The W2024 database of the water isotopologue ${\mathrm{H}}_2^{16}\mathrm{O}$

Abstract

The rovibrational spectrum of the water molecule is the crown jewel of high-resolution molecular spectroscopy. While its significance in numerous scientific and engineering applications and the challenges behind its interpretation have been well known, the extensive experimental analysis performed for this molecule, from the microwave to the ultraviolet, is admirable. To determine empirical energy levels for ${\mathbf{H}}_{\mathbf{2}}^{\mathbf{16}}\mathbf{O}$, this study utilizes an improved version of the MARVEL (Measured Active Rotational-Vibrational Energy Levels) scheme, which now takes into account multiplet constraints and first-principles energy-level splittings. This analysis delivers 19027 empirical energy values, with individual uncertainties and confidence intervals, utilizing 309 290 transition wavenumbers collected from 189 (mostly experimental) data sources. Relying on these empirical, as well as some computed, energies and first-principles intensities, an extensive composite line list, named CW2024, has been assembled. The CW2024 dataset is compared to lines in the canonical HITRAN 2020 spectroscopic database, providing guidance for future experimental investigations.

Background & Summary

Over the last half of a century, collecting accurate, line-by-line spectroscopic data for isotopologues of the water molecule has been a major research activity in a large number of spectroscopic laboratories (see, for instance, Refs. ^1–10 and references cited therein). An important contribution toward the detailed understanding of high-resolution spectra recorded for several water isotopologues, beyond selecting particular data for a particular database, started two decades ago, when a Task Group (TG) was set up by the International Union of Pure and Applied Chemistry (IUPAC) on “A Database of Water Transitions from Experiment and Theory” (Project No. 2004-035-1-100). This TG, formed by experimental and computational spectroscopists, published validated sets of measured rovibrational transitions and empirical energy levels on nine water isotopologues, ${\mathrm{H}}_2^x\mathrm{O}$^1–3, HD^xO², and ${\mathrm{D}}_2^x\mathrm{O}$⁴ (x = 16, 17, 18).

A significant update of the IUPAC TG water data⁵ was published by four of the five authors of this paper in 2020, in the form of the W2020 database^7,8, for the three ${\mathrm{H}}_2^x\mathrm{O}$ species. During the development of the W2020 datasets^7,8, the spectroscopic data of the three isotopologues were considered jointly⁸, allowing improvements to be made for the individual datasets. The W2020-${\mathrm{H}}_2^x\mathrm{O}$ line lists were successfully employed in the latest edition of HITRAN¹⁰, the canonical source of line-by-line spectroscopic information for species of atmoshperic interest, representing about 85% of the  ≈  233 000 lines with complete assignment in the HITRAN-${\mathrm{H}}_2^x\mathrm{O}$ catalogs.

Accurate, high-resolution spectroscopic information on various water isotopologues is required by numerous complex applications, including climate-change and atmospheric research, astronomy, combustion chemistry, metrology, planetary science, and remote sensing^6,11–13, all with vastly different environments. The experimental studies of water spectra have been aided by the development of high-resolution and ultrahigh-precision techniques, such as cavity ring-down spectroscopy (CRDS)¹⁴ and noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS)^15–22. Theoretical interpretation of complex (ultra)high-resolution spectra also had to be improved. With the arrival of the fourth age of quantum chemistry²³ came the ability to compute nearly complete line lists for molecules, like those constructed under the aegis of the ExoMol project^24–27. Novel algorithms have also been devised which can cope with experimental data of vastly different accuracy^23,28,29.

A notable theoretical advancement in high-resolution spectroscopy was the introduction of the concept of spectroscopic networks (SN)^30–34. SNs form the basis of the MARVEL (Measured Active Rotational-Vibrational Energy Levels) procedure^{30–32,34–37}, a global spectrum analysis tool^{7,33,34,38–40}. MARVEL inverts the information contained in experimental line positions and delivers empirical energy levels⁴¹ with individual uncertainties. MARVEL has been used to study the spectra of several diatomic^42–47, triatomic^48–52, tetratomic^53–55, and larger^56,57 species.

A number of developments since the publication of the extensive W2020-${\mathrm{H}}_2^{16}\mathrm{O}$ dataset have made its reexamination desirable. Most importantly, results from carefully designed precision-spectroscopy experiments have become available for four water isotopologues^29,58–63. These studies, in particular, yielded empirical energies, accurate to a few kHz, for a large number of lower states in the experimental SN of ${\mathrm{H}}_2^{16}\mathrm{O}$. Further new experimental studies have also appeared^64–69. In particular, experimentalists published a number of measured lines, with a typical uncertainty of 10⁻³–10⁻⁵ cm^{−164,66,68– 70}, challenging certain database entries of the W2020-${\mathrm{H}}_2^x\mathrm{O}$ lists. Avoiding the criticism directed towards a subset of empirical W2020-${\mathrm{H}}_2^x\mathrm{O}$ energy levels⁶⁶, some already refuted in ref. ⁶⁵, requires further improvements on how the experimental information is handled during a MARVEL-type analysis.

The research behind in this paper focused on the (a) refinement of the W2020-${\mathrm{H}}_2^{16}\mathrm{O}$ database via an improved MARVEL methodology, leading to the W2024 dataset, and (b) construction of a large composite line list, called CW2024, for the ${\mathrm{H}}_2^{16}\mathrm{O}$ molecule. The (C)W2024 datasets are compared to HITRAN 2020¹⁰, to expedite the inclusion of the (C)W2024 data in spectroscopic information systems.

Methods

Improved MARVEL methodology

During our studies devoted to MARVEL-based analyses of high-resolution rovibronic spectra of small, usually atmospherically and astronomically relevant molecules^42–57, novel aspects and analysis tools have constantly been introduced. These became essential features in later versions of the MARVEL approach and they are described in a number of publications^{7,33,34,38–40}. Nevertheless, the ever-expanding spectroscopic information available for ${\mathrm{H}}_2^{16}\mathrm{O}$ made it necessary to further improve our MARVEL-based analysis technique, as outlined below.

In what follows, the ${\mathrm{H}}_2^{16}\mathrm{O}$ energy levels are labelled as $\left(v_1{v}_2{v}_3\right)J_{K_a,K_c}$, whereby v₁, v₂, and v₃ are the normal-mode quantum numbers of the symmetric stretch, bend, and antisymmetric stretch motions, respectively, J is the overall rotational quantum number, while K_a and K_c symbolize the conventional prolate- and oblate-top rotational quantum numbers, respectively. As usual, $\left(v_1^{\prime }{v}_2^{\prime }{v}_3^{\prime }\right)J_{K_a^{\prime },K_c^{\prime }}^{\prime }\leftarrow \left(v_1^{″}{v}_2^{″}{v}_3^{″}\right)J_{K_a^{″},K_c^{″}}^{″}$ denotes a rovibrational transition, where ′ and ″ signify the upper and lower states, respectively⁷¹.

Multiplet constraints

Under favorable circumstances, a spectral line representing a transition between two states is well separated from all neighboring lines, yielding a unique position for it. If two or more transitions are closer to each other than can be resolved by a particular experiment, the lines form an unresolved multiplet. In such cases, (a) the spectral line shape might become distorted, (b) the observed intensity corresponds to the sum of intensities of the individual transitions, and (c) the measured position will be an intensity-weighted average of the unknown individual positions. How the treatment of these multiplets was introduced to MARVEL is described next.

Let the wavenumber of the ith line in the dataset, σ_i, be represented with the following expression:

$${\sigma }_i\approx S_i\equiv \sum _{j=1}^{N_{\mathrm{T}}}{u}_{ij}{s}_j,$$ 1

whereby N_T is the number of transitions within the database, s_i means the exact (unknown) position of the ith transition, and the u_ij entries are the relative weights satisfying $u_{11}+u_{12}+\dots +u_{1N_{\mathrm{T}}}=1$. When the (i, j) line pair is part of the same multiplet, u_ij will be the relative intensity of the jth line in this multiplet; otherwise, u_ij = 0. For example, if (1, 2) means an unresolved (ortho, para) doublet of ${\mathrm{H}}_2^{16}\mathrm{O}$, then σ₁ = σ₂ ≈ 0.75s₁ + 0.25s₂. Note that equation (1) holds for an isolated line, as well, then u_ii = 1 and u_ij = 0 for all j ≠ i.

According to quantum mechanics, the s_j wavenumber of a transition is subject to the Ritz principle⁷²,

$$s_j=E_{\mathrm{up}\left(j\right)}-E_{\mathrm{low}\left(j\right)},$$ 2

where up(j) and low(j) symbolize the indices of the upper and lower states of the jth line, respectively, and E_k is the (unknown) energy value of the kth quantum state within the transition dataset. Combining equation (1) and equation (2), the following least-squares objective function can be prescribed for MARVEL:

$$\mathit{\Omega }\left(\mathbf{E}\right)=\sum _{i=1}^{N_{\mathrm{T}}}{w}_i{\left[{\sigma }_i-S_i\left(\mathbf{E}\right)\right]}^2,$$ 3

whereby w_i is the statistical (MARVEL) weight of the ith transition and E is the vector of unknown energy values (variables) in the S_i(E) ≡ S_i sums. If an $\overline{\mathbf{E}}$ vector minimizes the (quadratic) objective function Ω(E), its entries are called empirical (MARVEL) energies [for details on how to calculate these MARVEL energies, see Supplementary Information (A)].

A drawback of applying multiplet constraints is that they reduce the number of statistical degrees of freedom, n_DOF, in the database. In effective Hamiltonian (EH) fits, where such constraints are often employed, e.g., within the SPFIT code⁷³, this is not a problem, as EH models contain much fewer fitting parameters than MARVEL; thus, they can tolerate a decreased n_DOF value. Accordingly, to make the MARVEL equations solvable, the input dataset must be complemented with accurate estimates for the relative positions of the individual lines within unresolved multiplets. A feasible way on how to find such estimates is proposed in the next subsection.

Use of computed energy-level splittings and relative positions

As evidenced multiple times, also for water isotopologues^{29,58,60–62}, energy differences of rovibrational state pairs pertaining to the same vibrational band can be accurately derived from first-principles solution of the nuclear Schrödinger equation. This favorable state of affairs is due to the utilization of exact kinetic energy operators and the fact that discrepancies arising from deficiencies, such as local inaccuracies in the model potential energy surface (PES) employed, are largely systematic, leading to considerable error cancellation when energy differences between highly similar state pairs are formed. The same holds for the relative position of two lines sharing their upper and lower vibrational parents, as it can be obtained from the (signed) splittings of their upper and lower states, $d_{ij}^{\prime }$ and $d_{ij}^{″}$, respectively:

$${\rho }_{ij}=s_i-s_j=\left[E_{\mathrm{up}\left(i\right)}-E_{\mathrm{low}\left(i\right)}\right]-\left[E_{\mathrm{up}\left(j\right)}-E_{\mathrm{low}\left(j\right)}\right]=\left[E_{\mathrm{up}\left(i\right)}-E_{\mathrm{up}\left(j\right)}\right]-\left[E_{\mathrm{low}\left(i\right)}-E_{\mathrm{low}\left(j\right)}\right]=d_{ij}^{\prime }-d_{ij}^{″}.$$ 4

Thus, computed energy-level splittings, which can be added as wavenumbers of “virtual” lines to the input file, are able to eliminate the underdeterminacy induced by multiplet constraints. Note that resonance interactions among closely-spaced levels in the same J/symmetry block may decrease the accuracy of these computed splittings at high Js, which must be accounted for in the final uncertainty budget.

Within the W2024 dataset, the virtual transitions defined above are placed into a segment called “24virt” and correspond only to energy splittings of ortho–para state pairs, whose assignments differ solely in their K_a or K_c quantum numbers (the error cancellation seems to work exceptionally well for these state pairs). The splitting values included in the 24virt segment are taken from the first-principles POKAZATEL⁷⁴ energy list, for which

$$\mathcal{U}\left(d^{\mathrm{POK}}\right)\approx \max \left[\mid d^{\mathrm{POK}}-d^{\mathrm{BT2}}\mid ,\min \left(0.1\mid d^{\mathrm{POK}}\mid ,0.025{\mathrm{cm}}^{-1}\right),10^{-6}{\mathrm{cm}}^{-1}\right]$$ 5

is employed as an (initial) uncertainty approximation, whereby d^POK and d^BT2 are the POKAZATEL⁷⁴ and BT2⁷⁵ estimates for the same splitting, respectively. A description of how uncertainties of relative positions are taken into account in the uncertainties of energies and predicted wavenumbers is offered in Supplementary Information (B).

Confidence intervals

To appreciate the importance of confidence intervals, a new concept introduced here to MARVEL, one needs to understand the limitations of network-based procedures for the recognition of outliers. During the analysis of SNs, outliers are lines with faulty wavenumbers, uncertainties, or assignments. As shown before⁴⁰, outlier-detection tools designed for SNs are built upon the notion of network cycles (that is, sequences of connected lines and states, where each state has exactly two neighboring states) and network (in)consistency. It must also be stressed that there are a few misconceptions surrounding outlier detection in high-resolution spectroscopy⁴⁰. One of them is related to latent outliers, which cannot be detected via network-theoretical means, as they do not violate the consistency of the SN.

Owing to potential error compensation in cycles, see misconception M5 in Ref. ⁴⁰, in principle any transition might be a latent outlier. In practice, a latent outlier is typically (a) a bridge (i,e., a line without cycles) or (b) a transition whose uncertainty is smaller than the threshold (that is, the sum of uncertainties) in all of its cycles. For instance, if a line has an uncertainty of 10⁻⁴ cm⁻¹, but it participates only in cycles with thresholds being 10⁻³ cm⁻¹, the accuracy of this transition can be validated by MARVEL only to 10⁻³ cm⁻¹.

Based on all these considerations, it is worth defining a measure of “validity”, what is called here a confidence interval (CI), characterizing each transition of the dataset. A CI value provides a lower limit, below which no error can be recognized by MARVEL in a line position or its uncertainty. The emphasis is on the lower-limit property of CI, because there is no upper limit for the magnitude of hidden errors (again, due to possible error cancellation). Intuitively, the CI of a line can be defined as the accuracy of its most accurate, non-trivial cycle (a trivial cycle has only two transitions with the same assignment). This specification leaves CI undefined for a line which is not part of any non-trivial cycles. Actually, for such transitions it is not meaningful to speak of a MARVEL-based validation. For a formal definition of the CI parameter and its extension to energy levels, see Supplementary Information (C).

Data sources and their treatment

There are only a limited number of data sources^{60,62–64,66– 69,76–81} which are available today but were not handled during the construction of the W2020-${\mathrm{H}}_2^{16}\mathrm{O}$ dataset. Apart from five publications^76–80, these sources were published after 2020. In the W2020 input file, 93GuRa⁸² was mistakenly referred to as ‘86GuRa’; this tag should refer to one of the new sources, Ref. ⁷⁸. Seven W2020 sources, 67HaDo⁸³, 73PuRa⁸⁴, 09GrBoRiMa⁸⁵, 12Boyarkin⁸⁶, 20virt⁷, 20extra⁷, and 20compl⁷, have been fully removed from the present analysis. Of these sources, 09GrBoRiMa⁸⁵ and 12Boyarkin⁸⁶, which utilize multiphoton techniques to probe highly-lying states of ${\mathrm{H}}_2^{16}\mathrm{O}$, may well be included in a future update of W2024, when more accurate first-principles energies will be available above 30 000 cm⁻¹, allowing a reliable validation of the 09GrBoRiMa⁸⁵ and 12Boyarkin⁸⁶ lines.

Table 1 contains segments constructed from the new sources considered during this study. While certain sources not divided up in W2020 into segments were divided into multiple segments in W2024, for the sake of simplicity these segments are not specified in Table 1. Furthermore, the 24virt segment, which substitutes its predecessors in W2020, 20virt, 20virt_S2, 20virt_S3, and 20virt_S4, is not given in Table 1 either. The 14 transitions which had to be deleted from the new sources are listed in Table 2.

Table 1.

List of segments involving data sources new to W2024 compared to W2020-${\mathrm{H}}_2^{16}\mathrm{O}$.

Segment tag	Range	A/D	ESU	MSU	LSU
22KaLaChCa63	7168.4–7168.4	1/0/0	3.34e-09	3.34e-09	3.34e-09
24ToDiCoUb62	6755.3–8670.3	135/0/0	1.90e-07	1.90e-07	1.28e-06
22DiToScCo60	7008.4–7346.8	71/0/0	2.54e-07	2.54e-07	1.33e-06
24MiKaKoCa256	51.434–705.36	679/0/0	3.00e-05	4.00e-05	1.08e-03
22ToKoMiPi81	51.435–718.43	1310/0/0	5.00e-05	5.00e-05	3.00e-03
24KaMiKoCa69	51.434–721.42	1130/0/0	5.00e-05	5.00e-05	1.50e-03
86GuRa78	25.085–349.76	265/0/0	1.20e-04	1.50e-04	7.70e-04
86GuRa_S278	501.57–713.79	85/0/1	2.30e-04	2.80e-04	8.20e-04
46ToMe76	0.741 75–0.741 75	1/0/0	1.67e-04	1.67e-04	1.67e-04
23KoMiKaCa68	8041.5–8633.4	3246/2/1	3.60e-04	4.30e-04	2.31e-01
23KoMiKaCa_S268	8058.5–8065.5	1/0/0	1.00e-02	1.00e-02	1.00e-02
82EsHuSaVa77	1601.2–2001.0	703/2/11	4.00e-04	9.65e-04	2.12e-02
20MiKaMoCa80	5789.5–7841.7	11/0/0	1.00e-03	1.00e-03	1.00e-03
21VaMiCa64	12 969–13 172	542/0/9	1.00e-03	1.00e-03	3.48e-02
21VaMiCa_S264	13 015–13 100	6/0/6	1.00e-03	1.00e-03	1.00e-03
22SoPeSoDe65	9344.3–9787.6	783/0/0	1.00e-03	1.00e-03	2.58e-02
97Lanqueti79	2373.3–3722.3	114/5/0	2.00e-03	2.00e-03	1.00e-02
22VaMiCa66	13 171–13 418	1265/5/15	3.00e-03	3.00e-03	3.00e-02
22VaMiCa_S266	13 191–13 193	3/0/3	3.00e-03	3.00e-03	3.00e-03
22YaCoLiGo67	24 064–24 124	21/0/1	3.00e-03	3.00e-03	9.20e-03

^a Tags designate data-source segments. The range, in cm⁻¹, corresponds to line positions of the segments. A, D, and R represent the number of assigned, deleted, and reassigned transitions within a segment, respectively. ESU, MSU, and LSU, in cm⁻¹, stand for the estimated, the median, and the largest segment uncertainties, respectively (the last two quantities characterize the final W2024 database). For the “main” segments, the ‘_S1’ identifier is neglected from the segment tags. The introduction of the ‘_S2’ segments is justified by two reasons: (i) the underlying transitions have been recorded with a different technique than those of the main segment (86GuRa_S2), and (ii) they contain lines which were left unassigned in the original publications (23KoMiKaCa_S2, 21VaMiCa_S2, and 22VaMiCa_S2). Note also that 23KoMiKaCa_S2 carries a line which is given only in the main text of the source 23KoMiKaCa⁶⁸, but not listed in the Supplementary Material of that paper. The R values in the third column include the originally unassigned transitions, as well. Some LSUs are large due to (i) two large original uncertainties reported in 22SoPeSoDe⁶⁵ and 23KoMiKaCa⁶⁸, as well as (ii) the disturbing presence of nearby lines in 21VaMiCa⁶⁴ and 22VaMiCa⁶⁴. For the source 82EsHuSaVa⁷⁷, an optimized recalibration factor of 0.999 999 598 was determined via the MARVEL protocol (for details, see ref. ³⁴).

Table 2.

Complete list of experimental lines deleted from the new sources given in Table 1.

Position/cm−1	Original assignment	Line tag	Comment
1648.265 237(400)	(0 2 0)32,2 ← (0 1 0)31,3	82EsHuSaVa.6177	—
1963.728 110(400)	(1 0 0)72,5 ← (0 1 0)73,4	82EsHuSaVa.62677	—
2629.948 4(20)	(0 0 1)300,30 ← (0 1 0)290,29	97Lanqueti.1979	—
3513.305 6(20)	(0 1 1)1515,1 ← (0 1 0)1515,0	97Lanqueti.5379	—
3490.139 6(20)	(0 1 1)1616,0 ← (0 1 0)1616,1	97Lanqueti.5479	—
3467.102 8(20)	(0 1 1)1717,1 ← (0 1 0)1717,0	97Lanqueti.5579	—
2649.340(10)	(1 0 0)2020,1 ← (0 0 0)2121,0	97Lanqueti.5679	—
8203.349 62(260)	(0 5 1)51,5 ← (0 2 0)41,4	23KoMiKaCa.78368	HD16O line
8529.592 96(29)	(0 5 0)114,8 ← (0 0 0)101,9	23KoMiKaCa.254768	bad multiplet
13 183.545 9(30)	(1 2 2)112,9 ← (0 0 0)127,6	22VaMiCa.4566	HD16O line
13 214.22 73(30)	(2 4 0)87,1 ← (0 0 0)98,2	22VaMiCa.14866	bad multiplet
13 280.38 49(30)	(0 6 1)63,4 ← (0 0 0)51,5	22VaMiCa.39366	bad multiplet
13 306.91 39(30)	(0 4 2)123,10 ← (0 0 0)132,11	22VaMiCa.53366	bad multiplet
13 324.30 68(30)	(0 6 1)55,0 ← (0 0 0)55,1	22VaMiCa.63666	bad multiplet

^aThe values in the first column are observed line positions, with the uncertainties of the last few digits in parentheses. The second column provides the original assignments of the deleted lines which could not be corroborated during the validation of the W2024-${\mathrm{H}}_2^{16}\mathrm{O}$ input dataset. In the third column, the line tags used within the W2024 input file are itemized. Where applicable, the last column contains comments on why these lines proved to be incorrect. ‘Bad multiplet’ means that the given line cannot be part of an unresolved multiplet, as incorrectly indicated in its data source.

During the construction of the W2024 database, it became necessary to add short comments to lines which, in certain aspects, must be distinguished from other transitions of the input dataset. Accordingly, the standard format of the line tags¹, which consists of the segment name and a serial number, has been extended in this study with so-called markers. The principal markers used in the W2024 input file are listed in Supplementary Information (D).

As to the MARVEL treatment of the input transitions dataset, it is worth emphasizing a few important aspects. First, an ortho–para doublet of a segment, observed under Doppler-limited conditions, was deemed to be unresolved if the separation of the reported experimental positions were smaller than one third of the associated Doppler half width at the actual measurement temperature. Second, when the ortho/para complement of a para/ortho line was not published, then it was added to the W2024 input [see also Supplementary Information (D)]. Third, lines within unresolved multiplets other than ortho–para doublets were not subject to multiplet constraints, as their relative positions are usually not known accurately from first-principles computations; their confidence intervals have been increased to reflect their potential inaccuracy. Fourth, a set of empirical positions and energy levels, taken from the literature^{64–66,68,70,80}, was used during the refinement of the wavenumber uncertainties, to reach better agreement, wherever possible, with these auxiliary data. Upon termination of the refinement process, MARVEL was re-executed by eliminating all but 11 lines of this auxiliary dataset from the final W2024 input. The 11 empirical transitions preserved come from the source 20MiKaMoCa⁸⁰, see Table 1, which seem to rely partially on accurate unpublished experimental lines.

Data Records

The W2024 database is available in an OSF (Open Science Framework) repository⁸⁷, which contains validated transitions, empirical energy levels, and an extensive line list for the ${\mathrm{H}}_2^{16}\mathrm{O}$ isotopologue. In the rest of this section, a brief summary is provided about the nine files located in the file “W2024.zip” within the W2024 repository.

As customary, the W2024 repository is accompanied with a “README.txt” file, including a concise description of the content of the other files. In “README.txt”, the file names are arranged in the order of their importance.

The entire collection of the 212 segments created from the 189 sources^{29,41,60,62–69,74,76–82,86,88–256} are presented in “W2024_segment_table.pdf”, where a couple of important statistical parameters are given for each segment, in a form similar to that of Table 1 (the difference is only that R is missing from “W2024_segment_table.pdf”).

The file “W2024_segments.txt” is the segment input file for the MARVEL code, where the unit of the line positions and their uncertainties are specified for each segment. The file “W2024_transitions.txt” contains the 309 290 input transitions collected for the MARVEL procedure. In this file, each input transition is associated with (a) a line position, (b) an initial and an adjusted line-position uncertainty, (c) a $\left(v_1^{\prime }{v}_2^{\prime }{v}_3^{\prime }\right)J_{K_a^{\prime },K_c^{\prime }}^{\prime }\leftarrow \left(v_1^{″}{v}_2^{″}{v}_3^{″}\right)J_{K_a^{″},K_c^{″}}^{″}$ rovibrational assignment, and (d) a line tag representing a unique identifier.

The empirical energy values, obtained for 19 027 rovibrational states in the 0–26 268 cm⁻¹ range, are placed in the file “W2024_energy_levels.txt”. Each state of this data file is supplied with (a) a $\left(v_1{v}_2{v}_3\right)J_{K_a,K_c}$ label, (b) an empirical (MARVEL) energy, (c) an energy uncertainty followed by a (relative) confidence interval in parentheses, (d) the number of transitions incident to this state, and (e) the index of the respective POKAZATEL⁷⁴ state.

The file “W2024-24MiVaCa_comparison.xls” lists 57 states, for which the W2024 and the 24MiVaCa⁷⁰ energies deviate by more than 0.005 cm⁻¹ or their assignments are different. For each line a short comment is given indicating a potential reason for the discrepancy.

Using empirical (W2024) and first-principles (POKAZATEL⁷⁴) energies, a composite line list, named CW2024, was constructed, forming part of the file “CW2024_line_list.txt”. This line list consists of more than 490 000 dipole-allowed transitions in the 0–41 200 cm⁻¹ range, with room-temperature intensities down to 10⁻³¹ cm molecule⁻¹. For almost half of the CW2024 entries, that is for about 231 000 lines in the 0.07–25681.5 cm⁻¹ region, empirical positions are reported; all of them are augmented with individual wavenumber uncertainties and (relative) confidence intervals. For all of the CW2024 lines, the intensities are taken from the POKAZATEL line list, complemented with their BT2⁷⁵ counterparts, whenever applicable. For the empirical transitions of this list, essential walks are also provided in “CW2024_walk_file.txt”. These walks help to understand how the empirical positions and their uncertainties can be approximately extracted from a handful of W2024 input lines [for details on the use of walks, see Supplementary Information (B)]. A line-by-line comparison between HITRAN 2020 and the (C)W2024 dataset is presented in the file “HITRAN_comparison.txt”, which will be discussed in the “Technical Validation” part of this paper.

Finally, the “MARVEL.zip” file contains a developer version of the MARVEL code, written in the C++ language. This version of the MARVEL code, distributed with the necessary input files, was used to generate the numerical data in the TXT files of the W2024 repository (except the input data listed in “W2024_transitions.txt”). The novel MARVEL features, implemented in this code version and described in the “Methods” section, will form part of the http://kkrk.chem.elte.hu/marvelonline/MARVELOnline web application in the future.

Technical Validation

Validation of the W2024 energy levels

The principal validation of the W2024 energy levels was performed via MARVEL, by checking the consistency of the input transitions in relation to their assignments, wavenumbers, and uncertainties. This process resulted in a self-consistent energy-level dataset with individual uncertainties and confidence intervals.

The W2024 energy levels have been matched with their first-principles (BT2⁷⁵, POKAZATEL⁷⁴, and VoTe²⁵⁷) counterparts, making use of the |E^W2024 − E^comp| ≤ 10⁻⁴E^comp criterion, where E^W2024 and E^comp denote empirical and computed energy values, respectively. Despite previous efforts^{74,75,257–259}, no unambiguous labelling scheme exists for water isotopologues and, indeed, it is unlikely that such a scheme could be developed^259,260. Owing to a number of notable differences in rovibrational assignments across the three datasets, only the J/symmetry labels were used during the formation of the (E^W2024, E^comp) pairs.

Consistency of the W2024 energy levels was also checked via the pair identity and smooth variation rules of Ma et al.²⁶¹. For each vibrational state and J, a plot was made of the energy versus the K_a quantum number. These plots were studied to insure their correct pairing structures and smooth variations. Everything checked out correctly, giving further confidence in the correctness of the W2024 assignments and empirical rovibrational energies.

Compared to W2020-${\mathrm{H}}_2^{16}\mathrm{O}$, the W2024-${\mathrm{H}}_2^{16}\mathrm{O}$ dataset deals with only a small number of new data sources. Nevertheless, it contains more than 500 new empirical rovibrational energy levels. How each new source contributes to the set of new energy levels is given in Table 3. Not too surprisingly, the largest contributor is the source 24virt, yielding empirical energies for the ortho/para complements of over 200 para/ortho state pairs. Note also that a few additional energy levels, not reflected in the numbers given in Table 3, were also obtained from the set of more than 1000 transitions reassigned during this study. Consideration of new sources has particular relevance when they provide new energy levels or help to determine improved empirical energy values and/or uncertainties for states already available. For the latter case, the highly accurate sources listed in the first few rows of Table 1 proved to be particularly useful.

Table 3.

Contribution of the new sources to the set of new W2024 energy levels.

Source tag	New states	Most accurate new state
	All [ortho/para]	Assignment	Energy / cm−1
20MiKaMoCa80	3 [2/1]	(0 6 0)11,0	9 004.617 0(10)
21VaMiCa64	119 [68/51]	(0 9 0)21,2	13 093.354 7(10)
22VaMiCa66	147 [67/80]	(0 4 2)21,1	13 566.440 4(30)
22YaCoLiGo67	18 [14/4]	(2 2 4)33,0	24 200.455 7(22)
22SoPeSoDe65	27 [11/16]	(0 6 0)73,5	10 158.506 0(10)
23KoMiKaCa68	27 [9/18]	(1 3 0)106,4	10 458.465 32(12)
24virt (Ref. 74; this work)	211 [15/196]	(1 4 0)97,3	12 148.651 8(10)

^a If a new state occurs in more than one of the new sources, it is attributed to the oldest source. When a source contains multiple matches for the most accurate new state, the lowest-energy match is given in the last two columns.

A detailed comparison of the W2024 energy levels with their W2020 counterparts reveals occasional significant shifts, displayed in Table 4, in previously known energy values. As shown there, not only the less accurate emission sources, like 08ZoShOvPo²⁰⁴ and 05CoBeCaCo¹⁸⁰, but some of the more dependable absorption sources, namely 08ToTe²⁰³, 08CaMiLi²⁰⁰, 11BeMiCa²⁰⁸, and 14ReOuMiWa²²⁵, produced a few unreliable energies, as well. All in all, there are only about 500 cases where the W2024 – W2020 deviations fall outside of the W2020/W2024 uncertainties.

Table 4.

Ten selected W2024 energy levels deviating by more than 0.1 cm⁻¹ from their W2020 counterparts.

Assignment	Rovibrational energy/cm−1			Source of the best incident line
	Theory74	W2024	W20208	W2024	W20208 (comment on best line)
(0 6 0)71,7	9 539.20	9 539.185 77(69)	9 539.368 7(18)[–0.18]	23KoMiKaCa68	14ReOuMiWa225	(deleted)
(0 6 0)71,6	9 714.54	9 714.490 67(86)	9 714.758 8(18)[–0.27]	23KoMiKaCa68	14ReOuMiWa225	(reassigned)
(0 6 0)112,9	10 826.06	10 825.988 1(10)	10 826.152 2(33)[–0.16]	22SoPeSoDe65	05CoBeCaCo180	(deleted, bmult)
(0 9 0)52,3	13 894.80	13 894.760 9(30)	13 895.032 4(40)[–0.27]	22VaMiCa66	08CaMiLi200	(reassigned)
(0 4 2)70,7	14 025.94	14 025.951 1(30)	14 026.309 3(18)[–0.36]	22VaMiCa66	08ToTe203	(reassigned)
(0 4 2)63,3	14 228.79	14 228.816 3(30)	14 229.149 9(40)[–0.33]	22VaMiCa66	08CaMiLi200	(reassigned)
(4 0 0)106,4	15 539.41	15 539.547 6(30)	15 539.209 1(80)[0.34]	22VaMiCa66	11BeMiCa208	(deleted)
(2 2 1)132,12	15 615.18	15 615.198 4(30)	15 615.497 2(40)[–0.30]	22VaMiCa66	08CaMiLi200	(deleted, typo)
(2 0 2)112,9	15 788.77	15 788.944 0(30)	15 789.218 7(80)[–0.27]	22VaMiCa66	11BeMiCa208	(deleted)
(3 0 1)117,5	15 905.13	15 905.325 7(30)	15 904.862(18)[0.46]	22VaMiCa66	08ZoShOvPo204	(deleted, bmult)

^aThe rovibrational energies, tabulated in columns 2–4, are extracted from the POKAZATEL⁷⁴, W2024, and W2020⁸ energy lists. In all cases, the POKAZATEL energies agree better with the W2024 estimates than with the W2020 ones. The values in parentheses represent last-two-digit uncertainties, whereas those in brackets are energy shifts characterizing the W2020 energies. In the last two columns, each state is associated with the source of its most accurate line in the W2024 and W2020 databases. As the comments in the last column suggest, the lines yielding the incorrect W2020 energies were either deleted or reassigned in W2024. In the last column, ‘typo’ indicates that the problematic 08CaMiLi²⁰⁰ entry probably contains a typographical error, while ‘bmult’ denotes that the lines leading to the incorrect W2020 energies form part of ‘bad’ (spurious) multiplets.

To provide a comprehensive picture about the collection of W2024 energy levels, their distributions are plotted against the rovibrational energies and their uncertainties in Fig. 1. For reference purposes, the energy distribution of the first-principles POKAZATEL⁷⁴ states used, forming a complete set in the 0–26 268 cm⁻¹ range investigated, is also given. As obvious from Fig. 1, (a) all states are known in W2024 up to 8995 cm⁻¹, (b) the number of missing empirical states increases rapidly as the energy increases, (c) for a significant number of states the energies are known with an accuracy better than 10⁻⁶ cm⁻¹, and (d) a few W2024 states, deduced from some less accurate 24virt lines, have relatively large, 0.1 – 0.7 cm⁻¹, uncertainties (such states could be targeted by future measurements).

Fig. 1.

Distribution of the W2024 and POKAZATEL states along the rovibrational energies and their uncertainties. The lower panel gives the distribution of the energy values for the W2024 and the POKAZATEL datasets. The upper panel provides the distribution of W2024 states by uncertainties. The range represented by a bin is given by the actual and the previous axis ticks (e.g., the blue bin at 10⁻⁴ cm⁻¹ contains empirical states with an uncertainty of 10⁻⁵–10⁻⁴ cm⁻¹). The state counts, that is the bin sizes, are plotted on a unified bi-directed vertical axis for both distributions. Note the logarithmic scale on both parts of the vertical axis.

Comparison with HITRAN 2020

Comparing the (C)W2024 and the HITRAN 2020-${\mathrm{H}}_2^{16}\mathrm{O}$ line catalogs is particularly important, as it allows additional validation of the W2024 database; furthermore, it might reveal HITRAN entries which require further verification/modification. Results of this comparison are discussed next, without reliance on BT2 intensities.

To facilitate the comparison of the HITRAN 2020-${\mathrm{H}}_2^{16}\mathrm{O}$ line list with the (C)W2024 dataset, an attempt was made to set up a simple quality-assessment scheme for six HITRAN data types which are also present in CW2024. A six-character quality sequence, Q₁-Q₂-Q₃-Q₄-Q₅-Q₆, has been introduced, where Q_p symbolizes the pth quality indicator (QI). Intuitive definitions for the four possible values, A–D, of the six QIs are included in Table 5. Briefly, (i) “A” is the best category, (ii) “B” means acceptable, given the present knowledge, (iii) “C” indicates a conflict between (C)W2024 and HITRAN, which is most probably due to the incorrectness of the HITRAN entry, and (iv) “D” means that no verification was possible for a data type. Table 5 also lists four “comments” attached by us to a few peculiar HITRAN 2020 lines. For further details on the QI values, see Supplementary Information (E).

Table 5.

Quality indicators and comment categories used to characterize the HITRAN-${\mathrm{H}}_2^{16}\mathrm{O}$ lines.

Indicator	Data type	Indicator value
		A	B	C	D
Q1	line match	unique	ambiguous	expt. only	none
Q2	position	approved	acceptable	change/conflict	unverified
Q3	uncertainty code (position)	approved	undefined	change/conflict	unverified
Q4	intensity	approved	acceptable	change/conflict	unverified
Q5	assignment	approved	undefined	change/conflict	unverified
Q6	lower-state energy	approved	acceptable	change/conflict	unverified
Comments to Q1b:		BADSYM	QUADRUP	SMALLINT	DUPSTATE

^aThe uncertainty codes of the positions are specified on the HITRANonline page. The term “expt. only” refers to a unique match with a measured W2024 line not present in the CW2024 list.

^bMeaning of comments made for certain ill-matched HITRAN lines: (i) BADSYM = symmetry violation, (ii) QUADRUP = quadrupole transition, (iii) SMALLINT = a line with too small (<10⁻³¹cm molecule⁻1) intensity, and (iv) DUPSTATE = a transition whose upper state is deleted from W2024.

During this comparison, more than 2500 “W2020” transitions have been identified in HITRAN 2020 for which either the position or the rovibrational assignment differ significantly from that contained in the W2020 database⁸. Table 6 gives six characteristic examples for such questionable HITRAN 2020 transitions. For these lines, certain parameter values were seemingly incorrectly transcribed from W2020.

Table 6.

Typical examples for problematic transitions in the HITRAN 2020-${\mathrm{H}}_2^{16}\mathrm{O}$ list.

HITRAN position/cm−1	Unc. code	Intensity/cm molecule−1	Lower-state energy / cm−1	Assignment	Quality sequence
$2388.0\underset{\_}{14241}$	$\underset{\_}{3}$	2.560e-27	3 266.76 4$\underset{\_}{3}$	(0 1 0)1311,3 ← (0 0 0)1210,2	A-B-C-A-A-B
9 055.06$\underset{\_}{3450}$	$\underset{\_}{4}$	1.312e-29	4 021.217 $\underset{\_}{9}$	$\underset{\_}{\left(111\right)20_{2,19}}\leftarrow \underset{\_}{\left(000\right)19_{2,18}}$	A-C-C-A-A-C
$\underset{\_}{12058.339388}$	$\underset{\_}{7}$	1.902e-29	816.694 2	$\underset{\_}{\left(160\right)6_{0,6}}\leftarrow \underset{\_}{\left(000\right)7_{3,5}}$	D-D-D-D-D-D(DUPSTATE)
13 990.042 460	3	3.138e-27	756.724 8	$\underset{\_}{\left(320\right)7_{5,2}}\leftarrow \left(000\right)6_{4,3}$	A-C-C-C-A-A
15 344.739 6$\underset{\_}{00}$	$\underset{\_}{7}$	3.236e-28	212.156 4	$\underset{\_}{\left(410\right)4_{-4,-4}}\leftarrow \left(000\right)3_{2,1}$	A-C-C-C-C-A
16 823.54$\underset{\_}{9400}$	$\underset{\_}{7}$	6.640e-25	285.219 3	(3 2 1)31,2 ← (0 0 0)33,1	A-C-C-A-C-A

^aThe first two columns give HITRAN positions and their associated uncertainty codes. The intensity values, the lower-state energies, and the assignments are taken from HITRAN. The negative subscripts in the penultimate row mean that those K_a and K_c values are undefined in HITRAN. For these six lines, the quality sequences, reflecting only the level of agreement between the CW2024 line list and HITRAN 2020, are also exhibited in the last column. Although all these lines are claimed to be extracted from W2020⁸, the underlined digits of this table do not correspond to W2020 information.

Figure 2 depicts the distribution of the 13 most common quality sequences, corresponding to 90% of the HITRAN 2020-${\mathrm{H}}_2^{16}\mathrm{O}$ lines. The good news is that the leading sequence is “6A”, see the dark green slice in Fig. 2, where all the six data types of Table 5 are corroborated by (C)W2024. Nevertheless, there is a considerable number of transitions which require particular attention, and may lead, after additional validation, to corrections of certain HITRAN 2020 entries. Lines falling into the “gray zone”, with a sequence “6D”, for which none of the six HITRAN parameters could be affirmed by (C)W2024, must be investigated carefully. Note in this respect that POKAZATEL intensities are highly accurate in the infrared, but get increasingly inaccurate as one moves toward visible wavelengths^262,263. For the full list of quality sequences attached to the HITRAN 2020-${\mathrm{H}}_2^{16}\mathrm{O}$ lines, see ref. ⁸⁷.

Fig. 2.

Distribution of the most frequent quality sequences, covering 90 % of the HITRAN 2020-${\mathrm{H}}_2^{16}\mathrm{O}$ lines. The color codes applied for the quality sequences are shown on the left-hand side of this figure, where slices formed by empirical, computed, or mixed (empirical plus computed) HITRAN 2020 transitions are clearly distinguished. The blue arrow indicates the direction whereby the slices follow the ordering utilized in the color legend. To highlight its most important characteristics, each slice is supplied with short stamps, displayed on the pie charts (notice that there are stamps shared between two slices). The relative fractions of the individual slices, with respect to the complete HITRAN 2020-${\mathrm{H}}_2^{16}\mathrm{O}$ database, are included in colored boxes. Those lines which could not be matched with the (C)W2024 line list appear in the gray slice. The remaining transitions, whose quality sequences differ from those exhibited in the color legend, are collected in the cyan slice. For improved transparency, slices with less than 10 000 lines are enlarged in an inset.

Usage Notes

The empirical energy levels derived during this study are associated with individual uncertainties and confidence intervals, giving numerical characterization of the trust we have in the W2024 energies, as well as in the predicted transition wavenumbers. These important statistical parameters must be taken into account in applications using the (C)W2024 datasets. The set of new and corrected empirical energies of this study could, for example, prove useful for adjusting existing potential energy surfaces of the ${\mathrm{H}}_2^{16}\mathrm{O}$ molecule, reducing the discrepancies between the results of variational nuclear-motion computations and experiment.

The CW2024 database could be helpful for experimental spectroscopists, who wish to (re)analyze their new and old spectra, especially when looking for new energy levels absent from the W2024 energy list. This CW2024 catalog would also provide support for the validation and occasional correction of ${\mathrm{H}}_2^{16}\mathrm{O}$ transitions present in line-by-line spectroscopic databases.

Updated versions of the database files will be made available at the website https://respecth.elte.hu/. Version history will be provided in a file called “NOTES.txt” under the W2024 repository⁸⁷.

Author contributions

T.F., A.G.C., and R.T. designed the study, with important contributions from J.T. and R.R.G. T.F., R.T., and R.R.G. checked different aspects of the datasets (transitions and energy levels). R.T. draw the figures and prepared the tables. T.F., R.T., and A.G.C. wrote the original version of the manuscript. All of the authors have contributed to writing the final manuscript.

Code availability

The developer version of the MARVEL code, used during the compilation and validation of the W2024 database, is freely available⁸⁷.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41597-024-03847-3.