An improved rovibrational linelist of formaldehyde, \spec{h212c16o}
Afaf R. Al-Derzi, Jonathan Tennyson, Sergei N. Yurchenko, Mattia Melosso, Ningjing Jiang, Cristina Puzzarini, Luca Dore, Tibor Furtenbacher, Roland Tóbiás, Attila G. Császár
aa r X i v : . [ a s t r o - ph . S R ] F e b An improved rovibrational linelist of formaldehyde, H C O Afaf R. Al-Derzi a , Jonathan Tennyson , Sergei N. Yurchenko a , Mattia Melosso b , Ningjing Jiang b ,Cristina Puzzarini b , Luca Dore b , Tibor Furtenbacher c , Roland T´obi´as c , Attila G. Cs´asz´ar c a Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UnitedKingdom b Dipartimento di Chimica “Giacomo Ciamician”, Universit`a di Bologna, Via F. Selmi 2, 40126 Bologna, Italy c ELKH-ELTE Complex Chemical Systems Research Group and ELTE E¨otv¨os Lor´and University, Institute ofChemistry, Laboratory of Molecular Structure and Dynamics, H-1117 Budapest, P´azm´any P´eter s´et´any 1/A,Hungary
Abstract
Published high-resolution rotation-vibration transitions of H C O, the principal isotopologue ofmethanal, are analyzed using the MARVEL (Measured Active Rotation-Vibration Energy Levels)procedure. The literature results are augmented by new, high-accuracy measurements of pure rota-tional transitions within the ground, ν , ν , and ν vibrational states. Of the 16 596 non-redundanttransitions processed, which come from 43 sources including the present work, 16 403 could bevalidated, providing 5029 empirical energy levels of H C O with statistically well-defined uncer-tainties. All the empirical rotational-vibrational energy levels determined are used to improve theaccuracy of ExoMol’s AYTY line list for hot formaldehyde. The complete list of collated experi-mental transitions, the empirical energy levels determined, as well as the extended and improvedline list are provided as Supplementary Material.
Keywords:
Formaldehyde; Line list; Ro-vibrational energy; MARVEL analysis. To whom correspondence should be addressed; email: [email protected]
Preprint submitted to Elsevier February 16, 2021 . Introduction
Formaldehyde, formally methanal, HCHO, is the simplest aldehyde. In the gas phase, formalde-hyde is considered to be carcinogenic [1, 2]. Since formaldehyde is a trace species in the earth’satmosphere, formed as a result of photo-oxidation or through incomplete biomass burning, its car-cinogenic nature motivated the development of spectroscopic techniques for detecting it in tracequantities [3].The photochemistry and photophysics of formaldehyde, involving several excited electronicstates and isomerization processes, leading to the products H + CO or H + HCO, have been in-vestigated in considerable detail [4–9], both experimentally and theoretically. The ˜X A and ˜A A 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 constantscharacterizing the ˜X A ground electronic state [10]. Thus, in 1984, Field and co-workers [10] couldlegitimately claim that the “unique characteristics of SEP have enabled us to describe the E − rotation-vibration structure of H CO ˜X A at a level of detail and completeness which webelieve is without precedent for a four-atomic polyatomic molecule”. Nevertheless, though a hugeachievement, the accuracy attainable through these spectroscopic constants was a mere 3 cm − ,hardly acceptable by present-day standards.Over the years, beyond anharmonic (quartic) force field representations [11] of the ˜X A PESof formaldehyde [12–15], several global PESs have been developed for the [C,H,H,O] system [9, 16–22]. Part of the interest in a global ground-electronic-state ( S ) PES stems from the existence offurther important minima, cis - and trans -H–C–OH, hydroxycarbene [17, 20], on it besides thatcorresponding to formaldehyde. The trans -hydroxycarbene minimum is some 18 000 cm − [23]above the H CO minimum (the global minimum on S belongs to CO + H [23]), separated bya transition state of 10 400 cm − , measured from the hydroxycarbene side, and through enhanceddeep tunneling trans -hydroxycarbene rearranges to formaldehyde with a half-life of only two hours[23]. The CO · H 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, firstby 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 havebeen performed [29].Formaldehyde is a trace species on Mars [30], as well, and it is well known in comets [31–34]. Its2nfrared absorption spectrum has been observed in protoplanetary disks [35]. Interstellar H C Owas detected 50 years ago [36] and many of the formaldehyde lines are now resolved routinelyin high-resolution spectroscopic studies of the interstellar medium [37]. Interstellar formaldehydemasers have been widely observed [38–42].Much of the laboratory spectroscopy of formaldehyde has focused on electronic (rovibronic)transitions. In fact, formaldehyde was the first polyatomic molecule where the rotational subbandstructure of vibronic transitions has been successfully analyzed [43]. It is probably fair to saythat the photodissociation of formaldehyde is one of the most thoroughly understood polyatomicunimolecular reactions [6, 44].The importance of line-by-line information on formaldehyde has led to a large number of lab-oratory studies of its high-resolution rotational-vibrational spectra [10, 28, 45–68, 68–107], involv-ing a diverse set of experimental techniques, such as SEP [10], dispersed laser-induced fluores-cence [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-Dopplerlaser Stark and Doppler-limited Fourier-transform spectroscopy [79]. Many of the results of thesehigh-resolution experimental studies will be addressed and discussed in detail below.During the present study our focus is exclusively on transitions within the electronic groundstate and we perform a MARVEL (Measured Active Rotational-Vibrational Energy Levels) [108–110] analysis of all the measured transitions of H C O in its ˜X A state. This involves collatingall available laboratory spectra and then validating the observed and assigned transitions. Thecollection of the validated lines is then inverted to give a set of accurate empirical energy levelswith statistically significant associated uncertainties. As part of this study, the published spectraare augmented by high-accuracy measurements of pure rotational transitions within the ground, ν , ν , and ν 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 compre-hensive 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 newmeasurements 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 empirically3djusted energy levels of the AYTY line list by the empirical rovibrational energy levels resultingfrom 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 determinedenergy levels wherever possible [116]; this work forms part of this effort.
2. Methods
Details about the MARVEL technique [108–110], built upon the theory of spectroscopic networks(SNs) [117–119], and the xMARVEL code, introduced in Refs. 110 and 120, have been given inseveral recent publications [110, 118, 120–122]. MARVEL has been used to treat the laboratory
OCH H OCH H OCH HOCH H OCH H OCH H ν a -typesymm. CH stretch (a ) CO stretch (a ) CH bend (a ) ν a -type ν a -typeout-of-plane bend (b ) CH rock (b )asymm. CH stretch (b ) ν c -type ν b -type ν b -type2782 cm -1 -1 -1 -1 -1 -1 Figure 1: Vibrational fundamentals of ˜X A H C O, with their symmetry species and the types of bands indi-cated. The arrows representing atomic motions during the normal-mode vibrations are not proportional to the actualdisplacements. C O. The corresponding xMARVEL input andoutput data files are provided as supplementary material.
The xMARVEL procedure requires as input spectral lines assigned with a unique set of quantumnumbers. The equilibrium structure of formaldehyde on its ground electronic state has C v point-group symmetry (with irreducible representations, irreps, A , A , B , and B , which are the samefor the isomorphic C v (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 vibrationalnormal-mode quantum numbers, [ v , v , v , v , v , v ], arranged in standard order [140], and the threestandard “rigid-rotor” quantum numbers [141], ( J, K a , K c ). See Fig. 1 for a pictorial representationof the normal modes of H C O. Rotational states with K a ≈ J which differ only in their K c quantum number lie close in energy and often give rise to two very-closely-spaced transitions, aphenomenon known as K -doubling or asymmetry splitting.The electric-dipole-allowed transitions are described as A ←→ A and B ←→ B . These selec-tion rules are derived from symmetry relations. Namely, the two nuclear-spin isomers of H C O, para and ortho , correspond to v + v + v + 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 + v + K a + K c is even or odd,respectively. Table 1: Pure rotational Lamb-dip lines of H C O measured during this study with 2 kHz (6 . × − cm − )accuracy. a new /cm − Line assignment σ lit /cm − ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , a In the first and last columns, the new, σ new , and the best previous, σ lit , transition wavenumbers arereported, respectively. The “Line assignment” column contains the assignments of the lines in the form of ν , J ′ K ′ a ,K ′ c ← J ′′ K ′′ a ,K ′′ c , where J K a ,K c denote the rotational assignment and ν refers to the ground vibrationalband both for the upper and the lower state. In order to increase the number of available experimental data of H C O, and thus improveour MARVEL treatment, we have recorded a set of 89 pure rotational transitions in the groundand some vibrationally excited states. The measurements were performed using the frequency-modulation millimeter/submillimeter spectrometer in Bologna. The instrumental setup of the spec-trometer is detailed elsewhere [142, 143]; here we provide only a brief description.
Table 2: Rotational lines of H C O measured during this study with 20 kHz (6 . × − cm − ) accuracy. a σ new /cm − Line assignment σ new /cm − Line assignment σ new /cm − Line assignment9.396 103 294 ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , Continued on next page able 2 – Continued from previous page σ new /cm − Line assignment σ new /cm − Line assignment σ new /cm − Line assignment12.604 848 190 ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , ν , , ← , a σ new denotes newly measured transition wavenumbers. The line assignments are given in the form of ν i , J ′ K ′ a ,K ′ c ← J ′′ K ′′ a ,K ′′ c , where ν i and J K a ,K c represent the vibrational and the rotational assignment of themeasured lines, respectively, ν refers to the ground vibrational state, while ν , ν , and ν designate vibra-tional 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 rubidiumatomic clock. Spectral coverage between 240 and 420 GHz is obtained by coupling the Gunn diode topassive multipliers. Then, the electromagnetic radiation is frequency-modulated at f = 48 kHz andfed into a glass absorption cell (3 m optical path length) filled with formaldehyde vapors at pressurebetween 0.5 and 20 µ bar; specifically, pressures between 0.5 and 20 µ bar were used to record groundstate lines with intensities spanning almost six orders of magnitude, while 10–15 µ bar were usedto observe the vibrational excited states. The sample of H CO was freshly obtained before eachmeasurement from the vapors of solid, room-temperature paraformaldehyde. The output radiationwas finally detected by a Schottky barrier diode, demodulated by a lock-in amplifier set at twicethe modulation-frequency (2 f 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 H CO was used (0.5 µ bar).The new measurements consist of (a) two complete a -type q R -branch transitions ( J = 4 ← ←
4) within the ν , ν , ν , and ν bands, where ν denotes the ground vibrational state,and (b) about 20 ground-state P and Q transitions with ∆ K a = 0 or 2, with integrated intensity7etween 10 − and 10 − cm molecule − .The uncertainty of our measurements is estimated to be in the range of 2–20 kHz. We labelthe seven Lamb-dip transitions of Table 1 with the reference tag 21AlTeYuMe, and the others as21AlTeYuMe S2, see Table 2. In the absence of transitions linking the ortho and para nuclear-spin isomers, the measuredtransitions form two distinct principal components (PC) [118] in the SN of H C O. As usual [111]during MARVEL analyses of experimental transitions of species having more than one nuclear-spinisomer, the ortho and para principal components are linked using artificial transitions, colloquiallycalled magic numbers, taken possibly from effective Hamiltonian fits (EH) [122].For H C O, the wavenumber of the unmeasurable rotational transition 1 , ← , , reportedin [146] as 10.539 039 1 cm − , is utilized as a magic number. Since the very accurate empirical(MARVEL) energies of the pure rotational states 1 , , 2 , , and 3 , , that is, 2.429 612 60(33),7.286 404 28(47), and 14.565 513 07(58) cm − , respectively, are reproduced by the EH-predictedvalues, see the fourth column of Table 3 of Ref. [146], with unsigned deviations of 8 . × − ,2 . × − , and 4 . × − cm − , respectively, the average of these unsigned deviations, 2 . × − cm − , is adopted as a conservative estimate for the uncertainty of the magic number. In addition, 11artificial transitions were used to link the largest floating components to the PCs. These transitionsform 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 Hamiltonianstudy of 88NaReDaJo [87].First-principles computations are capable of estimating small energy splittings very accurately[122]; thus, we decided to add the J J, / separations as virtual lines to the dataset. They are partof a source tagged as ‘21virt’ (see Table 3). These virtual transitions are distributed into four seg-ments, based on the magnitude of the splittings and thus on their assumed uncertainties. Throughthese virtual lines 199 further rovibrational energy levels could be determined: the experimentallyunavailable states of certain J J, / pairs. Table 3: Data source segments and their characteristics for the H C O molecule a egment tag Range A/N/V
ESU MSU LSU Recalib. Factor72TuToTh [61] 0.483 28–0.483 28 1/1/1 2.67e-09 2.67e-09 2.67e-0971TuToTh [58] 0.141 30–0.165 74 3/3/3 3.67e-09 3.67e-09 5.34e-0981ChMi [77] 0.002 373–0.064 793 14/9/9 6.67e-09 1.00e-08 3.34e-0873ChGu [62] 0.000 003–0.966 50 23/21/21 1.00e-08 1.00e-08 3.66e-0773ChGu S2 [62] 2.415 3–2.415 3 1/1/1 3.34e-05 3.34e-05 3.34e-0568Takami [53] 0.000 022–0.001 829 12/10/10 1.67e-08 3.15e-08 6.44e-0777ChGu [70] 0.000 005–0.002 305 39/39/35 3.34e-08 3.34e-08 1.33e-0796BoDePoLi [85] 0.000 003–0.708 93 68/35/35 3.34e-08 1.67e-07 1.67e-0796BoDePoLi S2 [85] 0.000 059–46.275 93/45/44 1.00e-06 1.00e-06 4.37e-0696BoDePoLi S3 [85] 0.165 27–63.241 91/72/72 3.34e-06 3.34e-06 1.67e-0596BoDePoLi S4 [85] 34.009–84.711 21/21/21 1.67e-04 1.67e-04 1.67e-0421AlTeYuMe 9.390 7–10.035 7/7/7 7.00e-08 7.00e-08 1.33e-0721AlTeYuMe S2 8.237 6–12.988 82/82/82 7.00e-07 7.00e-07 4.74e-0666TaEvSh [147] 0.000 153–0.000 610 2/2/2 1.00e-07 1.00e-07 1.33e-0777FaKrMu [71] 0.002 373–0.082 838 4/4/4 1.67e-07 1.67e-07 1.67e-0763Esterowi [50] 0.010 044–0.035 553 4/4/4 2.50e-07 4.17e-06 1.11e-0580CoWi [76] 0.000 003–15.039 127/95/94 3.34e-07 3.34e-07 3.15e-0617MuLe [106] 45.744–49.915 45/45/45 3.34e-07 3.34e-07 3.34e-0788NaReDaJo [87] 0.000 020–0.561 82 10/10/10 6.67e-07 6.67e-07 1.12e-0588NaReDaJo S2 [87] 0.000 005–10.029 64/64/63 1.00e-06 1.00e-06 1.72e-0588NaReDaJo S3 [87] 608.85–630.67 2/2/2 5.00e-05 1.00e-05 1.00e-0588NaReDaJo S4 [87] 922.63–1 578.4 314 9/312 8/312 8 2.00e-04 1.19e-04 2.01e-0288NaReDaJo S5 [87] 681.57–1 201.9 9/9/9 5.00e-03 1.00e-03 1.00e-0303ThCaRiMu [86] 0.163 02–0.164 96 3/3/2 8.34e-07 8.34e-07 1.00e-0603BrMuLeWi [91] 27.793–66.778 136/136/136 1.00e-06 1.00e-06 2.74e-0509MaPeJaBa [100] 5.016 8–30.115 172/172/171 1.00e-06 1.00e-06 1.33e-0512ElCuGuHi [104] 23.353–58.596 87/87/87 1.00e-06 1.00e-06 5.55e-0621virt 0.000 000–0.000 000 446/446/446 1.00e-06 1.00e-06 1.00e-0621virt S2 0.000 001–0.000 009 178/177/177 1.00e-05 1.00e-05 2.21e-0421virt S3 0.000 010–0.000 098 157/151/151 1.00e-04 1.00e-04 2.78e-0321virt S4 0.000 100–0.000 999 168/166/166 1.00e-03 1.00e-03 3.89e-0372Nerf [60] 1.610 6–10.035 15/15/15 1.47e-06 1.60e-06 7.86e-0651LaSt [45] 0.245 59–2.429 6 18/18/18 1.67e-06 2.40e-06 2.03e-0556Erlandss [148] 2.429 6–7.528 5 9/8/8 1.67e-06 1.60e-05 2.79e-0564OkTaMo [52] 0.000 153–2.437 1 70/49/44 1.67e-06 1.80e-06 2.66e-0597CaHaDe [146] 10.539–10.539 1/1/1 2.30e-06 2.30e-06 2.30e-0672JoLoKi [59] 0.000 153–4.504 1 37/10/10 3.34e-06 8.34e-07 5.00e-0678DaWiBe [72] 0.353 87–12.187 86/82/82 6.67e-06 6.67e-06 2.31e-0573ChFrJoOk [63] 0.553 19–1.979 8 7/7/7 1.67e-05 1.67e-05 2.67e-0585TiChKuHu [80] 4 211.6–5 224.0 100 5/100 5/100 3 1.00e-04 1.10e-04 1.16e-0282BrJoMcWo [79] 1 693.1–1 793.4 248/248/248 4.00e-04 1.59e-04 8.99e-0479BrHuPi [74] 2 700.0–3 000.2 319 3/318 9/318 3 5.00e-04 3.27e-04 4.52e-0281SwSa [78] 1 707.1–1 746.9 76/76/76 5.00e-04 6.50e-04 3.38e-0388ClVa [82] 2 867.2–2 880.4 24/22/22 5.00e-04 3.48e-04 7.23e-0394ItNaTa [84] 2 276.2–2 553.6 320/320/317 5.00e-04 3.91e-04 5.56e-0310JaLaTcGa [103] 1 675.0–3 089.8 782/781/781 5.00e-04 1.62e-04 6.45e-03 1.000 001 29307TcPeLa S2 [97] 0.000 006–10.029 84/84/84 2.10e-04 1.00e-04 2.25e-0307TcPeLa [97] 927.68–1 821.6 402 4/110 1/110 1 8.00e-04 1.04e-04 5.00e-0306PeBrUtHa [96] 2 756.6–2 864.6 147/143/143 1.00e-03 1.00e-03 3.82e-0209CiMaCi [102] 4 350.9–4 360.6 49/46/36 1.00e-03 1.00e-03 7.97e-03
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Segment tag Range
A/N/V
ESU MSU LSU Recalib. Factor07SaBaHaRi [99] 5 597.8–5 698.3 424/422/418 1.50e-03 1.00e-03 9.08e-0317TaAdNg [107] 3 398.7–3 529.7 786/786/785 1.60e-03 1.11e-03 3.66e-0273Toth [65] 3 428.2–3 507.4 299/299/298 3.00e-03 2.83e-03 4.23e-0277AlJoMc [68] 945.23–1 540.0 991/922/904 5.00e-03 3.94e-03 4.57e-0277AlJoMcb [69] 1 483.0–1 518.1 70/70/70 5.00e-03 4.14e-03 3.57e-0276NaYaKu [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 − ) correspondingto validated wavenumbers within the transition list. A is the number of assigned transitions, N is the numberof non-redundant lines (with distinct wavenumbers or labels), and V is the number of validated transitionsobtained at the end of the xMARVEL analysis. In the heading of this table, ESU, MSU, and LSU denote theestimated, the median, and the largest segment uncertainties, respectively, in cm − . Rows are arranged in theorder of the ESUs with the restriction that the segments of the same data source should be listed consecutively.
3. The rovibrational database
Tables 3 and 4 present an overview of the experimental information considered during thisproject. Each experimental source is given a tag composed of the last two digits of the year ofpublication 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 statisticalinformation about the transitions these sources contain. We utilized data from 43 literature sources,
Table 4: High-resolution spectroscopic studies on H C O which were considered but not utilized during the presentMARVEL analysis, with reason for the exclusion.
Tag Range / cm − Reason for exclusion17FjHeBaLe [28] 6230 – 6240 700 K emission spectrum, line parameters not provided15RuHeHeFi [105] 6547 – 7051 No line assignments09PeJaTcLa [101] 1600 – 3200 Calculated line positions only07ZhGaDeHu [98] 6351 – 6362 No line assignments06FlLaSaSh [94] 3096 – 5263 Data not made available by the authors06PeVaDa [95] 2600 – 3100 No line parameters provided, data not made available by the authors05StGaVeRu [93] 6547 – 6804 No line assignments03PeKeFl [92] 1000 – 2000 Data not made available by the authors02BaCoHaPe [90] 5600 – 5700 Data analysed by 07SaBaHaRi [99]96BoHaGrSt [88] – Dispersed fluorescence, vibrational state data only96LuCoFrCr [89] 7800 – 15200 No line parameters provided89ReNaDaJo [83] 890 – 1590 No line parameters provided87NaDaRe [81] 1148 – 1193 Lines included in 88NaReDaJo [87]78Pine [73] 2700 – 3000 No line parameters provided75Nerf [66] 1 – 10 Lines are given in 72Nerf [60]73JoMc [64] 1707 – 1767 No line parameters provided70TuThTo [56] 0.15 – 0.15 Line is given in 71TuToTh [58]60OkHiSh [48] 0.98 Lines are given in 64OkTaMo [52] not included in the final analysis. A number of other oldersources [47, 51, 55] were also excluded from the present analysis, as their measurements have beensuperseded by those of significantly more accurate studies.
A significant problem we faced during the data collection was that there are several publications[92, 94, 95] for H C O where the authors did not provide the direct measured data in their originalpaper and when approached they turned down our request to send the measured data forming thebasis of their existing publication. Similar problems, and related issues, have been highlighted ina recent paper by Gordon et al. [150]. Seemingly it is not straightforward to include old data intonew data compilations.Another significant issue is that a number of sources do not provide a clear statement about theuncertainties of the observed lines. Thus, we had to estimate them by various means, which includedcomparisons with other sources as well as combination-difference relations. Such sources include76NaYaKu [149], 82BrJoMcWo [79], 88ClVa [82], 85TiChKuHu [80], 77AlJoMc [68], 77AlJoMcb[69], and 10JaLaTcGa [103]. Since the authors of 10JaLaTcGa [103] state that their line positionswere not calibrated, we used an xMARVEL facility to calibrate the wavenumbers of the measuredtransitions 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 vibra-tionally excited states. However, 78DaWiBe [72] found it necessary to reassign the vibrationalstates 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 linesfor which no data are available from alternative sources. The authors used a compact, non-standardnotation for the assignments, which had to be unpicked. A total of 134 lines could not be validatedand 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 possibletransitions.88NaReDaJo [87] gives only calculated line positions and residuals; thus, the observed frequen-cies were reconstructed from this information. Similarly, the tables in 79BrHuPi [74] are of verypoor quality and their line frequency data could only be accurately reconstructed by extensive com-parison with HITRAN [151]; 88ClVa [82] also provides a small portion of 79BrHuPi’s spectrum inreadable form.02BaCoHaPe [90] recorded a spectrum of the 5 ν overtone; this spectrum was analyzed by07SaBaHaRi [99], who also provided the data (C.M. Western, private communication, 2020). Thesestudies did not contain an estimated uncertainty; thus, an estimated value of 0.0015 cm − wasadopted on the basis of combination difference relations.112JoLoKi [59] presents an extensive compilation of early microwave experiments on formalde-hyde. Although this is a secondary source, some data were taken from here as the primary sourcesare not available to us. 72JoLoKi also contains tabulations of hyperfine-resolved transitions for the1 − , 2 − , and 3 − rotational lines. Other microwave studies, including 59TaShSh [46],70TuThTo [56], 71TuToTh [58], 72TuToTh [61], and 17MuLe [106] also present hyperfine-resolveddata. Within the present study we completely neglect hyperfine effects and, where necessary, usecentral, hyperfine-unresolved line positions.
4. Results and Discussion
According to Table 3, we considered for validation 16 596 non-redundant transitions measuredfor H C O. 13 of these transitions are artificial (see above) and 143 do not attach to the principalcomponents, they remain parts of floating components. The transitions of the giant components ofthe SN of H C O 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 transitionswere 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 whichcould not be validated are retained in the transitions list given in the supplementary data, butthey are given there as a negative wavenumber entry, which means that they are ignored duringthe processing of the data by xMARVEL. Therefore, these transitions do not contribute to the finalempirical energy levels.The other important step in the validation process involved comparisons with the AYTY linelist [113]. These comparisons were performed iteratively between the empirical (xMARVEL) andthe AYTY energy levels. During the first phase, the comparisons identified a number of issues withthe original xMARVEL dataset in the form of incorrect quantum numbers or scanning errors. Oncethese were corrected, only a few lines were found for which the upper states had no reasonable matchwithin the AYTY list. These transitions are augmented with a comment ‘theoretical mismatch’ inthe MARVEL XML file given in the supplementary material. Those excluded lines which violatethe electric-dipole selection rules have the comment ‘wrong labels’ in the XML file. At the end, wewere able to validate 16 403 of the collected transitions.
The MARVEL analysis yielded, in total, 5029 validated empirical rovibrational energies forH C O; they are provided in the supplementary material. The highest rotational quantum num-ber is J max = 38 and the empirical energies go up to 6188 cm − ; there are transitions which probeenergy levels higher than this but none of them are assigned. Thus, all empirical energy levels are12 able 5: MARVEL-based vibrational band origins (VBO) of H C O and their literature counterparts obtainedfrom accurate effective Hamiltonian fits. a Band Symmetry VBO(MARVEL) N N RL J max VBO(lit) ν a ν b ν b ν a ν a ν a – 0 65 11 2327.523 9(5) [95]2327.1(8) [75] ν + ν a – 0 47 11 2422.970 1(50) [95]2 ν a – 0 81 25 2494.354 3(5) [95] ν + ν b – 0 15 10 2667.048 1(20) [95]2655.5 [74] ν + ν b ν a ν b ν + ν b – 0 153 24 2905.968 5(20) [95]2905(1) [74]2 ν a – 0 38 25 2998.987 3(5) [95]2999.5(5) [74] ν + ν b ν a ν + ν b ν + ν b ν + ν b ν + ν b – 0 207 26 4734.207 8(50) [94]3 ν a – 0 176 26 5177.759 52(70) [94]2 ν a a The VBO values are in cm − . The uncertainties of the last VBO digits are given in parentheses. Thesymmetry of the band, the number of transitions determining a particular VBO ( N ), as well as the numberof rotational levels ( N RL ) and the maximum J value within a given vibrational band are also displayed. well below the minima corresponding to hydroxymethylene. A large number of further experimentalstudies are needed to reach that dynamically important and interesting region. The region coveredis 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 empiricalrovibrational energy levels determined for each vibrational state. The bands 2 ν + ν , ν + ν , and ν + ν + ν , which have a combined total of only 6 levels, have been omitted from the table. Table 5also gives term values of this and previous studies, i.e. , levels with J = 0 for given vibrationalbands.Table 5 contains term values for 14 vibrational band origins, including the six fundamentals,determined during this study. The term values of the ν , ν , and ν fundamentals are determined13
500 1000 1500 20000100020003000400050006000 U ppe r s t a t e t e r m v a l ue s , c m - Lower state term values, cm -1 Figure 2: The upper-state energies of the experimental transitions used in this work, against corresponding lower-stateenergies of H C O. The vertical bars along the horizontal axis show the lower state energies, while the horizontalbars along the vertical axis give the upper state energies. Each circle represents a particular transition, with the sizeproportional to the number of transitions supporting the corresponding upper state. This value ranges from 1 (darkblue) to 102 (red). particularly accurately, in part because in these cases the upper states have been subjected to high-accuracy microwave studies; indeed, such studies for ν and ν form part of the present work. Wenote that these fundamentals are all characterized by at least three transitions. In contrast, theterm values of the combination bands are much less accurate and determined by only one, or in onecase 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 MARVELtreatment.Figure 2 provides a visual representation of the spectroscopic network of H C O, showing theupper-state energies of the measured transitions against corresponding lower-state energies, andgiving a visual impression of the number of transitions linking them.
The National Academies of Sciences, Engineering, and Medicine (NASEM) maintains a listof key long-wavelength transitions relevant for astrophysical studies [152]. This list contains six14 able 6: Pure rotational frequencies of H C O lines under protection by the National Academies of Sciences, Engi-neering, and Medicine (NASEM) [152] and their various experimental and empirical determinations. The uncertaintiesof the last few frequency digits are indicated in parentheses. f NASEM /GHz Line assignment f MARVEL /GHz f expt /GHz4.829 66 ν , , ← , ν , , ← , ν , , ← , ν , ← , ν , , ← , ν , , ← , transitions belonging to H C O; 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 theselines were known, to be improved using MARVEL. In the case of H C O, however, we find thatthe uncertainties in our MARVEL-determined frequencies simply match those of the current bestlaboratory 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-protectedlines.
5. Updated AYTY line list
The AYTY line list [113] of H C O contains approximately 10 billion transitions linking 10.3million rotational-vibrational states. These states are those with J ≤
70 which lie up to 10 000cm − above the ground state. As part of this study, we replace those energy levels of the AYTYlist which are determined empirically by the MARVEL process.The rovibrational states were matched on the basis of quantum numbers and then checkedusing the energies. As a result, 367 779 transition frequencies are determined using the empiricalenergy 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 the15 able 7: Extract from the H CO state file. The full table is available from . a I ˜ E /cm − g J δ /cm − τ /s Γ tot v v v v v v Γ vib K a K c | C i | n n n n n n K ˜ E AYTY /cm − a I : state identifier; ˜ E : state term value; g : state degeneracy; J : state rotational quantumnumber; δ : energy uncertainty; τ : lifetime; Γ tot : total symmetry in C ν ( M ); v − v : normal modevibrational quantum numbers; Γ vib : symmetry of vibrational contribution in C ν ( M ); K a and K c :rotational quantum numbers; | C i | : largest coefficient used in the assignment; n − n : TROVEvibrational quantum numbers; K: data kind indicating if the term value is based on the MARVEL(‘Ma’) or the AYTY energy list (‘Ca’); ˜ E AYTY : original AYTY state term value.input to the original spectroscopic network of H C O. It can be seen that this process leads to asignificant increase in the number of transitions whose line centres are determined to high-resolutionexperimental accuracy.Obviously, 5029 empirically determined energy levels is only a small portion of the 10.3 millionentries 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 itallows transitions that are predicted with high accuracy to be easily identified. We estimated theuncertainties, in cm − , of the computed levels using the formula δ = 0 . v + v + v + v + v + v ) + 0 . J ( J + 1) . This update is in line with the requirements of the 2020 release of ExoMol [154] and is designedto facilitate the use of the data in high-resolution studies by quantifying the uncertainty associatedwith each transition and hence identifying those known with the required accuracy for a given study.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 currentprocedure, can be obtained from the ExoMol website ( ).The HITRAN database [151] contains limited data on H C O: besides pure rotations it coversonly two bands. Figure 3 gives a comparison for these regions. The agreement is very good andshows that our MARVEL analysis is largely sufficient to cover these regions. Figure 4 illustrates therelative (in)completeness of the HITRAN and our MARVEL data sets when used to simulate theroom temperature spectra of H C O. There are a number of missing bands in HITRAN and the16 -20 -20 -20 -20 I n t en s i t y , c m / m o l e c u l e T = 296 K -20 -20 -20 -20 wavenumber/cm -1 MARVELHITRAN -20 -20 -20 -20 -19 I n t en s i t y , c m / m o l e c u l e T = 296 K -19 -20 -20 -20 -20 wavenumber/cm -1 MARVELHITRAN -20 -20 -20 -20 I n t en s i t y , c m / m o l e c u l e T = 296 K -20 -20 -20 -20 wavenumber/cm -1 MARVELHITRAN
Figure 3: Room-temperature ( T = 296 K) spectra of H C O 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 workand the Einstein- A coefficients from the AYTY line. The lower panels, in red, show the corresponding spectra takenfrom HITRAN 2016.Figure 4: Room temperature ( T = 296 K) spectra of H C O from three different sources: lower panel, HI-TRAN 2016 [151]; upper panel, this work (see caption of Fig. 3) where possible or else from the AYTY [113] linelist. database can now be supplemented with our synthetic line list constructed using the line positionsdetermined to experimental accuracy using MARVEL and AYTY transition intensities.The AYTY line list was actually designed for treating formaldehyde in hot environments. Inorder to illustrate the lack of experimental data at high temperatures, Fig. 5 shows T = 1000 Kabsorption spectra simulated with the same three source as above: MARVEL line positions (thiswork) combined with the AYTY Einstein- A coefficients, the HITRAN line list, and the AYTY linelist. It is clear that at high temperatures only AYTY provides the spectroscopic coverage requiredfor high temperatures and that known laboratory spectra are not capable of giving a completerepresentation of hot formaldehyde. 17 -24 -23 -22 -21 -20 -19 -18 AYTY MARVEL HITRAN wavenumber/cm -1 c r o ss s e c t i on s , c m m o l e c u l e - T = 1000 K
Figure 5: High temperature ( T = 1000 K) spectra of H C O from three different sources: HITRAN 2016 [151],AYTY [113], and this work (see also caption to Fig. 3).
6. Summary and Conclusions
Apart from a few sources where the authors were reluctant to provide their published results foranalysis [92, 94, 95], all literature lines measured for H C O have been collected and analyzedduring this study. The information contained in the literature sources was augmented by newmeasurements results, including seven Lamb-dip lines and 82 further lines, all corresponding torotational transitions within the ground, ν , ν , and ν vibrational states. The analysis utilizedthe 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 newsources, or even existing sources that have not been included, became available it is straightforwardto include them in a renewed analysis and produce an updated version of the line-by-line databaseassembled for H C O.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 C O with statisticallywell-defined uncertainties. Our newly-determined empirical rotational-vibrational energy levels areused to improve the accuracy of ExoMol’s AYTY line list for hot formaldehyde [113]. This improvedline list is available as supplementary material, along with the xMARVEL files, containing all therovibrational transitions collated, whether validated or not, as well as all the empirical energy levelsderived during this study.It is the stated plan of the ExoMol project [112] to update all ExoMol line lists to include both18mpirical energy levels, where available, and uncertainties for all the energy levels; thus allowingthe uncertainty associated with any transition to be estimated. These data will become a standardpart of the ExoMol line list and also form input for the new ExoMolHR database, which providesdata only on lines of higher intensity whose wavelengths are known to high accuracy. So far only alimited 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 thosefor water [156], AlH [157], C [131], C H [158] and Ti O [158], and recently produced linelists including those for CO [159], H O + [160] and NH [161]. The present study is the first in aseries 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 workat UCL is supported by STFC Projects No. ST/M001334/1 and ST/R000476/1, and the Euro-pean Research Council (ERC) under the European Union’s Horizon 2020 research and innovationprogramme through Advance Grant number 883830. The work performed in Budapest receivedsupport from NKFIH (grant no. K119658), from the grant VEKOP-2.3.2-16-2017-000, and fromthe ELTE Institutional Excellence Program (TKP2020-IKA-05) financed by the Hungarian Min-istry of Human Capacities. The work performed in Bologna was supported by the Universit`a diBologna (RFO funds) and by MIUR (Project PRIN 2015: STARS in the CAOS, Grant Number2015F59J3R).
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