IUPAC Critical Evaluation of the Rotational-Vibrational Spectra of Water Vapor, Part III: Energy Levels and Transition Wavenumbers for H216O

This is the third of a series of articles reporting critically evaluated rotational– vibrational line positions, transition intensities, and energy levels, with associated critically reviewed labels and uncertainties, for all the main isotopologues of water. This paper presents experimental line positions, experimental-quality energy levels


Introduction
Water is the most abundant polyatomic molecule in the universe and it is responsible for the majority of the greenhouse effect on Earth [1].As a result, the spectrum of water vapor is one of the most thoroughly studied [2].Topical reviews are available from both experimental [2,3] and theoretical [4,5] perspectives.The need for highly-accurate levels and lines of water vapor is emphasized by articles in the recent volume of ''Water in the gas phase'' [6].As for astrophysics, lines and levels are crucial to interpret maser sources [7][8][9], comets [10], planets [11], exoplanets [12], cool stars [13], carbon stars [14], and interstellar clouds [15,16].Critically-evaluated energy levels are also useful for a variety of scientific and engineering applications, see, for example, the introduction of Ref. [17], including determining partition functions and hence thermodynamic data [18],a n d the refinement of theoretical models.Water vapor also plays an important role in characterizing combustion systems [19].
The first 13 authors of this paper form a Task Group under the auspices of IUPAC (International Union of Pure and Applied Chemistry), with the aim of constructing a database of water transitions from experiment and theory, and with individual tasks described in Table 1 of the first paper in this series [20], henceforth referred to as Part I. Since absorption due to electronic excitation requires energies over 50 000 cm À 1 in the case of the water molecule, most of its physical properties are determined by its ground electronic state [21].This determines the nature of water spectroscopy [2], and thus the present effort concentrates on the pure rotational and rovibrational energy levels of water from within the ground electronic state.We consider all the corresponding transitions up to the first dissociation limit of the molecule simultaneously and on an equal basis.
This paper is the third in a series presenting our evolving methods for collecting and analyzing the experimental (spectroscopic) and quantum chemical information available as well as our validated data recommended for deposition in information systems.In Part I [20],w e derived labeled energy levels and transition wavenumbers for the water isotopologues H 2 17 O and H 2 18 O.In Part II [22], we derived labeled energy levels and transition wavenumbers for the partially deuterated water isotopologues HD 16 O, HD 17 O, and HD 18 O.These analyses were based on the concept of spectroscopic networks [23,24] and were executed using the MARVEL (Measured Active Rotational-Vibrational Energy Levels) protocol of Furtenbacher and Csa ´sza ´r [23,[25][26][27], which was considerably refined during the course of the present study [27] to allow for the treatment of large datasets.Within the MARVEL analysis, a significant amount of checking is performed in order to minimize inconsistencies and errors in the experimental transition data.
In this work we apply the MARVEL algorithm and code to the main H 2 16 O isotopologue of the water molecule.
Unlike the other isotopologues of water, the energy levels of H 2 16 O were already subjected to a comprehensive and systematic study by Tennyson et al. [28].We note, for example, that the recent release of the HITEMP database [29] used the energy levels of Ref. [28] to generate transition wavenumbers for hot-water spectroscopy.The present study significantly improves on the methodology used previously [28] and, given a decade of further collection of experimental data, considerably extends its scope.In this context we note in particular the multiresonance studies of Boyarkin, Rizzo, and co-workers, which have probed the energy levels of water up to [30][31][32][33][34] and even beyond [35] the first dissociation limit.These sophisticated experiments have sparked corresponding theoretical studies [34,36,37].Here we consider all the available experimental spectroscopic transition data linking rotation-vibration levels below the first dissociation limit of H 2 16 O.
As emphasized already in Parts I and II, a distinguishing feature of the present series of IUPAC-sponsored spectroscopic studies is the joint utilization of all available experimental and the best theoretical line (transition) and energy-level data, with a long-term aim of creating complete linelists for all water isotopologues.While determination of a complete linelist is outside the scope of present-day experiments, it can be determined by means of sophisticated first-principles quantum chemical computations.Studies on the spectroscopic networks of water isotopologues [24,38] also revealed that a large number of energy levels participate in some transitions strong enough to be observable.Thus, although only a small portion of all the allowed transitions will ever be observed experimentally, it seems likely that the majority of energy levels will eventually be connected to observed transitions.For the time being, as experimental line positions have a higher accuracy than those yielded by even the most advanced computations, complete linelists will necessarily contain a mixture of accurate experimental data and less accurate computational data.MARVEL-type efforts (a) replace as many computed lines as possible with their experimental counterparts, (b) validate and ideally reduce the uncertainty with which a transition has been determined, and (c) facilitate the assignment of experimental spectra.Unlike line positions, the overwhelming majority of one-photon, temperature-dependent absorption and Open access under CC BY license.

Methods, input data, and data treatment
The methods employed in this study for collecting and critically evaluating labeled experimental transition wavenumbers and their uncertainties and for inverting the wavenumbers in order to obtain the best possible energy levels with corresponding uncertainties are principally based on the concept of spectroscopic networks [23,24] and on the MARVEL procedure [23][24][25][26][27].During a MARVEL analysis we simultaneously process all the available assigned and labeled experimental lines to give the associated energy levels of the chosen isotopologue.We adopted a reweighting scheme [51] where uncertainties for selected line positions are changed (in practice increased) during iterationso ft h eM A R V E Lp r o c e d u r e [25].After removing outliers from the experimental transition data and applying the iterative robust reweighting algorithm, a database is created containing self-consistent and uniquely labeled transitions and related uncertainties.The procedure is such that the final energy levels and their uncertainties are guaranteed to be compatible with the (adjusted) uncertainties of the experimental line positions.This means that all transitions used in the MARVEL procedure agree, within their revised stated uncertainties, with the MARVEL predictions.This criterion for the error is therefore more stringent than the usual standard deviation used to represent statistical error and will usually lead to the quoted MARVEL errors being systematically larger.
The first step in the MARVEL procedure is to split the transition data into components of the spectroscopic network (SN) characterizing the molecule [24].C o m p o n e n t so fS N s contain all interconnected rotational-vibrational energy levels supported by the grand database of the labeled transitions.For H 2 16 O, the transitions must form two rooted components, an ortho and a para one (Table 1).Other components of the SN whose nodes are unattached to either of the two roots are designated as floating spectroscopic networks (FSNs) or, in the case of a single transition with no energy level in common with any of the other transitions in the compilation, orphans (ORPs).The selection rules for electric-dipole-allowed transitions are as follows: within the ortho or para SNs transitions are allowed if DJ ¼ 7 1 and Dp ¼ 0o rDJ ¼ 0a n dDp ¼ 1, where J is the quantum number describing the overall rotation of the molecule and p is defined in the footnote to Table 1.
For H 2 16 O, there exists an unusually large number of at least partially assigned experimental spectra [17,28,30,[33][34][35].The data from room-temperature spectra are augmented by data from a number of warm (400-700 K) and hot (up to 3000 K) H 2 16 Os p e c t r a [83,93,94,96-98,107, 124,135,148,164,166-177. Hot spectra are rich in high-J and hot-band transitions but often have significantly larger uncertainties and a much increased chance of misassignment and mislabeling.Another significant feature of water spectroscopy is that the spectrum has been probed by multiresonance experiments up to [30][31][32][33][34] and even beyond [35] the first dissociation limit of the molecule.These studies significantly extend our knowledge of the ground electronic state of the water molecule.Some of the papers on water vapor spectra report only intensity or lineshape data and are therefore not employed in a direct fashion in this study.
Measured spectra of H 2 16 O vapor are basically a superposition of two separate spectra, that of ortho-water and para-water (Table 1), the strongly forbidden transitions between the two spin isomers have never been observed [178].Lines of ortho-water (total spin of protons I¼1) exhibit, under ultrahigh resolution, hyperfine structure (hfs) due to the coupling of the proton nuclear spins with the rotational angular momentum (i.e., interaction between the magnetic moments of the protons and the magnetic field generated by the molecular rotation) and to a direct spin-spin interaction.All ortho-water levels with J 4 0 are split into three hfs components with F ¼ JÀ1,J, and J þ 1, where F ¼ J þ I is the total angular momentum of the system.The ''center of gravity'' of the levels and associated lines is not shifted due to this splitting.The spectrum of para-water (I¼0) exhibits no hfs.There are several papers on water spectroscopy which addressed the hfs splittings [7,61,145,159].The accurate and precise unperturbed frequencies reported in these papers are highly useful for the present study.However, since hfs-resolved data exist only for a few levels, we work with the (hfs-averaged) line centers.Since the separation between ortho and para states is not measured experimentally, it is necessary to fix it using a so-called magic number.This was done by setting the energy of the (0 0 0) [1 0 1] state to 23.794352 cm À 1 , the number determined by 01LaCoCa [124] using an effective Hamiltonian.This ''magic number'' was supported, to the given number of digits, by a MARVEL analysis of the final energy levels whereby the degeneracy between a large number of ortho and para levels was introduced to couple the two SNs.There are several papers, e.g., Refs.[28,135,137], which report many derived energy levels but no or few original experimental transitions.The energy levels of H 2 16 O have been studied using several theoretical schemes.As the usual effective Hamiltonian approach is problematic for water [179], several simple theoretical approaches have been developed and tried on water spectra, including the Pade ´approximation [179][180][181], the Borel approximation [90,137,179], generating functions [39,182,183], joint perturbational-variational approaches [184],a n d a Parity ¼ðÀ1Þ ðJ þ pÞ ¼ðÀ1Þ Kc , where p is defined as 0 for e and 1 for f states [50].The ortho and para labels are defined by ðÀ1Þ v3 þ Ka þ Kc .See Section 2 for the meaning of the approximate vibrational and rotational quantum numbers.[192] 5103.3-5547.1 234/234 96BrMa [106] 5206.3-5396.5 28 /28 RT FTS 94Tothb [103] 5750.9-7987.a four-dimensional bending-rotation Hamiltonian [185,186].Due to the assumed simplicity of the water molecule, it has also been a favorite subject of variational nuclear motion computations [34,36,37,39,[42][43][44][45][47][48][49]187].Table 2 provides, for each transition data source, experimental information related to the spectra.The number of originally measured and assigned (A) and validated (V) transitions for each data source is given there, as well.Due to the large amount of related experimental studies, a nearly continuous coverage has been achieved for H 2 16 Ou pt oa b o u t1 50 0 0c m À 1 , significantly above the barrier to linearity of the molecule [175,[188][189][190], with transitions sporadically observed beyond this value.
To be included in our tabulation, data sources must include original experimental line positions with uncertainties and line assignments with labels.Information on the conditions under which the experimental data were recorded is summarized in the column 'Physical conditions' in Table 2.As in Parts I and II, the data source is identified with a tag based on the year of publication and the names of the authors (see Part I for more details).
It is important to make a distinction between resolution of the spectrometer and uncertainty of the lines.Often uncertainties can be 10-15 times smaller than the resolution.
Most of the spectra were obtained in absorption by Fourier transform spectroscopy (FTS), which has allowed a wide spectral coverage from the microwave region through the visible to the near ultraviolet.In order to detect weak lines, FTS spectrometers have been equipped with long multipass cells.Absorption path lengths as large as 433, 801, and 1804 m have been achieved with the cells available at Kitt Peak, Rutherford Appleton Laboratory, and University of Reims, respectively, providing a large number of observed transitions in the near infrared and visible regions (up to 26 000 cm À 1 ).
Laser-based methods, such as CRDS (cavity ringdown spectroscopy) and ICLAS (intracavity laser absorption spectroscopy), are limited to certain spectral regions depending on the availability of tunable laser sources.These techniques have specific advantages in terms of sensitivity and spectral resolution, which make them particularly suitable for the characterization of spectral regions with weak absorption features.This is why extensive investigations with laser-based methods were mostly limited to transparency windows or to the visible region.The precision of CRDS is limited to about 0.001 cm À 1 .Emission spectra can provide large datasets of line positions.Emission spectra for water are available over a particularly extensive range of temperatures, the hottest being spectra recorded in an oxy-acetylene flame at about 3000 K [174].These spectra provide a rich source of information on states both with significant bending excitation [175], which are normally not probed in standard absorption spectra, and with high levels of rotational excitation.Hot spectra suffer from the disadvantage that it is not usually possible to obtain line positions with the same accuracy as spectra recorded at room temperature.This is due to the increased Doppler width of the transitions and, in the case of atmospheric pressure spectra such as those recorded in flames, significant pressure broadening and pressure shifts.The extended linewidths and high density of transitions also leads to very large numbers of blended transitions, which places a further constraint on the accuracy with which the positions of the individual lines can be determined.It should be noted that absorption spectra recorded in sunspots correspond to a temperature of about 3200 K [191], close to the hottest laboratory emission spectra.Sunspots provide a very rich source of spectroscopic data on water [168] and many lines, which are almost certainly due to hot water, still need to be assigned.

Pre-MARVEL validation
Similar to Parts I and II, the experimental database assembled based on literature data were first checked for simple transcription errors or other problems in the original data source.Checks were made for formatting incompatibilities, entries with zero uncertainties (not allowed in a MARVEL-type analysis), strongly forbidden (ortho to para) transitions, and transitions with impossible labels.As to invalid labels, the dataset of transitions was searched for cases where the K a þ K c sum, where K a and K c are the usual asymmetric-top rigid-rotor quantum numbers, did not equal J or J þ 1, where J is the rotational quantum number, where either K a or K c were greater than J.
A search was performed to identify obvious duplications in the dataset in order to avoid entering the same measurement twice.A warning was issued if this happened and only the earliest occurrence of the datum was kept (unless the line was reanalyzed in a later study).We also made sure that the dataset of transitions to be analyzed by MARVEL did not contain (a) entries where the two transitions had the same lower-level label but different upper-level labels with transition wavenumber differences less than 0.05 cm À 1 , and (b) where the transitions had the same lower-and upper-level labels and their wavenumbers differed beyond the related uncertainties.
Employing the parities detailed in Table 1 and the associated selection rules, transitions which had incompatible labels were identified and deleted.The rotational parity along with the J values helped in the search for correspondence between the experimental and the variational datasets.
Another pre-MARVEL validation procedure utilized the so-called BT2 linelist [194].If the BT2 variational nuclearmotion computations indicated the existence of a degeneracy between an ortho and a para transition within the BT2 linelist to better than 10 À 3 cm À 1 and one of the transitions was missing from the MARVEL input, the missing entry was added by hand and indicated as such in the input by adding the letter ''D'' to the tag of the duplicated transition entry.We added altogether 6589 duplicates to the original database.This step was important to link a number of what otherwise would have been FSNs.

Assignment, labels
It is a requirement of the MARVEL protocol that the dataset contains a single unique label for both the lower and the upper states involved in each transition.There is no requirement that labels have any physical significance beyond those needed to give selection rules.Due to the extent of the data for transitions of H 2 16 O, a large number of problems were expected for the approximate vibrational labels.For consistency and to maintain a single set of uniform labels for all levels, we chose to label vibrational states in the usual normal-mode notation, (v 1 v 2 v 3 ).There are strong physical arguments that the higher stretching states of water are better labeled using local modes [195,196].However, there is a one-to-one correspondence between local-and normal-mode labeling schemes [116], so the use of normal-mode labels leads to no loss of information.We have therefore translated assignments given in local modes into normal modes.We note that for a number of data sources it proved necessary to systematically re-label data assigned in normal mode notation as the labels did not map to the correct and very characteristic local mode energy level structure.This issue has been noted previously [116].In this work we retain, whenever possible, the ordering and the normal-mode labeling of the vibrational states of H 2 16 O presented and advocated in Ref. [197].Similar problems arise from the approximate standard asymmetric top quantum numbers [JK a K c ], used as part of the label of the rovibrational states.Driven by the required uniqueness of the labels, the rotation-vibration levels of H 2 16 O are identified in this study by altogether six quantum numbers: Before processing the published transition data, we checked, as thoroughly as possible, whether the transition labels were correct and consistent.For H 2 16 O several of the older sources permuted labels for a number of higherlying excited states compared to those advocated here.
Where possible, we corrected for this.Rovibrational labels, which could be used for checking the (v 1 v 2 v 3 ) [JK a K c ] labels of the experimental transitions, could be taken from computations based on the use of an effective Hamiltonian (EH).However, for the majority of the highlying energy levels, results from proper EH computations are not available.Validation of the labels attached to the observed transitions was thus performed as follows.
Transitions were examined for consistency of the upper levels derived from combination difference (CD) relations.This method is a simple and powerful tool for the assignment of rovibrational spectra; however, it cannot be applied to transitions not part of several CD relations.
All the transitions associated with a given rotational level of the (0 0 0) vibrational ground state have been considered for combination differences.At this stage, conflicting labels could be traced and corrected.Many CD relations for other rovibrational states have also been checked.An important check of the labels is provided by the normal-mode decomposition (NMD) and rigid-rotor decomposition (RRD) analyses [198,199] of the variationally computed rovibrational wave functions.This is a powerful technique to identify approximate vibrational and rotational labels based on the harmonic oscillator and rigid rotor formalisms and the rovibrational wavefunctions obtained from variational treatments.Validation of rovibrational labels with J less than 26 has been attempted by computing RRD tables.For obtaining RRD labels, a cut-off value of 0.7 was chosen for the largest RRD coefficient, i.e., only cases where the labeling is unambiguously provided by the RRD scheme were utilized.Rovibrational states with an energy larger than 25 000 cm À 1 were also not investigated since for them there appear to be very few states which can clearly be labeled via an RRD table.This means that for a large number of MARVEL energy levels no validation via the RRD scheme was attempted.The extent of validated labels for the different vibrational band origins (VBO) is given in Table 3, which also gives the J max values (the maximum J value on the particular VBO) for a large number of VBOs.Clearly, it is more problematic to provide unambiguous rotational labels for VBOs which contain a high level of bending excitation.In particular, starting from the (0 10 0) VBO no rotational labels on pure bending VBOs could be provided by the RRD methodology; this problem is almost certainly associated with the rearrangement of the energy level structure caused by the monodromy point at linearity [175,200].In general, as the energy of excitation increases, the highest J value where RRD can be used to validate the MARVEL labels decreases.
Finally, consistent labeling has been established for all the assigned transitions considered.We recommend that the labeling provided in this paper should be generally adopted, although in cases of strongly perturbed energy levels there is considerable remaining uncertainty.In particular, the approximate normal-mode labels, and at higher energies, and especially for high K a levels, the rigid-rotor labels [198,199], are not expected to provide a physically correct description.

Uncertainties
Within the MARVEL protocol, reasonable estimates for the accuracy of the observed transitions must be provided.Despite the adjustments by the robust reweighting scheme, due to the existence of a huge number of cycles of various size within the SN of H 2 16 O, false uncertainties attached to the transitions can noticeably deteriorate the accuracy of a large number of MARVEL energy levels.
In some of the data sources, approximate experimental uncertainties are not given for each individual transition.Often only the general accuracy of the transitions for the region investigated is provided.For a few publications we were forced to estimate the experimental uncertainties.If no values were presented in the original source, these uncertainties were based on average values characteristic of the experimental setup exploited in the measurement.We note also that sometimes the experimental uncertainty attached to a line in the original source reflects the quality of the line profile fit rather than the real accuracy with which the wavenumber was determined.For further important adjustments of the uncertainties of transitions of certain data sources see Section 2.7.
MARVEL may increase, via robust reweighting, the assumed experimental uncertainty of a transition when it is not consistent with the one derived from the MARVEL energy levels.For transitions with low J and K a values it is rather easy to evaluate a feasible experimental uncertainty if enough CDs are available.However, for increased values of J and K a , the CD relations become less accurate and instructive, a number of experimental lines represent unresolved multiplets, and it becomes more and more difficult to judge properly the actual experimental accuracy of the transitions.This in turn limits the accuracy of the MARVEL energy levels derived.This situation could be improved only by including additional accurate experimental information in the MARVEL input file.
For a number of transitions which proved to be clear outliers, the experimental uncertainties were increased manually (see Section 2.7).This was done when the energy of an upper state deviated far more from the corresponding mean value established by MARVEL than the stated experimental uncertainty.It is our hope that by a computerized search and a subsequent manual adjustment, at least the majority of clear outliers have been identified and either removed from further analysis or were included with a more reasonable uncertainty.

Hot transitions
The high-lying rotational levels probed by hot transitions are hard to validate due to the high density of both predicted and observed transitions.In addition, some of the transitions to a given upper level originate from lower (sometimes unknown) levels belonging to excited vibrational states.
The most significant problem with the existing hotwater transition data were their inconsistent labeling.Thus, the labels from these data sources were carefully checked once the MARVEL energy levels, and their labels, were well established based on cold water spectra.As the comments to Table 2 (Section 2.7) demonstrate, a large number required careful relabeling so that only a small number of assigned transitions were actually omitted during the final MARVEL runs.It must be emphasized again that (a) labels for the hot water transitions may not be physically correct but at least they are unique and consistent as far as the present database is concerned, and Table 3 Validation of the rotational labels of the rovibrational levels determined by the final MARVEL analysis via the rigid rotor decomposition (RRD) protocol [198,199] (b) MARVEL is not able to produce correct labels for assigned transitions; it can only point out inconsistencies within a given dataset.

Multiphoton transitions
Multiphoton spectra provide valuable information on the energy levels of water extending all the way to dissociation.By use of known intermediate levels and alternative routes to the same upper energy levels, the experiments yield assignment information on the rotational quantum numbers and nuclear spin parity of the upper state.
At present it is not really possible for us to independently validate the energy levels obtained from multiresonance spectra.Standard linelists such as BT2 [194] used here to validate the other levels only extend to about 30 000 cm À 1 .Although there are a number of ab initio computations which study levels, particularly ones with low J, all the way to dissociation [197,[201][202][203], even the most reliable of these [197] does not reproduce the observations near dissociation satisfactorily.Similarly, the near dissociation experiments are difficult and so far have only been successfully performed by one group [35].For these reasons, the levels obtained by multiresonance spectroscopy are all assessed as being among the energy levels which we deem the least well determined, as seen in Section 3.
Finally, we should also note that while it appears possible to provide meaningful vibrational quantum numbers of the high-lying levels of water probed by the multiresonance experiments, this is generally not true for all levels at high energy [197,204,205].These experiments particularly probed states of high stretching excitation which appear to keep their localized nature all the way to dissociation.However, it is clear that for many high-lying states neither of the two standard schemes for labeling vibrational levels of water, normal modes or local modes, appear to yield physically reasonable quantum number assignments [197].

Recalibration
When sets of experimental line positions measured over several decades are combined, systematic differences can be easily identified if several other groups reported high precision values for at least some of the same transitions.Some inconsistencies between studies occur because of mistakes, but others arise simply because the calibration standards changed over time.To correct this situation properly, the best available standards must be applied.
Calibration standards for the infrared were reviewed in 1985 [84], 1992 [207], and in a 1996 IUPAC study [206].For three decades, the standards involved high-accuracy, Doppler-limited heterodyne frequencies, but only up to 6563.3 cm À 1 .These were used to obtain positions of easily handled gases, but some lists required correction when better standards appeared [206].In addition to refined mid-IR standards (e.g., for CO 2 and N 2 O at 550-700 cm À 1 [208] and for CH 4 at 3000 cm À 1 [209]), new calibration standards now available at near-IR and visible wavelengths enable better scrutiny of the older measurements, especially above 7000 cm À 1 (see Ref. [210] for atomic potassium and Ref. [211] for 127 I 2 and the references therein).
Data from Fourier transform spectrometers are easily corrected by applying a multiplying factor.For the present study, we relied on one particular laboratory spectrum recorded with the Fourier transform spectrometer at Kitt Peak (FTS-KP) that spanned the 4000 to 14 000 cm À 1 region using low pressure mixtures of CO, C 2 H 2 ,H 2 O, and O 2 (see Ref. [118]).Originally, line centers of the O 2 Aband at 13 100 cm À 1 were calibrated to precisions of 0.0005 cm À 1 using CO and C 2 H 2 standards near 4250 and 6400 cm À 1 .However, absolute accuracies were estim a t e dt ob eo n l y0:0015ð 70:0007Þ cm À1 because the good CO and C 2 H 2 positions stopped at 6560 cm À 1 [118].Eight years later, Robichaud et al. [212] recalibrated the O 2 positions against two nearby 39 K lines [210], and this in turn permitted the earlier mix-gas spectrum [118] to be renormalized based on features of three widely spaced standards (CO, C 2 H 2 , and O 2 ).In the present study, line centers of H 2 O at 5300, 7400, 8300 and 10600 cm À 1 (obtained by peak-finding with the apodized spectrum) were compared with collected MARVEL values.Line centers separated by less than 0.05 cm À 1 from adjacent lines were omitted, as were lines that were too weak ( o 15% deep) or too strong ( 4 90% deep).Table 4 lists near-IR and visible studies identified for correction, along with the multiplicative calibration factors obtained (as was done in Ref. [206]).
During the MARVEL analysis it also became clear that there are other sources of data, obtained by FTS, which might suffer from calibration problems.It is straightforward to determine multiplicative calibration factors with MARVEL [20,22].The procedure involves the minimization of the root-mean-square (rms) deviation between the observed transitions including the source with wavenumbers scaled with a given calibration factor and those produced by MARVEL from the energy levels.The sources identified for recalibration include 73CaFlGuAm [62], 78KaKaKy [70], 80CaFlMa [74], 82KaJoHo [79], 96PoBuGuZh [107], and 99CaJeVaBe [116].It was found that no recalibration is needed for the following sources containing FTS data: 94Tothb [103], 95PaHo [105], and 05HoAnAlPi [138].This rms minimization was performed sequentially for all data sources identified as problematic.The multiplicative calibration factors that emerged from these analyses are given in Table 4.For all the sources identified in Table 4, the experimental data were recalibrated using the calibration factors determined and only the recalibrated transitions were included in the final MARVEL analysis.Recalibrated transitions are distinguished within the dataset by a letter ''R'' attached to the end of the transition entry.
While there are known calibration problems with some of the ICLAS data [213], during the analysis of ICLAS spectra different calibration lines have been used for every few cm À 1 .Thus, one cannot determine a unique calibration factor for the whole region covered or employ a constant shift value.Improvement of the ICLAS data via recalibration was thus not attempted here.

Comments on the data sources
Information on deleted and relabeled transitions given here use MARVEL format; only the new labels are given for the relabeled transitions.All relabeling was performed during the course of the present study.
(2a) 91PeAnHeDe [94].A hot cell (5 cm in diameter and 1.5 m long, which can be heated up to 1400 K) was employed with a tunable Fourier infrared spectrometer; details are given in Ref. [214].The typical uncertainty in the determination of the transition frequencies was about 100 kHz (3 Â 10 À6 cm À1 ).For the assigned transitions, the maximum value of J (J max ) and the maximum value of K a (K max a ) are 17 and 7, respectively.(2b) 00ChPePiMa [119].Coherently generated THz radiation lasers locked to a stabilized etalon, single-pass absorption with a flowing sample.The authors specified one standard deviation for the uncertainties of the lines.The accuracy of the spectrometer was validated by measuring the 4 2,2 23 3,1 ground-state transition and comparing it to previous measurements.
(2c) 06GoMaGuKn [145].Hyperfine structure measurement by the Lamb-dip technique in the millimeter wave and submillimeter wave regions.Frequencies linked to a 10 MHz GPS satellite clock.For the mm-wave absorption spectroscopy the radiation was generated by backward wave oscillators (BWOs).The BWOs were phase stabilized against a 100-GHz RF source, the synthesizers were phase stabilized against a satellite-controlled clock with relative frequency uncertainty of 10 À9 .InSb bolometer detection at T¼4.2 K. Second-harmonic detection at 2-5 kHz.
(2f) 12YuPeDrMa [164].Terahertz absorption spectroscopy and far-infrared Fourier-transform emission spectroscopy were employed to measure new rotational and rovibrational transitions and validate some of the lines of existing datasets.
(2j) 78KaKaKy [70].Pure rotational spectrum of H 2 O measured by a Michelson-type double beam FTS with 50 cm optical path difference.Pressure (Torr) in the different spectral ranges (cm À 1 ): 30-150: 1.4, 159-390: 1.8, and 390-720: 4.0.The authors quote a maximum resolution of 0.018 cm À 1 but with the optical path difference it should be more like 0.02 cm À 1 ; in fact, for a number of plots the authors quote 0.03 cm À 1 resolution.The authors quote accuracies of 0.001 cm À 1 under favorable conditions.Calibration was done using one line from a He-Ne or Ar ion laser.This assumes a linear scale.The authors also use six H 2 O lines from 73FlCa [63], as used by 75ToMa [67], having an accuracy on the order of 0.001 cm À 1 .Calculated wavenumbers for H 2 16 O lines use the energies of 73FlCa and at higher J levels the values reported by 73PuRa [64].For the assigned transitions, J max ¼ 17 and K max a ¼ 11.This study was recalibrated to remove systematic differences, see Table 4.
(2k) 04CoPiVeLa [135].Emission spectra from a RF discharge.The spectra were recorded on an FTS Bruker IFS 120.The wavenumbers were calibrated against data in previous papers.The accuracy of the wavenumbers was originally estimated to be 0:9 Â 10 À3 cm À1 .
(2l) 11DrYuPeGu [162].The measurements were performed in a water discharge cell using cascaded frequency multiplication, see Ref. [217] for details.Water pressures ranged from o 1:3 hPa (cell evacuated) to 400 hPa for the weakest features.One transition, at 2 527 953.387 (200) MHz, a well-known water laser line, was observed in emission.
(2m) 95PaHo [105].Pure rotational lines with J max ¼ 17 and K max a ¼ 11 were measured using a Bruker IFS 120 HR Fourier-transform spectrometer.A White-type cell that gives an absorption path length of 3.2 m was filled with a mixture of H 2 16 O, HD 16 O, and D 2 16 O to a total pressure of 0.50 hPa, and then 0.50 hPa of OCS was added for calibration.The cell, with white polyethylene windows, was at room temperature (295 K).Calibration was carried out using the n 2 band of OCS with values from Ahonen et al. [218].The signal-to-noise ratio in the best part of the spectrum was 100 with a spectral resolution of 0.0025 cm À 1 .The emphasis of the paper was deuterated species and many principal isotopologue lines were in fact saturated.
(2n) 05HoAnAlPi [138].The emphasis of this paper was to transfer the high accuracy of CO 2 and OCS standards to pure rotational lines of H 2  16 O.FTS study with globar or synchrotron sources.Range of J: 0-17, range of K a : 0211.Resolution ranging from 1 Â 10 À3 to 3:5 Â 10 À3 cm À1 .Uncertainties are given for each transition, the range is 1022000 Â 10 À6 cm À1 .(2o) 97PoZoViTe [168].Reports laboratory emission spectra recorded with an FTS and reanalysis of sunspot absorption spectrum reported by Wallace et al. [219,220] and initially analyzed by 97PoZoViTea [221].
(2p) 97PoTeBe [167].Based on data from 96PoBu-GuZh [107], which had measurement problems of an unknown origin that made the wavenumber errors higher than expected in a rather erratic fashion.Treatment of these data agrees with those of 96PoBuGuZh [107].
(2q) 96PoBuGuZh [107].The spectra were recorded on a Bruker IFS 120 HR spectrometer in emission at 1823 K, with a resolution of 0.01 cm À 1 and a pressure of 20 hPa.A calibration factor of 1.000 048 707 68 was used.The dataset contains transitions up to J max ¼ 24 and K max a ¼ 23.This study was recalibrated to remove systematic differences, see Table 4.
(2r) 82KaJoHo [79].65 lines corresponding to 71 transitions.Only positions are given.The goal of the paper was to transfer the high accuracy of the n 1 band of OCS (rms of 3:6 Â 10 À5 cm À1 ) to the pure rotational lines of H 2  16 O.Range of J: 6-14, range of K a : 129.Resolution of the measurements is 0.0045 cm À 1 .The precision of the measurements is high, 7 4 Â 10 À5 cm À1 .This study was recalibrated to remove systematic differences, see Table 4. (2s) 05CoBeCaCo [174].Emission spectrum recorded on a Bruker IFS 120 M FTS in an oxyacetylene torch at atmospheric pressure.The calibration used water lines from 02TeBeZoSh [173] checked using a CO standard.The precision of the measurements is 0.02 cm À 1 .The spectrum was recorded between 500 and 13 000 cm À 1 but only the low-frequency region was analyzed here; other regions were analyzed in subsequent work [176,177].
(2y) 97PoZoTeLo [169].Laboratory emission spectrum calibrated using 91Toth [95] and sunspot data.The paper concentrates entirely on hot bending transitions and only g i v e sd a t af o rt r a n s i t i o n so ft h ef o r mðn þ 1Þn 2 Ànn 2 for n ¼ 1,2,3a n d4 .
(2ff) 85BrTo [84].In the 1100-2200 cm À 1 region, the positions of the 2n 2 and n 1 bands of N 2 O and the n 2 band of H 2 O calibrated previously against the fundamental of CO are readily corrected to achieve absolute accuracies of 0.0001 cm À 1 or better.The 'precision' was a bit better (about 0.00006 cm À 1 ) for lines well separated from other transitions.
(2gg) 97MiTyKeWi [110].The spectra were recorded on a FT-IR Bruker IFS 120 HR spectrometer at room temperature with resolution limited by pressure broadening; the pressure-path length product is up to 8700 mbar.The pressure changed between 0.6 and 29.7 mbar.There are OCS and CO 2 lines in the spectrum.The energy levels involved have J max ¼ 17 and K max a ¼ 11.The uncertainty of the line position determination is estimated to be about 10 À4 cm À1 or better for good isolated lines and is on average one order of magnitude worse for weaker and overlapping lines.
(2ii) 73CaFlGuAm [62].This paper reports lines obtained from FTS measurements of water vapor at low pressure is in the spectral range of 2903-4255 cm À 1 .A 2 m path difference was used with a home-made FTS.For the principal isotopologue of water, values of line positions were obtained for the n 1 , n 3 , and 2n 2 bands as well as the hot band n 2 þ n 3 Àn 2 .The precision on wavenumbers relative to each other was estimated to be 0.0005 cm À 1 , but the absolute accuracy of positions was estimated to be only 0.003 cm À 1 .This study was recalibrated to remove systematic differences, see Table 4.
(2jj) 73PuRa [64].The authors used an old grating spectrometer with lower spectral resolution (0.03 cm À 1 ) than the FTS experiments that took over about this time.73PuRa implemented a ''proto-MARVEL'' technique to obtain the energy levels.The thesis by Pugh, used in this work, was the source of water-vapor lines in this region (2500-5915 cm À 1 ) in the first HITRAN edition of 1973 [224].However, a caveat was issued: ''the observed contamination of the sample by deuterium to give the HDO abundances varying from 10-200 times normal was not reported.''The accuracy of the measurements was estimated to be 70.005cm À 1 .Seven vibrational bands were treated: (0 0 0), (0 2 0), (1 0 0), (0 0 1), (0 3 0), (1 1 0), and (0 1 1).Naturally, the highest J value was fitted for the ground state and for the asymmetric bend mode, J max ¼ 15.The fit was sparse for (0 3 0).Intensity values were also obtained; while the actual number of line intensities measured was probably very large, they are not provided except for wavenumbers of 35 lines that exhibit ''intensity anomalies''.
(2oo) 80CaFlMa [74].The region studied was 5930-6440 cm À 1 , using an FTS with a resolution of 0.070 cm À 1 .Positions and intensities for 78 water lines were reported; 31 of them were assigned to the weak third overtone of water, 4n 2 (with J max ¼ 10 and K max a ¼ 2 for this band and J max ¼ 14 and K max a ¼ 9 for the n 2 þ n 3 band).The results of this work were incorporated into HITRAN, although there was some criticism of the work by the late Prof. William Benedict concerning the analysis of the intensities for this band.This study was recalibrated to remove systematic differences, see Table 4.
(2qq) 75ToMa [67].Measurement in the 1:3321:45 mm region using a 1.8 m Jarrel-Ash grating spectrometer.Data are reported for five vibrational states, (0 1 0), (0 2 1), (2 0 0), (0 0 2), and (1 2 0), extracted using the groundstate energies of 73FlCa [63].FTIR with a resolution of 0.005 cm À 1 between 2930 and 4255 cm À 1 .Calibration was done using the super-radiant line of Xe at 2850.6396 cm À 1 .These lines have uncertainties of 0.001 cm À 1 .S a m p l e pressure 1-9 mm Hg, path 8-32 m, at T¼295 K. Calibrations were made with 2n 1 þ n 3 and n 1 þ 2n 2 þ n 3 bands of N 2 O by observing the N 2 O lines in second order and the H 2 O lines in third order of the grating.Further calibration was done by observing the 2:9 mmH 2 O lines in first and second order and the 1:4 mmH 2 O lines in second and fourth order.Accuracies: 0.007 cm À 1 for unblended lines, 0.01 and 0.015 cm À 1 for blended, resolved lines and very weak, observed, and unblended absorptions.The uncertainties are not given for each line and had to be assigned by inspection.Spectral resolution: 0.07 cm À 1 in the 1:4 mm region.J max ¼ 15 and K max a ¼ 7. (2rr) 11MiKaWaCa [163].Based on an absorption spectrum recorded using water at natural isotopic abundance.
(2tt) 88MaChFlCa [91].The authors used the Kitt Peak FTIR to measure water vapor spectra in the 7900-9500 cm À 1 region.The measurements were made at 300 K under the following conditions: It appears that some of these spectra are the same as those used in 86MaChCaFl [88].However, for 18 O and 17 O the resolution given above is different.S/N was between 500 and 2300.There is no discussion about the frequency calibration of the spectra.441 rotational levels with J max ¼ 14 and K a max ¼8 are reported.For each energy state, the tables report the uncertainty as 1 standard deviation but systematic error is not accounted for.
(2vv) 05ToNaZoSh [142].Spectra were recorded in a Bruker IFS 120M spectrometer and calibrated using the I 2 visible spectrum.The resolution is 0.03-0.06cm À 1 .The calibrated line positions were shown to agree with previous studies and with HITRAN to better than 5 Â 10 À4 cm À1 .422 lines were relabeled.
(2ww) 08ToTe [155].These authors undertook a comprehensive reanalysis of the FTS data of 02ScLeCaBr [128], some of whose results were reported by 02ToTeBrCa [130], who only reported long pathlength data.08ToTe results involved a simultaneous fit to data reported at several path lengths.The fitting method employed constrained the line positions of transitions with the same upper energy levels.The results were calibrated using HITRAN (i.e., 02BrToDu [126], 03MeJeHeVa [131], and 02CoFaCaCl [129]).60 lines have been relabeled.This study was recalibrated to remove systematic differences, see Table 4.
(2fff) 08CaMiLi [157].30 lines were relabeled.(2ggg) 99CaJeVaBe [116].Calibration was performed with the I 2 line positions with the values of Gerstenkorn and Luc [226].The raw spectra were shifted by þ0.002 353 cm À 1 , corrected for the refractive index of air (the spectrometer was not evacuated), and then multiplied by a factor of 1.000 001 7 8 .N ok n o w np r o b l e m sw i t ht h el i n e s ,a l t h o u g has m a l l , further overall calibration factor may be needed if the I 2 calibration described in the paper was not quite right.494 lines were relabeled and 20 deleted.This study was recalibrated to remove systematic differences, see Table 4.
(2hhh) 98PoZoViTe [115].Theoretical analysis of measurement is taken from Mandin et al. [87] who used an FTS built by Brault.Originally, the uncertainty in line positions was estimated to vary between 0.002 to 0.015 cm À 1 .110 lines were relabeled.
(2iii) 05KaMaNaCa [143].The goal of this paper was the detection of very weak lines in a region where atmospheric detection of water dimer was claimed.I 2 lines were used as references for the calibration.Range of J: 2-15, range of K a : 028.No uncertainty values are given in the paper for the lines.High sensitivity, a min ¼ 3 Â 10 À10 cm À1 .
Five lines were relabeled.
(2jjj) 08GrMaZoSh [33].This paper reports two photon spectra of H 2 16 O performed under conditions whereby the water can collisionally relax between the first and second photon absorption.The work also contains lines measured by 07MaMuZoSh [32] who did not allow for collisional relaxation.Both works probe energy levels between 26 000 and 34 200 cm À 1 .Only data on transitions involving the second photon, which is in the frequency range 13 531-17 448 cm À 1 , were included since the route to the final state involves a non-radiative process.(2kkk) 11BeMiCa [161] Four lines were relabeled.(2lll) 85CaFlMaCh [85].FTIR measurements at Kitt Peak Solar Observatory and a grating and a Bomem FT with a 33 m path.Measurement conditions were as follows: The signal to noise ratio was greater than 500:1.Part of the atmospheric path is in the laboratory so the path area was purged with dry N 2 to reduce the effect of water lines in the lab.Resolution is 23240 Â 10 À3 cm À1 ,s p e c t r am e a s u r e di n Ottawa had 0.04 cm À 1 unapodized resolution.Calibration for the wavenumber scale was a problem, solved by recording spectra in regions.The first spectrum was recorded between 4000 and 9000 cm À 1 with N 2 Op r e s e n ta st h e calibration standard.Higher wavenumber spectra had overlap with the lower so that lines present in two spectra allowed the upper wavenumber spectra to be calibrated.Estimated errors in line position was 2 Â 10 À3 cm À1 for intense well-isolated lines down to 25 Â 10 À3 cm À1 for the weakest lines.The authors measured 1174 line positions from which 539 vibrational-rotational energies were deduced.J max ¼ 11 and K max a ¼ 7.For each energy state, the tables report the uncertainty as 1 standard deviation.There is no accounting for systematic error in the experimental uncertainties.The following three lines were relabeled: 8436.5910.005 1 1 1 14 1 14 0 0 0 15 1 15 75FlCaNaCh.00228;8818.77460.005 2 1 0 8 3 5 0 0 0 8 2 6 75FlCaNaCh.00745;8892.06880.020 1 1 1 10 3 7 0 0 0 10 3 8 75FlCaNaCh.00862(2mmm) 00ZoBePoTe [120].This paper presents an analysis of the high-frequency data originally measured with a Bruker IFS 120 M FTS by 99CaJeVaBe [116] and uses the same calibration procedure.The original estimated uncertainty in the line positions is 0.004 cm À 1 .However, during the MARVEL analysis this proved to be too optimistic and had to be increased.
(2nnn) 05DuGhZoTo [144].There are 43 reported transitions extending into the near ultraviolet; this is the highest frequency one-photon vibration-rotation spectrum of water available.No information is given about calibration.

Variational validation
As an independent validation of the experimental transition wavenumbers and the derived MARVEL energy levels and their labels, systematic and mostly automated comparisons were made with the results of state-of-theart variational nuclear motion computations.For this comparison, the so-called BT2 linelist [194] was principally used; this linelist was computed using a spectroscopically determined PES [43],anab initio DMS [47], and the DVR3D [227] nuclear motion program suite.
Those measured transitions involving a MARVEL energy level which did not have a matching variational counterpart within 1.0 cm À 1 , with proper rotational parity (Table 1), were investigated individually and the MARVEL process was repeated until all MARVEL levels had variational counterparts within the chosen cut-off value.There were only a couple of cases where the deviation between the BT2 and MARVEL energy levels was larger than 1.0 cm À 1 .F o r transitions removed at this stage, see the appropriate comments in Section 2.7 and the supplementary data.
Any MARVEL rotational-vibrational energy level obtained as part of this work which differed by more than 0.5 cm À 1 from its variational counterpart was subject to further scrutiny.When variational results are used for validation, we can rely on a well-known feature of such calculations: the smooth and slow variation of obs À calc residuals for the energy levels of a particular vibrational state having the same K a and increasing J values [228].Thelongest obs À calc sequences could be investigated for the hot spectra, where transitions involving J as high as 42 have been detected.The obs À calc residues for levels with a given K a but different K c diverge as J increases, hindering the assignment of the dense observed spectrum without detailed consideration of near degeneracies.The obs À calc trends for the highly excited vibrational states are not particularly smooth as they can be strongly perturbed by nearby states.Cases with erratic obs À calc trends were additionally checked to see whether the calculated energy level set includes the resonance partner, whose energy level has to be close to the level under investigation and whose quantum numbers should satisfy the conventional Coriolis-, Fermi-, or Darling-Dennison-type resonance interaction rules, or some combination of them.Where necessary, labels were changed assuming similar increases in rotational energies as a function of J and K a for similar vibrational states with the same v 2 quantum number as well as quasi-degeneracy of rotational levels with K a close to J or K a equal to 0 or 1.
At the end of a MARVEL analysis cycle, the MARVEL energy levels obtained were distributed into bins having different J values and parities (thus we used only exact quantum numbers for the matching).These bins were checked against ones derived variationally from the best possible PES [46], which was also refined during this study taking into account the best MARVEL energy levels up to J¼15.For each J up to J¼15, MARVEL energy levels which deviated by more than 3s from their variational counterparts were checked individually.Transitions which proved to be inconsistent with this type of information were removed or their uncertainties were adjusted to reflect the knowledge gained.As the results of Table 5 show, at the end all MARVEL energy levels up to J¼15 are reproduced by the variational energy levels with an accuracy better than 0.05 cm À 1 .Of course, as J increases the average deviation between the variational and the MARVEL levels grows.The success of these comparisons reflect simultaneously the high quality of the PES employed for the nuclear motion computations and of the MARVEL energy levels derived.

Post-MARVEL validation
For a number of transitions which proved to be outliers by combination difference relations, the experimental uncertainties were increased manually (see Section 2.7).This extra validation and the subsequent adjustment were done when the energy of an upper rotationalvibrational state deviated far more from the corresponding mean value established by the lower MARVEL energy levels plus the transition wavenumbers than the stated experimental uncertainty.Using the MARVEL protocol can result in similar adjustments automatically if the error associated with a transition is an outlier and all the data have similar accuracy.However, a problem arises if an erroneously small experimental uncertainty is attached to what is actually a much less accurate experimental datum, the same level is involved in several measurements, and other transitions in the combination difference relations, though consistent, have formally much larger uncertainties.In this case the MARVEL energy level will be determined by the formally most accurate transition which, in fact, represents an outlier.We attempted to check carefully all such cases but it is extremely hard to ensure that all problematic cases were properly identified and treated.
At this stage we also checked whether the rovibrational MARVEL energy lies below the corresponding vibrational band origin (VBO).If this happened, the corresponding labeling was investigated and adjusted to comply with the majority of the data.At the end of this process we were left with a list of 182 156 (68 027 para and 114 129 ortho) validated transitions.The 2511 transitions deleted from the initial list may be incorrectly measured or assigned; however, they may also be correct but have a large uncertainty and are thus dropped when higher accuracy data are available from other measurements.In the end only 60 transitions proved to be orphans or part of FSNs.From these transitions we derive a final set of 18 486 energy levels.One can observe that, due to the large number of measured transitions, the list of VBOs of the main isotopologue is much more complete than in Parts I and II for the minor isotopologues of water.If a polyad number P ¼ 2v 1 þv 2 þ 2v 3 is defined, all VBOs are determined experimentally up to P ¼5.The first three missing VBOs are (1 4 0), (0 7 0), and (1 5 0) at about 10 000 cm À 1 ; otherwise the coverage is complete up to P¼7.There are eight measured VBOs out of 10 for P¼7.

MARVEL energy levels
There are very few VBOs below about 18 000 cm À 1 for which rotational-vibrational levels have not been determined at all.The first VBO is (0 0 5) (the highest-lying P¼10 VBO) which has no measured and assigned rovibrational levels.For P ¼11 and beyond there are more and more VBOs which have no measured rovibrational lines.
A comparison can be made between the original set of observed transitions and those calculated from the ''experimental'' energy levels determined by MARVEL; this is presented in Fig. 1.About 36.0%and 83.7% of all transitions are reproduced within 0.001 and 0.01 cm À 1 , respectively.
A comparison of the experimental (MARVEL) and variational (BT2 [194]) energy-level values is given in Fig. 2. Clear trends are visible in Fig. 2 showing the systematic nature of the errors of most of the computed rotational-vibrational transitions.
The set of MARVEL energy levels derived from processing the validated observed transitions can be used to predict a large number of rovibrational transitions often with positions at a level of experimental accuracy.These line positions were augmented with variational, onephoton absorption intensities corresponding to T¼296 K and obtained from BT2 [194].The resulting list can be considered as one of the key results of the present investigation.The total number of predicted transitions with intensities larger than 1:0 Â 10 À28 and 1:0 Â 10 À32 cm molecule À 1 is 66 582 and 211 489, respectively.Observed, MARVEL predicted, and variational O.It is important to point out that in the low end of the spectrum, below about 10 000 cm À 1 , there is a huge number of ''observed'' lines with predicted intensities down to 1 Â 10 À30 cm molecule À1 .
Such a complete coverage by the ''observed'' transitions is due to the existence of a large number of energy levels derived from the analysis of hot emission spectra, and these transitions have never been measured directly in absorption.
The accuracy and precision of the MARVEL energy levels determined in this study depend upon quite a large number of factors.Perhaps most important among these is the accuracy of the measured transitions.Experimental uncertainties provided in the original sources are often overly optimistic, by an order of magnitude for some of the lines (weak, blended, etc.).Consequently, in a large number of cases the published uncertainties had to be increased substantially during the course of the MARVEL analyses, either before or during the robust reweighting procedure.Since our analyses attempted to utilize all the experimental information available for the Task Group in the form of results published in scientific journals, there are many energy levels which are involved in multiple transitions measured by several experimental groups utilizing different spectrometers, different setups, and different experimental conditions.We consider an energy level particularly well determined, i.e., accurate and precise, if it is involved in more than 12 transitions and there are at least five independent experimental investigations which determined this energy level.These energy levels are graded as A þ , see supplementary data.This means that the value of the energy level, within the stated uncertainty, and the label of the energy level are completely dependable.These energy levels should be particularly useful for future studies and in modeling work.Energy levels which are involved in more than 10 transitions and are also part of transitions published in at least four independent experimental investigations are graded as A À .These energy levels and their labels should still be considered dependable.Energy levels which are involved in at least eight transitions and are also part of transitions published in at least three independent experimental investigations are graded as B þ .Energy levels which are involved in at least six transitions and are also part of transitions published in at least two independent experimental investigations are graded as B À .All other energy levels are graded as C.This means that there are energy levels with the worst grade, C, which may be accurately known.Nevertheless, it is expected that since many of the grade C energy levels come from a single source, their uncertainty may not be dependable.Our recommendation for transitions is that they should be graded using the grade of the lower graded energy level involved.

Status of highly accurate transitions
The agreement between the MARVEL predicted and the experimental pure rotational transitions improved slightly by the recalibration of the experimental transitions.Another source of inaccuracy when combining measured line positions from several sources is due to pressure effects, not corrected for in this study.In order to measure the weakest spectral features in several experiments the pressure had to be increased beyond 20 hPa (for details, see Table 2).Analysis of some of the experiments have also used different line profiles and line profile parameters.This could result in small shifts in the line centers.Furthermore, since both the ortho and para components of the SN of H 2 16 Oc o n t a i na n extremely large number of cycles of widely varying size, even the inclusion of a few seemingly inconsequential transitions with incorrect uncertainties can distort the value and the uncertainty of MARVEL energy levels which would be determined accurately and precisely by a subset of the experiments.The effect of all th e s ef a c t o r si sr e fl e c t e di n MARVEL uncertainties larger than otherwise expected for several ''highly accurate'' rotational-vibrational levels (see Figs. 4 and 5).
The MARVEL uncertainties of the pure rotational levels are uniformly larger, perhaps by an order of magnitude, than is usual for lines coming from microwave determinations.To show that this is due to the (inappropriate) uncertainties of the upper states we performed a MARVEL analysis of the pure rotational states.MARVEL can reproduce the microwave uncertainties very nicely, down to the level of the experimental uncertainties, as also observed before [26].
As seen in Table 7, MARVEL can reproduce recent accurate THz measurements [162] rather well, usually better than the stated uncertainty of the MARVEL transitions would suggest.This is a pleasing result as the energy levels participating in the THz transitions are involved in a large number of other transitions of lower accuracy which could distort their prediction.Nevertheless, for many observed transitions in the THz region the difference between the MARVEL and the 11DrYuPeGu [162] transitions is outside the original experimental uncertainty limits.Reproduction of other measurements [104,135,146] of the same transitions is also excellent as the data collected in Table 7 demonstrate.

Comparison with previous sets of energy levels
There are several labeled energy-level lists available for H 2 16 O.Those which were available to us in an electronic form were used for a comparison with MARVEL energy levels.Note that no attempt was made to relabel the energy levels of the original sources.This means that in a number of cases where the present study relabeled the transitions a discrepancy is kept between the two data sources.For example, most of the more than 500 mismatches between the MARVEL and Ref. [28] energy levels are due to this. a The VBOs are reported in the order of the polyad number P defined as All the VBOs are listed up to P¼ 12 but only selected ones holding measured rovibrational states beyond it.The uncertainties (Unc.) are given in units of 10 À 6 cm À 1 .For VBOs not determined by the available experimental data, approximate computed VBOs, based on J 4 0 rotational levels as reported in Ref. [36], are given in brackets.These values should only be used for guidance about the VBOs, although their accuracy is expected to be better than 0.1 cm À 1 .No uncertainties are given for these VBOs.For completeness, some of the lower-P VBOs which are involved in none of the observed rovibrational transitions are also given.The corresponding labels are printed in italics for guidance.The VBOs are ordered according to their formal labels within a given P, which corresponds to their energy order up to P ¼ 23 and thus they appear in increasing energy order.b The value of the vibrational ground state was fixed to zero with zero uncertainty.
One of the most accurate effective-Hamiltonian-based energy-level lists available for H 2 16 Oi sp u b l i s h e di n Ref. [135].The eight panels of Fig. 6 present a comparison between the accurate, EH-based energy levels of Ref. [135] and their MARVEL counterparts for the first eight VBOs of H 2

16
O. The agreement is outstanding for (0 0 0): there is no deviation larger than 0.005 cm À 1 .The agreement is somewhat less perfect for a few rovibrational levels of the excited vibrational states.Nevertheless, no systematic deviations can be observed.It is possible that the energy levels of Ref. [135] are still more accurate than the present MARVEL levels but this would require further studies based on a new and larger set of available experimental results to derive new parameters for the effective Hamiltonian and the inclusion of extensive, more accurate new measurements into the MARVEL analysis.
Comparison between the present MARVEL energy levels and those of a previous large set of ''measured'' energy levels compiled in Ref. [28] are shown in Fig. 7.The agreement is excellent up to about 5000 cm À 1 .Beyond this, the scatter is considerably larger but still lower than the accuracy of many of the underlying rovibrational transitions.
Further figures which are given in the supplementary data show the agreement between the present MARVEL and the literature energy levels [127,130,132,135-137, 141,143,144,147,163,174-176] is excellent in almost all cases.There are a number of problematic energy levels, which we are not able to reproduce within 0.04 cm À 1 .I n nearly all cases these differences are caused by changes in labeling used to describe particular energy levels between the original article and the present work.We have not attempted to relabel published energy level data.The agreement seen on all figures confirms the high accuracy of the previous determinations of the energy levels and that of the present MARVEL analysis.
Finally, the present MARVEL results are compared to those of Toth [229] in Fig. 8.As observed repeatedly, the agreement is impressive below about 5000 cm À 1 .Above this, the agreement becomes somewhat poorer due probably at least partially to the inclusion of emission results in the present MARVEL analysis.Comparisons of the MARVEL energy levels with other (partial) compilations of energy levels are given as a dynamic figure in the supplementary data.Fig. 2. Differences between the present MARVEL energy levels and their counterparts present in the BT2 linelist [194].second panel-MARVEL predicted but omitting the above observed line; third panel-sum of the first two; and bottom panel-all lines predicted by variational computations (BT2 [194]).The weakest measured intensities come from emission spectra of hot water which are difficult to obtain directly in absorption.

A comparison with HITRAN and HITEMP
The relevant features of the HITRAN database [225,235] are summarized in the original publication and in Part I; thus, they are not repeated here in detail, just a few remarks are made.
For a discussion of comparisons of the results of the present IUPAC-sponsored work (transition wavenumbers and energy levels) with data archived in HITRAN, it is worth recalling the structure of and quantities in the current edition of the HITRAN database [225].Table 2 of Ref. [225] presents the parameters (fields) that are contained in each transition (record) of the HITRAN linelist.The structure is in a text-file format of fixed-length records.The parameters contained in each transition are those that have been deemed necessary for input for calculating high-resolution absorption or radiance spectra through room-temperature gaseous media.There are 36 550 assigned transitions of H 2  16 O (and an additional 882 unassigned transitions for this isotopologue) in the current edition of the HITRAN database [225] from 0 to 25 300 cm À 1 .The values for the line positions of different transitions originate from a wide variety of sources which are documented by means of a special reference index.Unlike the MARVEL database, HITRAN contains both experimental and calculated values and also provides only one chosen value for each transition.Table 8 provides statistics on the number of transitions in the MARVEL and HITRAN databases and how many of them are unique for the former set.One can see that there are 4599 assigned transitions in HITRAN that are not present in the MARVEL database or, in other words, have never been measured directly (or at least not published in the refereed works).This is not surprising because HITRAN contains data that are often calculated using (1) spectroscopic constants that are derived from fitting experimental spectra (for instance data in the 0-500 cm À 1 region originate from Ref. [124]), and (2) energy differences between empirically determined levels.For instance, there are 4976 transitions in HITRAN that originate from the SISAM database [229] which are calculated by the second method.Unlike the case of some other water isotopologues, there are no line positions in HITRAN for the parent isotopologue that are computed variationally using ab initio PESs.One of the most valuable results of this IUPAC effort is that by using validated empirical energy levels, in the present case MARVEL ones, one can generate an extensive database of line positions of all allowed transitions between such levels.Of course, the number of such transitions is extremely large, and although they are allowed, some of them will be very weak even under extreme thermodynamic conditions and are unlikely to be observed experimentally even in emission.Table 9 compares the database created from MARVEL energy-level differences with HITRAN.In general there is a very good agreement, although some differences exist and they need to be studied carefully.One should note that there is a problem of correspondence between quantum assignments of energy levels given in the HITRAN and MARVEL lists.This is due to the fact that only the rotational quantum number J and symmetry (cf.Table 1) can be unambiguously identified using theoretical approaches.The different experimental works that contributed to HITRAN have used techniques and conventions to aid in the assignments of the observed transitions, which can result in labeling ambiguities.Therefore, although Table 9 shows that there are 1068 transitions that differ by more than 0.1 cm À 1 , a few hundred of such transitions are in a category of different assignments and do not necessarily indicate problems in the HITRAN line positions.It is planned to revisit the labeling of energy levels in a future release of HITRAN; this issue is of particular importance since the algorithm used by HITRAN to generate pressurebroadening parameters is based on the quantum numbers [231].
It is worth noting that the database of transitions generated from MARVEL energy levels can prove extremely useful in updating the HITEMP database [29] and the ab initio line positions adopted from the BT2 line list [194].In fact, when the HITEMP database was put together, a similar effort had been carried out but the database of transitions generated from experimental energy levels was significantly less complete (by more than a million transitions) and had not undergone a rigorous validation procedure.That being said, there are 114 209 395 transitions of H 2 16 O in the current edition of the HITEMP database [29], and the majority of line positions will still remain to be of ab initio origin.

Conclusions
Among many other applications of such data, an extreme quantity of high-quality molecular data are needed to understand properties of spectroscopic measurements related to different stars and the atmospheres of planets and exoplanets [232].A tt h es a m e time, non-thermodynamic equilibrium spectroscopic sources, such as water masers [9], are sensitive to thermally strongly suppressed transitions and probe a few, selected, high-lying energy levels.Such applications require knowledge of precise positions and often times intensities and line shapes to extract information such as chemical composition and pressure-temperature profiles.Thus, studies of the complete spectra of the water isotopologues are of prime importance.Furthermore, the high-resolution rovibrational spectra of the isotopologues of the water molecule form a fertile test ground for different experimental and theoretical approaches, like the present IUPAC effort, yielding the required information.
While the ambitious task of the IUPAC TG partially responsible for this work is to obtain a complete linelist for all isotopologues of water, a first step is to determine energy levels and line positions.This paper provides a dependable and carefully validated set of energy levels and transition wavenumbers, all with dependable and self-consistent uncertainties and labels, for the parent isotopologue, H 2 16 O.The uncertainties produced by this work, due to the algorithm used, are usually larger for the energy levels than the underlying errors.This is something that should be investigated in future work.O.The uncertainties, given in parentheses, are in kHz.
An important aspect of the work presented here relies on recent advances in the ab initio calculation of dipole moment surfaces and hence accurate transition intensities [49,233].Transition intensities computed using a high quality ab initio dipole moment surface have been proven to be reliable and competitive with the best laboratory measurements in nearly all cases, although possible issues still remain [160].This situation is certainly not true for the ab initio calculation of transition frequencies Fig. 6.Differences between MARVEL and empirical, effective-Hamiltonian-based energy levels from 04CoPiVeLa [135] for the eight lowest-energy vibrational states of H 2 [187].This means that the combination of MARVEL line positions and ab initio line intensities can be used to give highly accurate spectroscopic parameters.A first study using this approach has recently been completed for H 2 18 O and H 2 17 O [234]; the present data provides the starting point for a similar study on the main isotopologue, H 2 16 O, for which there remains a number of issues obtaining reliable spectroscopic data for purposes such as atmospheric monitoring [235].The distributed information system W@DIS [236,237], one of the intended end products of the effort of this IUPAC TG, can be accessed via http://wadis.saga.iao.ru/ and contains the data forming the basis of this paper.O transition found in HITRAN [225] and used in the present compilation for 298 K.

Table 9
Comparison of H 2 16 O MARVEL-based one-photon absorption transition data and those found in HITRAN [230].
Total number of transitions generated from MARVEL database of energy levels 5 046 272 Concordant transitions with HITRAN a 36 369 Transitions differing by 4 0:001 cm À1  10 772 Transitions differing by 4 0:01 cm À1 2021 Transitions differing by 4 0:1cm À1 1068 HITRAN transitions absent in database of transitions generated from MARVEL database of energy levels 181 a Unique transitions which are present both in HITRAN [225] and in the present validated IUPAC database.

H 2 16 O
transitions are shown in the panels of Fig. 3.This figure is especially important for future experimental studies of the high-resolution spectra of H 2 16

Fig. 8 .
Fig.8.Differences between the present MARVEL and a previous large set of ''mixed'' energy levels compiled in Ref.[229] (SISAM) for H 216 O.

a
Unique transitions which are present both in HITRAN and in the present validated IUPAC database.bThese transitions within HITRAN most likely correspond to computed and not to measured results.

Table 2
Data sources and their characteristics for H 2

Table 2
, the vibrational band origins (VBOs) are listed in order of increasing energy.a

Table 3 (
RRD labels have been determined only for states with J less than 26.VBO ¼vibrational band origin.J max gives the maximum J value for rovibrational MARVEL states determined on the particular VBO.J v is the maximum J value for which all labels have been validated. a

Table 4
Recalibration factors determined during the present study for selected data sources reporting H 2 16 O transitions.See text for a discussion of the methods used.

Table 6 contains
MARVEL vibrational band origins (VBO) for H 2 16 O.For each VBO, Table 6 also gives the number of rovibrational energy levels validated within this work and based on the original 184 667 transitions.

Table 5
[46]dard deviation, in cm À 1 , of the H 216O MARVEL energy levels with respect to energies obtained from variational nuclear motion computations executed with a PES based on Ref.[46]and improved as part of this study and an exact kinetic energy operator.

Table 6
MARVEL vibrational band origins (VBO) for H 216O, with normal-mode (v 1 v 2 v 3 ) labels, MARVEL uncertainties, and the number of validated rotational-vibrational levels (RL) associated with the vibrational levels in the present database.a

Table 7
Reproduction of experimental transition data in the THz region, given in MHz, by MARVEL energy levels of A þ quality for H 2 16

Table 8
Comparison of H 2 16