#  An improved rovibrational linelist of formaldehyde, H${}_{2}^{~{}12}$C16O

Afaf R. Al-Derzi  Jonathan Tennyson111To whom correspondence should be
addressed; email<EMAIL_ADDRESS>Sergei N. Yurchenko  Mattia Melosso
Ningjing Jiang  Cristina Puzzarini  Luca Dore  Tibor Furtenbacher  Roland
Tóbiás  Attila G. Császár  Department of Physics and Astronomy, University
College London, Gower Street, London WC1E 6BT, United Kingdom  Dipartimento di
Chimica “Giacomo Ciamician”, Università di Bologna, Via F. Selmi 2, 40126
Bologna, Italy  ELKH-ELTE Complex Chemical Systems Research Group and ELTE
Eötvös Loránd University, Institute of Chemistry, Laboratory of Molecular
Structure and Dynamics, H-1117 Budapest, Pázmány Péter sétány 1/A, Hungary

###### Abstract

Published high-resolution rotation-vibration transitions of
H${}_{2}^{~{}12}$C16O, the principal isotopologue of methanal, are analyzed
using the MARVEL (Measured Active Rotation-Vibration Energy Levels) procedure.
The literature results are augmented by new, high-accuracy measurements of
pure rotational transitions within the ground, $\nu_{3}$, $\nu_{4}$, and
$\nu_{6}$ vibrational states. Of the 16 596 non-redundant transitions
processed, which come from 43 sources including the present work, 16 403 could
be validated, providing 5029 empirical energy levels of H${}_{2}^{~{}12}$C16O
with statistically well-defined uncertainties. All the empirical rotational-
vibrational energy levels determined are used to improve the accuracy of
ExoMol’s AYTY line list for hot formaldehyde. The complete list of collated
experimental transitions, the empirical energy levels determined, as well as
the extended and improved line list are provided as Supplementary Material.

Keywords: Formaldehyde; Line list; Ro-vibrational energy; MARVEL analysis.

##  1 Introduction

Formaldehyde, formally methanal, HCHO, is the simplest aldehyde. In the gas
phase, formaldehyde is considered to be carcinogenic [1, 2]. Since
formaldehyde is a trace species in the earth’s atmosphere, formed as a result
of photo-oxidation or through incomplete biomass burning, its carcinogenic
nature motivated the development of spectroscopic techniques for detecting it
in trace quantities [3].

The photochemistry and photophysics of formaldehyde, involving several excited
electronic states and isomerization processes, leading to the products H2 \+
CO or H + HCO, have been investigated in considerable detail [4, 5, 6, 7, 8,
9], both experimentally and theoretically. The $\tilde{\rm X}\,^{1}A_{1}$ and
$\tilde{\rm A}\,^{1}A_{2}$ electronic states of formaldehyde have been
utilized in two-photon stimulated emission pumping (SEP) experiments to
derive, for example, a complete set of 27 normal-mode vibrational constants
characterizing the $\tilde{\rm X}\,^{1}A_{1}$ ground electronic state [10].
Thus, in 1984, Field and co-workers [10] could legitimately claim that the
“unique characteristics of SEP have enabled us to describe the $E\leqslant
9300$ cm-1 rotation-vibration structure of H2CO $\tilde{\rm X}\,^{1}A_{1}$ at
a level of detail and completeness which we believe is without precedent for a
four-atomic polyatomic molecule”. Nevertheless, though a huge achievement, the
accuracy attainable through these spectroscopic constants was a mere 3 cm-1,
hardly acceptable by present-day standards.

Over the years, beyond anharmonic (quartic) force field representations [11]
of the $\tilde{\rm X}\,^{1}A_{1}$ PES of formaldehyde [12, 13, 14, 15],
several global PESs have been developed for the [C,H,H,O] system [9, 16, 17,
18, 19, 20, 21, 22]. Part of the interest in a global ground-electronic-state
($S_{0}$) PES stems from the existence of further important minima, $cis$\-
and $trans$-H–C–OH, hydroxycarbene [17, 20], on it besides that corresponding
to formaldehyde. The $trans$-hydroxycarbene minimum is some 18 000 cm-1 [23]
above the H2CO minimum (the global minimum on $S_{0}$ belongs to CO + H2
[23]), separated by a transition state of 10 400 cm-1, measured from the
hydroxycarbene side, and through enhanced deep tunneling _trans_
-hydroxycarbene rearranges to formaldehyde with a half-life of only two hours
[23]. The CO$\cdot$H2 complex has also been the topic of interesting dynamical
and spectroscopic studies [24, 25].

Formaldehyde has been observed in different flames [26, 27], most famously in
cold flames, first by Sir Humphry Davy [26] in 1817. The spectrum of
formaldehyde has been used for its time-resolved monitoring in combustion
engines [28], for which supporting spectroscopic studies have been performed
[29].

Formaldehyde is a trace species on Mars [30], as well, and it is well known in
comets [31, 32, 33, 34]. Its infrared absorption spectrum has been observed in
protoplanetary disks [35]. Interstellar H${}_{2}^{~{}12}$C16O was detected 50
years ago [36] and many of the formaldehyde lines are now resolved routinely
in high-resolution spectroscopic studies of the interstellar medium [37].
Interstellar formaldehyde masers have been widely observed [38, 39, 40, 41,
42].

Much of the laboratory spectroscopy of formaldehyde has focused on electronic
(rovibronic) transitions. In fact, formaldehyde was the first polyatomic
molecule where the rotational subband structure of vibronic transitions has
been successfully analyzed [43]. It is probably fair to say that the
photodissociation of formaldehyde is one of the most thoroughly understood
polyatomic unimolecular reactions [6, 44].

The importance of line-by-line information on formaldehyde has led to a large
number of laboratory studies of its high-resolution rotational-vibrational
spectra [10, 28, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77,
78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96,
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107], involving a diverse set
of experimental techniques, such as SEP [10], dispersed laser-induced
fluorescence [75], conventional infrared spectroscopy, with some of the oldest
results contained in Refs. [54, 57, 65, 67, 68], tunable infrared difference
frequency laser spectroscopy [74], and sub-Doppler laser Stark and Doppler-
limited Fourier-transform spectroscopy [79]. Many of the results of these
high-resolution experimental studies will be addressed and discussed in detail
below.

During the present study our focus is exclusively on transitions within the
electronic ground state and we perform a MARVEL (Measured Active Rotational-
Vibrational Energy Levels) [108, 109, 110] analysis of all the measured
transitions of H${}_{2}^{~{}12}$C16O in its $\tilde{\rm X}\,^{1}A_{1}$ state.
This involves collating all available laboratory spectra and then validating
the observed and assigned transitions. The collection of the validated lines
is then inverted to give a set of accurate empirical energy levels with
statistically significant associated uncertainties. As part of this study, the
published spectra are augmented by high-accuracy measurements of pure
rotational transitions within the ground, $\nu_{3}$, $\nu_{4}$, and $\nu_{6}$
vibrational states.

Accurate energy levels have a number of uses in spectroscopy, kinetics, and
thermochemistry. One of the opportunities is to improve theoretical prediction
of rotation-vibration spectra [111]. Recently, as part of the ExoMol project
[112], Al-Refaie et al. [113] computed an extensive rotation-vibration line
list for hot formaldehyde, which they called AYTY. While this line list is
comprehensive and reasonably accurate, its predicted line centers do not meet
high resolution standards. The empirical energy levels generated, based
partially on literature results and partially on the new measurements
performed as part of this study, are used to make this line list suitable for
high-resolution observational studies. This is achieved by a one-by-one
replacement of the empirically adjusted energy levels of the AYTY line list by
the empirical rovibrational energy levels resulting from our MARVEL-based
investigation. Such high-accuracy line lists are required, for example, by
astronomers studying exoplanets using high-resolution Doppler-shift
spectroscopy [114]. Indeed, experience shows that failed detections can in
some cases be attributed to inaccurate line lists [115]. For this reason the
ExoMol project decided to refactor its line lists to include empirically
determined energy levels wherever possible [116]; this work forms part of this
effort.

Figure 1: Vibrational fundamentals of $\tilde{\rm X}\,^{1}A_{1}$
H${}_{2}^{~{}12}$C16O, with their symmetry species and the types of bands
indicated. The arrows representing atomic motions during the normal-mode
vibrations are not proportional to the actual displacements.

##  2 Methods

###  2.1 MARVEL

Details about the MARVEL technique [108, 109, 110], built upon the theory of
spectroscopic networks (SNs) [117, 118, 119], and the xMARVEL code, introduced
in Refs. 110 and 120, have been given in several recent publications [110,
118, 120, 121, 122]. MARVEL has been used to treat the laboratory
spectroscopic data of nine isotopologues of water [111, 123, 124, 125, 126],
as well as of diatomics [127, 128, 129, 130, 131], triatomics [132, 133, 134,
135], tetratomics [136, 137, 138], and beyond [139]. Thus, it is sufficient to
give only a relatively brief discussion about MARVEL here.

Of importance to the present study, the MARVEL protocol yields empirical
rovibronic energies with well-defined provenance and statistically-sound
uncertainties. It always starts with the construction of a SN using the
dataset of measured and assigned transitions collated from the literature.
Then, a sophisticated inversion of the transitions is performed, yielding
empirically determined rovibrational energy levels, with associated
uncertainties, within each component of the SN. The inversion process
expedites the validation of the experimental information through utilization
of several elements of network theory.

The xMARVEL procedure [110, 120] has been used extensively during the present
study to treat experimental rovibrational transitions of
H${}_{2}^{~{}12}$C16O. The corresponding xMARVEL input and output data files
are provided as supplementary material.

###  2.2 Quantum numbers

The xMARVEL procedure requires as input spectral lines assigned with a unique
set of quantum numbers. The equilibrium structure of formaldehyde on its
ground electronic state has $C_{2v}$ point-group symmetry (with irreducible
representations, irreps, A1, A2, B1, and B2, which are the same for the
isomorphic $C_{2v}$(M) molecular-symmetry group); it is an asymmetric-top,
semirigid molecule. The vibration-rotation motions of formaldehyde can be well
represented by a set of six vibrational normal-mode quantum numbers,
$[v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}]$, arranged in standard order [140], and
the three standard “rigid-rotor” quantum numbers [141], $(J,K_{a},K_{c})$. See
Fig. 1 for a pictorial representation of the normal modes of
H${}_{2}^{~{}12}$C16O. Rotational states with $K_{a}\approx J$ which differ
only in their $K_{c}$ quantum number lie close in energy and often give rise
to two very-closely-spaced transitions, a phenomenon known as $K$-doubling or
asymmetry splitting.

The electric-dipole-allowed transitions are described as ${\rm
A}_{1}\longleftrightarrow{\rm A}_{2}$ and ${\rm B}_{1}\longleftrightarrow{\rm
B}_{2}$. These selection rules are derived from symmetry relations. Namely,
the two nuclear-spin isomers of H${}_{2}^{~{}12}$C16O, _para_ and _ortho_ ,
correspond to $v_{4}+v_{5}+v_{6}+K_{a}$ being even (irreps A) or odd (irreps
B), respectively, while the subscripts of the irreps are 1 or 2 depending on
whether $v_{5}+v_{6}+K_{a}+K_{c}$ is even or odd, respectively.

Table 1: Pure rotational Lamb-dip lines of H${}_{2}^{~{}12}$C16O measured during this study with 2 kHz ($6.67\times 10^{-8}$ cm-1) accuracy.a $\sigma_{\rm new}$/cm-1 | Line assignment |  $\sigma_{\rm lit}$/cm-1
---|---|---
9.390 727 127 | $\nu_{0},4_{1,4}\leftarrow 3_{1,3}$ | 9.390 727 53(33) [76]
9.694 153 448 | $\nu_{0},4_{0,4}\leftarrow 3_{0,3}$ | 9.694 153 31(33) [76]
9.714 646 159 | $\nu_{0},4_{2,3}\leftarrow 3_{2,2}$ | 9.714 646 7(67) [72]
|  | 9.714 646 0(67) [76]
9.719 405 248 | $\nu_{0},4_{3,2}\leftarrow 3_{3,1}$ | 9.719 407(10) [76]
9.719 536 101 | $\nu_{0},4_{3,1}\leftarrow 3_{3,0}$ | 9.719 532 8(33) [85]
9.738 339 080 | $\nu_{0},4_{2,2}\leftarrow 3_{2,1}$ | 9.738 339 0(67) [72]
10.034 829 847 | $\nu_{0},4_{1,3}\leftarrow 3_{1,2}$ | 10.034 830 00(33) [76]

a In the first and last columns, the new, $\sigma_{\rm new}$, and the best
previous, $\sigma_{\rm lit}$, transition wavenumbers are reported,
respectively. The “Line assignment” column contains the assignments of the
lines in the form of
$\nu_{0},J^{\prime}_{K^{\prime}_{a},K^{\prime}_{c}}\leftarrow
J^{\prime\prime}_{K^{\prime\prime}_{a},K^{\prime\prime}_{c}}$, where
$J_{K_{a},K_{c}}$ denote the rotational assignment and $\nu_{0}$ refers to the
ground vibrational band both for the upper and the lower state.

###  2.3 New rotational measurements

In order to increase the number of available experimental data of
H${}_{2}^{~{}12}$C16O, and thus improve our MARVEL treatment, we have recorded
a set of 89 pure rotational transitions in the ground and some vibrationally
excited states. The measurements were performed using the frequency-modulation
millimeter/submillimeter spectrometer in Bologna. The instrumental setup of
the spectrometer is detailed elsewhere [142, 143]; here we provide only a
brief description.

Table 2: Rotational lines of H${}_{2}^{~{}12}$C16O measured during this study with 20 kHz ($6.67\times 10^{-7}$ cm-1) accuracy.a $\sigma_{\rm new}$/cm-1 | Line assignment |  $\sigma_{\rm new}$/cm-1 | Line assignment |  $\sigma_{\rm new}$/cm-1 | Line assignment
---|---|---|---|---|---
9.396 103 294 |  $\nu_{3},4_{1,4}$ |  $\leftarrow$ |  $3_{1,3}$ | 12.098 165 470 |  $\nu_{4},5_{3,3}$ |  $\leftarrow$ | $4_{3,2}$ | 9.694 153 398 |  $\nu_{0},4_{0,4}$ |  $\leftarrow$ | $3_{0,3}$
9.720 532 400 |  $\nu_{3},4_{0,4}$ |  $\leftarrow$ |  $3_{0,3}$ | 12.098 598 600 |  $\nu_{4},5_{3,2}$ |  $\leftarrow$ | $4_{3,1}$ | 9.714 646 044 |  $\nu_{0},4_{2,3}$ |  $\leftarrow$ | $3_{2,2}$
9.745 845 447 |  $\nu_{3},4_{2,3}$ |  $\leftarrow$ |  $3_{2,2}$ | 12.129 667 850 |  $\nu_{4},5_{2,3}$ |  $\leftarrow$ | $4_{2,2}$ | 9.719 405 501 |  $\nu_{0},4_{3,2}$ |  $\leftarrow$ | $3_{3,1}$
9.753 700 377 |  $\nu_{3},4_{3,2}$ |  $\leftarrow$ |  $3_{3,1}$ | 12.432 459 250 |  $\nu_{4},5_{1,4}$ |  $\leftarrow$ | $4_{1,3}$ | 9.719 535 726 |  $\nu_{0},4_{3,1}$ |  $\leftarrow$ | $3_{3,0}$
9.753 859 167 |  $\nu_{3},4_{3,1}$ |  $\leftarrow$ |  $3_{3,0}$ | 9.366 709 450 |  $\nu_{6},4_{1,4}$ |  $\leftarrow$ | $3_{1,3}$ | 9.738 339 087 |  $\nu_{0},4_{2,2}$ |  $\leftarrow$ | $3_{2,1}$
9.773 007 973 |  $\nu_{3},4_{2,2}$ |  $\leftarrow$ |  $3_{2,1}$ | 9.682 101 847 |  $\nu_{6},4_{0,4}$ |  $\leftarrow$ | $3_{0,3}$ | 9.834 452 846 |  $\nu_{0},22_{2,20}$ |  $\leftarrow$ | $23_{0,23}$
10.088 575 460 |  $\nu_{3},4_{1,3}$ |  $\leftarrow$ |  $3_{1,2}$ | 9.697 356 479 |  $\nu_{6},4_{2,3}$ |  $\leftarrow$ | $3_{2,2}$ | 10.034 829 710 |  $\nu_{0},4_{1,3}$ |  $\leftarrow$ | $3_{1,2}$
11.739 822 940 |  $\nu_{3},5_{1,5}$ |  $\leftarrow$ |  $4_{1,4}$ | 9.696 151 729 |  $\nu_{6},4_{3,2}$ |  $\leftarrow$ | $3_{3,1}$ | 10.248 367 590 |  $\nu_{0},9_{2,7}$ |  $\leftarrow$ | $10_{0,10}$
12.129 667 850 |  $\nu_{3},5_{0,5}$ |  $\leftarrow$ |  $4_{0,4}$ | 9.696 255 862 |  $\nu_{6},4_{3,1}$ |  $\leftarrow$ | $3_{3,0}$ | 10.364 365 470 |  $\nu_{0},18_{3,15}$ |  $\leftarrow$ | $19_{1,18}$
12.178 445 400 |  $\nu_{3},5_{2,4}$ |  $\leftarrow$ |  $4_{2,3}$ | 9.718 870 594 |  $\nu_{6},4_{2,2}$ |  $\leftarrow$ | $3_{2,1}$ | 10.448 342 040 |  $\nu_{0},37_{4,33}$ |  $\leftarrow$ | $37_{4,34}$
12.192 149 190 |  $\nu_{3},5_{4,2}$ |  $\leftarrow$ |  $4_{4,1}$ | 10.029 447 240 |  $\nu_{6},4_{1,3}$ |  $\leftarrow$ | $3_{1,2}$ | 10.555 863 330 |  $\nu_{0},11_{1,10}$ |  $\leftarrow$ | $11_{1,11}$
12.192 149 190 |  $\nu_{3},5_{4,1}$ |  $\leftarrow$ |  $4_{4,0}$ | 11.704 713 210 |  $\nu_{6},5_{1,5}$ |  $\leftarrow$ | $4_{1,4}$ | 10.896 074 830 |  $\nu_{0},19_{2,17}$ |  $\leftarrow$ | $19_{2,18}$
12.194 013 170 |  $\nu_{3},5_{3,3}$ |  $\leftarrow$ |  $4_{3,2}$ | 12.086 508 690 |  $\nu_{6},5_{0,5}$ |  $\leftarrow$ | $4_{0,4}$ | 11.212 427 070 |  $\nu_{0},28_{3,25}$ |  $\leftarrow$ | $28_{3,26}$
12.194 571 510 |  $\nu_{3},5_{3,2}$ |  $\leftarrow$ |  $4_{3,1}$ | 12.120 488 600 |  $\nu_{6},5_{2,4}$ |  $\leftarrow$ | $4_{2,3}$ | 11.733 738 380 |  $\nu_{0},5_{1,5}$ |  $\leftarrow$ | $4_{1,4}$
12.232 626 840 |  $\nu_{3},5_{2,3}$ |  $\leftarrow$ |  $4_{2,2}$ | 12.109 817 660 |  $\nu_{6},5_{4,2}$ |  $\leftarrow$ | $4_{4,1}$ | 12.073 937 490 |  $\nu_{0},8_{2,6}$ |  $\leftarrow$ | $9_{0,9}$
12.604 848 190 |  $\nu_{3},5_{1,4}$ |  $\leftarrow$ |  $4_{1,3}$ | 12.109 817 660 |  $\nu_{6},5_{4,1}$ |  $\leftarrow$ | $4_{4,0}$ | 12.099 570 770 |  $\nu_{0},5_{0,5}$ |  $\leftarrow$ | $4_{0,4}$
9.384 944 381 |  $\nu_{4},4_{1,4}$ |  $\leftarrow$ |  $3_{1,3}$ | 12.121 054 530 |  $\nu_{6},5_{3,3}$ |  $\leftarrow$ | $4_{3,2}$ | 12.139 927 050 |  $\nu_{0},5_{2,4}$ |  $\leftarrow$ | $4_{2,3}$
9.649 643 518 |  $\nu_{4},4_{0,4}$ |  $\leftarrow$ |  $3_{0,3}$ | 12.121 422 250 |  $\nu_{6},5_{3,2}$ |  $\leftarrow$ | $4_{3,1}$ | 12.145 176 040 |  $\nu_{0},5_{4,2}$ |  $\leftarrow$ | $4_{4,1}$
9.670 558 195 |  $\nu_{4},4_{2,3}$ |  $\leftarrow$ |  $3_{2,2}$ | 12.163 422 430 |  $\nu_{6},5_{2,3}$ |  $\leftarrow$ | $4_{2,2}$ | 12.145 176 040 |  $\nu_{0},5_{4,1}$ |  $\leftarrow$ | $4_{4,0}$
9.677 770 970 |  $\nu_{4},4_{3,2}$ |  $\leftarrow$ |  $3_{3,1}$ | 12.532 770 930 |  $\nu_{6},5_{1,4}$ |  $\leftarrow$ | $4_{1,3}$ | 12.150 912 170 |  $\nu_{0},5_{3,3}$ |  $\leftarrow$ | $4_{3,2}$
9.677 893 765 |  $\nu_{4},4_{3,1}$ |  $\leftarrow$ |  $3_{3,0}$ | 8.237 600 872 |  $\nu_{0},21_{2,19}$ |  $\leftarrow$ | $22_{0,22}$ | 12.151 369 510 |  $\nu_{0},5_{3,2}$ |  $\leftarrow$ | $4_{3,1}$
9.693 290 175 |  $\nu_{4},4_{2,2}$ |  $\leftarrow$ |  $3_{2,1}$ | 8.600 996 014 |  $\nu_{0},35_{3,32}$ |  $\leftarrow$ | $36_{1,35}$ | 12.187 211 040 |  $\nu_{0},5_{2,3}$ |  $\leftarrow$ | $4_{2,2}$
9.949 961 352 |  $\nu_{4},4_{1,3}$ |  $\leftarrow$ |  $3_{1,2}$ | 8.615 829 497 |  $\nu_{0},10_{2,8}$ |  $\leftarrow$ | $11_{0,11}$ | 12.441 481 610 |  $\nu_{0},12_{1,11}$ |  $\leftarrow$ | $12_{1,12}$
11.726 529 220 |  $\nu_{4},5_{1,5}$ |  $\leftarrow$ |  $4_{1,4}$ | 8.798 468 478 |  $\nu_{0},36_{4,32}$ |  $\leftarrow$ | $36_{4,33}$ | 12.538 447 510 |  $\nu_{0},5_{1,4}$ |  $\leftarrow$ | $4_{1,3}$
12.045 134 760 |  $\nu_{4},5_{0,5}$ |  $\leftarrow$ |  $4_{0,4}$ | 8.815 102 825 |  $\nu_{0},10_{1,9}$ |  $\leftarrow$ | $10_{1,10}$ | 12.795 269 680 |  $\nu_{0},20_{2,18}$ |  $\leftarrow$ | $20_{2,19}$
12.084 311 370 |  $\nu_{4},5_{2,4}$ |  $\leftarrow$ |  $4_{2,3}$ | 9.160 256 575 |  $\nu_{0},18_{2,16}$ |  $\leftarrow$ | $18_{2,17}$ |  |  |  |
12.093 976 360 |  $\nu_{4},5_{4,2}$ |  $\leftarrow$ |  $4_{4,1}$ | 9.390 727 004 |  $\nu_{0},4_{1,4}$ |  $\leftarrow$ | $3_{1,3}$ |  |  |  |
12.093 936 030 |  $\nu_{4},5_{4,1}$ |  $\leftarrow$ |  $4_{4,0}$ | 9.461 197 252 |  $\nu_{0},27_{3,24}$ |  $\leftarrow$ | $27_{3,25}$ |  |  |  |

a $\sigma_{\rm new}$ denotes newly measured transition wavenumbers. The line
assignments are given in the form of
$\nu_{i},J^{\prime}_{K^{\prime}_{a},K^{\prime}_{c}}\leftarrow
J^{\prime\prime}_{K^{\prime\prime}_{a},K^{\prime\prime}_{c}}$, where $\nu_{i}$
and $J_{K_{a},K_{c}}$ represent the vibrational and the rotational assignment
of the measured lines, respectively, $\nu_{0}$ refers to the ground
vibrational state, while $\nu_{3},~{}\nu_{4},{\rm and~{}}\nu_{6}$ designate
vibrational fundamentals (see Fig. 1).

The radiation source is a Gunn diode (80–115 GHz) that is frequency- and
phase-stabilized _via_ a phase-lock-loop and is driven by a centimeter-wave
synthesizer referenced to a 5 MHz rubidium atomic clock. Spectral coverage
between 240 and 420 GHz is obtained by coupling the Gunn diode to passive
multipliers. Then, the electromagnetic radiation is frequency-modulated at
$f=48$ kHz and fed into a glass absorption cell (3 m optical path length)
filled with formaldehyde vapors at pressure between 0.5 and 20 $\mu$bar;
specifically, pressures between 0.5 and 20 $\mu$bar were used to record ground
state lines with intensities spanning almost six orders of magnitude, while
10–15 $\mu$bar were used to observe the vibrational excited states. The sample
of H2CO was freshly obtained before each measurement from the vapors of solid,
room-temperature paraformaldehyde. The output radiation was finally detected
by a Schottky barrier diode, demodulated by a lock-in amplifier set at twice
the modulation-frequency ($2f$ scheme), filtered, and analog-to-digitally
converted.

A small sub-set of measurements have been performed exploiting the Lamb-dip
technique [144]. In this respect, the optics of the spectrometer were
adequately set up in a double-pass configuration (as detailed in [145]), $f$
was set to 1 kHz, and a low-pressure of H2CO was used (0.5 $\mu$bar).

The new measurements consist of (a) two complete $a$-type ${}^{q}R$-branch
transitions ($J=4\leftarrow 3$ and $5\leftarrow 4$) within the $\nu_{0}$,
$\nu_{3}$, $\nu_{4}$, and $\nu_{6}$ bands, where $\nu_{0}$ denotes the ground
vibrational state, and (b) about 20 ground-state $P$ and $Q$ transitions with
$\Delta K_{a}=0$ or $2$, with integrated intensity between $10^{-23}$ and
$10^{-27}$ cm molecule-1.

The uncertainty of our measurements is estimated to be in the range of 2–20
kHz. We label the seven Lamb-dip transitions of Table 1 with the reference tag
21AlTeYuMe, and the others as 21AlTeYuMe_S2, see Table 2.3.

###  2.4 Artificial transitions

In the absence of transitions linking the _ortho_ and _para_ nuclear-spin
isomers, the measured transitions form two distinct principal components (PC)
[118] in the SN of H${}_{2}^{~{}12}$C16O. As usual [111] during MARVEL
analyses of experimental transitions of species having more than one nuclear-
spin isomer, the _ortho_ and _para_ principal components are linked using
artificial transitions, colloquially called magic numbers, taken possibly from
effective Hamiltonian fits (EH) [122].

For H${}_{2}^{~{}12}$C16O, the wavenumber of the unmeasurable rotational
transition $1_{1,1}\leftarrow 0_{0,0}$, reported in [146] as 10.539 039 1
cm-1, is utilized as a magic number. Since the very accurate empirical
(MARVEL) energies of the pure rotational states $1_{0,1}$, $2_{0,2}$, and
$3_{0,3}$, that is, 2.429 612 60(33), 7.286 404 28(47), and 14.565 513 07(58)
cm-1, respectively, are reproduced by the EH-predicted values, see the fourth
column of Table 3 of Ref. [146], with unsigned deviations of $8.26\times
10^{-7}$, $2.20\times 10^{-6}$, and $4.37\times 10^{-6}$ cm-1, respectively,
the average of these unsigned deviations, $2.3\times 10^{-6}$ cm-1, is adopted
as a conservative estimate for the uncertainty of the magic number. In
addition, 11 artificial transitions were used to link the largest floating
components to the PCs. These transitions form bridges between the principal
components and the floating components of disconnected higher-$J$ transitions.
In fact, these transitions link the ground state to series of states with
$K_{a}=8$, $K_{a}=10$, and $K_{a}=11$. The values of these artificial
transitions are taken from the effective Hamiltonian study of 88NaReDaJo [87].

First-principles computations are capable of estimating small energy
splittings very accurately [122]; thus, we decided to add the $J_{J,0/1}$
separations as virtual lines to the dataset. They are part of a source tagged
as ‘21virt’ (see Table 3). These virtual transitions are distributed into four
segments, based on the magnitude of the splittings and thus on their assumed
uncertainties. Through these virtual lines 199 further rovibrational energy
levels could be determined: the experimentally unavailable states of certain
$J_{J,0/1}$ pairs.

Table 3: Data source segments and their characteristics for the H${}_{2}^{~{}12}$C16O moleculea Segment tag | Range |  $A/N/V$ | ESU | MSU | LSU | Recalib. Factor
---|---|---|---|---|---|---
72TuToTh [61] | 0.483 28– |  0.483 28 |  1/1/1 | 2.67e-09 | 2.67e-09 | 2.67e-09 |
71TuToTh [58] | 0.141 30– |  0.165 74 |  3/3/3 | 3.67e-09 | 3.67e-09 | 5.34e-09 |
81ChMi [77] | 0.002 373– |  0.064 793 |  14/9/9 | 6.67e-09 | 1.00e-08 | 3.34e-08 |
73ChGu [62] | 0.000 003– |  0.966 50 |  23/21/21 | 1.00e-08 | 1.00e-08 | 3.66e-07 |
73ChGu_S2 [62] | 2.415 3– |  2.415 3 |  1/1/1 | 3.34e-05 | 3.34e-05 | 3.34e-05 |
68Takami [53] | 0.000 022– |  0.001 829 |  12/10/10 | 1.67e-08 | 3.15e-08 | 6.44e-07 |
77ChGu [70] | 0.000 005– |  0.002 305 |  39/39/35 | 3.34e-08 | 3.34e-08 | 1.33e-07 |
96BoDePoLi [85] | 0.000 003– |  0.708 93 |  68/35/35 | 3.34e-08 | 1.67e-07 | 1.67e-07 |
96BoDePoLi_S2 [85] | 0.000 059– |  46.275 |  93/45/44 | 1.00e-06 | 1.00e-06 | 4.37e-06 |
96BoDePoLi_S3 [85] | 0.165 27– |  63.241 |  91/72/72 | 3.34e-06 | 3.34e-06 | 1.67e-05 |
96BoDePoLi_S4 [85] | 34.009– |  84.711 |  21/21/21 | 1.67e-04 | 1.67e-04 | 1.67e-04 |
21AlTeYuMe | 9.390 7– |  10.035 |  7/7/7 | 7.00e-08 | 7.00e-08 | 1.33e-07 |
21AlTeYuMe_S2 | 8.237 6– |  12.988 |  82/82/82 | 7.00e-07 | 7.00e-07 | 4.74e-06 |
66TaEvSh [147] | 0.000 153– |  0.000 610 |  2/2/2 | 1.00e-07 | 1.00e-07 | 1.33e-07 |
77FaKrMu [71] | 0.002 373– |  0.082 838 |  4/4/4 | 1.67e-07 | 1.67e-07 | 1.67e-07 |
63Esterowi [50] | 0.010 044– |  0.035 553 |  4/4/4 | 2.50e-07 | 4.17e-06 | 1.11e-05 |
80CoWi [76] | 0.000 003– |  15.039 |  127/95/94 | 3.34e-07 | 3.34e-07 | 3.15e-06 |
17MuLe [106] | 45.744– |  49.915 |  45/45/45 | 3.34e-07 | 3.34e-07 | 3.34e-07 |
88NaReDaJo [87] | 0.000 020– |  0.561 82 |  10/10/10 | 6.67e-07 | 6.67e-07 | 1.12e-05 |
88NaReDaJo_S2 [87] | 0.000 005– |  10.029 |  64/64/63 | 1.00e-06 | 1.00e-06 | 1.72e-05 |
88NaReDaJo_S3 [87] | 608.85– |  630.67 |  2/2/2 | 5.00e-05 | 1.00e-05 | 1.00e-05 |
88NaReDaJo_S4 [87] | 922.63– |  1 578.4 |  314 9/312 8/312 8 | 2.00e-04 | 1.19e-04 | 2.01e-02 |
88NaReDaJo_S5 [87] | 681.57– |  1 201.9 |  9/9/9 | 5.00e-03 | 1.00e-03 | 1.00e-03 |
03ThCaRiMu [86] | 0.163 02– |  0.164 96 |  3/3/2 | 8.34e-07 | 8.34e-07 | 1.00e-06 |
03BrMuLeWi [91] | 27.793– |  66.778 |  136/136/136 | 1.00e-06 | 1.00e-06 | 2.74e-05 |
09MaPeJaBa [100] | 5.016 8– |  30.115 |  172/172/171 | 1.00e-06 | 1.00e-06 | 1.33e-05 |
12ElCuGuHi [104] | 23.353– |  58.596 |  87/87/87 | 1.00e-06 | 1.00e-06 | 5.55e-06 |
21virt | 0.000 000– |  0.000 000 |  446/446/446 | 1.00e-06 | 1.00e-06 | 1.00e-06 |
21virt_S2 | 0.000 001– |  0.000 009 |  178/177/177 | 1.00e-05 | 1.00e-05 | 2.21e-04 |
21virt_S3 | 0.000 010– |  0.000 098 |  157/151/151 | 1.00e-04 | 1.00e-04 | 2.78e-03 |
21virt_S4 | 0.000 100– |  0.000 999 |  168/166/166 | 1.00e-03 | 1.00e-03 | 3.89e-03 |
72Nerf [60] | 1.610 6– |  10.035 |  15/15/15 | 1.47e-06 | 1.60e-06 | 7.86e-06 |
51LaSt [45] | 0.245 59– |  2.429 6 |  18/18/18 | 1.67e-06 | 2.40e-06 | 2.03e-05 |
56Erlandss [148] | 2.429 6– |  7.528 5 |  9/8/8 | 1.67e-06 | 1.60e-05 | 2.79e-05 |
64OkTaMo [52] | 0.000 153– |  2.437 1 |  70/49/44 | 1.67e-06 | 1.80e-06 | 2.66e-05 |
97CaHaDe [146] | 10.539– |  10.539 |  1/1/1 | 2.30e-06 | 2.30e-06 | 2.30e-06 |
72JoLoKi [59] | 0.000 153– |  4.504 1 |  37/10/10 | 3.34e-06 | 8.34e-07 | 5.00e-06 |
78DaWiBe [72] | 0.353 87– |  12.187 |  86/82/82 | 6.67e-06 | 6.67e-06 | 2.31e-05 |
73ChFrJoOk [63] | 0.553 19– |  1.979 8 |  7/7/7 | 1.67e-05 | 1.67e-05 | 2.67e-05 |
85TiChKuHu [80] | 4 211.6– |  5 224.0 |  100 5/100 5/100 3 | 1.00e-04 | 1.10e-04 | 1.16e-02 |
82BrJoMcWo [79] | 1 693.1– |  1 793.4 |  248/248/248 | 4.00e-04 | 1.59e-04 | 8.99e-04 |
79BrHuPi [74] | 2 700.0– |  3 000.2 |  319 3/318 9/318 3 | 5.00e-04 | 3.27e-04 | 4.52e-02 |
81SwSa [78] | 1 707.1– |  1 746.9 |  76/76/76 | 5.00e-04 | 6.50e-04 | 3.38e-03 |
88ClVa [82] | 2 867.2– |  2 880.4 |  24/22/22 | 5.00e-04 | 3.48e-04 | 7.23e-03 |
94ItNaTa [84] | 2 276.2– |  2 553.6 |  320/320/317 | 5.00e-04 | 3.91e-04 | 5.56e-03 |
10JaLaTcGa [103] | 1 675.0– |  3 089.8 |  782/781/781 | 5.00e-04 | 1.62e-04 | 6.45e-03 | 1.000 001 293
07TcPeLa_S2 [97] | 0.000 006– |  10.029 |  84/84/84 | 2.10e-04 | 1.00e-04 | 2.25e-03 |
07TcPeLa [97] | 927.68– |  1 821.6 |  402 4/110 1/110 1 | 8.00e-04 | 1.04e-04 | 5.00e-03 |
06PeBrUtHa [96] | 2 756.6– |  2 864.6 |  147/143/143 | 1.00e-03 | 1.00e-03 | 3.82e-02 |
09CiMaCi [102] | 4 350.9– |  4 360.6 |  49/46/36 | 1.00e-03 | 1.00e-03 | 7.97e-03 |
07SaBaHaRi [99] | 5 597.8– |  5 698.3 |  424/422/418 | 1.50e-03 | 1.00e-03 | 9.08e-03 |
17TaAdNg [107] | 3 398.7– |  3 529.7 |  786/786/785 | 1.60e-03 | 1.11e-03 | 3.66e-02 |
73Toth [65] | 3 428.2– |  3 507.4 |  299/299/298 | 3.00e-03 | 2.83e-03 | 4.23e-02 |
77AlJoMc [68] | 945.23– |  1 540.0 |  991/922/904 | 5.00e-03 | 3.94e-03 | 4.57e-02 |
77AlJoMcb [69] | 1 483.0– |  1 518.1 |  70/70/70 | 5.00e-03 | 4.14e-03 | 3.57e-02 |
76NaYaKu [149] | 2 643.9– |  3 011.8 |  184 8/184 5/171 1 | 3.00e-02 | 1.68e-02 | 9.89e-02 |

a Tags denote segments used in this study. The column ‘Range’ indicates the
range (in cm-1) corresponding to validated wavenumbers within the transition
list. $A$ is the number of assigned transitions, $N$ is the number of non-
redundant lines (with distinct wavenumbers or labels), and $V$ is the number
of validated transitions obtained at the end of the xMARVEL analysis. In the
heading of this table, ESU, MSU, and LSU denote the estimated, the median, and
the largest segment uncertainties, respectively, in cm-1. Rows are arranged in
the order of the ESUs with the restriction that the segments of the same data
source should be listed consecutively.

Table 4: High-resolution spectroscopic studies on H${}_{2}^{~{}12}$C16O which
were considered but not utilized during the present MARVEL analysis, with
reason for the exclusion.

Tag | Range / cm-1 | Reason for exclusion
---|---|---
17FjHeBaLe [28] | 6230 – 6240 | 700 K emission spectrum, line parameters not provided
15RuHeHeFi [105] | 6547 – 7051 | No line assignments
09PeJaTcLa [101] | 1600 – 3200 | Calculated line positions only
07ZhGaDeHu [98] | 6351 – 6362 | No line assignments
06FlLaSaSh [94] | 3096 – 5263 | Data not made available by the authors
06PeVaDa [95] | 2600 – 3100 | No line parameters provided, data not made available by the authors
05StGaVeRu [93] | 6547 – 6804 | No line assignments
03PeKeFl [92] | 1000 – 2000 | Data not made available by the authors
02BaCoHaPe [90] | 5600 – 5700 | Data analysed by 07SaBaHaRi [99]
96BoHaGrSt [88] | – | Dispersed fluorescence, vibrational state data only
96LuCoFrCr [89] | 7800 – 15200 | No line parameters provided
89ReNaDaJo [83] | 890 – 1590 | No line parameters provided
87NaDaRe [81] | 1148 – 1193 | Lines included in 88NaReDaJo [87]
78Pine [73] | 2700 – 3000 | No line parameters provided
75Nerf [66] | 1 – 10 | Lines are given in 72Nerf [60]
73JoMc [64] | 1707 – 1767 | No line parameters provided
70TuThTo [56] | 0.15 – 0.15 | Line is given in 71TuToTh [58]
60OkHiSh [48] | 0.98 | Lines are given in 64OkTaMo [52]

##  3 The rovibrational database

###  3.1 Overview

Tables 3 and 4 present an overview of the experimental information considered
during this project. Each experimental source is given a tag composed of the
last two digits of the year of publication and letters of the names of up to
the first four co-authors.

Table 3 summarizes the sources included in our final MARVEL analysis and gives
statistical information about the transitions these sources contain. We
utilized data from 43 literature sources, including this work, yielding 19 831
transitions, of which 16 596 are non-redundant.

Table 4 lists sources that were _not_ included in the final analysis. A number
of other older sources [47, 51, 55] were also excluded from the present
analysis, as their measurements have been superseded by those of significantly
more accurate studies.

###  3.2 Source-specific comments

A significant problem we faced during the data collection was that there are
several publications [92, 94, 95] for H${}_{2}^{~{}12}$C16O where the authors
did not provide the direct measured data in their original paper and when
approached they turned down our request to send the measured data forming the
basis of their existing publication. Similar problems, and related issues,
have been highlighted in a recent paper by Gordon et al. [150]. Seemingly it
is not straightforward to include old data into new data compilations.

Another significant issue is that a number of sources do not provide a clear
statement about the uncertainties of the observed lines. Thus, we had to
estimate them by various means, which included comparisons with other sources
as well as combination-difference relations. Such sources include 76NaYaKu
[149], 82BrJoMcWo [79], 88ClVa [82], 85TiChKuHu [80], 77AlJoMc [68], 77AlJoMcb
[69], and 10JaLaTcGa [103]. Since the authors of 10JaLaTcGa [103] state that
their line positions were not calibrated, we used an xMARVEL facility to
calibrate the wavenumbers of the measured transitions published in 10JaLaTcGa.
This gave a calibration factor of 1.000 001 148 for this source.

64OkTaMo [52] provides high-resolution pure rotational transitions between
levels in vibrationally excited states. However, 78DaWiBe [72] found it
necessary to reassign the vibrational states in this source; we have adopted
and in fact validated the assignments of 78DaWiBe.

76NaYaKu [149] is an older and relatively low-resolution source, which
nonetheless contains lines for which no data are available from alternative
sources. The authors used a compact, non-standard notation for the
assignments, which had to be unpicked. A total of 134 lines could not be
validated and thus were removed. Transitions involving high $K_{a}$ states did
not resolve the $K$-doublets, i.e., the $K_{c}$ quantum number. For these
transitions we used two lines corresponding to both possible transitions.

88NaReDaJo [87] gives only calculated line positions and residuals; thus, the
observed frequencies were reconstructed from this information. Similarly, the
tables in 79BrHuPi [74] are of very poor quality and their line frequency data
could only be accurately reconstructed by extensive comparison with HITRAN
[151]; 88ClVa [82] also provides a small portion of 79BrHuPi’s spectrum in
readable form.

02BaCoHaPe [90] recorded a spectrum of the $5\nu_{2}$ overtone; this spectrum
was analyzed by 07SaBaHaRi [99], who also provided the data (C.M. Western,
private communication, 2020). These studies did not contain an estimated
uncertainty; thus, an estimated value of 0.0015 cm-1 was adopted on the basis
of combination difference relations.

72JoLoKi [59] presents an extensive compilation of early microwave experiments
on formaldehyde. Although this is a secondary source, some data were taken
from here as the primary sources are not available to us. 72JoLoKi also
contains tabulations of hyperfine-resolved transitions for the
$1_{01}-1_{11}$, $2_{11}-2_{12}$, and $3_{12}-3_{13}$ rotational lines. Other
microwave studies, including 59TaShSh [46], 70TuThTo [56], 71TuToTh [58],
72TuToTh [61], and 17MuLe [106] also present hyperfine-resolved data. Within
the present study we completely neglect hyperfine effects and, where
necessary, use central, hyperfine-unresolved line positions.

##  4 Results and Discussion

###  4.1 Validation

According to Table 3, we considered for validation 16 596 non-redundant
transitions measured for H${}_{2}^{~{}12}$C16O. 13 of these transitions are
artificial (see above) and 143 do not attach to the principal components, they
remain parts of floating components. The transitions of the giant components
of the SN of H${}_{2}^{~{}12}$C16O were validated using several techniques.

193 transitions, including a few which did not obey the selection rules
governing electric-dipole-allowed transitions, were removed at the first stage
of validation. In almost all cases these transitions were ones for which
alternative, validated wavenumber entries were available from other sources.
The number of validated transitions within each segment is given in Table 3.
The transitions which could not be validated are retained in the transitions
list given in the supplementary data, but they are given there as a negative
wavenumber entry, which means that they are ignored during the processing of
the data by xMARVEL. Therefore, these transitions do not contribute to the
final empirical energy levels.

The other important step in the validation process involved comparisons with
the AYTY line list [113]. These comparisons were performed iteratively between
the empirical (xMARVEL) and the AYTY energy levels. During the first phase,
the comparisons identified a number of issues with the original xMARVEL
dataset in the form of incorrect quantum numbers or scanning errors. Once
these were corrected, only a few lines were found for which the upper states
had no reasonable match within the AYTY list. These transitions are augmented
with a comment ‘theoretical mismatch’ in the MARVEL XML file given in the
supplementary material. Those excluded lines which violate the electric-dipole
selection rules have the comment ‘wrong labels’ in the XML file. At the end,
we were able to validate 16 403 of the collected transitions.

Table 5: MARVEL-based vibrational band origins (VBO) of H${}_{2}^{~{}12}$C16O
and their literature counterparts obtained from accurate effective Hamiltonian
fits.a

Band | Symmetry | VBO(MARVEL) | $N_{0}$ | $N_{\rm RL}$ | $J_{\rm max}$ | VBO(lit)
---|---|---|---|---|---|---
$\nu_{0}$ |  a1 | 0.0 | 31 | 525 | 38 | 0.0
$\nu_{4}$ |  b1 | 1167.256 76(10) | 6 | 346 | 30 |  1167.258(2) [68]
|  |  |  |  |  |  1167.256 28(2) [92]
$\nu_{6}$ |  b2 | 1249.094 42(10) | 3 | 405 | 30 |  1249.091(2) [68]
|  |  |  |  |  |  1249.094 68(2) [92]
$\nu_{3}$ |  a1 | 1500.174 50(10) | 4 | 382 | 30 |  1500.176(3) [68]
|  |  |  |  |  |  1500.174 74(12) [92]
$\nu_{2}$ |  a1 | 1746.009 14(10) | 4 | 446 | 37 |  1746.008 9(1) [79]
|  |  |  |  |  |  1746.008 86(13) [92]
$2\nu_{4}$ |  a1 | – | 0 | 65 | 11 |  2327.523 9(5) [95]
|  |  |  |  |  |  2327.1(8) [75]
$\nu_{4}+\nu_{6}$ |  a2 | – | 0 | 47 | 11 |  2422.970 1(50) [95]
$2\nu_{6}$ |  a1 | – | 0 | 81 | 25 |  2494.354 3(5) [95]
$\nu_{3}+\nu_{4}$ |  b1 | – | 0 | 15 | 10 |  2667.048 1(20) [95]
|  |  |  |  |  |  2655.5 [74]
$\nu_{3}+\nu_{6}$ |  b2 | 2719.156 04(50) | 1 | 232 | 25 |  2719.155 9(10) [74]
$\nu_{1}$ |  a1 | 2782.456 91(50) | 2 | 440 | 36 |  2782.456 9(10) [74]
$\nu_{5}$ |  b2 | 2843.323 54(12) | 4 | 388 | 35 |  2843.325 6(10) [74]
$\nu_{2}+\nu_{4}$ |  b1 | – | 0 | 153 | 24 |  2905.968 5(20) [95]
|  |  |  |  |  |  2905(1) [74]
$2\nu_{3}$ |  a1 | – | 0 | 38 | 25 |  2998.987 3(5) [95]
|  |  |  |  |  |  2999.5(5) [74]
$\nu_{2}+\nu_{6}$ |  b2 | 3000.065 74(50) | 1 | 100 | 21 |  3000.065 6(10) [74]
$2\nu_{2}$ |  a1 | 3471.720 8(78) | 2 | 373 | 31 |  3471.718(4) [65]
$\nu_{1}+\nu_{4}$ |  b1 | 3941.530 8(10) | 1 | 30 | 10 | –
$\nu_{4}+\nu_{5}$ |  b2 | 3996.518 6(10) | 1 | 35 | 9 | –
$\nu_{3}+\nu_{5}$ |  b2 | 4335.0989 4(10) | 1 | 170 | 24 | –
$2\nu_{2}+\nu_{6}$ |  b2 | – | 0 | 207 | 26 |  4734.207 8(50) [94]
$3\nu_{2}$ |  a1 | – | 0 | 176 | 26 |  5177.759 52(70) [94]
$2\nu_{5}$ |  a1 | 5651.196 4(15) | 1 | 179 | 20 | –

aThe VBO values are in cm-1. The uncertainties of the last VBO digits are
given in parentheses. The symmetry of the band, the number of transitions
determining a particular VBO ($N_{0}$), as well as the number of rotational
levels ($N_{\rm RL}$) and the maximum $J$ value within a given vibrational
band are also displayed.

###  4.2 Final energies

The MARVEL analysis yielded, in total, 5029 validated empirical rovibrational
energies for H${}_{2}^{~{}12}$C16O; they are provided in the supplementary
material. The highest rotational quantum number is $J_{\rm max}=38$ and the
empirical energies go up to 6188 cm-1; there are transitions which probe
energy levels higher than this but none of them are assigned. Thus, all
empirical energy levels are well below the minima corresponding to
hydroxymethylene. A large number of further experimental studies are needed to
reach that dynamically important and interesting region. The region covered is
also considerably more limited than that covered by SEP measurements in the
1980s [10].

Table 5 presents the vibrational band origins (VBO) and a summary of the
number of empirical rovibrational energy levels determined for each
vibrational state. The bands $2\nu_{3}+\nu_{6}$, $\nu_{2}+\nu_{5}$, and
$\nu_{2}+\nu_{3}+\nu_{4}$, which have a combined total of only 6 levels, have
been omitted from the table. Table 5 also gives term values of this and
previous studies, i.e., levels with $J=0$ for given vibrational bands.

Table 5 contains term values for 14 vibrational band origins, including the
six fundamentals, determined during this study. The term values of the
$\nu_{2}$, $\nu_{3}$, and $\nu_{4}$ fundamentals are determined particularly
accurately, in part because in these cases the upper states have been
subjected to high-accuracy microwave studies; indeed, such studies for
$\nu_{3}$ and $\nu_{4}$ form part of the present work. We note that these
fundamentals are all characterized by at least three transitions. In contrast,
the term values of the combination bands are much less accurate and determined
by only one, or in one case two, transitions. It is important to note that the
fundamentals of Ref. [68] result from effective-Hamiltonian calculations,
which may yield higher apparent accuracy than the present MARVEL treatment.

Figure 2: The upper-state energies of the experimental transitions used in
this work, against corresponding lower-state energies of
H${}_{2}^{~{}12}$C16O. The vertical bars along the horizontal axis show the
lower state energies, while the horizontal bars along the vertical axis give
the upper state energies. Each circle represents a particular transition, with
the size proportional to the number of transitions supporting the
corresponding upper state. This value ranges from 1 (dark blue) to 102 (red).

Figure 2 provides a visual representation of the spectroscopic network of
H${}_{2}^{~{}12}$C16O, showing the upper-state energies of the measured
transitions against corresponding lower-state energies, and giving a visual
impression of the number of transitions linking them.

Table 6: Pure rotational frequencies of H${}_{2}^{~{}12}$C16O lines under protection by the National Academies of Sciences, Engineering, and Medicine (NASEM) [152] and their various experimental and empirical determinations. The uncertainties of the last few frequency digits are indicated in parentheses. $f_{\rm NASEM}$/GHz | Line assignment |  $f_{\rm MARVEL}$/GHz |  $f_{\rm expt}$/GHz
---|---|---|---
4.829 66 | $\nu_{0},1_{1,0}\leftarrow 1_{1,1}$ | 4.829 659 960(50) |  4.829 659 960(50) [58]
|  |  |  4.829 660 0(10) [85]
|  |  |  4.829 649 0(50) [62]
|  |  |  4.829 660(10) [59]
14.488 | $\nu_{0},2_{1,1}\leftarrow 2_{1,2}$ | 14.488 478 810(80) |  14.488 478 810(80) [61]
|  |  |  14.488 479 0(10) [85]
|  |  |  14.488 650(20) [52]
|  |  |  14.488 650(50) [45]
|  |  |  14.488 65(10) [62]
140.84 | $\nu_{0},2_{1,2}\leftarrow 1_{1,1}$ | 140.839 502(10) |  140.839 502(10) [85]
|  |  |  140.839 526(32) [60]
|  |  |  140.839 300(50) [148]
145.603 | $\nu_{0},2_{02}\leftarrow 1_{0,1}$ | 145.602 949(10) |  145.602 949(10) [85]
|  |  |  145.602 966(34) [60]
|  |  |  145.603 100(50) [148]
150.498 | $\nu_{0},2_{1,1}\leftarrow 1_{1,0}$ | 150.498 334(10) |  150.498 334(10) [85]
|  |  |  150.498 355(34) [60]
|  |  |  150.498 200(50) [148]
218.222 | $\nu_{0},3_{0,3}\leftarrow 2_{0,3}$ | 218.222 192(10) |  218.222 192(10) [85]
|  |  |  218.222 186(49) [60]
|  |  |  218.221 600(50) [148]

###  4.3 Protected formaldehyde lines

The National Academies of Sciences, Engineering, and Medicine (NASEM)
maintains a list of key long-wavelength transitions relevant for astrophysical
studies [152]. This list contains six transitions belonging to
H${}_{2}^{~{}12}$C16O; they are given in Table 6. Some previous MARVEL
studies, notably on water [110, 120] and ammonia [138], allowed the accuracy,
to which some of these lines were known, to be improved using MARVEL. In the
case of H${}_{2}^{~{}12}$C16O, however, we find that the uncertainties in our
MARVEL-determined frequencies simply match those of the current best
laboratory determinations; for a discussion how these uncertainties are
determined, see Ref. [110]. Nevertheless, Table 6 is still useful as it shows
all highly-accurate studies of the NASEM-protected lines.

Table 7: Extract from the H2CO state file. The full table is available from
www.exomol.com.a

$I$ |  $\tilde{E}$/cm-1 |  $g$ |  $J$ |  $\delta$/cm-1 |  $\tau$/s |  $\mathit{\Gamma}_{\rm tot}$ | $v_{1}$ | $v_{2}$ | $v_{3}$ | $v_{4}$ | $v_{5}$ | $v_{6}$ |  $\mathit{\Gamma}_{\rm vib}$ | $K_{a}$ | $K_{c}$ |  $|C_{i}^{2}|$ | $n_{1}$ | $n_{2}$ | $n_{3}$ | $n_{4}$ | $n_{5}$ | $n_{6}$ |  K |  $\tilde{E}_{\rm AYTY}$/cm-1
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 |  0.000000 |  1 |  0 |  0.000001 |  Inf |  A1 | 0 | 0 | 0 | 0 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 0 | 0 | 0 | 0 |  Ma | 0.000000
2 |  1500.174503 |  1 |  0 |  0.000100 |  3.1423E-01 |  A1 | 0 | 0 | 1 | 0 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 0 | 0 | 1 | 0 |  Ma | 1500.120955
3 |  1746.009136 |  1 |  0 |  0.000100 |  3.1935E-02 |  A1 | 0 | 1 | 0 | 0 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 1 | 0 | 0 | 0 | 0 | 0 |  Ma | 1746.045388
4 |  2327.497142 |  1 |  0 |  0.200000 |  4.7496E-01 |  A1 | 0 | 0 | 0 | 2 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 0 | 0 | 0 | 2 |  Ca | 2327.497142
5 |  2494.322937 |  1 |  0 |  0.200000 |  2.0931E-01 |  A1 | 0 | 0 | 0 | 0 | 0 | 2 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 0 | 1 | 1 | 0 |  Ca | 2494.322937
6 |  2782.456913 |  1 |  0 |  0.000500 |  1.5673E-02 |  A1 | 1 | 0 | 0 | 0 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 1 | 0 | 0 | 0 |  Ma | 2782.410921
7 |  2999.006647 |  1 |  0 |  0.200000 |  1.0513E-01 |  A1 | 0 | 0 | 0 | 0 | 0 | 2 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 0 | 1 | 1 | 0 |  Ca | 2999.006647
8 |  3238.937891 |  1 |  0 |  0.200000 |  2.9533E-02 |  A1 | 0 | 1 | 1 | 0 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 1 | 0 | 0 | 0 | 1 | 0 |  Ca | 3238.937891
9 |  3471.720843 |  1 |  0 |  0.007770 |  1.5448E-02 |  A1 | 0 | 2 | 0 | 0 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 2 | 0 | 0 | 0 | 0 | 0 |  Ma | 3471.719306
10 |  3825.967015 |  1 |  0 |  0.300000 |  1.6623E-01 |  A1 | 0 | 0 | 1 | 2 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 0 | 0 | 1 | 2 |  Ca | 3825.967015
11 |  3936.435541 |  1 |  0 |  0.300000 |  3.3267E-02 |  A1 | 0 | 0 | 3 | 0 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 0 | 0 | 0 | 3 | 0 | 0 |  Ca | 3936.435541
12 |  4058.101422 |  1 |  0 |  0.300000 |  3.0046E-02 |  A1 | 0 | 1 | 0 | 2 | 0 | 0 |  A1 | 0 | 0 |  1.00 | 1 | 0 | 0 | 0 | 0 | 2 |  Ca | 4058.101422

a $I$: state identifier; $\tilde{E}$: state term value; $g$: state degeneracy;
$J$: state rotational quantum number; $\delta$: energy uncertainty; $\tau$:
lifetime; $\mathit{\Gamma}_{\rm tot}$: total symmetry in $C_{2\nu}(M)$;
$v_{1}-v_{6}$: normal mode vibrational quantum numbers; $\mathit{\Gamma}_{\rm
vib}$: symmetry of vibrational contribution in $C_{2\nu}(M)$; $K_{a}$ and
$K_{c}$: rotational quantum numbers; $|C_{i}^{2}|$: largest coefficient used
in the assignment; $n_{1}-n_{6}$: TROVE vibrational quantum numbers; K: data
kind indicating if the term value is based on the MARVEL (‘Ma’) or the AYTY
energy list (‘Ca’); $\tilde{E}_{\rm AYTY}$: original AYTY state term value.

##  5 Updated AYTY line list

The AYTY line list [113] of H${}_{2}^{~{}12}$C16O contains approximately 10
billion transitions linking 10.3 million rotational-vibrational states. These
states are those with $J\leq 70$ which lie up to 10 000 cm-1 above the ground
state. As part of this study, we replace those energy levels of the AYTY list
which are determined empirically by the MARVEL process.

The rovibrational states were matched on the basis of quantum numbers and then
checked using the energies. As a result, 367 779 transition frequencies are
determined using the empirical energy levels of this study. Of these, 183 673
lie above the dynamic HITRAN intensity cutoff [153]. These numbers should be
compared to the 16 596 non-redundant transitions which form the input to the
original spectroscopic network of H${}_{2}^{~{}12}$C16O. It can be seen that
this process leads to a significant increase in the number of transitions
whose line centres are determined to high-resolution experimental accuracy.

Obviously, 5029 empirically determined energy levels is only a small portion
of the 10.3 million entries in the original AYTY line list. We therefore had
to estimate the (significantly larger) uncertainties associated with the
calculated energy levels. This is important, not least because it allows
transitions that are predicted with high accuracy to be easily identified. We
estimated the uncertainties, in cm-1, of the computed levels using the formula

$\delta=0.1\,(v_{1}+v_{2}+v_{3}+v_{4}+v_{5}+v_{6})+0.005J(J+1).$

This update is in line with the requirements of the 2020 release of ExoMol
[154] and is designed to facilitate the use of the data in high-resolution
studies by quantifying the uncertainty associated with each transition and
hence identifying those known with the required accuracy for a given study.

Figure 3: Room-temperature ($T=296$ K) spectra of H${}_{2}^{~{}12}$C16O in
three different regions covered by HITRAN 2016 [151]. The upper panels, in
blue, show stick spectra simulated using the MARVEL energy term values from
this work and the Einstein-$A$ coefficients from the AYTY line. The lower
panels, in red, show the corresponding spectra taken from HITRAN 2016.

Table 7 gives a small portion of the updated ExoMol States file in standard
ExoMol format [155, 154]. The complete file along with the AYTY Trans file,
which is unchanged by the current procedure, can be obtained from the ExoMol
website (www.exomol.com).

Figure 4: Room temperature ($T=296$ K) spectra of H${}_{2}^{~{}12}$C16O from
three different sources: lower panel, HITRAN 2016 [151]; upper panel, this
work (see caption of Fig. 3) where possible or else from the AYTY [113] line
list.

The HITRAN database [151] contains limited data on H${}_{2}^{~{}12}$C16O:
besides pure rotations it covers only two bands. Figure 3 gives a comparison
for these regions. The agreement is very good and shows that our MARVEL
analysis is largely sufficient to cover these regions. Figure 4 illustrates
the relative (in)completeness of the HITRAN and our MARVEL data sets when used
to simulate the room temperature spectra of H${}_{2}^{~{}12}$C16O. There are a
number of missing bands in HITRAN and the database can now be supplemented
with our synthetic line list constructed using the line positions determined
to experimental accuracy using MARVEL and AYTY transition intensities.

Figure 5: High temperature ($T=1000$ K) spectra of H${}_{2}^{~{}12}$C16O from
three different sources: HITRAN 2016 [151], AYTY [113], and this work (see
also caption to Fig. 3).

The AYTY line list was actually designed for treating formaldehyde in hot
environments. In order to illustrate the lack of experimental data at high
temperatures, Fig. 5 shows $T=$ 1000 K absorption spectra simulated with the
same three source as above: MARVEL line positions (this work) combined with
the AYTY Einstein-$A$ coefficients, the HITRAN line list, and the AYTY line
list. It is clear that at high temperatures only AYTY provides the
spectroscopic coverage required for high temperatures and that known
laboratory spectra are not capable of giving a complete representation of hot
formaldehyde.

##  6 Summary and Conclusions

Apart from a few sources where the authors were reluctant to provide their
published results for analysis [92, 94, 95], all literature lines measured for
H${}_{2}^{~{}12}$C16O have been collected and analyzed during this study. The
information contained in the literature sources was augmented by new
measurements results, including seven Lamb-dip lines and 82 further lines, all
corresponding to rotational transitions within the ground, $\nu_{3}$,
$\nu_{4}$, and $\nu_{6}$ vibrational states. The analysis utilized the
Measured Active Rotational-Vibration Levels (MARVEL) approach and the xMARVEL
code. A characteristic of MARVEL is that it is an “active” approach, which
means that should new sources, or even existing sources that have not been
included, became available it is straightforward to include them in a renewed
analysis and produce an updated version of the line-by-line database assembled
for H${}_{2}^{~{}12}$C16O.

Of the total of 19 831 transitions processed, which come from 43 sources
including this work, 16 403 could be validated, providing 5029 empirical
energy levels of H${}_{2}^{~{}12}$C16O with statistically well-defined
uncertainties. Our newly-determined empirical rotational-vibrational energy
levels are used to improve the accuracy of ExoMol’s AYTY line list for hot
formaldehyde [113]. This improved line list is available as supplementary
material, along with the xMARVEL files, containing all the rovibrational
transitions collated, whether validated or not, as well as all the empirical
energy levels derived during this study.

It is the stated plan of the ExoMol project [112] to update all ExoMol line
lists to include both empirical energy levels, where available, and
uncertainties for all the energy levels; thus allowing the uncertainty
associated with any transition to be estimated. These data will become a
standard part of the ExoMol line list and also form input for the new ExoMolHR
database, which provides data only on lines of higher intensity whose
wavelengths are known to high accuracy. So far only a limited number of ExoMol
line lists are available with specified uncertainties in the energy levels.
These include important line lists for which MARVEL data were already
available, namely those for water [156], AlH [157], 12C2 [131], 12C2H2 [158]
and 48Ti16O [158], and recently produced line lists including those for CO2
[159], H3O+ [160] and NH3 [161]. The present study is the first in a series
where a MARVEL analysis is performed with the intention of updating an
available line list.

## Acknowledgments

We thank Dr. Colin Western for providing the data generated by 07SaBaHaRi
[99]. The work at UCL is supported by STFC Projects No. ST/M001334/1 and
ST/R000476/1, and the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme through Advance Grant
number 883830. The work performed in Budapest received support from NKFIH
(grant no. K119658), from the grant VEKOP-2.3.2-16-2017-000, and from the ELTE
Institutional Excellence Program (TKP2020-IKA-05) financed by the Hungarian
Ministry of Human Capacities. The work performed in Bologna was supported by
the Università di Bologna (RFO funds) and by MIUR (Project PRIN 2015: STARS in
the CAOS, Grant Number 2015F59J3R).

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