A Network Approach for the Accurate Characterization of Water Lines Observable in Astronomical Masers and Extragalactic Environments

The water molecule, crucial to the chemical composition and dynamics of the universe, is typically identified in its gas phase via radio and submillimeter transitions, with frequencies up to a few THz. To understand the physicochemical behavior of astronomical objects, accurate transition frequencies are required for these lines. From a set of 26 new and 564 previous Lamb dip measurements, utilizing our ultrasensitive laser-based spectrometers in the near-infrared region, ultrahigh-precision spectroscopic networks were set up for H216O and H218O, augmented with 40 extremely accurate frequencies taken from the literature. Based on kHz-accuracy paths of these networks, considerably improved line-center frequencies have been obtained for 35 observed or predicted maser lines of H216O, as well as for 14 transitions of astronomical significance of H218O. These reference frequencies, attached with 5–25 kHz uncertainties, may help future studies in various fields of astrochemistry and astrophysics, in particular when precise information is demanded about Doppler-velocity components, including the gas flows of galactic cores, the kinematics of planetary nebulae, or the motion in exoplanetary atmospheres.


INTRODUCTION
Water is a key molecular ingredient of the chemical universe, ubiquitous on Earth, in our planetary system, in interstellar clouds inside our Milky way, and in far-distant galaxies.The large dipole moment of the water molecule supports its effective cooling in the interstellar medium, playing a fundamental role in the physical development of galaxies, as well as in the formation of planetary systems. 1 Furthermore, water acts as a key player in the evolution of life on Earth and hence it is a target species for the spectroscopic investigation of exoplanets.
−4 The surface of these waterbased ices facilitates the production of several interstellar components, ranging from small species 5 to complex organic molecules. 6The chemistry involving water ices, irradiated by ultraviolet radiation, was extensively investigated by Harold Linnartz, to whom this article is dedicated.
The presence, the amount, and the distribution of water molecules can be studied via spectroscopic means.A wide range of astronomical objects and phenomena can be explored by measuring water transitions in the gas phase, either in emission or in absorption, which fall into the radio frequency domain.Radio astronomy from Earth-bound observatories focuses on line frequencies below 1 THz, representing an atmospheric window.For low-altitude radio telescopes, the observation range is limited to below 100 GHz.
The first identification of water in the interstellar medium, in fact through a maser transition around 22 GHz, was established at the Hat Creek Observatory. 7A decade later, 8 high-velocity molecular outflows were found in Orion-KL by probing this 22 GHz line via interferometric radio astronomy.In 1993, the Nobeyama radio station discovered water maser emission of extreme velocity in a distant galaxy. 9The near-to-sea-level Effelsberg 100-m telescope allowed the study of water in the early Universe at redshift z = 2.64 or f = 6 GHz. 10 The improved sensitivity of the Atacama Large Millimeter Array (ALMA), positioned at an altitude above 5 km, enabled the investigation of water lines at z = 6.9 or f = 3 GHz. 11ALMA made it possible to scrutinize the role of water in planet formation via water masers at 183, 321, and 658 GHz. 12,13The ALMA observatory, currently the most sensitive radio telescope in the mm and submm regions, aims at quadrupling the system bandwidths in its ALMA 2030 project, 14 further improving its spectral resolution.
For frequencies above 1 THz, satellite-based observations are required, due to the opacity of water in the Earth's atmosphere.Utilization of the HIFI (Heterodyne Instrument for the Far Infrared) device aboard the Herschel observatory ensured the detection of rotational water lines at 540−1700 GHz, used for the analysis of the physicochemical conditions in the water emitting region toward the high-mass protostar AFGL 2591. 15t even higher frequencies, water absorption lines were identified around 37 THz with the Spitzer telescope, finding a large amount of water in the atmosphere of a transiting exoplanet. 16Using the 120−500 THz range of the James Webb Space Telescope (JWST), water could also be detected in hot exoplanetary atmospheres. 17Water lines in the visible range (585−600 nm) were also employed to study the Earth's atmosphere. 18ith the advent of outer-atmospheric spectral devices, such as the (by now inactive) Herschel and SOFIA (Stratospheric Observatory for Infrared Astronomy) instruments, the need for refined line frequencies in the sub-mm region has been stressed in advanced astronomical investigations. 19,20This equally holds for ALMA, which covers a wide frequency range below 1 THz (more specifically, 35−950 GHz).Hence, it is an important task to increase the accuracy of water lines applied in radio and submm astronomy.A list of maser transitions detectable below 2 THz has been been recently compiled by Gray et al. for H 2 16 O, 21 helping future astrophysical applications.In addition, two studies 22,23 reported a couple of H 2 18 O lines observed via the PACS (Photoconductor Array Camera and Spectrometer) and HIFI instruments on Herschel.
The principal goal of this work is to show how the spectroscopic-network-assisted precision spectroscopy (SNAPS) approach 24 can be applied to deduce accurate line positions for astronomically relevant transitions of two water isotopologues H 2

16
O and H 2   18   O.The present SNAPS analysis is based on ultraprecise experimental results of previous studies, 24−32

SNAPS-BASED LINE SELECTION
As advocated in our previous studies, 24,30−32 the SNAPS protocol is a particularly useful tool when the aim is to extract the maximal amount of accurate spectroscopic information from a limited number of precision-spectroscopy experiments.SNAPS facilitates the selection of connected transition sequences (t 1 , t 2 , ..., t N ), where t i is incident to the (s i , s i+1 ) state pair and the intermediate (s 2 , s 3 , ..., s N ) states are pairwise distinct.A connected transition sequence may be a path/cycle, depending on whether the exterior (s 1 and s N+1 ) states are distinct/identical.A path can be applied to obtain an accurate energy difference between its starting (s 1 ) and ending (s N+1 ) state, whereas a cycle helps to confirm the internal accuracy of its underlying transitions through the analysis of its discrepancy. 33,34For further details on SNAPS, see refs 24 and 32.
The SNAPS scheme has been used to derive accurate rovibrational energies for H 2

16
O 24,31,3224,31,32 and H 2 18 O, 30,32 within the ground and highly excited vibrational states.These accurate results relied on more than 500 Lamb dips detected, with 1.5−38.9kHz accuracy, via two NICE−OHMS (noiseimmune cavity-enhanced optical-heterodyne molecular spectroscopy) setups. 35,36For both species, the lowest ortho energies, which cannot be extracted purely from experiments due to the lack of observed ortho ↔ para lines, 37 could be deduced with 6−8 kHz uncertainty.These two energies were taken from effective Hamiltonian (EH) fits, as well as from network paths where the ortho and para subpaths were concatenated with (exceedingly small) accurate, first-principles ortho−para splittings as special links. 24,30In the following section, the sets of high-precision H O and H 2 18 O energy levels are represented with , where v 1 , v 2 , and v 3 are the normal-mode vibrational quantum numbers of the symmetric stretch, bend, and asymmetric stretch motions, respectively, J is the overall rotational quantum number, whereas K a and K c are the conventional prolate-and oblate-top rotational quantum numbers, respectively.For a specific state, (a) the labels ortho/ para and even/odd correspond to (−1) v 3 +K a +K c = +1/−1 and (−1) K c = +1/−1, respectively, and (b) the polyad number is g i v e n a s designates a rovibrational line, where ′ and ″ distinguish between its upper and lower states, respectively. 38Unless otherwise noted, the words "transition" and "line" indicate a one-photon, dipole-allowed, rovibrational transition measured under absorption conditions.

THE ULTRAPRECISE H 2 16
O AND H 2

O NETWORKS
As part of the SNAPS procedure, ultraprecise spectroscopic networks were formed for H 2 16 O and H 2

18
5][26][27][28][29]39 To ensure connectivity among (0 0 0) states within the ortho-H 2 X O and para-H 2 X O subnetworks (X = 16, 18), 0 ⇐ 0 and 4 ⇐ 0 lines have been concatenated, where P′ ⇐ P″ denotes a transition between polyads P′ and P″. Th utilization of a few 0 ⇐ 0 lines, taken from ultrahigh-precision microwave measurements, 25−27 is required, because the subsets of even-and odd-parity (0 0 0) states cannot be linked with near-infrared dipole transitions.Note that (0 1 0) states are also included in the H 2 16 O network, whose ortho/para subnetworks become connected via highly accurate 5 ⇐ 1 and 5 ⇐ 0 lines.In the remainder of this section, a brief description is given about the data sources which were employed during the compilation of the (hyperfine-free) ultraprecise H 2 16 O and H 2

18
O networks.In an early beam-maser study by Kukolich, 25 the hyperfine and Zeeman structure of the 22 GHz maser line was probed with 50 Hz accuracy, yielding so far the most accurate line center for water.Later, Golubiatnikov et al. 26 analyzed the spectrum of water in the 180−560 GHz frequency region, identifying 13 and 6 Lamb-dip transitions, with 1−20 kHz accuracy, for H 2

16
O and H 2 18 O, respectively.In the case of the ortho transitions, some well-separated hyperfine components could also be resolved. 26azzoli et al. 27 conducted an analysis for seven ortho-H 2 16 O lines, via an ultrahigh-resolution spectrometer in the 320−620 GHz range.For the seven hyperfine-free rotational lines, derived from the hyperfine components, sub-kHz accuracy could be attained.−32 In ref 24, 156 carefully chosen transitions were accurately measured, which produced kHz-accuracy absolute energies for all but two (0 0 0) rotational states up to J = 8.Later, 30 195 lines were detected for H 2 18 O under saturation and then were used to derive empirical energies for all J (0 0 0)  1, were observed with our newer NICE−OHMS setup, 36 to reach as low frequency uncertainties as feasible.The new H 2 16 O transitions serve as the basis for a refined determination of the maser frequencies taken from Gray et al. 21(see Sec. 4), while those measured for H 2 18 O are used to improve the frequencies of some less accurate 0 ⇐ 0 lines in the full experimental H 2 18 O network. 41For the newly recorded H 2

16
O and H 2 18 O lines, the 1σ uncertainties were obtained via where u stat , u day , u pow , and u pres are the statistical, day-to-day, power-shift, and pressure-shift uncertainties, respectively.Four typical recordings, yielding regular and inverted 31 Lamb-dip profiles, are plotted in Figure 1.
−44,46−48 This comparison reveals significant shifts, exceeding 4σ, for six Doppler-broadened observations, while the rest of the former frequencies agree within 2σ with the NICE−OHMS values.Overall, the NICE−OHMS measurements yield a considerable improvement for the 26 line frequencies, corresponding to 3 orders of magnitude, when compared to their previous determinations. 42 −44,46−48

EXTRACTION OF FREQUENCY PREDICTIONS FROM NETWORK PATHS
To derive a prediction for a transition frequency within the SNAPS approach, it is necessary to establish an uninterrupted connection between the upper and lower states of the predicted line.This connection must be secured by a path, whose starting and ending states correspond to the upper and lower states of the desired transition, respectively.In the ultraprecise H 2

16
O and H 2 18 O networks, such paths mostly involve sequential Λ schemes, ensuring a kind of "spectroscopic triangulation" via up and down jumps between polyads.Among certain Λ schemes, pure rotational transitions must also be inserted on a path, producing seamless connection between opposite-parity lower states.If there are multiple (line-disjoint) paths between the same starting and ending states, they represent independent predictions for the same transition, warranting a comparison among the alternative frequencies and their uncertainties.Two paths form one or more cycles, depending on whether they have two ore more common states, respectively.A few characteristic paths and cycles, employed during the determination of accurate frequencies for astronomically important H 2 16 O and H 2 18 O lines, are visualized in Figure 2, guiding our analysis in the remaining part of this section.
To understand how a frequency prediction can be obtained from a path, one must use the Ritz principle 49 in a successive way. 24,32This process yields the following expression for the predicted frequency: whereby N T is the number of transitions in the network, and f i is the experimental frequency of the ith line preceded by a pathdependent "ternary" parameter, τ i .If the ith transition does not participate in this path, then τ i = 0, otherwise τ i is +1 or −1, depending on whether it points toward the upper or the lower state of the predicted line, respectively.For instance, the lines of Figure 2a have the following signs (from left to right): +1, −1, +1, −1, +1, +1, and −1.Supposing uncorrelated experimental errors, a well-defined 1σ uncertainty estimate can be formulated for f pred : where u( f i ) is the 1σ uncertainty of the f i frequency.For two independent predictions, f pred I and f pred II , their discrepancy and its uncertainty, respectively, can be calculated as and If D ≤ 2u(D), then the two predictions are statistically identical at the 95% significance level.
To minimize the u( f pred ) uncertainty, one must find a shortest path, called here a best path, between the upper and lower states of the predicted line within the ultraprecise H 2 16 O/H 2 18 O network.For this purpose, the Dijkstra algorithm 50 can be invoked, using the u 2 (f i ) values as edge weights.With the aid of best paths, one can bypass less accurate transitions, like the noisy transition of Figure 1c with 51.2 kHz uncertainty.Some specific examples for best paths are denoted with solid arrows in Figure 2, accompanied by alternative (dashed) paths in its last two panels.As obvious from Figures 2c and 2d, the best and the alternative predictions for the two 0 ⇐ 0 line frequencies agree well with each other, exhibiting discrepancies within the 2σ limit.Similarly good agreement is seen for a 1 ⇐ 1 line, (0 1 0)6 2,5 ← (0 1 0)5 3,2 , expressed with two long paths in Figure 3.

IMPROVED FREQUENCIES FOR ASTRONOMICAL WATER LINES
Built upon the best paths taken from the ultrahigh-accuracy H 2 16 O/H  18 O.−27,51−63 The best paths yielding the recommended frequencies, augmented with a line-by-line comparison to multiple experimental positions existing for the same astronomical line, are provided as Supporting Information.These comparison files also contain SNAPS values derived without using the new lines of Table 1, showing full agreement between the two kinds of SNAPS predictions.O maser transitions was composed in the 0− 1910 GHz frequency range, most of which play an essential role in the radiative-transfer models of various astrophysical environments.From that list, an excerpt was made, see Table 2, covering all the lines for which SNAPS-predicted frequencies are available.This excerpt does not include transitions pertaining to the P = 2 polyad, nor those with large J or K a The columns carry the same meaning as in Table 2.In the last column, "O−I" and "O−II" indicate that a line was observed in astronomical environments I (NGC 7129) and II (NGC 4418/Arp 220), respectively, while "U" means that a transition has not yet been identified in astronomical sources.Comment "U" is always followed by the respective HITRAN intensity, 78 multiplied by 0.002 (that is, the terrestrial relative abundance of H 2 18 O) and given in cm molecule −1 .The boldfaced (and asterisked) frequencies of the lines with comment "U" exploit the newly measured Lamb-dips presented in O.More specifically, the lines belonging to the (0 0 0) and (0 1 0) vibrational states span the 0.05−9.7 and 9.5−27.5 kHz accuracy ranges, respectively.The reason behind the larger uncertainties of the 1 ⇐ 1 frequencies is that (a) their best paths are typically longer than those of the 0 ⇐ 0 lines, and (b) the 5 ⇐ 1 transitions demanded for the 1 ⇐ 1 predictions were recorded at higher (usually 0.25 Pa) pressure values due to their increased line widths, leading to elevated total uncertainties for the 5 ⇐ 1 Lamb dips.Of the SNAPS-based frequencies, 13 benefit from the new transitions presented in Table 1.In another 13 cases, typeset in italics, our SNAPS predictions coincide with those measured in refs 25−27 at the (sub-)kHz level; thus, no further improvement could be carried out in this study for them.It must be stressed, however, that these 13 highly accurate frequencies are confirmed, within 10−15 kHz, via network cycles (see, for example, the cycle displayed in Figure 2c, involving two new Lamb-dip lines).
Table 2 also provides a comparison with the most accurate previous laboratory measurements (see columns 4−6).Except for the 13 cases with very low (<3 kHz) uncertainties, the SNAPS approach delivers significantly more accurate frequencies: in several cases, the improvement reaches a factor of 10 over previous results.For a maser transitions, large deviations are found, even above 100 kHz for six lines 53,56,57,60,61 and outgrowing the uncertainty values by 4σ for three examples. 57,60In ref 52, Kuze reported overly conservative uncertainty estimates: the deviations remain well within 0.5σ for the four lines taken from it.

18
O Lines.For the less abundant H 2 18 O species, no maser action has been detected and the astronomical observations are mostly related to absorption lines among low-lying (0 0 0) rotational states.These transitions fall typically into the sub-mm range, outside the transmission window of the Earth's atmosphere.As an application of the SNAPS approach to H 2 18 O lines of astronomical interest, a sample of Herschel-based observations have been collected from the literature.Eleven of these transitions were probed with the PACS device in the luminous NGC 4418 and Arp 220 galaxies, 22 while three via the HIFI instrument targeting the NGC 7129 star-forming region. 23hese 14 transitions, plus 7 extra lines with boldfaced SNAPS frequencies, can be found in Table 3.
Utilizing the best paths extracted from the ultraprecise H 2 18 O network, accurate SNAPS predictions could be determined for the 21 transition frequencies of Table 3.In the second column of this table, the 14 "plain" frequencies, obtained for the lines observed by Herschel, are characterized by 5−8 kHz accuracy.The uncertainties of these SNAPS-based predictions are lower, in all cases, than those arising from direct measurements. 26,55,58,63Except for three cases, the deviations of the literature positions from our predicted frequencies are smaller than the 2σ uncertainty limits.
During the experimental campaign of the present work, it was noticed that there remained only six 0 ⇐ 0 transitions in the experimental data sets of Belov et al. 55 and Matsushima et al. 58 whose upper or lower states had not been connected to the ultraprecise H 2   18   O network.This inspired us to record further Lamb-dip lines, given in Table 1, for H 2 18 O.The six additional 0 ⇐ 0 transitions, along with a microwave line around 6 GHz, are listed in Table 3 with boldfaced SNAPS predictions.For the 6 GHz transition, which possesses the same assignment as the well-studied maser line of H 2 16O at 22 GHz (see Table 3), the frequency uncertainty could be halved via SNAPS, with respect to an old laboratory measurement. 77The boldfaced frequencies of Table 3, especially those with large attached intensities, may prove useful in future astronomical investigations.

DISCUSSION AND CONCLUSIONS
In the present study, the SNAPS method 24 was utilized to obtain ultrahigh-precision frequency predictions for selected H 2  O transitions of astronomical significance.Ultraprecise H 2 16 O and H 2 18 O networks, lying at the heart of this investigation, were built with the help of new near-infrared Lamb-dip lines, observed using our second-generation NICE− OHMS spectrometer. 36The increased sensitivity of this upgraded NICE−OHMS setup enables the measurement of molecular transitions at a very low pressure (0.1 Pa or even less), thus decreasing the pressure shifts to an almost negligible amount.This stringent pressure condition, coupled with frequency-comb-based calibration, leads to kHz accuracy for the retrieved positions.Unfortunately, this is not the case for HD 16 O, another water isotopologue relevant in outer space, as the near-infrared Lamb dips of semiheavy water are significantly shifted/distorted during the NICE-OHMS measurements due to laser-induced Stark mixing, especially for min (K a ′,K a ″)>3. 79y measuring nearly 600 NICE−OHMS lines, chosen via the SNAPS protocol and combined with extremely accurate literature transitions, [25][26][27][28][29]39 a large number of (0 0 0) 24,30 and (0 1 0) 32 states could be included in the ultraprecise H 2

16
O and H 2 18 O networks.For the exploration of the (0 0 0) rotational states, a single probe laser at 1.4 μm proved to be sufficient to form serial Λ schemes, whereby both lower states belong to (0 0 0). 24Nevertheless, to attain the (0 1 0) states from the ortho/ para ground state, an extra laser, operating around 1.2 μm, had to be involved, ensuring the construction of Λ schemes where one of the lower states pertains to (0 0 0) and the other to (0 1 0) . 32These design principles were kept in mind when the new Lamb-dip lines of the present work were selected for detection, closing most of them into network cycles to verify their internal consistency (see, e.g., the two green transitions shown in Figure 2c).This procedure led to accurate predictions for nine rotational frequencies, whose upper/lower states were not covered in our previous analysis. 32rom the smallest-uncertainty paths of the ultraprecise H O networks, predicted frequencies could be extracted, with a few kHz uncertainty, for a collection of 68 astronomical lines.−27 For somewhat floppy molecules such as water, the SNAPS method is clearly superior to an EH model, the usual representation of quantum states in high-resolution spectroscopy.The issues with EH models are even more pronounced for states in the highly excited P = 4 and P = 5 polyads, where there are strong interactions among closely spaced states of the same symmetry.For example, while there was an attempt to reach high accuracy via an EH fit for the P = 4 polyad of H 2 16 O, 80 the fitting error could not be decreased below 4 GHz, an unacceptably large value in light of the kHz-level uncertainties achieved by today's precision-spectroscopy techniques.
It is truly remarkable that the high precision of near-infrared Lamb-dip spectroscopy could be transferred, via the SNAPS method, to astronomical transitions at a competitive level.Apart from 13 pure rotational lines, which are also included in the ultrahigh-accuracy H 2 16 O network, our frequency predictions turned out to be more accurate than the direct microwave and sub-mm spectroscopy measurements (see Tables 2 and 3).In several cases, large deviations were found, up to 100−200 kHz, exceeding the claimed uncertainties 53,56,57,60,61 even by 5σ and demonstrating the need for updated line positions.These considerable deviations may partly arise from the higher pressures employed in some of the data sources.For example, the transitions of Matsushima et al. 57 were recorded at 4.7 Pa, whereas the lines behind our predictions were investigated at much lower pressures, 0.01−0.55Pa.Taking a pressure slope of ±20 kHz Pa −1 , an effective value ascertained for near-infrared Lamb dips, 24,30−32 4.7 Pa may be translated to a pressure shift of ±94 kHz, which is reasonably close to the 100 kHz level reflected by the problematic deviations.
Due to electric-dipole selection rules, the SNAPS protocol hinges on the inclusion of a few highly accurate rotational transitions in the network, needed to attach opposite-parity states within the same vibrational manifold.However, if quadrupole lines were available and combined with dipole transitions, such connections could be made without reliance on pure rotational lines.In fact, quadrupole transitions have been detected for H 2 16 O in Doppler-broadened CRDS spectra with 60−90 MHz uncertainty. 81,82However, high-quality Lamb dips would be demanded for our purposes, such that measurement might be possible if suggested by the recent detection of a Lambdip feature probed for a quadrupole transition in the first overtone of H 2 , yielding a highly accurate position for this line. 36lternatively, two-photon transitions could also be applied as direct links among clusters of one-photon lines with differentparity lower states.Double resonance techniques 83,84 bear promise for intracavity observations of two-photon lines in water, securing the desired kHz accuracy.
Most of the purely rotational water lines derived in this study have immediate astronomical relevance: (a) 48 transitions of H 2 16 O (may) act as masers in evolved-star envelopes, 21 and (b) 14 lines of H 2 18 O have been detected 22,23 in extragalactic regions.The accurate frequencies determined for these transitions could be valuable in numerous applications, including the analysis of kinematics, Doppler motions, and redshifts in astronomical objects, as well as the investigation of inflows and outflows characterizing these celestial sources.Moreover, due to the omnipresence of water, the recommended frequencies of Tables 2 and 3 can be employed as reference values to calibrate new high-resolution spectra of astronomically relevant molecules in the 0−5 THz frequency region.
As to the applicability of the SNAPS method to other molecules, it is noted that our current NICE−OHMS setup is designed for probing stable closed-shell molecules.Similar optical technologies have been developed to probe molecular ions 85 and open-shell molecular radicals, 86 but the accuracy obtained is still insufficient to extract competitive frequencies in the radio domain.As an another molecule, acetylene has been investigated via cavity-enhanced techniques within a network approach, 87 but this species may be less relevant from an astronomical perspective.A molecule suitable for future SNAPS studies is methanol, which exhibits important maser action on a multitude of lines in the Milky Way and in extragalactic sources. 88In the near-infrared region, there are numerous vibrational bands of methanol which could be subject to a SNAPS analysis for the extraction of ultraprecise radio-line frequencies. 89These accurate frequency predictions could play a decisive role in the quest for probing variation of fundamental constants, like the proton−electron mass ratio, on a cosmological time scale, 90,91 as well as to test the weak equivalence principle. 92or the extensive study of starless cores, 93 space-related fundamental physics, 91,92 as well as hyperfine-resolved maser observations, 94 the experimental resolution and accuracy of existing radio observatories is well suited.The upcoming upgrade of the ALMA observatory, 14 allowing a unique spectral resolution of 1−30 kHz over the entire ALMA bandwidth, will open up new territories within the realm of astronomical spectroscopy.In this situation, the arrival of ultraprecise radio lines, such as those provided here for water, is well-timed to address the demands of contemporary astronomy.

■ ASSOCIATED CONTENT
* sı Supporting Information The Supporting Information is available free of charge at h t t p s : / / p u b s .a c s .o r g / d o i / 1 0 . 1 0 2 1 / a c s e a r t h s p a c echem.4c00161.
Detailed descriptions of Supporting Information tables (PDF) (a) Machine-readable list of new H 2 16 O and H 2 18 O transitions recorded with our upgraded NICE-OHMS setup and (b) detailed comparison of previous laboratory measurement results with the SNAPS-predicted frequencies obtained for the H 2 16 O and H 2

18
O lines of Tables 2  and 3, respectively (ZIP) ■ AUTHOR INFORMATION Corresponding Authors and innovation programme.The work performed in Budapest received support from NKFIH (grant no.K138233 to A.G.C. and grant no.PD145972 to R.T.).At the Amsterdam side, support was obtained from a NWO program (16MYSTP).
. 2). c The most accurate Doppler-broadened experimental results are taken from the literature (see column "Ref.").The column "Dev."lists the deviations of the Doppler-limited positions from their Lamb-dip counterparts.The column "Unc."contains uncertainty estimates provided in the references.Except ref 46, these sources report only average uncertainties for the observed lines (thus, it is not surprising that some of the respective deviations grow above 4σ).d Transitions not measured via Doppler spectroscopy.e Lines, three in total, characterized by very low (<10 −26 cm molecule −1 ) absorption intensities.f Transitions, eight in total, with inverted Lamb-dip profiles. 31g Line forming part of an unresolved ortho−para doublet in Doppler-limited spectra at room temperature.0.01−0.55Pa.Taking an effective pressure-shift coefficient, 20 kHz Pa −1 , 24,30 into account, a pressure-shift uncertainty of 0.2− 11 kHz is included in the uncertainty budget.The three less accurate lines with >10 kHz uncertainties are characterized by small (<10 −26 molecule −1 ) intensities, leading to somewhat lower signal-to-noise ratios.A low-intensity H 2 18

Figure 1 .
Figure 1.Typical Lamb dips detected during this study for H 2 16 O and H 2 18 O.Panels (a) and (d) present two regular Lamb-dip profiles, characterized by a single dip.Panel (b) displays an inverted (double-dip) profile, 31 which occurs for transitions with large (>0.5 s −1 ) Einstein-A coefficients.Panel (c) exhibits an H 2 16 O line with very low (9.7 × 10 −28 molecule −1 ) intensity, obtained via averaging over 20 scans.To facilitate their visual comparison, these spectra are mapped onto the (I min − I)/I min relative intensity scale, where I means the intensity at a detuning point for a specific transition, and I min is the lowest intensity in the ±3 MHz detuning region.

Figure 2 .
Figure 2. Typical short paths and cycles used for the characterization of astronomical H 2 16O and H 218  O lines.The ortho and para states of this figure are symbolized with circles and squares, respectively.For these states, the J K K , a c labels are written out explicitly, whereas the (v 1 v 2 v 3 ) triplets are shown in the left-side color legend.The green arrows illustrate new Lamb-dip transitions, while those with dark blue, orange, purple, brown, cyan, and light blue colors are ultrahigh-accuracy lines taken from refs 24, 26, 27, 30, 31, and 32, respectively.For the pure rotational transitions included on the paths, thicker arrows are used.The numbers on the arrows designate frequencies in kHz, with 1σ uncertainties of the last digits in parentheses.The solid arrows constitute the "best" (lowest-uncertainty) paths between their starting and ending states, distinguished with dotted magenta and mint boxes, respectively.The dashed arrows form alternative paths, producing cycles with the solid ones.The approximate positions, related to transitions between the starting and ending states of the best paths, are shown at the top of the panels.The yellowish-green boxes provide predicted frequencies and discrepancies, with their 1σ uncertainties, for paths [panels a and b] and cycles [panels c and d], respectively.For further details, see the text.

Figure 3 .
Figure 3. Example for a long cycle formed by two line-independent paths between two (0 1 0) states.The notation of this figure is the same as in Figure 2, with the extension that the gray arrow denotes a transition taken from ref 25.

Table 1 .
16List of New Lamb-Dip Lines Recorded for H 216O and H 2

Table 2 .
Recommended Frequencies for H 2 16 O Maser Lines Collected from Ref 21 21ost accurate laboratory measurements from the literature.The column "Dev."contains the deviations of the measured positions from the recommended values of this table.The column "Unc."includes the uncertainties taken from the individual data sources.The italicized frequencies of the second column coincide with the measured literature values, leading to zero deviations.dShortcomments:"O" means an observed maser line reported in the cited reference, "P" is a predicted maser transition, and "(abs)" indicates that a maser line is not (easily) observable due to strong terrestrial absorption.eThe380.197GHzline was listed as a predicted maser line by Gray et al.,21but in fact it has been observed in ref70.

Table 3 .
Recommended Frequencies for Selected H 218O Lines of (Potential) Astronomical Relevance a Table 1, involving new rotational states compared to ref 32.values, as they are not accessible from the ultraprecise H 2