$. Among a number of complex organic molecules
studied (CH3COH, CH3CONH2, CH3OCOH, CH3COOH in [13]; CH3NH2 in [19]; and CH3SH
in [20]), suitable transitions were found in methanol isotopologues ([21]):
for 8 lines of deuterated methanol CD3OH from the frequency interval
$1.202<\nu<20.089$ GHz, the sensitivity coefficients222Note that
$Q_{\mu}=-K_{\mu}$, where $K_{\mu}$ is used by [21] who defined $\mu$ as the
$proton$-$to$-$electron$ mass ratio. lie between $-330$ and +88; 2 lines of
CD3OD at 2.238 GHz and 2.329 GHz have, respectively, $Q_{\mu}=-45$ and +80; 2
lines of CH318OH at 2.605 GHz and 11.630 GHz have, respectively, $Q_{\mu}=-93$
and +34; and $Q_{\mu}=+63$ for one line of 13CH3OH at 1.990 GHz.

A list of lines with high sensitivity coefficients of the 13C- and 18O-bearing
methanol has recently been expanded to 27 transitions of 13CH3OH from the
frequency interval $1.989<\nu<75.415$ GHz with $Q_{\mu}$ ranging from $-32$ to
+78, and to 30 transitions of CH318OH with $2.604<\nu<105.181$ GHz and
$-109\leq Q_{\mu}\leq+33$ ([22]). We have used these newly calculated
$Q_{\mu}$ values for 13CH3OH to set constraint on $\Delta\mu/\mu$ $<3\times
10^{-8}$ from the thermal emission lines of 13CH3OH detected in the star-
forming region NGC 6334I by [23].

Astrophysical observations of deuterated methanol in Galactic molecular clouds
have been performed previously and are still an important goal of current
studies. The CH3OD lines were detected in the high-mass star-forming regions
Sgr B2 ([24], [25], [26]) and Orion-KL ([27], [28], [29], [30]) which are both
the reference sources to search for complex organic molecules. Besides the
isotopologue CH3OD was observed in the high-mass star-forming regions NGC
7538-IRS1 ([31]), NGC 6334I ([32]), DR 21 ([33]), in the intermediate-mass
star-forming regions CepE-mm and OMC2-FIR4 ([34]), and in the low-mass star-
forming regions IRAS 16293-2422 ([35], [36]), IRAS 4A, IRAS 4B, and IRAS 2
([37]), and SVS 13A ([38]), and HH 212 ([39]). Triply deuterated methanol,
CD3OH, was observed in the low-mass protostar IRAS 16293-2242 ([40]). The
search for quadruply deuterated methanol, CD3OD, towards this object has not
yet led to a reliable result but the research is currently ongoing ([41]).

The present work is a continuation of our calculations of sensitivity
coefficients for deuterated methanol CH3OD, CD3OH, and CD3OD in a wider
spectral range as compared to the previous work by [21].

##  II Calculations

Each of the deuterated methanol molecules (CH3OD, CD3OH, and CD3OD) has, along
with the overall rotation, internal torsion motions, the energy spectrum of
which is similar to that of methanol CH3OH. The geometrical structure of these
molecules is characterized by slight asymmetry which is the same for CH3OH and
CD3OH, but increases for CH3OD and CD3OD resulting from the off-axis
deuterium. Small differences in the degree of asymmetry between CH3OH and
deuterated methanol molecules do not, however, affect the results of
calculations of the energy spectra of deuterated molecules using the torsion-
rotation Hamiltonian models developed for the parent methanol, as shown by
[42] who demonstrated that for a particular molecule CH3OD a global fit of the
observed transitions lies within the experimental uncertainties (a root-mean-
square deviation of 0.098 MHz for $J\leq 10$). Therefore, when calculating the
sensitivity coefficients $Q_{\mu}$ for lines in CH3OD, CD3OH, and CD3OD, we
used the developed method from our previous works in which the sensitivity
coefficients were calculated for CH3OH ([12]) and for two isotopologues
13CH3OH and CH318OH ([22]).

Figure 1: Partial rotational level diagrams of $A$\- and $E$-type methanol
CH3OH in the torsional ground state ($v_{t}=0$). Numbers below the horizontal
bars are the values of the rotational angular momenta $J$.  Figure 2: Same as
Fig. 1 but for the deuterated isotopologue CH3OD.  Figure 3: Same as Fig. 1
but for the deuterated isotopologue CD3OH.  Figure 4: Same as Fig. 1 but for
the deuterated isotopologue CD3OD.

###  II.1 Sensitivity coefficients and hyperfine corrections

Methanol in nature occurs in two types – $A$ and $E$, which differ in the
values of the total nuclear spin of the three hydrogen atoms in the methyl
group CH3. The $A$-methanol has parallel proton spins and the total nuclear
spin $I_{123}=3/2$, whereas the $E$-methanol has one of the protons of the
methyl group an antiparallel spin with respect to the others and
$I_{123}=1/2$. These non-zero total nuclear spins, interacting with the
magnetic fields generated by the overall and torsion rotations of the
molecule, lead to a hyperfine splitting of energy levels.

As noted by [43], the hyperfine splitting of the torsion-rotation levels of
molecules with hindered internal rotation such as methanol and its deuterated
isotopologues may affect values of the sensitivity coefficients $Q_{\mu}$,
since hyperfine effects shift the centre of torsion-rotation lines which was
not accounted for by effective torsion-rotation Hamiltonians used in previous
calculations of $Q_{\mu}$ in [12], [21]. Below we outline conditions when the
hyperfine structure of the torsion-rotation levels can be neglected.

The frequency of a specific hyperfine transition, $\nu$, between upper and
lower states consists of the rotation, $\nu_{\scriptscriptstyle r}$, torsion,
$\nu_{\scriptscriptstyle t}$, and hyperfine, $\nu_{\scriptscriptstyle h}$,
parts

$\nu=\nu_{\scriptscriptstyle r}+\nu_{\scriptscriptstyle
t}+\nu_{\scriptscriptstyle h}\ ,$ (2)

which are scaled with $\mu$ as $\mu^{1}$, $\mu^{\tau}$, and $\mu^{2}$,
respectively, where $\tau$ is a number attributed to a given transition.

The sensitivity coefficient $Q_{\mu}$ is determined by two derivatives

$\frac{d\nu}{\nu}=Q_{\mu}\frac{d\mu}{\mu}\ .$ (3)

From (2) it follows that

$\frac{d\nu}{d\mu}=\frac{1}{\mu}(\nu_{\scriptscriptstyle
r}+\tau\nu_{\scriptscriptstyle t}+2\nu_{\scriptscriptstyle h})\ ,$ (4)

and then

$\frac{d\nu}{\nu}=\frac{d\mu}{\mu}\left[1+(\tau-1)\frac{\nu_{\scriptscriptstyle
t}}{\nu}+\frac{\nu_{\scriptscriptstyle h}}{\nu}\right]\ ,$ (5)

or, using definition of $Q_{\mu}$, we have

$Q_{\mu}\equiv Q^{(\scriptscriptstyle r)}_{\mu}+Q^{(\scriptscriptstyle
t)}_{\mu}+Q^{(\scriptscriptstyle
h)}_{\mu}=1+(\tau-1)\frac{\nu_{\scriptscriptstyle
t}}{\nu}+\frac{\nu_{\scriptscriptstyle h}}{\nu}\ .$ (6)

Therefore, for a pure rotation transition, the sensitivity coefficient
$Q^{(\scriptscriptstyle r)}_{\mu}=1$. The energy difference between torsion-
rotation levels of methanol in the microwave range is of the order of $\sim
10$s GHz, whereas the hyperfine splitting occurs at $\sim 10$s kHz, so that
$|\nu_{\scriptscriptstyle h}/\nu|\ll 1$ and, thus, we can neglect the
hyperfine corrections in the effective Hamiltonian.

If, however, the hyperfine splitting is large compared to the transition
frequency, then the 3rd term in (2) and (6) must be included in calculations
of the sensitivity coefficients. In this case the torsion-rotation transition
splits into several lines with different frequencies and different $Q_{\mu}$
values.

As for the 2nd term in (2) and (6), it shows that transitions with enhanced
sensitivities occur when $\nu_{\scriptscriptstyle t}$ and
$\nu_{\scriptscriptstyle r}$ have different signs (i.e., energies associated
with the hindered internal rotation and the overall rotation are canceled) and
$|\nu_{\scriptscriptstyle t}/\nu|\gg 1$. We note in passing that different
methanol isotopologues have slightly different rotation
$\nu_{\scriptscriptstyle r}$ and torsion $\nu_{\scriptscriptstyle t}$
frequencies, but significantly different $\nu$. This results in different
$Q_{\mu}$ values for transitions with identical sets of quantum numbers in
methanol isotopologues.

The microwave torsion-rotation transition can be defined by the quantum
numbers $J$ and $K$ which are, respectively, the total angular momentum and
its projection on the axis of the molecule. If subscripts $u$ and $\ell$ will
denote the upper and lower energy levels, then the microwave torsion-rotation
transition in emission can be written as $J_{{\scriptscriptstyle
u}K_{\scriptscriptstyle u}}\to
J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$. Figures 1-4 show
diagrams of the methanol and its deuterated isotopologues energy levels in the
torsional ground state ($v_{t}=0$) which range up to an excitation temperature
of 300 K above ground. The allowed transitions occur in accord with selection
rules which have the following forms for $A$\- and $E$-type methanol (e.g.,
[44]):

$A$-type
---
$\Delta J=0$ | $\Delta K=0,\pm 1$ |  $\pm\leftrightarrow\mp$
$\Delta J=\pm 1$ | $\Delta K=0,\pm 1$ |  $\pm\leftrightarrow\pm$

$E$-type
---
$\Delta J=0$ | $\Delta K=\pm 1$
$\Delta J=\pm 1$ | $\Delta K=0,\pm 1$

It is interesting to note the inversion of energy levels in some transitions
with identical sets of quantum numbers ($J$, $K$) when passing from one
isotopologue to another which affects the values of $Q_{\mu}$ considerably.
For instance, the transition $J_{{\scriptscriptstyle u}K_{\scriptscriptstyle
u}}\to
J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}=2_{1}-3_{0}A^{+}$ in
CH3OH has $Q_{\mu}=+6.3$, but its inverted analog $J_{{\scriptscriptstyle
u}K_{\scriptscriptstyle u}}\to
J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}=3_{0}-2_{1}A^{+}$ in
CH3OD has $Q_{\mu}=-19.3$, or $J_{{\scriptscriptstyle u}K_{\scriptscriptstyle
u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}=4_{-1}-3_{0}E$
in CH3OH has $Q_{\mu}=-9.6$, but $J_{{\scriptscriptstyle
u}K_{\scriptscriptstyle u}}\to
J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}=3_{0}-4_{-1}E$ in CD3OH
has $Q_{\mu}=+73$, etc.

###  II.2 Calculating procedure

The procedure for calculating sensitivity coefficients of methanol lines from
microwave range was developed in [12]. It is based on an approach by [45] who
suggested a convenient form of the effective Hamiltonian (hereafter RF
Hamiltonian) which contains six spectroscopic constants having clear
dependence on $\mu$. Comparison with the results of calculations using a more
complex form of the Hamiltonian from [21] showed that the RF Hamiltonian gives
quite accurate values of the sensitivity coefficients ([12], [22]).

Seven parameters of the RF Hamiltonian are three rotation parameters $A$, $B$,
$C$, one parameter $D$, describing interaction of the internal rotation with
overall rotation, the kinetic coefficient $F$, and the depth of the three-fold
symmetric torsion potential $V_{3}$,

$V(\omega)=\frac{V_{3}}{2}(1-\cos 3\omega)\ ,$ (7)

where $0\leq\omega\leq 2\pi$ is the torsion angle of the internal rotation of
the CH3 group relative to the OH radical.

The values of all parameters, except $V_{3}$, were calculated from the moments
of inertia of CH3OD, CD3OH and CD3OD listed in Table 1. The spectroscopic
constants are defined as follows:

$A=\frac{1}{2}\hbar^{2}\left(\frac{I_{a}+I_{b}}{I_{a}I_{b}-I^{2}_{ab}}-\frac{I_{b}}{I^{2}_{b}+I^{2}_{ab}}\right),$
(8)

$B=\frac{1}{2}\hbar^{2}\frac{I_{b}}{I^{2}_{b}+I^{2}_{ab}},$ (9)

$C=\frac{1}{2}\hbar^{2}\frac{1}{I_{c}},$ (10)

$D=\frac{1}{2}\hbar^{2}\frac{I_{ab}}{I^{2}_{b}+I^{2}_{ab}},$ (11)

$F=\frac{1}{2}\hbar^{2}\frac{I_{a}I_{b}-I^{2}_{ab}}{I_{a2}(I_{a1}I_{b}-I^{2}_{ab})}.$
(12)

Here $\hbar=h/2\pi$, $I_{a}$, $I_{b}$, $I_{c}$ are the moments of inertia,
$I_{ab}$ is the product of inertia about the $a$\- and $b$-axes in the
$a,b,c$-axis system whose $a$-axis is parallel to the internal rotation axis
(assumed to be that of the methyl top), the $c$-axis perpendicular to the COH
plane. $I_{a2}$ is the axial moment of inertia of the methyl group, and
$I_{a1}$ is that of the framework (the hydroxyl group), the sum of which
determines $I_{a}$: $I_{a}=I_{a1}+I_{a2}$ (for details, see [46]). The RF
Hamiltonian also includes a dimensionless parameter $\rho$ – the internal
rotation interaction constant, which is given for asymmetric molecules by

$\rho=\frac{I_{a2}\sqrt{I^{2}_{b}+I^{2}_{ab}}}{I_{a}I_{b}-I^{2}_{ab}}.$ (13)

Table 1: Moments of inertia (in units amu$\cdot$Å2) for the parent methanol molecule and its deuterated isotopologues from [47]. | CH3OD | CD3OH | CD3OD | CH3OH
---|---|---|---|---
$I_{a1}$ | 1. | 3946 | 0. | 7514 | 1. | 4082 | 0. | 7493
$I_{a2}$ | 3. | 2249 | 6. | 3872 | 6. | 3939 | 3. | 2122
$I_{a}^{*}$ | 4. | 6195 | 7. | 1386 | 7. | 8021 | 3. | 9615
$I_{b}$ | 21. | 5304 | 25. | 4621 | 26. | 7242 | 20. | 4771
$I_{ab}$ | 0. | 850 |  $-0$. | 125 | 0. | 8047 |  $-$0. | 065
$I_{c}$ | 22. | 9880 | 26. | 2203 | 28. | 1529 | 21. | 2614
${}^{*}I_{a2}$ is the axial moment of inertia of the methyl group,
$I_{a1}$ is that of the hydroxyl group, $I_{a}$ = $I_{a1}$ \+ $I_{a2}$.
Table 2: Spectroscopic parameters of the RF Hamiltonian and the asymmetry parameter $\kappa$ for CH3OD, CD3OH, CD3OD, and CH3OH. | CH3OD | CD3OH | CD3OD | CH3OH
---|---|---|---|---
$A$ (cm-1) | 3. | 679363 | 2. | 361733 | 2. | 169903 | 4. | 255604
$B$ (cm-1) | 0. | 7818695 | 0. | 6620463 | 0. | 6302237 | 0. | 8232282
$C$ (cm-1) | 0. | 7333172 | 0. | 6429177 | 0. | 5987836 | 0. | 7928686
$D$ (cm-1) | 0. | 02923331 |  $-0$. | 003250156 | 0. | 01897685 |  $-$0. | 002613155
$F$ (cm-1) | 17. | 58161 | 25. | 09238 | 14. | 81704 | 27. | 75181
$V_{3}$ (cm-1) | 370. | 3 | 371. | 8 | 362. | 8 | 375. | 6
$\rho$ | 0. | 7031751 | 0. | 8948290 | 0. | 8224358 | 0. | 8109008
$\kappa$ |  $-0$. | 967 |  $-0$. | 978 |  $-0$. | 960 |  $-0$. | 982
Values of $V_{3}$ are taken from [47].

The degree of asymmetry is defined by the parameter $\kappa$:

$\kappa=\frac{2B-A-C}{A-C}\ ,$ (14)

which is equal to 1 or $-1$ when the molecule is symmetric ([48]). Table 2
contains the calculated spectroscopic parameters for CH3OD, CD3OH and CD3OD
with the potential barriers $V_{3}$ and the asymmetry parameter $\kappa$.

The calculated spectroscopic parameters were used to determine the Hamiltonian
matrix, the non-vanishing elements of which are listed in [22] for specific
values of the rotational angular momentum $J$. Diagonalization of this matrix
gives simultaneously the eigen-energies and eigenfunctions for $A$\- and
$E$-type methanol. The dependence of the found eigenvalues $E_{i}$ on
$\Delta\mu/\mu$ defines the sensitivity coefficient $Q_{\mu}$ through the
following relations:

$\Delta E_{i}=q_{i}\frac{\Delta\mu}{\mu},$ (15)

and

$Q_{\mu}=\frac{q_{\scriptscriptstyle u}-q_{\scriptscriptstyle\ell}}{\nu}.$
(16)

Here $q_{i}$ is the so-called $q$-factor, individual for each level, which
shows a response of the level $E_{i}$ to a small change of $\mu$, when
$|\Delta\mu/\mu|\ll 1$. This functional dependence is assessed through the
diagonalization of the Hamiltonian matrix for three sets of parameters that
correspond to $\mu=\mu_{\scriptscriptstyle 0}$ and
$\mu=\mu_{\scriptscriptstyle 0}(1\pm\varepsilon)$, where $\varepsilon$ is
equal to 0.001 or 0.0001, and $\mu_{\scriptscriptstyle
0}\equiv\mu_{\scriptscriptstyle\rm lab}$ ([12]). The subscripts $u$ and $\ell$
denote the upper and lower levels, respectively.

For calculations we selected molecular transitions with $0\leq J\leq 13$ and
$|\Delta K|=1$, since transitions with $\Delta K=0$ are purely rotation and
have $Q_{\mu}\approx 1$. Our calculations for the three deuterated
isotopologues of methanol CH3OD, CD3OH and CD3OD cover the low frequency range
$1-50$ GHz which contains transitions with the highest sensitivity to
$\mu$-variations.

The results of our calculations are reported in Table 3 (CH3OD)333The
sensitivity coefficients for CH3OD are calculated for the first time., Table 4
(CD3OH), and Table 5 (CD3OD). The structure of these tables is the following.
The first column contains the quantum numbers $J$ and $K$ of the upper and
lower levels. The second and third columns show the rest frequencies. The
fourth column lists the sensitivity coefficients $Q_{\mu}$ and their
uncertainties, estimated according to the scheme described in Section III.
Included in these tables are molecular transitions with large sensitivity
coefficients, $|Q_{\mu}|>3$, except for the $8_{-1}-8_{0}E$ transition in
CD3OD with $Q_{\mu}=-0.9$ which is added to Table 5 for comparison with [13].

Tables 3–5 show that deuterated isotopologues of methanol have high
sensitivity coefficients of both signs. The calculated values of $Q_{\mu}$
range from $-32$ to +25 for CH3OD, from $-300$ to +73 for CD3OH, and from
$-44$ to +38 for CD3OD. The results obtained indicate that the most attractive
molecule for testing $\mu$-variations is CD3OH, for which the maximum
difference between the sensitivity coefficients (denominator in Eq. 1) reaches
the value of $\Delta Q_{\mu}\approx 370$ in the narrow frequency range $1-5$
GHz which by itself minimizes possible systematic errors in the position
measuring of spectral lines from a wide frequency range.

Table 3: Numerical calculation of the sensitivity coefficients $Q_{\mu}$ for the low-frequency torsion-rotation transitions ($\Delta K=\pm 1$) in CH3OD. Given in parenthesis are error estimates in the last digits.  Transition | Frequency (MHz) | $Q_{\mu}$ | Transition | Frequency (MHz) | $Q_{\mu}$
---|---|---|---|---|---
$J_{{\scriptscriptstyle u}K_{\scriptscriptstyle u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$ | Exper. | Theor. |  | $J_{{\scriptscriptstyle u}K_{\scriptscriptstyle u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$ | Exper. | Theor. |
$11_{4}-12_{3}\ A^{-}$ | 3983. | 678∗ | 4839. | 8 | $-32(5)$ | $7_{-1}-7_{0}\ E$ | 17889. | 040 | 17982. | 9 | $-6.4(3)$
$3_{0}-2_{1}\ A^{+}$ | 4812. | 471∗ | 4468. | 3 | $-19.3(1.0)$ | $6_{-1}-6_{0}\ E$ | 18454. | 760 | 18525. | 9 | $-6.5(3)$
$10_{-3}-11_{-2}\ E$ | 6346. | 731∗ | 7012. | 6 | $25(2)$ | $5_{-1}-5_{0}\ E$ | 18792. | 970 | 18844. | 2 | $-6.6(3)$
$9_{2}-8_{3}\ A^{-}$ | 6362. | 584∗ | 7195. | 5 | $15(2)$ | $4_{-1}-4_{0}\ E$ | 18957. | 170 | 18992. | 2 | $-6.6(3)$
$9_{2}-8_{3}\ E$ | 6761. | 206∗ | 5619. | 9 | $11(2)$ | $1_{-1}-1_{0}\ E$ | 18957. | 950 | 18965. | 0 | $-6.8(3)$
$9_{2}-8_{3}\ A^{+}$ | 10279. | 780 | 11079. | 8 | $9.3(1.4)$ | $2_{-1}-2_{0}\ E$ | 18991. | 670 | 19004. | 8 | $-6.8(3)$
$12_{-1}-12_{0}\ E$ | 11367. | 779∗ | 11580. | 5 | $-6.4(6)$ | $3_{-1}-3_{0}\ E$ | 19005. | 640 | 19028. | 0 | $-6.7(3)$
$11_{-1}-11_{0}\ E$ | 13033. | 065∗ | 13225. | 9 | $-6.4(5)$ | $1_{1}-2_{0}\ E$ | 19518. | 790 | 19404. | 9 | $3.67(17)$
$10_{-1}-10_{0}\ E$ | 14588. | 875∗ | 14758. | 9 | $-6.4(4)$ | $6_{2}-7_{1}\ A^{+}$ | 23407. | 520 | 23200. | 3 | $6.6(6)$
$3_{2}-4_{1}\ E$ | 14920. | 430 | 14716. | 3 | $-8.8(6)$ | $5_{1}-4_{2}\ E$ | 30839. | 200 | 31093. | 0 | $5.7(3)$
$7_{1}-6_{2}\ A^{-}$ | 15467. | 910 | 15510. | 4 | $-7.5(8)$ | $5_{2}-6_{1}\ A^{-}$ | 34537. | 290 | 34559. | 4 | $4.8(3)$
$8_{1}-7_{2}\ A^{+}$ | 15720. | 990 | 16043. | 7 | $-7.3(1.0)$ | $1_{1}-2_{0}\ A^{+}$ | 41858. | 774∗ | 42261. | 2 | $3.34(11)$
$9_{-1}-9_{0}\ E$ | 15948. | 140 | 16094. | 5 | $-6.4(3)$ | $9_{-3}-10_{-2}\ E$ | 51926. | 768∗ | 52658. | 7 | $3.9(3)$
$8_{-1}-8_{0}\ E$ | 17057. | 090 | 17176. | 5 | $-6.4(3)$ |  |  |  |  |  |
Notes: Experimental frequencies are taken from [49]; predicted frequencies are
marked by asterisk.
Table 4:  Numerical calculation of the sensitivity coefficients $Q_{\mu}$ for the low-frequency torsion-rotation transitions ($\Delta K=\pm 1$) in CD3OH. Given in parenthesis are error estimates in the last digits.  Transition | Frequency (MHz) | $Q_{\mu}$ | Transition | Frequency (MHz) | $Q_{\mu}$
---|---|---|---|---|---
$J_{{\scriptscriptstyle u}K_{\scriptscriptstyle u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$ | Exper. | Theor. |  | $J_{{\scriptscriptstyle u}K_{\scriptscriptstyle u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$ | Exper. | Theor. |
$2_{2}-1_{1}\ E$ | 1202. | 3089∗ | 1344. | 8 | $-300(12)^{a}$ | $11_{3}-11_{2}\ A^{-+}$ | 17918. | 90 | 18210. | 8 | $-20.2(9)$
$1_{1}-2_{0}\ E$ | 1424. | 1693∗ | 1330. | 4 | $47(2)^{b}$ | $9_{3}-9_{2}\ A^{+-}$ | 18081. | 20 | 18289. | 3 | $-20.4(9)$
$12_{-7}-13_{-6}\ E$ | 2656. | 1884∗ | 2032. | 2 | $-103(9)$ | $12_{3}-12_{2}\ A^{-+}$ | 18097. | 00 | 18436. | 6 | $-19.8(9)$
$10_{-2}-9_{-3}\ E$ | 2754. | 4417∗ | 3247. | 8 | $-84(5)$ | $13_{3}-13_{2}\ A^{-+}$ | 18317. | 15 | 18706. | 1 | $-19.4(9)$
$8_{3}-9_{2}\ E$ | 2827. | 2446∗ | 2367. | 6 | $11(5)^{c}$ | $10_{3}-10_{2}\ A^{+-}$ | 18395. | 60 | 18650. | 2 | $-19.9(9)$
$6_{0}-5_{1}\ A^{+}$ | 2971. | 0289∗ | 3360. | 0 | $-96(4)^{d}$ | $8_{6}-9_{5}\ A^{+}$ | 18988. | 36 | 18429. | 6 | $-16.9(1.1)$
$6_{1}-5_{2}\ A^{-}$ | 3130. | 6032∗ | 3417. | 2 | $-46(3)$ | $8_{6}-9_{5}\ A^{-}$ | 18988. | 36 | 18429. | 6 | $-16.9(1.1)$
$3_{0}-4_{-1}\ E$ | 4011. | 3609∗ | 3734. | 5 | $73(3)^{e}$ | $12_{3}-12_{2}\ A^{+-}$ | 19294. | 50 | 19651. | 5 | $-18.7(8)$
$11_{-5}-12_{-4}\ E$ | 4608. | 4386∗ | 4030. | 4 | $5(3)$ | $13_{3}-13_{2}\ A^{+-}$ | 19905. | 8 | 20318. | 1 | $-17.9(8)$
$5_{5}-6_{4}\ E$ | 6723. | 8853∗ | 6293. | 0 | $-56(3)^{f}$ | $10_{5}-9_{6}\ A^{+}$ | 20088. | 47 | 20678. | 1 | $17.9(1.1)^{g}$
$5_{2}-6_{1}\ A^{+}$ | 7662. | 4811∗ | 7379. | 8 | $20.2(1.3)$ | $10_{5}-9_{6}\ A^{-}$ | 20088. | 47 | 20678. | 1 | $17.9(1.1)^{h}$
$8_{-1}-7_{-2}\ E$ | 9945. | 16 | 10418. | 8 | $-22.0(1.1)$ | $12_{4}-13_{3}\ E$ | 24612. | 81 | 23996. | 1 | $13.5(7)$
$11_{3}-10_{4}\ A^{+}$ | 11125. | 37 | 11672. | 4 | $-6.5(1.2)$ | $9_{4}-10_{3}\ A^{-}$ | 27981. | 73 | 27515. | 2 | $4.0(5)$
$11_{3}-10_{4}\ A^{-}$ | 11215. | 14 | 11764. | 1 | $-6.5(1.2)$ | $9_{4}-10_{3}\ A^{+}$ | 28032. | 94 | 27567. | 5 | $4.0(5)$
$3_{3}-3_{2}\ A^{-+}$ | 17328. | 19 | 17340. | 1 | $-21.9(1.0)$ | $6_{-2}-7_{-1}\ E$ | 28811. | 25 | 28420. | 8 | $8.9(4)$
$3_{3}-3_{2}\ A^{+-}$ | 17335. | 14 | 17347. | 0 | $-21.9(1.0)$ | $7_{1}-6_{2}\ A^{+}$ | 29550. | 25 | 29899. | 6 | $-4.0(4)$
$4_{3}-4_{2}\ A^{-+}$ | 17353. | 94 | 17387. | 0 | $-21.8(1.0)$ | $7_{4}-6_{5}\ E$ | 32352. | 307 | 32816. | 1 | $12.8(6)$
$4_{3}-4_{2}\ A^{+-}$ | 17374. | 42 | 17407. | 7 | $-21.8(1.0)$ | $5_{-1}-4_{0}\ E$ | 35748. | 571 | 36099. | 7 | $-6.2(3)$
$5_{3}-5_{2}\ A^{-+}$ | 17388. | 63 | 17447. | 6 | $-21.7(9)$ | $8_{-3}-9_{-2}\ E$ | 36663. | 008 | 36258. | 2 | $7.4(4)$
$6_{3}-6_{2}\ A^{-+}$ | 17433. | 99 | 17523. | 4 | $-21.5(9)$ | $4_{1}-5_{0}\ A^{+}$ | 37322. | 977 | 37013. | 0 | $8.7(3)$
$5_{3}-5_{2}\ A^{+-}$ | 17435. | 92 | 17495. | 7 | $-21.6(9)$ | $4_{2}-5_{1}\ A^{-}$ | 37463. | 512 | 37242. | 4 | $5.0(2)$
$7_{3}-7_{2}\ A^{-+}$ | 17492. | 27 | 17616. | 0 | $-21.3(9)$ | $3_{2}-2_{1}\ E$ | 40649. | 426 | 40858. | 1 | $-7.1(3)$
$6_{3}-6_{2}\ A^{+-}$ | 17527. | 61 | 17618. | 3 | $-21.4(9)$ | $11_{-7}-12_{-6}\ E$ | 41737. | 585 | 41149. | 7 | $-5.6(6)$
$8_{3}-8_{2}\ A^{-+}$ | 17566. | 15 | 17727. | 7 | $-21.1(9)$ | $2_{0}-3_{-1}\ E$ | 43453. | 48 | 43245. | 6 | $8.5(3)$
$7_{3}-7_{2}\ A^{+-}$ | 17658. | 70 | 17784. | 6 | $-21.1(9)$ | $7_{0}-6_{1}\ A^{+}$ | 43501. | 125 | 43975. | 9 | $-5.6(3)$
$9_{3}-9_{2}\ A^{-+}$ | 17659. | 00 | 17861. | 6 | $-20.8(9)$ | $4_{2}-5_{1}\ A^{+}$ | 45160. | 765 | 44941. | 7 | $4.3(2)$
$10_{3}-10_{2}\ A^{-+}$ | 17774. | 89 | 18021. | 1 | $-20.5(9)$ | $9_{-1}-8_{-2}\ E$ | 48567. | 661 | 49130. | 7 | $-3.7(2)$
$8_{3}-8_{2}\ A^{+-}$ | 17839. | 40 | 18004. | 7 | $-20.8(9)$ |  |  |  |  |  |
Notes: Experimental frequencies are taken from [50]; predicted frequencies are
marked by asterisk.
$Q_{\mu}$ from [21]: $-330\pm 20^{a}$, $42\pm 2^{b}$, $-43\pm 2^{c}$, $-93\pm
5^{d}$, $73\pm 4^{e}$, $-167\pm 8^{f}$, $88\pm 4^{g}$, $88\pm 4^{h}$.
Table 5:  Numerical calculation of the sensitivity coefficients $Q_{\mu}$ for the low-frequency torsion-rotation transitions ($\Delta K=\pm 1$) in CD3OD. Given in parenthesis are error estimates in the last digits.  Transition | frequency (MHz) | $Q_{\mu}$ | Transition | Frequency (MHz) | $Q_{\mu}$
---|---|---|---|---|---
$J_{{\scriptscriptstyle u}K_{\scriptscriptstyle u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$ | Exper. | Theor. |  | $J_{{\scriptscriptstyle u}K_{\scriptscriptstyle u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$ | Exper. | Theor. |
$1_{-1}-1_{0}\ E$ | 2237. | 8224∗ | 2229. | 4 | $-44.1(1.7)^{a}$ | $4_{3}-5_{2}\ A^{-}$ | 5276. | 6275∗ | 5020. | 4 | $-19.7(1.6)$
$8_{-3}-7_{-4}\ E$ | 2328. | 9943∗ | 2699. | 6 | $38(5)^{b}$ | $12_{0}-12_{-1}\ E$ | 5432. | 1218∗ | 5456. | 3 | $9.3(3)$
$2_{-1}-2_{0}\ E$ | 2476. | 7516∗ | 2472. | 2 | $-35.3(1.3)$ | $13_{0}-13_{-1}\ E$ | 5898. | 7376∗ | 5913. | 7 | $9.5(3)$
$3_{-1}-3_{0}\ E$ | 2777. | 7033∗ | 2778. | 4 | $-27.0(1.0)$ | $1_{1}-2_{0}\ A^{+}$ | 8147. | 610 | 8072. | 4 | $11.5(4)$
$4_{-1}-4_{0}\ E$ | 3100. | 6415∗ | 3107. | 6 | $-20.0(8)$ | $5_{1}-4_{2}\ A^{+}$ | 11755. | 04 | 11985. | 1 | $-4.4(6)$
$5_{-1}-5_{0}\ E$ | 3413. | 586∗ | 3427. | 9 | $-14.4(6)$ | $3_{2}-4_{1}\ A^{-}$ | 14172. | 74 | 14028. | 4 | $5.5(4)$
$9_{4}-10_{3}\ E$ | 3561. | 2687∗ | 3225. | 4 | $32(3)$ | $3_{2}-4_{1}\ A^{+}$ | 22792. | 07 | 22615. | 8 | $3.8(3)$
$6_{-1}-6_{0}\ E$ | 3695. | 4856∗ | 3717. | 7 | $-9.5(4)$ | $11_{5}-12_{4}\ A^{+}$ | 24130. | 44 | 23768. | 0 | $5.7(6)$
$7_{-1}-7_{0}\ E$ | 3940. | 3679∗ | 3970. | 3 | $-5.1(3)$ | $11_{5}-12_{4}\ A^{-}$ | 24141. | 87 | 23779. | 3 | $5.7(6)$
$8_{-1}-8_{0}\ E$ | 4161. | 7206∗ | 4197. | 5 | $-0.9(2)^{c}$ | $6_{2}-5_{3}\ A^{-}$ | 31381. | 11 | 31673. | 5 | $4.5(3)$
$13_{5}-12_{6}\ E$ | 4254. | 1923∗ | 4971. | 5 | $23(4)$ | $6_{2}-5_{3}\ A^{+}$ | 31980. | 59 | 32269. | 2 | $4.4(3)$
$4_{3}-5_{2}\ A^{+}$ | 4977. | 3938∗ | 4723. | 0 | $-20.9(1.7)$ | $9_{-3}-8_{-4}\ E$ | 39152. | 561 | 39556. | 0 | $3.2(3)$
$11_{0}-11_{-1}\ E$ | 5012. | 0901∗ | 5044. | 5 | $8.3(3)$ | $8_{4}-9_{3}\ E$ | 40645. | 642 | 40372. | 7 | $3.8(3)$
$11_{4}-10_{5}\ E$ | 5049. | 9782∗ | 5477. | 1 | $11(3)$ |  |  |  |  |  |
Notes: Experimental frequencies are taken from [41]; predicted transition
frequencies are marked by asterisk.
$Q_{\mu}=-45\pm 2^{a},80\pm 4^{b}$ ([21]), $Q_{\mu}=-0.7^{c}$ ([13]).
Table 6: The sensitivity coefficients for CH3OH calculated with BELGI code, $Q_{\mu}^{\scriptscriptstyle\rm BELGI}$, and “toy”, $Q_{\mu}^{\scriptscriptstyle\rm toy}$, model from JKX11, and, for comparison, the present paper $Q_{\mu}$ values. Given in parentheses are errors in the last digits. Transition | Frequency∗ | $Q_{\mu}^{\scriptscriptstyle\rm BELGI}$ | $Q_{\mu}^{\scriptscriptstyle\rm toy}$ | $Q_{\mu}$
---|---|---|---|---
$J_{{\scriptscriptstyle u}K_{\scriptscriptstyle u}}\to J_{{\scriptscriptstyle\ell}K_{\scriptscriptstyle\ell}}$ | (MHz) |  |  |
$5_{1}-6_{0}A^{+}$ | 6668.52 | 42(2) | 46 | 43(2)
$9_{-1}-8_{-2}E$ | 9936.20 |  $-$11.5(6) |  $-$16.7 |  $-$14.6(1.3)
$4_{3}-5_{2}A^{+}$ | 9978.69 |  $-$53(3) |  $-$35 |  $-$42(2)
$4_{3}-5_{2}A^{-}$ | 10058.26 |  $-$52(3) |  $-$35 |  $-$42(2)
$2_{0}-3_{-1}E$ | 12178.59 | 33(2) | 32 | 32.5(1.3)
$2_{1}-3_{0}E$ | 19967.40 | 5.9(3) | 5.0 | 6.3(3)
$9_{2}-10_{1}A^{+}$ | 23121.02 | 11.7(6) | 10.8 | 13.6(1.0)
$3_{2}-3_{1}E$ | 24928.72 |  $-$17.9(9) |  $-$15.2 |  $-$16.7(7)
$2_{2}-2_{1}E$ | 24934.38 |  $-$17.9(9) |  $-$15.2 |  $-$16.7(7)
$8_{2}-9_{1}A^{-}$ | 28969.95 | 9.5(6) | 8.8 | 11.0(7)
$4_{-1}-3_{0}E$ | 36169.29 |  $-$9.7(5) | 9.6 |  $-$9.6(4)
$6_{2}-5_{3}A^{+}$ | 38293.29 | 15.1(8) | 10.4 | 12.2(6)
$6_{2}-5_{3}A^{-}$ | 38452.65 | 15.0(8) | 10.4 | 12.2(6)
$7_{0}-6_{1}A^{+}$ | 44069.48 |  $-$5.2(3) |  $-$5.9 |  $-$5.4(3)
$1_{0}-2_{-1}E$ | 60531.49 | 7.4(4) | 7.3 | 7.3(3)
$1_{1}-2_{0}E$ | 68305.68 | 2.4(1) | 2.2 | 2.55(7)
Note: ∗Rest frequencies are taken from [51].

##  III Estimation of errors of the sensitivity coefficients

Some of the sensitivity coefficients presented in Tables 4 and 5 were
calculated earlier in [13], [21], which makes it possible to compare the
results of calculations using different forms of the Hamiltonian. The
footnotes to these tables show that the $Q_{\mu}$ coefficients from [13], [21]
differ significantly from our calculations for a number of lines. The most
striking cases are marked by superscripts $c,f,g,h$ in Table 4 and by $b$ in
Table 5. For the case $c$, we even found $Q_{\mu}$ of opposite signs.

Figure 5 illustrates the degree of agreement between common sensitivity
coefficients calculated in [13], [21] (red points) and in the present work
(black points). To understand large differences marked by the blue arrows in
this figure, we checked the errors of our procedure following the approach
developed in Refs. [12], [22]

Molecular transition frequencies depend on $\mu$ via the parameters (7)-(13)
of the effective Hamiltonian. Five of these parameters, $A$, …$F$, are
inversely proportional to the moments of inertia, which scales almost linearly
with $\mu^{-1}$. Thus, these parameters scale linearly with $\mu$. The
deviation from the linearity was estimated to be on the level 1-2% ([12]). It
is mostly caused by the weak dependence of the internuclear distances on $\mu$
due to the vibrational and centrifugal distortions. Up to the similar small
corrections, the remaining two parameters ($V_{3}$ and $\rho$) of the
effective Hamiltonian do not depend on $\mu$. All these corrections change the
scaling of the three terms in Eq. (2). This, in turn, changes the sensitivity
coefficients in Eq. (6). In order to estimate the scaling error for the
sensitivity coefficients $Q_{\mu}$ we independently change the scaling of each
of the parameters of the effective Hamiltonian by 2% and calculate sensitivity
coefficients using Eqs. (15, 16).

Another possible source of theoretical errors comes from the interaction of
the closely-lying torsion-rotation levels with the same exact quantum numbers.
Such interaction leads to the well-known level repulsion. Varying $\mu$
changes the distance between the levels and, thus, changes repulsion. This may
affect the sensitivity coefficients of the interacting levels ([52]). It is
important that the $q$-factors change in the opposite directions leaving their
sum unchanged. Clearly, this source of errors is most important when there are
close levels with significantly different $q$-factors. Numerically it leads to
the dependence of $q_{i}$, calculated from Eq. (16), on $\mu$ ([53]). To
estimate this error, we calculated $q_{i}$ using $\Delta\mu=\pm\epsilon$ and
estimated the derivative $\partial q_{i}/\partial\mu$. Our analysis shows that
this source of errors is subdominant for all levels considered in the present
paper. The estimated theoretical errors are given in parenthesis in Tables
3-6.

Coming back to the comparison of different sets of the $Q_{\mu}$ values shown
in Fig. 5, we note that in general there is a smooth dependence of $Q_{\mu}$
on frequency, despite some scatter of points at low frequencies caused by
interference of nearby energy levels. The highest $Q_{\mu}$ values are found
for transitions with the smallest differences between the energy levels,
$\Delta E\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;0.1$ K. The
results presented in Fig. 5 show a good concordance between [21] calculations
and ours for the parent molecule CH3OH, but essential deviations in some cases
for its isotopologues.

Figure 5: Comparison of the sensitivity coefficients for methanol and its
deuterated isotopologues: red dots – calculated by Jansen et al. [13], [21],
black dots – current results. The shadowed area shows the $\pm 3\sigma$
uncertainty zone. The blue arrows indicate maximum deviations in the $Q_{\mu}$
values between Jansen et al. and the present calculations. Note that $Q_{\mu}$
even changes sign in one case at 2.827 GHz for CD3OH (see Table 4). The
isotopologue CH3OD is considered for the first time in the present paper. In
general, $Q_{\mu}$ decreases with increasing frequencies of molecular
transitions in all four cases.

Among them, the most striking difference occurs at 20.088 GHz
($10_{5}-9_{6}A$) in CD3OH (Table 4) where Jansen’s point lies too far from
the $\pm 3\sigma$ uncertainty zone shown by the shadowed area, taking into
account that $\Delta E=0.96$ K for this transition.

The red point at 6.723 GHz ($5_{5}-6_{4}E$) also deviates from the general
trend, while the black one does not. However, this deviation is comparable to
the overall spread at $\Delta E\lesssim 0.3$ K. As for the $Q_{\mu}$ values at
2.827 GHz ($8_{3}-9_{2}E$), which have even opposite signs in our and Jansen’s
calculations, we cannot state with certainty for the intermediate value of
$\Delta E=0.14$ K for this transition whose result is more reliable since the
found values of $Q_{\mu}$ can belong either to the upper or to the lower
curve. The isotopologue CD3OD (Table 5) also demonstrates one point at 2.328
GHz ($8_{-3}-7_{-4}E$) where our and Jansen’s results differ significantly.

To illustrate the dependence of some values of the sensitivity coefficients on
the Hamiltonian model used, we compare in Table 6 the computational results
obtained by three models of the effective Hamiltonian which are outlined
below.

As mentioned above, [21] used the complex model of the Hamiltonian based on
119 parameters and calculated numerically the torsion-rotation levels with the
BELGI code. Despite the fact that this model calculates the transition
frequencies with high accuracy, it provides limited insight.

For a clearer understanding of the physical meaning, [13] used a “toy” model
with six parameters: $A,B,C,F,\rho$ and $V_{3}$. Using this model, the authors
calculated the sensitivity coefficients for some transitions in CH3OH. To
compare with our results, we recalculated the sensitivity coefficients
$Q_{\mu}$ for the same transitions using the seven spectroscopic parameters
from the last column in Table 2. The result of such comparison is presented in
Table 6. As seen, there are some discrepancies between the listed $Q_{\mu}$
values, the most pronounced of which are found for the following transitions:
$9_{-1}-8_{-2}E$, $4_{3}-5_{2}A^{+}$, $4_{3}-5_{2}A^{-}$, $6_{2}-5_{3}A^{+}$,
and $6_{2}-5_{3}A^{-}$.

The 119 parameters of the BELGI model have different dependence on $\mu$. Many
of the higher-order parameters are products of operators and can be quite
correlated. Therefore, the exact relationship between the higher-order
parameters of moments of inertia (and hence masses) is poorly known. It can
give some effects in calculating the sensitivity coefficients which were
tested for CH3OH in [13] by using different forms of the effective
Hamiltonian. The authors showed that the calculated sensitivity coefficients
coincide within a few percent. However, it is possible that the effects
produced by the high-order parameters may provide a more significant
contribution for methanol deuterated isotopologues. In any case, care should
be taken with those $Q_{\mu}$ values where large deviations have been detected
when different forms of the effective Hamiltonian were used. Elucidating the
nature of the discrepancies found requires more detailed analysis.

##  IV Conclusions

In this paper we calculated the sensitivity coefficients, $Q_{\mu}$, for
different molecular transitions from the microwave range 1–50 GHz to small
changes in the values of the electron-to-proton mass ratio, $\mu=m_{\rm
e}/m_{\rm p}$. The considered molecular species include CH3OD, CD3OH, and
CD3OD. The computational procedure is based on the effective Hamiltonian in
the form suggested by [45]. The obtained main results are as follows:

  1. 1.

For CH3OD, the $Q_{\mu}$ values lie in the interval from $-32$ to +25.

  2. 2.

For CD3OH, the sensitivity coefficients vary in a wide range from $-300$ at
1.202 GHz to +73 at 4.011 GHz.

  3. 3.

For CD3OD, the two neighboring transitions at 2.237 ($Q_{\mu}=-44$) and 2.328
GHz ($Q_{\mu}=+38$) confine the $Q_{\mu}$ values from the whole sample in the
frequency range 2–40 GHz.

  4. 4.

We also found that along with good agreement of the $Q_{\mu}$ values for the
parent methanol CH3OH calculated with different Hamiltonians in the present
and previous works, there are several pronounced outliers of unclear nature in
the isotopologues CD3OH and CD3OD.

## Acknowledgements

J.S.V. and S.A.L. are supported in part by the Russian Science Foundation
under grant No. 23-22-00124.

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