Accurate empirical rovibrational energies and transitions of H216O

Roland Tóbiás a, Tibor Furtenbacher a, Jonathan Tennyson b and Attila G. Császár *c
aLaboratory of Molecular Structure and Dynamics, Institute of Chemistry, ELTE Eötvös Loránd University and MTA-ELTE Complex Chemical Systems Research Group, P.O. Box 32, H-1518 Budapest 112, Hungary
bDepartment of Physics and Astronomy, University College London, London WC1E 6BT, UK
cMTA-ELTE Complex Chemical Systems Research Group, P.O. Box 32, H-1518 Budapest 112, Hungary. E-mail: csaszarag@caesar.elte.hu

Received 14th August 2018 , Accepted 21st November 2018

First published on 11th January 2019


Several significant improvements are proposed to the computational molecular spectroscopy protocol MARVEL (Measured Active Rotational–Vibrational Energy Levels) facilitating the inversion of a large set of measured rovibrational transitions to energy levels. The most important algorithmic changes include the use of groups of transitions, blocked by their estimated experimental (source segment) uncertainties, an inversion and weighted least-squares refinement procedure based on sequential addition of blocks of decreasing accuracy, the introduction of spectroscopic cycles into the refinement process, automated recalibration, synchronization of the combination difference relations to reduce residual uncertainties in the resulting dataset of empirical (MARVEL) energy levels, and improved classification of the lines and energy levels based on their accuracy and dependability. The resulting protocol, through handling a large number of measurements of similar accuracy, retains, or even improves upon, the best reported uncertainties of the spectroscopic transitions employed. To show its advantages, the extended MARVEL protocol is applied for the analysis of the complete set of highly accurate H216O transition measurements. As a result, almost 300 highly accurate energy levels of H216O are reported in the energy range of 0–6000 cm−1. Out of the 15 vibrational bands involved in accurately measured rovibrational transitions, the following three have definitely highly accurate empirical rovibrational energies of 8–10 digits of accuracy: (v1v2v3) = (0 0 0), (0 1 0), and (0 2 0), where v1, v2, and v3 stand for the symmetric stretch, bend, and antisymmetric stretch vibrational quantum numbers. The dataset of experimental rovibrational transitions and empirical rovibrational energy levels assembled during this study, both with improved uncertainties, is considerably larger and more accurate than the best previous datasets.


1 Introduction

In the era of optical frequency combs and frequency-comb spectroscopy1,2 it has become increasingly realistic to determine experimental line positions with an accuracy on the order of 10 kHz or even better. Previously, such experimental accuracy was reserved to pure rotational transitions measured in the microwave (MW) region of the electromagnetic spectrum but nowadays there is growing evidence that this accuracy can be extended all the way from the MW to the near infrared (NIR) region3–5 and beyond.6,7

Complementing the experimental advances in high-resolution molecular spectroscopy in the theoretical front is the introduction of spectroscopic networks,8–10 and the Measured Active Rotational–Vibrational Energy Levels (MARVEL) scheme.8,9,11–13 The MARVEL algorithm and code, originally presented in ref. 11 and 12, allows the efficient determination of accurate empirical energy levels with well-defined uncertainties from a set of measured and assigned rovibronic transitions of known uncertainty, the global analysis of measured spectra,14 and the validation of related spectroscopic datasets and databases.13 MARVEL has been employed for a number of molecular systems, including 12C2,15 nine isotopologues of water,16–20 three isotopologues of SO2,21 three isotopologues of H3+,22,2314NH3,24 and parent ketene.25 A web-based version of the MARVEL code has allowed the involvement of high-school students26 in spectroscopic research projects, leading to published studies on 48Ti16O,2790Zr16O,28 H232S,29 and 12C2H2,30 molecules of considerable astronomical interest.

Energy levels derived using the MARVEL procedure are increasingly being used to improve not only transition wavenumbers31 but also partition functions32 present in standard information systems. Just considering water, MARVEL energy levels have been used extensively by the ExoMol project33 to improve predicted line positions in line lists for H2nO (n = 16, 17, 18).34,35 Similarly, the most recent edition of the HITRAN database, HITRAN2016,31 makes use of MARVEL energy levels for H2nO (n = 16, 17, 18),36,37 as well as for the deuterated isotopologues.38 MARVEL energy levels formed a key part of a procedure used to greatly improve predicted energy levels for water isotopologues.34 Empirical (MARVEL) rovibrational energies have also been used to generate the most accurate partition functions available for H216O,39 heavy water,40 and its three constituent isotopologues.40 As they are present in a large number of spectra, water lines can also be used for calibration in different spectral regions.41,42

The second-generation MARVEL code12 runs extremely fast, treats transition data and the related matrices (vide infra) on the order of 100[thin space (1/6-em)]000 in less than a minute on a single CPU, and it is platform independent. This efficiency of the MARVEL code facilitates enhancements, such as those discussed below, of the MARVEL algorithm.

In all the MARVEL studies cited above accurate empirical rovibronic energy levels have been determined from the simultaneous treatment of all measured transitions. Nevertheless, it has repeatedly been observed that when transitions based on the empirical (MARVEL) energy levels and their uncertainties were compared to the best measurements, slight distortions and occasionally unnecessarily large, but sometimes too small, uncertainties characterize the MARVEL energy levels. These inconsistencies, at least in part, are the result of the inclusion of many transitions of orders of magnitude lower accuracy than the best measurements in the MARVEL inversion and refinement procedure. In this study we attempt to devise a MARVEL-based protocol which retains the accuracy of the best measurements while still working with as complete spectroscopic networks as feasible during the inversion and refinement process.

Next, let us review the state-of-the-art of high-resolution spectroscopy for the H216O isotopologue of water, the subject molecule of our feasibility study in which the extended MARVEL (extMARVEL) protocol is employed to gain highly accurate rovibrational energy levels meeting or even exceeding the accuracy of the best measurements.

The highest-quality database of accurate rovibrational energy levels and transitions of water vapor is the spectroscopic information system maintained at the Jet Propulsion Laboratory (JPL).43,44 Nowadays the H216O dataset of JPL energy levels is based on a study of Lanquetin et al.45 In particular, the experimental JPL energy levels of H216O are exactly those reported in ref. 45. The JPL transitions are the results of an effective Hamiltonian fit of a considerable number of transitions, as detailed on the JPL website.44 The most comprehensive evaluation of measured water transitions was performed by an IUPAC Task Group.16–20Ref. 20 gives a summary of this work. Part III of this series18 contains a validated and recommended set of measured H216O transitions (about 200[thin space (1/6-em)]000) and empirical energy levels (about 20[thin space (1/6-em)]000), based on experimental data available prior to 2013. These sets were used to update the water data in HITRAN2016.31 Since 2013, there have been many new studies of water spectra and, in particular, the use of optical-frequency-comb-based measurements to determine very accurate wavenumbers for selected transitions in the infrared, see, e.g., ref. 5, 46 and 47. The advent and availability of new, precise experimental techniques act as an impetus to further improve the MARVEL treatment of measured transitions allowing the determination of highly accurate empirical energy levels.

As of today no orthopara rovibrational transitions have been observed in water vapor.67 This has the consequence that water spectra define two principal components10 within the measured spectroscopic network of H216O, corresponding to ortho- and para-H216O. Note that the number q = v3 + Ka + Kc is even for para and odd for ortho rovibrational states, where v3 is the vibrational quantum number corresponding to the antisymmetric stretch motion, while Ka and Kc are the usual rigid-rotor quantum numbers of an asymmetric-top molecule.

As emphasized in ref. 42, despite the fact that ortho lines may have three times higher intensity than para lines (and thus about two times more of them have been determined experimentally18), the best frequency standards correspond to para-H216O, as there is no hyperfine splitting of the para-H216O lines and the minimum-energy level of the para PC can properly be set to zero with zero uncertainty.

High-resolution radial-velocity-shift transition measurements used to detect molecular species in the atmospheres of exoplanets68,69 has greatly increased the need for accurate laboratory transition frequencies. While this technique has been used successfully to identify water in exoplanets70,71 with high confidence, standard water line lists used for exoplanet modelling appear to be not sufficiently accurate for this task.72 It is also true that many of the most important line positions have been measured sufficiently accurately. In the extended list73 of the astrophysically most important spectral lines, the International Astronomical Union listed 13 H216O lines, with assignments and rest frequencies recalled in Table 1. For all these frequencies several independent, highly accurate experimental determinations are available (see Table 1). Nevertheless, not all of these transitions are part of accurate cycles satisfying the law of energy conservation13 to the accuracy of the best measurements, which calls for optical-frequency-comb-based remeasurement of certain rovibrational transitions on the ground vibrational state of water.

Table 1 The 13 astrophysically most important water (H216O) lines recommended by the International Astronomical Union and their different experimental determinations. The experimental (Expt.) frequencies of multiple measurements follow the order of their increased uncertainties. In the assignment column, the standard spectroscopic notation image file: c8cp05169k-t144.tif is used, where image file: c8cp05169k-t145.tif and image file: c8cp05169k-t146.tif are the rotational labels for the upper and lower energy levels, respectively. All transitions, with AF = approximate frequency, belong to the ground vibrational state
AF/GHz Component Assignment extMARVEL/kHz 01LaCoCa45/kHz Expt./kHz
22.235 para 616 ← 523 22[thin space (1/6-em)]235[thin space (1/6-em)]079.85(6) 22[thin space (1/6-em)]235[thin space (1/6-em)]007(4240) 22[thin space (1/6-em)]235[thin space (1/6-em)]079.85(6)48
22[thin space (1/6-em)]235[thin space (1/6-em)]080(29)49
22[thin space (1/6-em)]235[thin space (1/6-em)]200(600)50
22[thin space (1/6-em)]235[thin space (1/6-em)]220(154)51
22[thin space (1/6-em)]235[thin space (1/6-em)]000(190)52
183.310 ortho 313 ← 220 183[thin space (1/6-em)]310[thin space (1/6-em)]090.4(1) 183[thin space (1/6-em)]309[thin space (1/6-em)]897(2544) 183[thin space (1/6-em)]310[thin space (1/6-em)]090.6(1)53
183[thin space (1/6-em)]310[thin space (1/6-em)]087(4)54
183[thin space (1/6-em)]310[thin space (1/6-em)]117(29)55
183[thin space (1/6-em)]310[thin space (1/6-em)]150(65)49
183[thin space (1/6-em)]310[thin space (1/6-em)]200(190)52
183[thin space (1/6-em)]311[thin space (1/6-em)]300(1330)56
325.153 para 515 ← 422 325[thin space (1/6-em)]152[thin space (1/6-em)]899(2) 325[thin space (1/6-em)]153[thin space (1/6-em)]101(3392) 325[thin space (1/6-em)]152[thin space (1/6-em)]899(2)54
325[thin space (1/6-em)]152[thin space (1/6-em)]888(13)57
325[thin space (1/6-em)]152[thin space (1/6-em)]919(22)55
325[thin space (1/6-em)]153[thin space (1/6-em)]700(882)52
380.197 para 414 ← 321 380[thin space (1/6-em)]197[thin space (1/6-em)]359.8(6) 380[thin space (1/6-em)]197[thin space (1/6-em)]395(3610) 380[thin space (1/6-em)]197[thin space (1/6-em)]359.8(6)58
380[thin space (1/6-em)]197[thin space (1/6-em)]356(4)54
380[thin space (1/6-em)]197[thin space (1/6-em)]365(13)57
380[thin space (1/6-em)]197[thin space (1/6-em)]372(16)55
380[thin space (1/6-em)]196[thin space (1/6-em)]800(621)52
439.151 para 643 ← 550 439[thin space (1/6-em)]150[thin space (1/6-em)]794.8(6) 439[thin space (1/6-em)]151[thin space (1/6-em)]582(5528) 439[thin space (1/6-em)]150[thin space (1/6-em)]794.8(6)58
439[thin space (1/6-em)]150[thin space (1/6-em)]795(2)54
439[thin space (1/6-em)]150[thin space (1/6-em)]812(19)55
448.001 para 423 ← 330 448[thin space (1/6-em)]001[thin space (1/6-em)]077.5(6) 448[thin space (1/6-em)]001[thin space (1/6-em)]155(3610) 448[thin space (1/6-em)]001[thin space (1/6-em)]077.5(6)58
448[thin space (1/6-em)]001[thin space (1/6-em)]075(16)55
448[thin space (1/6-em)]000[thin space (1/6-em)]300(904)52
474.689 ortho 533 ← 440 474[thin space (1/6-em)]689[thin space (1/6-em)]108(2) 474[thin space (1/6-em)]689[thin space (1/6-em)]879(3816) 474[thin space (1/6-em)]689[thin space (1/6-em)]108(2)54
474[thin space (1/6-em)]689[thin space (1/6-em)]127(21)55
556.936 para 110 ← 101 556[thin space (1/6-em)]935[thin space (1/6-em)]987.6(6) 556[thin space (1/6-em)]935[thin space (1/6-em)]841(1824) 556[thin space (1/6-em)]935[thin space (1/6-em)]987.7(6)58
556[thin space (1/6-em)]935[thin space (1/6-em)]985(3)59
556[thin space (1/6-em)]935[thin space (1/6-em)]995(8)54
556[thin space (1/6-em)]936[thin space (1/6-em)]002(16)55
556[thin space (1/6-em)]935[thin space (1/6-em)]819(185)60
556[thin space (1/6-em)]935[thin space (1/6-em)]800(190)52
556[thin space (1/6-em)]935[thin space (1/6-em)]800(206)61
620.700 para 532 ← 441 620[thin space (1/6-em)]700[thin space (1/6-em)]954.9(6) 620[thin space (1/6-em)]701[thin space (1/6-em)]697(4664) 620[thin space (1/6-em)]700[thin space (1/6-em)]954.9(6)58
620[thin space (1/6-em)]700[thin space (1/6-em)]950(35)62
620[thin space (1/6-em)]700[thin space (1/6-em)]844.1(150)63
620[thin space (1/6-em)]700[thin space (1/6-em)]807(163)55
752.033 ortho 211 ← 202 752[thin space (1/6-em)]033[thin space (1/6-em)]113(15) 752[thin space (1/6-em)]032[thin space (1/6-em)]978(2544) 752[thin space (1/6-em)]033[thin space (1/6-em)]113(15)64
752[thin space (1/6-em)]033[thin space (1/6-em)]104(19)60
752[thin space (1/6-em)]033[thin space (1/6-em)]227(125)55
752[thin space (1/6-em)]033[thin space (1/6-em)]300(190)52
916.172 ortho 422 ← 331 916[thin space (1/6-em)]171[thin space (1/6-em)]581(21) 916[thin space (1/6-em)]171[thin space (1/6-em)]448(3392) 916[thin space (1/6-em)]171[thin space (1/6-em)]580(21)65
916[thin space (1/6-em)]171[thin space (1/6-em)]582(21)66
916[thin space (1/6-em)]171[thin space (1/6-em)]405(194)60
970.315 ortho 524 ← 431 970[thin space (1/6-em)]315[thin space (1/6-em)]021(21) 970[thin space (1/6-em)]315[thin space (1/6-em)]165(3392) 970[thin space (1/6-em)]315[thin space (1/6-em)]022(21)66
970[thin space (1/6-em)]315[thin space (1/6-em)]020(21)65
970[thin space (1/6-em)]314[thin space (1/6-em)]968(58)60
987.927 ortho 202 ← 111 987[thin space (1/6-em)]926[thin space (1/6-em)]762(21) 987[thin space (1/6-em)]926[thin space (1/6-em)]473(2341) 987[thin space (1/6-em)]926[thin space (1/6-em)]764(21)66
987[thin space (1/6-em)]926[thin space (1/6-em)]760(21)65
987[thin space (1/6-em)]926[thin space (1/6-em)]743(23)60


The present study has been executed with two principal goals in mind. First, we wanted to improve the MARVEL protocol,11,12 with particular emphasis on a better reproduction of the most accurately measured rovibrational transitions. Second, to test the utility of the improved protocol, we chose H216O, the admittedly most important polyatomic molecule for high-resolution spectroscopy, for which the accurate knowledge of the rovibrational energy levels and transitions is important in a number of scientific and engineering applications. Therefore, first a detailed description of the extMARVEL scheme is given in Section 2. To aid the reader of the methodological section, some of the more technical details are moved to appendices. Those interested only in the spectroscopic results of this study, and not the way they were determined, can skip Section 2. Then, Section 3 discusses the H216O experimental input data used, while Section 4 presents results of our extMARVEL analysis of a large number of old and new sources.46–66,74–103 This analysis also includes comparisons with previously determined accurate water levels and lines.45,60,95 The paper ends with some concluding remarks in Section 5.

2 Methodology

2.1 Spectroscopic networks

Sets of rovibrational transitions, whether they are measured or computed, can be treated as building blocks of spectroscopic networks (SN, see ref. 8–13), whereby (a) the vertices are energy levels, (b) the edges correspond to transitions, oriented from the lower energy level to the upper one (independently whether the transition was recorded in absorption, emission, or by means of an action spectroscopy), and (c) the (positive) edge weights are the wavenumbers of the transitions. Note that many other weighting schemes (e.g., weighting by line intensities) can be utilized in practical applications of SNs.

If SNs are formed by experimental transitions, called experimental SNs, the edge weights should be associated with measurement uncertainties. However, in many data sources, instead of providing line-by-line uncertainties, only an average “expected” accuracy is provided, usually corresponding to an intense unblended line, i.e., to a best-case measurement scenario. Based on these experimental line positions and approximate uncertainties—together with assignments to the lower and upper energy levels of the transition—one is able to derive empirical (MARVEL) energy levels with uncertainties via the MARVEL (Measured Active Rotational–Vibrational Energy Levels) procedure,11,12 utilizing a weighted least-squares technique (e.g., robust reweighting104) and the Rydberg–Ritz combination principle.105

The detailed analysis of experimental SNs is very important as one would like to treat all the measured rovibrational lines of a molecule simultaneously and make the validated lines and their energy levels available to spectroscopists and spectroscopic database developers. For this purpose, it is necessary to explore (a) the components (sets of energy levels not connected by any measured transition), (b) the bridges (transitions whose deletion increases the number of components), and (c) the cycles (collections of connected edges within which every vertex has two neighboring vertices) characterizing the given experimental SN.

Since the energy levels of the different components are not connected, during the MARVEL analysis we are forced to set the lowest-lying energy level (core) of each component to zero. If the core of a component corresponds to the lowest-energy level of a particular nuclear-spin isomer of the molecule examined, this component is a principal component (PC), otherwise it is called a floating component (FC). Clearly, one is most interested in the detailed characterization of those energy levels and transitions which are part of PCs.

Due to the fact that bridges of experimental SNs may compromise the accuracy of the empirical energy levels determined during a MARVEL analysis, they require special attention. That is, if a bridge is determined incorrectly or inaccurately, the energy levels connected to the core of their component through this bridge will be shifted or scattered. Energy levels “behind” a bridge cannot be known more accurately than the bridge itself, a considerable hindrance in the derivation of highly accurate empirical energy levels.

Cycles are extremely useful when compatibility of the transitions and their associated uncertainties obtained in different groups under often widely different experimental conditions are examined.106 It is important to check, for all the cycles of the SN, whether the law of energy conservation (LEC, see ref. 13) is satisfied within the experimental accuracy. Accordingly, in the cases where (a) a transition is assigned improperly, (b) its wavenumber is measured inaccurately, or (c) the uncertainty of this wavenumber is underestimated, the discrepancy (absolute signed sum of the line positions) becomes higher than the threshold (the sum of the uncertainties) in the given cycle, indicating a conflict among the related transitions.

All the cycles of an experimental SN can be expressed in a cycle basis (CB) with the symmetric differences of the basic cycles in the CB, which contains all non-bridge lines of this SN.13 Accordingly, construction of such CBs and evaluation of their entries could be sufficient to test the compatibility of the measured rovibrational lines in light of the LEC. For details about the use of CBs to explore inconsistencies in SNs, see ref. 13. Note also that in contrast to ref. 13, cycles of length 2, which correspond to repeated measurements, are also permitted here, as this extension makes the present formalism simpler than the previous one.13

Subnetworks are of great importance during the treatment of SNs. These derived graph structures are (a) represented with a participation matrix, P = diag(P1, P2,…, PNT), where NT is the number of lines in the SN, and Pi = 1 if the ith transition of the SN is inserted in the subnetwork, otherwise Pi = 0, and (b) filled with all the energy levels of the SN, among which there may also be isolated nodes (vertices without inserted lines).

Due to the presence of some outliers, which should be excluded from the database of transitions during our analyses (see also Section 2.3.2), it is necessary to introduce the leading subnetwork of the SN, denoted with image file: c8cp05169k-t147.tif, which contains all the non-excluded lines of the database. In the present description, P will always denote the participation matrix assigned to the image file: c8cp05169k-t148.tif subnetwork.

2.2 The MARVEL procedure

To understand the improvements proposed in this study better, first the traditional MARVEL approach,11,12 upon which our novel MARVEL algorithm is built, is recalled.

During a MARVEL analysis of measured rovibronic transitions the following objective function is minimized:

 
Ω(E) = (σRE)TPW(σRE),(1)
where (a) E = {E1, E2,…, ENL}T is the vector of energies of the NL energy levels, (b) σ = {σ1, σ2,…, σNT}T is the vector of wavenumbers of the NT transitions, (c) R is the Ritz matrix (see also eqn (3) of ref. 11), (d) W = diag(w1, w2,…, wNT) is a diagonal weight matrix with wi = δηi, (e) δi is the uncertainty of σi, and (f) η is a nonpositive exponent, set to −2 in our calculations.

Since the minimum of Ω(E), denoted by Ē = {Ē1, Ē2,…, ĒNL}T, is not unique,

 
Ēcore(1) = Ēcore(2) = ⋯ = Ēcore(Nc) = 0(2)
is set for the cores indexed as core(1), core(2),…, core(Nc), where Nc is the number of components in image file: c8cp05169k-t149.tif. Furthermore, Ē must also satisfy the relation
 
= F,(3)
where G = RTPWR is the weighted Gram–Schmidt matrix of R and F = RTPWσ is the vector of free terms. Note that eqn (2) and (3) can be solved with Cholesky decomposition, using, for example, the standard Eigen package,107 and can be reduced by transforming G into a block diagonal form, which consists of Nc independent diagonal blocks. Note that during the testing of the code we experimented with several decomposition procedures, but no significant differences were found.

To eliminate the linear dependencies of the block-diagonal form of G, one has to leave out a (row, column) pair from each block. Although, in principle, these pairs could be chosen arbitrarily, in practice it is best to select those (row, column) combinations which are associated with the largest diagonal entries of the corresponding blocks; thus, improves the numerical stability of the solution.

In practice, the core indices of image file: c8cp05169k-t150.tif are not necessarily known to the user. Then, from any solution, image file: c8cp05169k-t151.tif, of eqn (3), the core(i) index can be obtained for all 1 ≤ iNc as follows:

 
image file: c8cp05169k-t1.tif(4)
where comp(j) is the component index of the jth energy level, i.e., the index of the component containing this energy level. (In eqn (4), the operation ‘argmin’ returns the j index of the smallest Ej′ value for which comp(j) = i.) Based on the core indices, the Ēj values can be expressed for all 1 ≤ jNL as
 
image file: c8cp05169k-t291.tif(5)
In what follows, we always assume that the core indices of image file: c8cp05169k-t289.tif are available.

One must note in passing that measurement uncertainty is a combination of precision and absolute accuracy. Precision of measured line positions varies considerably according to a number of factors: (a) type and quality of the spectrometer, (b) the spectral resolution relative to the observed line width, (c) the signal to noise ratio, (d) the choice and especially the control of experimental conditions, (e) the complexity of the spectrum, and (f) the line retrieval methods employed. Since all of these factors must be considered as carefully as possible, it is not at all surprising that experimentalists measuring a large number of lines often refrain from reporting line-by-line uncertainties.

As soon as Ē is determined, it is necessary to examine the Δi = σi[small sigma, Greek, macron]i (fitting) residuals and the di = |Δi| − δi (fitting) defects, where [small sigma, Greek, macron]i = Ēup(i) + Ēlow(i) is the ith wavenumber estimate, while up(i) and low(i) are the indices of the upper and lower energy levels of the ith transition, respectively. image file: c8cp05169k-t152.tif is called consistent if the largest defect of image file: c8cp05169k-t153.tif, dmax, is not positive. In the case of dmax > 0, we have to either exclude certain lines or increase their uncertainties in order to ensure the consistency of image file: c8cp05169k-t154.tif.

2.3 The extended MARVEL (extMARVEL) approach

The approach outlined here is an extension of the MARVEL procedure,11,12 which proved to be a powerful tool to obtain correct and reliable empirical rovibrational energy levels and associated uncertainties in a number of applications, as detailed in the Introduction. Our improved protocol is depicted in Fig. 1, the following subsections explain each of the schemes shown in Fig. 1.
image file: c8cp05169k-f1.tif
Fig. 1 The extended MARVEL (extMARVEL) approach, see text for an explanation of each step.
2.3.1 Isolation of source segments (ISS). Having collated the transitions into a SN, good estimates for the uncertainties of the line positions are required for the extMARVEL analysis. Fortunately, in practice it is often sufficient to find a suitable uncertainty for each subset of transitions of the same (data) source which are chosen to have identical uncertainties, at least as a first approximation.

Then, one needs to divide the transitions of the sources into segments, in which the uncertainties of the wavenumbers are considered to be (approximately) within a factor of 10. These approximate uncertainties are referenced as estimated segment uncertainties (ESU). A particular segment is denoted in this study with a character sequence obtained by the concatenation of the source tag with one of the strings ‘_S1’, ‘_S2’, ‘S3’, etc. By convention, (a) segments are indexed in increasing order of their ESUs, and (b) ‘_S1’ is never written out explicitly in the first segment.

It is important to establish the ESU values with caution to avoid the problem of under- and overutilization of the data. If explicit information on the accuracy is not available from the sources, one can use the simple approximate relations ‘uncertainty = resolution/10’ and ‘uncertainty = resolution’ for the unblended and blended lines, respectively. (It would help spectroscopic database developers if experimentalists published their lines partitioned at least into blended and unblended transitions and reported average uncertainty estimates for at least these two categories.)

At the end of this process, two input files need to be created for further analysis. The first one is the transition database in MARVEL format11,12 with the distinction that the “uncertainty” column is not evaluated. The second one is the list of the source segments with their ESUs and a ternary flag for each segment. The value of this flag is (a) 0, if the given segment must not be subjected to recalibration using a single recalibration factor, (b) −1, if this segment should be recalibrated with the same factor as the segment in the previous record of the segment file, or (c) 1, otherwise.

2.3.2 Prior cleansing. Before the empirical energy levels are determined, it is mandatory to perform an extensive cleansing of the collated dataset to weed out clear outliers. As to inconsistencies concerning the dataset, both errors (misprints or transcription errors) and inaccuracies may occur (a) in the wavenumbers, (b) in the line assignments, and (c) in the ESU values of the segments. All these problems need to be identified. For this purpose, several checks must be carried out, including (a) a test whether the selection rules hold properly for the input transitions, (b) use of the ECART (Energy Conservation Analysis of Rovibronic Transitions, see ref. 13) algorithm to list and delete the incorrect cycles, or (c) an analysis of the fitting residuals derived from the MARVEL procedure.
2.3.3 Integrated MARVEL (intMARVEL) analysis. The integrated MARVEL (intMARVEL) algorithm (see Fig. 1) is a fully automated procedure able to (a) generate refined segment uncertainties (RSU) from the ESU values, (b) determine the energy levels of the SN based on their most accurate lines, (c) exclude outlier transitions from the SN, (d) calculate the individual wavenumber uncertainties of the lines originating from the RSU values, (f) recalibrate ill-calibrated segments, (e) synchronize the combination difference (CD) relations, and (g) assess the lines, segments, and energy levels on the basis of their accuracy and dependability.

This procedure was programmed into a code, written in the C++ language, called intMARVEL, which requires the two input files mentioned in Section 2.3.1. The output of the intMARVEL code contains the transitions, the energy levels, and the segments characterized, all in separate text files. A detailed description of the theoretical background of the intMARVEL algorithm is given in Section 2.4 and the corresponding appendices.

2.3.4 Recovering procedure. Once the intMARVEL analysis has been completed, the transitions excluded during certain stages of the extMARVEL algorithm need to be re-assessed and, if possible, corrected. In order to achieve data correction, the wavenumbers and assignments of the excluded lines are to be checked carefully based on the original sources.

After this revision, reassignments can be made for the uncorrected, but still excluded lines, using (a) the wavenumber-sorted experimental transition dataset, and (b) a MARVEL or an effective Hamiltonian (EH) linelist ignoring weak and forbidden transitions. At the end of the recovering step, the previous stages of the extMARVEL procedure should be repeated with the corrected lines until no transitions can be “repaired”.

2.4 Background of the intMARVEL analysis

2.4.1 Hiearachical segment perturbation (HSP). To obtain the RSU values, the hierarchical segment perturbation (HSP) scheme is introduced. During HSP, for all the s segments, (a) a image file: c8cp05169k-t155.tif perturbation subnetwork is constructed by including all the non-excluded lines from s and those segments which are much more accurate than s (hierarchical perturbation), and (b) the RSU value is provided for s from the discrepancies and thresholds of those basic cycles in a given CB of image file: c8cp05169k-t156.tif which contain transitions from s (representative cycles). If there are too few representative cycles for the refinement of a particular ESU, then the corresponding RSU will be identical to this ESU value. The consecutive stages of the HSP procedure are described in Appendix A.
2.4.2 BlockMARVEL refinement. Since inaccurately measured wavenumber entries may deteriorate the accuracy of the estimated energy levels, a novel procedure, called the constrained MARVEL scheme, was designed with which these harmful effects can be avoided or at least substantially reduced. This technique ensures that the wavenumber estimates derived from a given image file: c8cp05169k-t157.tif (previous) subnetwork of image file: c8cp05169k-t158.tif, via the MARVEL algorithm, should remain unaffected when the MARVEL analysis is repeated for a image file: c8cp05169k-t159.tif (actual) subnetwork of image file: c8cp05169k-t160.tif, which also contains image file: c8cp05169k-t161.tif. By definition, the cores of image file: c8cp05169k-t162.tif are the cores of image file: c8cp05169k-t163.tif, as well (see also Section 2.1).

Designate the core indices in image file: c8cp05169k-t164.tif and image file: c8cp05169k-t165.tif with corep(1), corep(2),…, corep(Np,c) and corea(1), corea(2),…, corea(Na,c), respectively, where Np,c and Na,c are the number of components in image file: c8cp05169k-t166.tif and image file: c8cp05169k-t167.tif, respectively. Let image file: c8cp05169k-t168.tif be the participation matrix of image file: c8cp05169k-t169.tif. With the notation introduced, eqn (2) and (3) can be reformulated for image file: c8cp05169k-t170.tif as follows:

 
Ēcorea(1) = Ēcorea(2) = ⋯ = Ēcorea(Na,c) = 0(6)
and
 
image file: c8cp05169k-t171.tif(7)
where image file: c8cp05169k-t172.tif and image file: c8cp05169k-t173.tif. Since the wavenumber estimates produced from image file: c8cp05169k-t174.tif with the MARVEL technique should remain unchanged, the following constraints must be enforced for all 1 ≤ iNL:
 
ĒiĒcorep(compp(i)) = βi,(8)
where compp(i) is the component index of the ith energy level in image file: c8cp05169k-t175.tif and βi is the bound for Ēi. (Bounds correspond to empirical energy levels derived from image file: c8cp05169k-t176.tif using the MARVEL analysis.) In other words, only the cores of image file: c8cp05169k-t177.tif are linearly independent variables, all the other energy levels can be expressed from eqn (8). Accordingly, these constraints can be written in a matrix-vector form as
 
Ē = C[H with combining macron] + β,(9)
where [H with combining macron] = {[H with combining macron]1, [H with combining macron]2,…, [H with combining macron]Np,c}T and C = {cij} are specified as [H with combining macron]j = Ēcorep(j) and
 
image file: c8cp05169k-t2.tif(10)
respectively. Applying eqn (7)–(9) and after some algebraic manipulations, the following formulae (constrained MARVEL equations) can be deduced:
 
[H with combining macron]1 = [H with combining macron]2 = ⋯ = [H with combining macron]Na,c = 0,(11)
 
image file: c8cp05169k-t3.tif(12)
where image file: c8cp05169k-t4.tif and image file: c8cp05169k-t5.tif.

Note that image file: c8cp05169k-t6.tif is a much smaller matrix than G; thus, solving eqn (11) and (12) is much less expensive than solving eqn (2) and (3). In fact, for a database of approximately 260[thin space (1/6-em)]000 H216O transitions, the traditional MARVEL and the constrained MARVEL codes were executed with fixed uncertainties. For the latter procedure, transitions were divided into eight blocks, indexed with −9, −8, −7, −6, −5, −4, −3, and −2, which contained 1, 9, 45, 666, 8304, 71[thin space (1/6-em)]990, 174[thin space (1/6-em)]965, and 3626 lines, respectively. The computational time concerning the older and the newer approaches was 17.8 and 4.5 s on a single processor, respectively, which corresponds to a speed-up of about four. Nevertheless, in contrast to the G matrix, image file: c8cp05169k-t7.tif is not diagonally dominant; thus, a pivoting strategy must be employed to yield the solution of eqn (11) and (12). Once [H with combining macron] is determined, Ē can easily be calculated by means of eqn (9).

Utilizing eqn (11) and (12), a divide-and-conquer-style algorithm (blockMARVEL procedure) was designed, whose successive steps are listed in Appendix B. During this refinement, (a) lines are divided into blocks according to the orders of magnitude of the associated RSUs, (b) image file: c8cp05169k-t178.tif is extended block by block through linking the transitions from the upcoming block to this subnetwork, and (c) the uncertainties of the added lines are adjusted to achieve consistency within image file: c8cp05169k-t179.tif. At the end of this process, accurate constrained empirical energy levels and reliable wavenumber uncertainties are obtained, provided that (a) all the segments are well calibrated, and (b) the CD relations are synchronized (see the next two subsections).

2.4.3 Accuracy-based recalibration (ABR). Having the constrained energy levels derived, each ill-calibrated source segment, for which the average of the residuals related to its transitions significantly differs from zero, has to be recalibrated.

To recalibrate the segments, they have to be placed into the R(1), R(2),…, R(NR) recalibration classes, where NR is the number of recalibration classes. This is achieved the following way: (a) those segments which must be recalibrated with the same factor are grouped into the same class, and (b) all the segments for which simple recalibration (recalibration with a single factor) is not permitted, are distributed into separate classes. For the R(i) class, a regional subnetwork, image file: c8cp05169k-t180.tif, which contains the non-bridge lines of the segments in R(i), is introduced. To assign a recalibration factor to R(i), the following objective function should be optimized:

 
image file: c8cp05169k-t8.tif(13)
where (a) pj is the index of the jth line in image file: c8cp05169k-t181.tif, (b) wreg,j = δreg,j−2, (c) δreg,j is an approximate uncertainty of the pjth transition during recalibration, and (d) Nreg is the number of lines in image file: c8cp05169k-t182.tif. The best estimate of the recalibration factor, fbest, can be expressed as
 
image file: c8cp05169k-t9.tif(14)

Starting from proper initial values, the δreg,j uncertainties are iteratively adjusted to provide an fbest value satisfying δreg,j ≤ |Δreg,j|, where Δreg,j = fbestσpj[small sigma, Greek, macron]pj for all 1 ≤ jNreg. If (a) the recalibration of R(i) is permitted, (b) Nreg is sufficiently large, and (c) the fbest parameter significantly reduces the distance of the residuals from zero in image file: c8cp05169k-t183.tif, then fbest is used as the recalibration factor of the R(i) recalibration class, otherwise R(i) is not recalibrated.

Eqn (14) leads to an improved recalibration technique, called accuracy-based recalibration (ABR). Its sequential steps are presented in Appendix C. It is planned that the intMARVEL code will handle input recalibration factors to speed up its running.

If there are no segments that can be recalibrated, then our SN is fully recalibrated. In the case that the investigated SN is not recalibrated fully, the blockMARVEL analysis is re-executed (only one more time) at the termination of the ABR method.

2.4.4 Local synchronization procedure. It may occur that those constrained empirical energy levels which were determined only by very few transitions during the blockMARVEL refinement must be substituted with more dependable energy values using the CD relations of the energy levels. For the ith energy level, a local subnetwork, image file: c8cp05169k-t184.tif is defined, where all the non-bridge transitions incident to this energy level are included. To calculate the best energy value, ebest, available from the lines of image file: c8cp05169k-t185.tif, the following objective function is minimized:
 
image file: c8cp05169k-t10.tif(15)
where (a) ej is the trial energy of the jth line in image file: c8cp05169k-t186.tif, (b) image file: c8cp05169k-t187.tif, (c) δloc,j is an approximate uncertainty of the jth transition inserted into image file: c8cp05169k-t188.tif, and (d) image file: c8cp05169k-t189.tif is the number of lines in image file: c8cp05169k-t190.tif. The ej energy is specified as
 
image file: c8cp05169k-t11.tif(16)
where uj is the index of the jth line in image file: c8cp05169k-t191.tif. Similarly to fbest in eqn (14), the ebest value is expanded in a closed form:
 
image file: c8cp05169k-t12.tif(17)

As in Section 2.4.3, an iterative refinement can be performed for the δloc,j uncertainties for which reasonable guesses are employed. Obviously, the final ebest value should fulfill the relation δloc,j ≤ |Δloc,j|, where Δloc,j = ejebest, for all 1 ≤ jNloc.

If (a) Nloc is sufficiently large, and (b) the |Δuj| absolute residuals are systematically larger than the corresponding |Δloc,j| values, then (a) the ith energy level is synchronized with image file: c8cp05169k-t192.tif (i.e., Ēi is substituted with ebest), (b) all the transitions of image file: c8cp05169k-t193.tif are excluded from image file: c8cp05169k-t194.tif whose recalculated fitting defects are positive, and (c) those lines of image file: c8cp05169k-t195.tif which were incorrectly excluded from image file: c8cp05169k-t196.tif are reincluded in image file: c8cp05169k-t197.tif. Based on eqn (17), a so-called local synchronization procedure is constructed. The stages of this algorithm are detailed in Appendix D.

If there is no energy level left which should be synchronized with its image file: c8cp05169k-t198.tif, then the SN is said to be perfectly synchronized. If our SN is not perfectly synchronized, the blockMARVEL procedure has to be repeated upon completion of the local synchronization process.

2.4.5 Full network characterization (FNC). As the closing step of the intMARVEL procedure, the energy levels, the transitions, and the segments are given quality labels via the so-called full network characterization (FNC) technique. During this step, described in detail in Appendix E, (a) the energy levels are equipped with uncertainties, dependability grades, and useful connectivity parameters, (b) the transitions are rated using the grades of their upper and lower energy levels, and (c) segments are associated with statistical data concerning their wavenumber ranges and their accuracy. The FNC completes the intMARVEL process.

3 Experimental data sources

An almost complete set of data sources, available up to 2013, about recorded and assigned high-resolution spectra of H216O is given in the IUPAC compilation, ref. 18. Since then, partly helped by the data of ref. 18, results of a number of new spectroscopic measurements and a number of new analyses have been published on H216O, some highly relevant for this study.46,47,63,99–103 Note that a complete update of the IUPAC Part III paper18 is in progress.108

In Table 2 segments of experimental sources, old and new, are collected reporting highly accurate measured transitions, defined as having ESU ≤ 10−4 cm−1. These are the segments utilized during this study. For the experimental data sources we follow the naming convention of a IUPAC study.16 The representation of the segment accuracies can be found in Table 2. It is comfortable to note that good agreement is seen among the ESU, RSU, ASU, and MSU values.

Table 2 H216O data source segments employed during this work and some of their most important characteristicsa
Segment tag Range/cm−1 A/V/E ESU/cm−1 RSU/cm−1 ASU/cm−1 MSU/cm−1
a Tags denote experimental data-source segments used in this study. The column ‘Range’ indicates the range corresponding to validated wavenumber entries within the experimental linelist. ‘A/V/E’ is an ordered triplet with A = the number of assigned transitions in the source segments, V = the number of validated transitions, and E = the number of exploited transitions (see also Appendix E). In the heading of this table, ESU, RSU, ASU, and MSU designate the estimated, the refined, the average, and the maximum segment uncertainties, respectively. Rows of this table are arranged in the order of the ESUs.
69Kukolich48 0.74168–0.74168 1/1/1 2.000 × 10−9 2.000 × 109 2.000 × 10−9 2.000 × 10−9
09CaPuHaGa58 10.715–20.704 7/7/7 2.000 × 10−8 2.000 × 10−8 2.000 × 10−8 2.000 × 10−8
71Huiszoon53 6.1146–6.1146 1/1/1 4.000 × 10−8 4.000 × 10−8 4.000 × 10−8 4.000 × 10−8
95MaKr59 18.577–18.577 1/1/1 7.000 × 10−8 7.000 × 10−8 9.475 × 10−8 9.475 × 10−8
06GoMaGuKn54 6.1146–18.577 12/12/11 2.000 × 10−7 1.145 × 10−7 8.827 × 10−8 2.724 × 10−7
18KaStCaCa102 7164.9–7185.6 8/8/8 4.000 × 10−7 4.000 × 10−7 4.000 × 10−7 4.000 × 10−7
09CaPuBuTa96 36.604–53.444 9/9/9 5.000 × 10−7 5.000 × 10−7 5.000 × 10−7 5.000 × 10−7
11Koshelev64 25.085–25.085 1/1/1 5.000 × 10−7 5.000 × 10−7 5.000 × 10−7 5.000 × 10−7
07KoTrGoPa57 10.715–12.682 3/3/3 6.000 × 10−7 8.507 × 10−7 8.584 × 10−7 1.725 × 10−6
83HeMeLu65 13.013–32.954 7/7/7 7.000 × 10−7 7.000 × 10−7 7.000 × 10−7 7.000 × 10−7
83MeLuHe66 16.797–32.954 5/5/5 7.000 × 10−7 7.000 × 10−7 7.000 × 10−7 7.000 × 10−7
71StBe49 0.74168–6.1146 3/3/3 9.000 × 10−7 1.941 × 10−6 5.186 × 10−6 1.241 × 10−5
87BaAlAlPo77 14.199–19.077 6/6/5 1.000 × 10−6 1.000 × 10−6 4.244 × 10−6 2.046 × 10−5
95MaOdIwTs60 18.577–162.44 139/138/135 1.000 × 10−6 1.281 × 10−6 2.520 × 10−6 3.638 × 10−5
18ChHuTaSu47 12.622–12.665 6/6/6 1.000 × 10−6 1.000 × 10−6 1.000 × 10−6 1.000 × 10−6
54PoSt51 0.74169–0.74169 1/1/1 2.000 × 10−6 4.673 × 10−6 5.142 × 10−6 5.142 × 10−6
80Kuze75 0.40057–4.0026 5/5/4 2.000 × 10−6 2.000 × 10−6 9.021 × 10−6 3.440 × 10−5
91AmSc79 8.2537–11.835 5/5/5 2.000 × 10−6 2.000 × 10−6 2.000 × 10−6 2.000 × 10−6
97NaLoInNo88 118.32–119.07 5/5/4 2.000 × 10−6 2.000 × 10−6 6.032 × 10−6 2.216 × 10−5
91PeAnHeLu80 4.6570–19.804 30/30/30 3.000 × 10−6 3.000 × 10−6 3.907 × 10−6 1.652 × 10−5
83BuFeKaPo62 16.797–21.545 5/5/5 4.000 × 10−6 2.316 × 10−6 2.120 × 10−6 5.070 × 10−6
95Pearson85 4.3300–17.220 9/8/8 4.000 × 10−6 4.000 × 10−6 4.000 × 10−6 4.000 × 10−6
06MaToNaMo95 28.685–165.31 104/104/102 4.000 × 10−6 1.745 × 10−6 2.637 × 10−6 3.066 × 10−5
12YuPeDrMa98 9.8572–90.767 103/102/101 4.000 × 10−6 4.000 × 10−6 6.643 × 10−6 2.337 × 10−4
72LuHeCoGo55 6.1146–25.085 14/14/13 5.000 × 10−6 1.062 × 10−6 1.451 × 10−6 5.427 × 10−6
13YuPeDr63 17.690–67.209 182/182/181 5.000 × 10−6 5.000 × 10−6 5.992 × 10−6 6.152 × 10−5
11DrYuPeGu97 82.862–90.843 26/25/25 6.000 × 10−6 6.000 × 10−6 6.734 × 10−6 1.156 × 10−5
00ChPePiMa90 28.054–52.511 17/17/17 8.000 × 10−6 1.600 × 10−6 3.184 × 10−6 1.422 × 10−5
79HeJoMc74 0.072049–0.072049 1/1/1 1.000 × 10−5 1.000 × 10−5 1.000 × 10−5 1.000 × 10−5
51Jen50 0.74169–0.74169 1/1/0 2.000 × 10−5 2.000 × 10−5 2.000 × 10−5 2.000 × 10−5
72FlCaVa52 0.74168–25.085 7/7/0 2.000 × 10−5 1.266 × 10−5 1.509 × 10−5 3.015 × 10−5
06JoPaZeCo94 1485.1–1486.2 2/2/2 2.000 × 10−5 2.000 × 10−5 3.116 × 10−5 4.232 × 10−5
70StSt61 18.577–18.577 1/1/0 3.000 × 10−5 6.241 × 10−6 6.883 × 10−6 6.883 × 10−6
87BeKoPoTr78 7.7616–19.850 5/5/5 3.000 × 10−5 4.103 × 10−5 2.858 × 10−5 4.663 × 10−5
96Belov86 28.054–30.792 5/5/5 3.000 × 10−5 7.376 × 10−6 7.849 × 10−6 1.765 × 10−5
05HoAnAlPi92 212.56–594.95 166/164/60 3.000 × 10−5 1.640 × 10−5 2.668 × 10−5 3.643 × 10−4
13LuLiWaLi99 12.573–12.752 73/73/65 3.000 × 10−5 9.660 × 10−5 6.039 × 10−5 5.243 × 10−4
54KiGo56 6.1146–6.1146 1/1/0 5.000 × 10−5 3.710 × 10−5 4.437 × 10−5 4.437 × 10−5
96BrMa87 5206.3–5396.5 28/28/28 5.000 × 10−5 1.327 × 10−5 2.534 × 10−5 9.346 × 10−5
91Toth81 1072.6–2265.3 740/738/722 6.000 × 10−5 2.117 × 10−5 5.183 × 10−5 1.460 × 10−3
97MiTyKeWi89 2507.2–4402.8 935/935/190 6.000 × 10−5 1.639 × 10−4 4.774 × 10−4 1.050 × 10−2
95PaHo84 177.86–519.59 246/246/5 7.000 × 10−5 1.072 × 10−4 2.200 × 10−4 9.827 × 10−3
93Totha82 1316.1–4260.4 587/587/575 8.000 × 10−5 3.317 × 10−5 8.755 × 10−5 9.449 × 10−4
93Tothb83 1881.1–4306.7 1076/1076/1061 8.000 × 10−5 3.428 × 10−5 8.151 × 10−5 1.485 × 10−3
05Toth93 2926.5–7641.9 1895/1895/1832 8.000 × 10−5 6.031 × 10−5 1.396 × 10−4 4.447 × 10−3
85BrTo76 1323.3–1992.7 71/71/69 1.000 × 10−4 2.042 × 10−5 8.836 × 10−5 1.877 × 10−3
03ZoVa91 3010.2–4044.9 469/469/456 1.000 × 10−4 9.533 × 10−5 2.385 × 10−4 2.465 × 10−3
15SiHo101 7714.8–7919.9 71/71/62 1.000 × 10−4 1.000 × 10−4 2.266 × 10−4 4.984 × 10−3


There is one source which reports hyperfine transitions characterizing the spectra of ortho-H216O, 09CaPuHaGa.58 Not the hyperfine split transitions but their weighted averages, also reported in 09CaPuHaGa,58 have been utilized in this study.

Whenever possible, the available experimental information was used to determine the ESU values of the source segments. In some cases, when the accuracy of the source segment was not explicitly given, an educated guess had to be employed for the ESU value. When line-by-line uncertainties were reported, e.g., in the cases of 95MaOdIwTs,60 12YuPeDrMa,98 11DrYuPeGu,97 13YuPeDr,63 and 06MaToNaMo,95 we formed an average from these data. The extended MARVEL protocol requires that less accurate segments of sources are ignored from the analysis of more accurate data. In the case of the sources 93Toth,109 93Tothb,83 and 91Toth,81 only the transitions given with five digits after the decimal point were included in our initial dataset.

Our extMARVEL refinements indicated that certain sources, namely 82KaJoHo,110 83Guelachv,111 14ReOuMiWa,100 17MoMiKaBe,46 and 18MiMoKaKa,103 are not sufficiently accurate for the purposes of the present study. Therefore, the transitions these sources contain were omitted from our analysis.

4 Results and discussion

Let us start by providing some statistical information about the present analysis of the H216O measured transitions based on the highly accurate data source segments reported in Table 2. The number of transition entries in the leading subnetwork is 3099 and 3988 for ortho- and para-H216O, respectively. As to transitions of A+/A grade (the detailed definition of the grades of the empirical energy levels determined in this study is given in Appendix E), their number is 462/650 for ortho/para-H216O. The number of lines which had to be excluded from our extMARVEL analysis from the source segments selected is only 8, three of which are in the THz region (vide infra). For ortho-H216O, the number of energy levels determined is 725, out of which 97 have A+/A grades. For para-H216O, the extMARVEL analysis resulted in 857 energy levels, of which 117 received A+/A grades. The largest J value, Jmax, is 17/18 for para/ortho-H216O. The range of highly accurate H216O energy levels is quite extended, it is 0–3000 cm−1 for para-H216O and somewhat narrower, 0–1800 cm−1, for ortho-H216O.

One of the important practical results of this study, a set of highly accurate empirical energy levels of H216O, is reported in Table 3. Only empirical energy levels of grade A+ and A quality are included in Table 3. The highly accurate rovibrational energy levels belong to the vibrational states (v1v2v3) = (0 0 0), (0 1 0), and (0 2 0), where v1, v2, and v3 stand for the symmetric stretch, bend, and antisymmetric stretch vibrational quantum numbers. All energy levels reported have at least an 8-digit accuracy, often considerably better (up to 10 digits of accuracy). Jmax is 13/14 for the most dependable ortho/para rovibrational energy levels presented in Table 3.

Table 3 Accurate empirical (extMARVEL) energy levels, and their assignments, for para-H216O (first four columns) and ortho-H216O (last four columns), of grade A+ and A quality. All of the energy levels are of grade A+ quality, except those with an asterisk, which are of grade A quality
Energy/cm−1 (v1v2v3)JKa,Kc Energy/cm−1 (v1v2v3)JKa,Kc Energy/cm−1 (v1v2v3)JKa,Kc Energy/cm−1 (v1v2v3)JKa,Kc
0 (0 0 0)00,0 1806.671529(30) (0 0 0)131,13 0 (0 0 0)10,1 1765.248467(64) (0 0 0)88,1
37.13712384(68) (0 0 0)11,1 1810.583280(79) (0 0 0)97,3 18.57738488(39) (0 0 0)11,0 1782.875649(18) (0 0 0)130,13
70.09081349(65) (0 0 0)20,2 1813.787601(17) (0 1 0)32,2 55.7020277(99) (0 0 0)21,2 1786.7935598(52) (0 0 0)97,2
95.1759380(31) (0 0 0)21,1 1817.451194(21) (0 1 0)40,4 111.1072838(60) (0 0 0)22,1 1789.429033(13) (0 0 0)113,8
136.1639195(12) (0 0 0)22,0 1843.029604(33) (0 0 0)114,8 112.967305(10) (0 0 0)30,3 1795.5407549(85) (0 1 0)32,1
142.2784859(10) (0 0 0)31,3 1875.461821(19) (0 0 0)106,4 149.5714542(92) (0 0 0)31,2 1797.802452(11) (0 1 0)41,4
206.301428(21) (0 0 0)32,2 1875.469719(18) (0 1 0)41,3 188.36200986(47) (0 0 0)32,1 1851.178614(11) (0 0 0)106,5
222.052757(13) (0 0 0)40,4 1907.451421(19) (0 1 0)33,1 201.04402869(42) (0 0 0)41,4 1875.213801(11) (0 0 0)114,7
275.497042(18) (0 0 0)41,3 1922.829071(12) (0 1 0)51,5 261.6242179(88) (0 0 0)33,0 1883.821404(23) (0 1 0)33,0
285.219339(10) (0 0 0)33,1 1922.901125(11) (0 1 0)42,2 276.56792527(55) (0 0 0)42,3 1884.221976(10) (0 1 0)42,3
315.77953341(82) (0 0 0)42,2 1960.207413(33) (0 0 0)122,10 301.5535473(94) (0 0 0)50,5 1896.972180(16) (0 1 0)50,5
326.62546666(58) (0 0 0)51,5 1985.784894(12) (0 0 0)115,7 358.722528(11) (0 0 0)43,2 1938.712538(10) (0 0 0)123,10
383.842515(13) (0 0 0)43,1 2005.917050(16) (0 1 0)43,1 375.663155(11) (0 0 0)51,4 1975.200945(19) (0 0 0)115,6
416.2087402(12) (0 0 0)52,4 2024.152654(39) (0 1 0)52,4 422.71630882(88) (0 0 0)52,3 1977.068674(18) (0 1 0)51,4
446.696589(13) (0 0 0)60,6 2041.780551(28) (0 1 0)60,6 423.4579912(56) (0 0 0)61,6 1981.0213341(95) (0 1 0)43,2
488.134178(15) (0 0 0)44,0 2042.374098(77) (0 0 0)132,12 464.313340(12) (0 0 0)44,1 1986.010725(72) (0 0 0)98,1
503.9681027(59) (0 0 0)53,3 2054.368667(66) (0 0 0)107,3 485.01769568(72) (0 0 0)53,2 2018.516215(38) (0 0 0)131,12
542.905778(13) (0 0 0)61,5 2105.867908(44) (0 0 0)123,9 529.117030(21) (0 0 0)62,5 2018.958981(16) (0 1 0)61,6
586.4791835(77) (0 0 0)71,7 2126.407724(37) (0 1 0)53,3 562.449182(22) (0 0 0)70,7 2030.174344(14) (0 1 0)52,3
602.7734936(28) (0 0 0)62,4 2129.618682(26) (0 1 0)44,0 586.546806(40) (0 0 0)54,1 2030.550848(17) (0 0 0)107,4
610.114430(14) (0 0 0)54,2 2142.597661(55) (0 0 0)116,6 625.184327(15) (0 0 0)63,4 2101.157030(17) (0 0 0)124,9
661.5489133(67) (0 0 0)63,3 2146.263726(36) (0 1 0)61,5 680.4196881(12) (0 0 0)71,6 2105.804845(43) (0 1 0)44,1
709.608213(12) (0 0 0)72,6 2205.652716(69) (0 0 0)124,8 718.281923(24) (0 0 0)55,0 2106.699939(14) (0 1 0)53,2
742.073025(29) (0 0 0)55,1 2211.1906371(89) (0 1 0)62,4 720.368303(20) (0 0 0)81,8 2120.251898(17) (0 0 0)116,5
744.063661(23) (0 0 0)80,8 2248.064567(84) (0 0 0)133,11 732.93041648(96) (0 0 0)64,3 2137.4916811(42) (0 1 0)62,5
757.78018789(88) (0 0 0)64,2 2300.685002(61) (0 0 0)125,7 758.6154628(87) (0 0 0)72,5 2223.090456(41) (0 0 0)132,11
816.694236(52) (0 0 0)73,5 2321.813015(61) (0 0 0)117,5 818.562227(10) (0 0 0)73,4 2228.068156(19) (0 1 0)54,1
882.890327(21) (0 0 0)81,7 2327.883775(49) (0 0 0)141,13 861.805844(21) (0 0 0)82,7 2230.489516(24) (0 0 0)108,3
888.632650(14) (0 0 0)65,1 2399.165477(17) (0 1 0)64,2 864.804377(26) (0 0 0)65,2 2247.917879(18) (0 1 0)63,4
920.210001(25) (0 0 0)91,9 2426.19618(21) (0 0 0)134,10 896.373983(24) (0 0 0)90,9 2251.578491(52) (0 0 0)125,8
927.743902(11) (0 0 0)74,4 2522.261331(64) (0 0 0)118,4* 907.442739(12) (0 0 0)74,3 2285.935851(16) (0 1 0)71,6
982.911714(13) (0 0 0)82,6 2613.104573(35) (0 0 0)127,5* 982.321572(21) (0 0 0)83,6 2298.111383(77) (0 0 0)117,4
1045.0583403(41) (0 0 0)66,0 2670.789689(15) (0 1 0)83,5 1021.2635806(10) (0 0 0)66,1 2304.119654(21) (0 0 0)142,13
1050.157663(19) (0 0 0)83,5 2748.099560(76) (0 0 0)136,8* 1036.0410822(48) (0 0 0)75,2 2313.872430(23) (0 1 0)81,8
1059.646655(15) (0 0 0)75,3 2920.132087(14) (0 1 0)85,3 1055.285214(19) (0 0 0)91,8 2368.7982056(85) (0 1 0)72,5
1080.385444(19) (0 0 0)92,8 1090.755547(36) (0 0 0)101,10 2374.587138(32) (0 1 0)64,3
1114.532248(50) (0 0 0)100,10 1098.914168(13) (0 0 0)84,5 2390.929063(14) (0 0 0)133,10
1131.775573(22) (0 0 0)84,4 1178.127134(11) (0 0 0)92,7 2410.006047(55) (0 0 0)126,7
1216.189769(53) (0 0 0)76,2 1192.4001374(68) (0 0 0)76,1 2439.0809341(94) (0 1 0)73,4
1216.231260(21) (0 0 0)93,7 1231.3723839(61) (0 0 0)85,4 2471.371510(47) (0 1 0)82,7
1255.911545(14) (0 0 0)85,3 1259.12474186(98) (0 0 0)93,6 2498.470791(42) (0 0 0)118,3
1293.018138(13) (0 0 0)101,9 1269.8396771(11) (0 0 0)102,9 2509.998800(16) (0 0 0)134,9
1327.117604(24) (0 0 0)111,11 1303.315623(28) (0 0 0)110,11 2527.689134(62) (0 0 0)143,12
1340.884880(14) (0 0 0)94,6 1336.440968(13) (0 0 0)94,5 2548.3448242(97) (0 1 0)74,3
1394.814159(54) (0 0 0)77,1 1371.0198349(43) (0 0 0)77,0 2589.005435(18) (0 0 0)127,6
1411.641890(12) (0 0 0)86,2 1387.8170727(57) (0 0 0)86,3 2605.540121(60) (0 0 0)135,8
1437.968586(14) (0 0 0)102,8 1422.333875(12) (0 0 0)103,8 2606.398257(16) (0 1 0)83,6
1474.980787(12) (0 0 0)95,5 1453.502995(12) (0 0 0)95,4 2607.474510(66) (0 0 0)151,14
1525.135991(43) (0 0 0)112,10 1501.053523(22) (0 0 0)111,10 2664.285572(38) (0 1 0)91,8*
1538.149477(27) (0 0 0)103,7 1534.053335(22) (0 0 0)121,12 2700.372781(12) (0 1 0)75,2
1557.844418(26) (0 0 0)120,12 1557.541614(18) (0 0 0)104,7 2794.603744(30) (0 1 0)92,7
1590.69071(14) (0 0 0)87,1 1566.89570(11) (0 0 0)87,2 2894.450561(85) (0 0 0)145,10*
1616.453054(22) (0 0 0)104,6 1594.762769(66) (0 1 0)10,1 2895.8386511(81) (0 1 0)85,4
1631.245487(18) (0 0 0)96,4 1607.588639(18) (0 0 0)96,3 3368.954987(18) (0 2 0)32,1
1634.967095(11) (0 1 0)11,1 1616.711503(44) (0 1 0)11,0 4383.251954(52) (0 2 0)66,1
1664.964587(11) (0 1 0)20,2 1653.267093(20) (0 1 0)21,2


4.1 Highly accurate sources

Fig. 2 shows the unsigned residuals (differences between the observed wavenumbers and their extMARVEL estimates) for those segments having RSU < 10−5 cm−1. Clearly, the extMARVEL refinement retained the high accuracy of the original measurements.
image file: c8cp05169k-f2.tif
Fig. 2 Unsigned residuals (differences between the observed and the calculated wavenumbers) for the source segments with RSU ≈ 10−7 cm−1 and RSU ≈ 10−6 cm−1, where RSU stands for “refined segment uncertainty”. Residuals below 10−9 cm−1 are not plotted as these are artificial results.

Some of the extMARVEL rovibrational energy levels may have an accuracy considerably lower than that of the transitions determining it. The simplest example that highlights the difficulties of providing uncertainties to energy levels is as follows. Let us have two separate 4-cycles, both formed by highly accurate transitions, connected by a bridge of lower measured accuracy. If one of the 4-cycles contains a core of the leading subnetwork, then all its energy levels have the same high accuracy as the measured transitions. However, this is not true for the other 4-cycle, where the transitions are known with high accuracy, but the overall accuracy of the energy levels is determined by the accuracy of the bridge. This also means that if the transitions are reconstructed from MARVEL energy levels, they may have considerably higher uncertainties than the directly measured transitions. This characteristics of the MARVEL protocol cannot be easily circumvented without new, accurate measurements. Thus, in MARVEL determinations of empirical rovibronic energy levels it can happen that the reconstructed lines should have a better uncertainty than indicated by the energy-level-based uncertainties. For the present study this means that uncertainties of transitions determined from uncertainties of rovibrational energy levels may not be fully realistic, they may provide inflated uncertainties, which must be considered when these transitions are used in an application.

For para-H216O one of the most important lines is the 51,5 ← 42,2 pure rotational transition at about 325.153 GHz (this is how this transition is usually reported) within the (0 0 0) ground vibrational state. Our final 8-digit-accuracy determination gives 325[thin space (1/6-em)]152[thin space (1/6-em)]899(2) kHz, based basically upon the experimental result of 06GoMaGuKn,54 but including the effects of all other relevant measured transitions.

Next, let us turn our attention to the THz region. In the THz region the following transitions had to be excluded from the extMARVEL analysis (all wavenumbers in cm−1): 39.003 0,98 82.638 9,60 and 86.467 2.97 95MaOdIwTs60 and 06MaToNaMo95 are the two most dominant sources which report accurately measured purely rotational transitions in the THz region. The unsigned residuals, i.e., the unsigned differences of the observed and the extMARVEL-predicted wavenumbers, are plotted in Fig. 3. The estimated ESU of the transitions in 95MaOdIwTs60 is 1 × 10−6 cm−1, i.e., 30 kHz, while the RSU, according to our study, is slightly higher, 38 kHz. Fig. 3 shows that extMARVEL is basically able to confirm the claimed accuracy of the measurements when they are refined together with all the other relevant source segments. Only very few transitions have absolute residuals higher than 100 kHz. This means that the extMARVEL procedure is able to retain the accuracy of the original measurements.


image file: c8cp05169k-f3.tif
Fig. 3 Unsigned residuals for 95MaOdIwTs60 and 06MaToNaMo,95 denoted with blue and red dots, respectively. Residuals below 1 kHz are not plotted as these are artificial results.

It should prove beneficial for future studies to know more and more highly accurate transitions in a wider and wider spectral range. For example, for ortho-H216O, two often quoted pure rotational transitions are 321.23 GHz for 102,9 ← 93,6 on (0 0 0) and 336.23 GHz for 52,3 ← 61,6 on (0 1 0). The extMARVEL counterpart of the first transition determined in this study is 321[thin space (1/6-em)]225[thin space (1/6-em)]677(44) kHz, while the frequency of the second transition is determined to be 336[thin space (1/6-em)]228[thin space (1/6-em)]131(140) kHz, both of which can be considered dependable due to the fact that they are based on energy levels of A+ grade.

4.2 IAU lines

For all 13 H216O lines recommended by the International Astronomical Union (IAU) empirical estimates have been derived in this study. These estimates are listed in Table 1. Note that all the IAU-selected frequencies are below 1 THz. This is due partly to the fact that when these lines were selected THz spectroscopy measurements were rarely available.

The data presented in Table 1 allows an assessment of the extMARVEL frequencies in relation to those deemed to be most important by IAU. As evident from the last column of Table 1, all these lines have been determined experimentally via multiple, independent, highly accurate measurements, contributing to an improved confidence in the astronomically important frequencies.

4.3 Comparison with the energy levels of Lanquetin et al.

Fig. 4 shows the unsigned differences of the extMARVEL and the IUPAC energy levels with respect to the experimental JPL data, which are the same as those given by Lanquetin et al.45
image file: c8cp05169k-f4.tif
Fig. 4 Unsigned deviations of the extMARVEL (this work) and IUPAC Part III energy levels [see eqn (18)],18 denoted with blue squares and red dots, respectively, referenced to those determined by Lanquetin et al.45

The first observation one can make about Fig. 4 is that the extMARVEL protocol is a significant improvement over the standard MARVEL protocol employed to obtain the IUPAC Part III energy levels.18 Furthermore, it is expected that the relation

 
image file: c8cp05169k-t13.tif(18)
should be (approximately) satisfied for the unsigned deviations (UD(M)i), where (a) ĒM,i is the energy of the ith energy level obtained in this paper (M = extMARVEL) and in the IUPAC Part III paper18 (M = IUPAC) with εM,i uncertainty, and (b) Ē01LaCoCa,i is the counterpart of ĒM,i estimated in the source 01LaCoCa45 with an ε01LaCoCa,i uncertainty. As a result, the corresponding weighted unsigned deviation (WUD), defined as
 
image file: c8cp05169k-t14.tif(19)
must not be larger than 1. These weighted unsigned deviations are plotted in Fig. 5.


image file: c8cp05169k-f5.tif
Fig. 5 Weighted unsigned deviations [see eqn (19)] of the extMARVEL (this work) and the IUPAC Part III18 energy levels, denoted with blue squares and red dots, respectively, from those obtained in 01LaCoCa.45

The substantial improvement achieved during the present study as compared to the IUPAC Part III study is obvious from Fig. 5. The relatively large WUDs related to the IUPAC Part III data (in fact, the uncertainties are smaller, by an order of magnitude, than the corresponding unsigned deviations), are attributed to the inaccuracy and the too optimistic energy uncertainties of some of the IUPAC Part III energy levels.

It is also worth examining whether the conditions

 
|Ē01LaCoCa,iĒextMARVEL,i| ≤ εextMARVEL,i(20)
are (approximately) satisfied. Accordingly, the relative unsigned deviations (RUD),
 
image file: c8cp05169k-t15.tif(21)
are formed and illustrated in Fig. 6. When these deviations are larger than 1, the 01LaCoCa45 energies are not within the ĒextMARVEL,i ± εextMARVEL,i intervals. The protocol used in this study for the calculation of the uncertainties of the rovibrational energy levels is reliable; thus, relative deviations greater than 1 indicate that our energy levels are more accurate than those derived by Lanquetin et al.,45 as also supported by the data of Table 1.


image file: c8cp05169k-f6.tif
Fig. 6 Relative unsigned deviations [see eqn (21)] of the energy levels found in 01LaCoCa45 from the extMARVEL energy levels.

Fig. 7 shows the 01LaCoCa45 uncertainties relative to their extMARVEL counterparts. Clearly, the present study represents a significant improvement over the uncertainties of the energies given in the source 01LaCoCa.45


image file: c8cp05169k-f7.tif
Fig. 7 Uncertainties of the energy levels in 01LaCoCa45 relative to their extMARVEL counterparts.

Finally, let us support the conclusions based on the figures with some statistical data related to the UD, WUD, and RUD values. The average and the maximum UD decreased from 4.4 × 10−4 and 2.3 × 10−3 to 1.1 × 10−4 and 5.2 × 10−4 cm−1, respectively. The average and the maximum WUD decreased from 2.21 to 0.59 and from 9.99 to 1.46, respectively. These substantial decreases in the deviations clearly prove that the extMARVEL treatment allows the full utilization of the most accurate spectroscopic measurements, as now we exceed the internal accuracy of the data presented by Lanquetin et al.45

5 Conclusions

During this study the standard Measured Active Rotational–Vibrational Energy Levels (MARVEL) algorithm11,12 has been extended, in order to improve its performance toward allowing the automatic determination of highly accurate empirical energy levels, matching or exceeding the accuracy of the best underlying spectroscopic measurements. This is an especially important undertaking in the era of optical-frequency-comb spectroscopies, as they can consistently yield orders of magnitude more accurate transitions than traditional high-resolution spectroscopic techniques.

There are several important algorithmic changes introduced in this study, resulting in what we call the extended MARVEL (extMARVEL) protocol. It is worth reiterating these improvements one by one as they all contributed to increase the utility of the MARVEL analysis of experimental transitions.

First, unlike in the standard version, in the new, extended protocol MARVEL-type analyses are performed based on the use of groups of transitions blocked by their estimated experimental uncertainties. This requires that the user segments the input sources based on assumed uncertainties of the different groups of transitions (a line-by-line analysis yielding individual initial line uncertainties would be ideal but this appears to be unrealistic). Second, the inversion and weighted least-squares refinement procedure is now based on sequential addition of blocks of decreasing accuracy. Wavenumber estimates determined in a given block are not allowed to be changed by the inclusion of less accurate measurements. Third, spectroscopic cycles are introduced during the refinement process. This is a particularly important advancement as due to the law of energy conservation13 one can detect straightforwardly the best as well as the worst transitions in the collated set of experimentally measured and assigned transitions. Fourth, automated recalibration of the segments requiring this adjustment is performed. As shown before,18 MARVEL is able to perform this job quite reliably. Fifth, synchronization of the combination difference relations is performed to reduce residual uncertainties in the resulting dataset of empirical (MARVEL) energy levels. Sixth, an improved classification scheme, providing seven grades decreasing in quality from A+ to D, of the empirical energy levels is introduced, the grading is based on the assumed accuracy and dependability of the energy levels. This grading of the energy levels directly results in a grading of the measured transitions, as the lower grade of the lower and the upper energy level is attached to the line they define.

We used H216O as the molecule of choice for our feasibility study testing the extMARVEL protocol. Since the International Astronomical Union selected 13 water lines as highly important for astrophysical applications, it is important to note that all these transitions are reproduced now perfectly well by the extMARVEL protocol. From an application point of view it is also important to note that all these transitions are measured extremely accurately, though they are not part of cycles of similar high accuracy. This calls for new measurements, most likely involving optical-frequency-comb spectroscopic techniques. For ortho- and para-H216O, we determined 97 and 117 energy levels with grades of at least A quality, meaning an accuracy better than 10−4 cm−1. The range covered by these highly accurate rovibrational energies is quite substantial, 0–3000 cm−1 for para-H216O and 0–1800 cm−1 for ortho-H216O.

The present dataset of highly accurate energy levels is larger than the experimental energy-level dataset maintained at JPL, which is the same as the one published by Lanquetin et al.45 Furthermore, on average our data have an accuracy about an order of magnitude better than the data of Lanquetin et al.45 Note also that the highly accurate energy levels are part of a large number of cycles, as clear from Fig. 8.


image file: c8cp05169k-f8.tif
Fig. 8 The para- and ortho-H216O principal components of the experimental spectroscopic network of highly accurate measured transitions, including only those energy levels and transitions which have a grade of at least A quality. Note that the energy levels were determined by a much larger number of transitions, all those which have an uncertainty of better than 10−4 cm−1.

Conflicts of interest

There are no conflicts to declare.

Appendix A: stages of the HSP method (see Section 2.4.1)

S.1 for all image file: c8cp05169k-t16.tif: set image file: c8cp05169k-t17.tif, where
 (a) image file: c8cp05169k-t18.tif is the estimated segment uncertainty (ESU) of the image file: c8cp05169k-t19.tif segment,
 (b) image file: c8cp05169k-t20.tif is the (actual) refined segment uncertainty (RSU) of image file: c8cp05169k-t21.tif, and
 (c) image file: c8cp05169k-t22.tif is the number of segments in the SN;
S.2 for all 1 ≤ inmh, where nmh (= 10) is the maximum number of HSP iterations:
S.2.1 save the indices of the segments into image file: c8cp05169k-t23.tif so that image file: c8cp05169k-t24.tif is enforced for all image file: c8cp05169k-t25.tif;
S.2.2 for all image file: c8cp05169k-t26.tif:
  S.2.2.1 set image file: c8cp05169k-t27.tif;
  S.2.2.2 initialize RSU[s]prev as RSU[s];
  S.2.2.3 if n[s] = 0, then continue S.2.2, where n[s] is the number of rovibrational lines in image file: c8cp05169k-t199.tif and s;
  S.2.2.4 create a image file: c8cp05169k-t200.tif perturbation subnetwork[I] with its image file: c8cp05169k-t201.tif participation matrix from image file: c8cp05169k-t202.tif as follows: image file: c8cp05169k-t28.tif if Pm = 1 and either image file: c8cp05169k-t29.tif or image file: c8cp05169k-t30.tif, else image file: c8cp05169k-t31.tif for all 1 ≤ mNT, where
   (a) image file: c8cp05169k-t32.tif is the mth segment index for which image file: c8cp05169k-t33.tif includes the mth transition, and
   (b) ϕper ∈ (0,1] is the perturbation factor (ϕper = 0.1 is the default value);
  S.2.2.5 construct a cycle basis (CB) for image file: c8cp05169k-t203.tif, designated with image file: c8cp05169k-t204.tif, using the breadth-first search (BFS) method;112
  S.2.2.6 calculate the reduced discrepancy[II] of every image file: c8cp05169k-t205.tif as
 
image file: c8cp05169k-t34.tif (22)
   where
   (a) n[s]C is the number of transitions in C from s, and
   (b) image file: c8cp05169k-t206.tif is the discrepancy of C;
  S.2.2.7 collect the image file: c8cp05169k-t207.tif representative cycles, for which RD[s]C > ϕhinRSU[s], into the set ρ, where ϕhin (= 0.01)[III] is the reduction hindrance factor;
  S.2.2.8 if |ρ| = 0, then continue S.2.2, otherwise determine the average and maximum reduced discrepancy of the representative cycles for s:
 
image file: c8cp05169k-t35.tif (23)
 
MRD[s] = maxCρRD[s]C, (24)
  where |ρ| is the cardinality of the set ρ;
  S.2.2.9 if MRD[s] > ϕterRSU[s], where ϕter (= 200) is the termination factor, then exit from intMARVEL;[IV]
  S.2.2.10 if n[s]nb < ⌊ϕrepn[s]⌋,[V] then set RSU[s] = RSU[s]prev, otherwise use RSU[s] = max(ARD[s],ϕsepESU[s]), where
   (a) ⌊⌋ is the floor operation,
   (b) n[s]nb is the number of non-bridge transitions from s in image file: c8cp05169k-t208.tif, and
   (c) ϕrep (= 0.1) and ϕsep (= 0.2) are the representativity and the separation factors, respectively;
S.2.3 if rmax < ϕconv, then break S.2, where
  (a) ϕconv (= 3) is the convergence factor,
  (b) image file: c8cp05169k-t36.tif, and
  (c) image file: c8cp05169k-t37.tif;
S.3 for all 1 ≤ jNT: in the case that image file: c8cp05169k-t38.tif, apply image file: c8cp05169k-t39.tif, else set image file: c8cp05169k-t40.tif, where
 (a) δ0,j is the initial uncertainty of the jth transition, and
 (b) ϕus (= 0.5) is the uncertainty scaling factor;
S.4 for all 1 ≤ jNT: if Pj = 1, then set image file: c8cp05169k-t41.tif, otherwise use Bj = 100,[VI] where Bj is the block index of the jth line;
S.5 set image file: c8cp05169k-t42.tif and image file: c8cp05169k-t43.tif;
S.6 identify the components of image file: c8cp05169k-t209.tif;
S.7 save the core(j)[VII] and comp(k) indices for all 1 ≤ jNc and 1 ≤ kNL, respectively, by means of eqn (4);
S.8 end procedure.
Notes:

IThe image file: c8cp05169k-t210.tif subnetwork allows the study of the lines arising from s and included in cycles where there is no transition from those s* segments with ESU[s*] > ESU[s].

IIThe reduced discrepancy of a cycle C related to s represents the inaccuracy of the lines from s in C. Note that nonpositive reduced discrepancies cannot be used for refinement purposes.

IIIIn what follows, the default values of the so-called control parameters, needed for the extMARVEL procedure, are mostly given in parentheses (see, e.g., ‘ϕhin (= 0.01)’ in stage S.2.2.7).

IVIn this case, the prior cleansing should be continued to decrease the reduced discrepancies.

VIf there are at least ⌊ϕrepn[s]⌋ lines from s in the cycles of image file: c8cp05169k-t211.tif, then its RSU is modified by the corresponding ARD value, else this RSU will remain unchanged in the given iteration.

VIIf the block index of a transition is 100, this line will not participate in the blockMARVEL refinement (see Appendix B).

VIIThe core indices are determined via a traditional MARVEL analysis, using eqn (4). To accelerate this identification process, a logical variable, uST, can be introduced: if uST = 1, then only lines related to a BFS-based ST of the LS are employed in eqn (4); otherwise, all the transitions of the LS are utilized in eqn (4). Obviously, uST = 0 is a more stable choice than uST = 1.

Appendix B: stages of the blockMARVEL refinement (see Section 2.4.2)

S.1 for all 1 ≤ iNL: set bi = 100, where bi is the block index[I] of the ith energy level;
S.2 set image file: c8cp05169k-t212.tif to an empty subnetwork, where image file: c8cp05169k-t44.tif for all 1 ≤ jNT;
S.3 assign Np,c = NL;
S.4 for all 1 ≤ iNL: set corep(i) = compp(i) = i and βi = 0;
S.5 for all 1 ≤ jNT: initialize δj as δ0,j;
S.6 for all image file: c8cp05169k-t45.tif:
S.6.1 construct a image file: c8cp05169k-t213.tif subnetwork of image file: c8cp05169k-t214.tif in the following fashion: if image file: c8cp05169k-t46.tif, then image file: c8cp05169k-t47.tif, otherwise image file: c8cp05169k-t48.tif for all 1 ≤ jNT;
S.6.2 if image file: c8cp05169k-t215.tif, then continue S.6;
S.6.3 find the components of image file: c8cp05169k-t216.tif;
S.6.4 archive the compa(1), compa(2),…,compa(NL) component indices of the energy levels in image file: c8cp05169k-t217.tif;
S.6.5 ‘infinite’ loop:
  S.6.5.1 solve eqn (11) and (12)[II] and determine Ē using eqn (9);
  S.6.5.2 for all 1 ≤ jNT with image file: c8cp05169k-t49.tif: calculate Δj and dj;
  S.6.5.3 search for image file: c8cp05169k-t50.tif;[III]
  S.6.5.4 if dmaxϕnoi, then break S.6.5, where ϕnoi (= 10−10) designates the numerical noise factor;
  S.6.5.5 for all 1 ≤ jNT: if image file: c8cp05169k-t51.tif and dj > ϕdiscdmax, then substitute δj with ϕinc|Δj|, where
   (a) ϕdisc (= 0.1) is the discrimination factor, and
   (b) ϕinc (= 1.1) denotes the increase factor;
  S.6.5.6 create a image file: c8cp05169k-t218.tif filtered subnetwork with its image file: c8cp05169k-t219.tif participation matrix as follows: if image file: c8cp05169k-t52.tif and image file: c8cp05169k-t53.tif, then set image file: c8cp05169k-t54.tif, otherwise use image file: c8cp05169k-t55.tif for all 1 ≤ jNT, where
  (a) ϕfil (= 10) is the filtration factor, and
  (b) ϕco (= 0.05 cm−1) is the cut-off factor;
  S.6.5.7 save the indices of the transitions in image file: c8cp05169k-t287.tif into l1, l2,, lNF in the order[IV,V] that either log10(dli) > log10(dlj) or log10(dli) = log10(dlj) and BliBlj holds for all 1 ≤ ijNF, where NF is the number of lines in image file: c8cp05169k-t220.tif (exclusion sort);
  S.6.5.8 for all 1 ≤ jNT: set image file: c8cp05169k-t56.tif to image file: c8cp05169k-t57.tif;
  S.6.5.9 for all 1 ≤ iimage file: c8cp05169k-t221.tif:
   S.6.5.9.1 identify the components of image file: c8cp05169k-t222.tif;
   S.6.5.9.2 if the lith line connects components in image file: c8cp05169k-t223.tif, then reset image file: c8cp05169k-t58.tif, otherwise assign Pli = 0 and Bli = 100;
  S.6.5.10 for all 1 ≤ jNT: set image file: c8cp05169k-t59.tif to image file: c8cp05169k-t60.tif;
  S.6.5.11 if image file: c8cp05169k-t224.tif, then set δj = δ0,j for every 1 ≤ jNT with image file: c8cp05169k-t61.tif, whereby image file: c8cp05169k-t225.tif is the trace of image file: c8cp05169k-t226.tif;
S.6.6 for all 1 ≤ iNL: whenever bi = 100 and corea(compa(i)) = core(comp(i)) together with image file: c8cp05169k-t227.tif, then set the bi index to image file: c8cp05169k-t62.tif, where
  (a) corea(j) is the jth core index of image file: c8cp05169k-t228.tif, and
  (b) image file: c8cp05169k-t229.tif is the jth component size;[VI]
S.6.7 replace Np,c with Na,c, where Na,c is the number of components in image file: c8cp05169k-t230.tif;
S.6.8 for all 1 ≤ iNp,c: overwrite corep(i) with corea(i);
S.6.9 for all 1 ≤ iNL: set compp(i) = compa(i) and βi = Ēi;
S.7 for all 1 ≤ jNT with Pj = 0:
S.7.1 recalculate Δj and dj;
S.7.2 set δj = max(δ0,j,ϕinc|Δj|);
S.8 end procedure.
Notes:

IAt the end of the blockMARVEL process, the bi block index denotes the image file: c8cp05169k-t63.tif value when (a) the ith energy level becomes a non-isolated node in image file: c8cp05169k-t231.tif, and (b) the component of this energy level contains at least two vertices, among which at least one core of image file: c8cp05169k-t232.tif can also be found. In the case that bi remains 100, the ith energy level became an isolated node of image file: c8cp05169k-t233.tif during the course of prior cleansing.

IIThe core indices of image file: c8cp05169k-t234.tif are set in analogy to eqn (4).

IIISince the residuals of the lines in image file: c8cp05169k-t235.tif cannot be changed, it is sufficient to restrict dmax to those lines satisfying image file: c8cp05169k-t64.tif.

IVAt this point, the values of the defects are the same as in stage S.6.5.2.

VFor convenience, steps S.6.5.7S.6.5.10 will be referred to as the exclusion procedure and denoted with image file: c8cp05169k-t236.tif in the remaining of this Appendix. During the exclusion procedure, lines are excluded one by one from image file: c8cp05169k-t237.tif so as to avoid increasing the number of components in image file: c8cp05169k-t238.tif.

VIIt is the number of energy levels in the jth component of image file: c8cp05169k-t239.tif.

Appendix C: stages of the ABR method (see Section 2.4.3)

S.1 find the bridges of the (whole) SN via the BFS method;
S.2 for all 1 ≤ iNT: if the ith transition is a bridge of the SN, then set image file: c8cp05169k-t240.tif, otherwise use image file: c8cp05169k-t241.tif;
S.3 for all 1 ≤ iNR: determine the image file: c8cp05169k-t65.tif values as
 
image file: c8cp05169k-t66.tif (25)
  where ℜj is the recalibration class index of the image file: c8cp05169k-t67.tif segment, for which R(ℜj) contains image file: c8cp05169k-t68.tif;
S.4 rearrange the recalibration classes such that image file: c8cp05169k-t69.tif should be satisfied for all 1 ≤ ijNR;[I]
S.5 set the image file: c8cp05169k-t70.tif flag[II] to 1;
S.6 for 1 ≤ iNR:
S.6.1 assign image file: c8cp05169k-t71.tif to R(mi);
S.6.2 set image file: c8cp05169k-t72.tif, where image file: c8cp05169k-t73.tif is the recalibration factor for image file: c8cp05169k-t74.tif;
S.6.3 in the case that simple recalibration is not permitted for image file: c8cp05169k-t75.tif, continue S.6;[III]
S.6.4 for all 1 ≤ jNT: if image file: c8cp05169k-t76.tif and image file: c8cp05169k-t77.tif, then set image file: c8cp05169k-t78.tif, otherwise use image file: c8cp05169k-t79.tif;[IV]
S.6.5 save the indices of the lines in image file: c8cp05169k-t242.tif into p1, p2,, pNreg;
S.6.6 if image file: c8cp05169k-t80.tif, then continue S.6;[III,V], where
  (a) image file: c8cp05169k-t81.tif is the number of transitions in the segments of image file: c8cp05169k-t82.tif,
  (b) μrec ≥ 2 is the recalibration margin with the default value of μrec = 5, and
  (c) ϕcdd(= 0.2) is the critical data-density factor;
S.6.7 for all 1 ≤ jNreg: initialize δreg,j as image file: c8cp05169k-t83.tif;
S.6.8 ‘infinite’ loop:[VI]
  S.6.8.1 determine fbest by means of eqn (14);
  S.6.8.2 for all 1 ≤ jNreg: calculate the Δreg,j residual and the dreg,j = |Δreg,j| − δreg,j defect;
  S.6.8.3 search for image file: c8cp05169k-t84.tif;
  S.6.8.4 if dreg,maxϕnoi, then break S.6.8;
  S.6.8.5 for all 1 ≤ jNreg: if dreg,j > ϕrddreg,max, where ϕrd(= 0.1) is the regional discrimination factor, then substitute δreg,j with ϕincδreg,j;
S.6.9 specify the recalibrated and non-recalibrated absolute median residuals (AMR) as follows:
 
image file: c8cp05169k-t85.tif (26)
 
image file: c8cp05169k-t86.tif (27)
S.6.10 whenever image file: c8cp05169k-t87.tif holds for image file: c8cp05169k-t88.tif, then continue S.6,[III,VII] where
  (a) ϕacc (= 0.8) is the acceptability factor, and
  (b) ϕass (= 3) is the assimilation factor;
S.6.11 reassign image file: c8cp05169k-t89.tif;
S.6.12 reset image file: c8cp05169k-t90.tif;
S.6.13 for 1 ≤ jNT with image file: c8cp05169k-t91.tif: replace σj with image file: c8cp05169k-t92.tif;[VIII]
S.6.14 for 1 ≤ jNreg: exchange δpj with δreg,j;
S.6.15 build an empty image file: c8cp05169k-t243.tif subnetwork in the following form: image file: c8cp05169k-t93.tif for all 1 ≤ jNT;[IX]
S.6.16 set Np,c = NL;
S.6.17 for every 1 ≤ jNL: assign bj = 100, corep(j) = compp(j) = j, and βj = 0;
S.6.18 for all image file: c8cp05169k-t94.tif:
  S.6.18.1 construct a image file: c8cp05169k-t244.tif subnetwork of image file: c8cp05169k-t245.tif according to the following scheme: if image file: c8cp05169k-t95.tif, then image file: c8cp05169k-t96.tif, else image file: c8cp05169k-t97.tif for all 1 ≤ jNT;
  S.6.18.2 if image file: c8cp05169k-t246.tif, then continue S.6.18;
  S.6.18.3 find the components of the image file: c8cp05169k-t247.tif subnetwork;
  S.6.18.4 save the compa(1), compa(2),…,compa(NL) component indices in image file: c8cp05169k-t248.tif;
  S.6.18.5 yield the solution for eqn (11) and (12);
  S.6.18.6 calculate Ē with the help of eqn (9);
  S.6.18.7 for each 1 ≤ jNL: if bj = 100, image file: c8cp05169k-t292.tif, and corea(compa(j)) = core(comp(j)), then set image file: c8cp05169k-t98.tif;
  S.6.18.8 overwrite Np,c with Na,c;
  S.6.18.9 for all 1 ≤ jNp,c: exchange corep(j) with corea(j);
  S.6.18.10 for all 1 ≤ jNL: set compp(j) = compa(j) and βj = Ēj;
  S.6.18.11 for all 1 ≤ jNT: update Δj;
S.6.19 for all 1 ≤ jNT: replace δj with max(δj,ϕinc|Δj|);
S.6.20 for all image file: c8cp05169k-t99.tif with R(ℜj) = image file: c8cp05169k-t100.tif:
  S.6.20.1 set image file: c8cp05169k-t101.tif;
  S.6.20.2 place the indices of the non-bridge transitions in s into q1, q2,…,qN[s], where N[s] is the number of non-bridge lines in s;
  S.6.20.3 define the recalibated median absolute residual (MAR) for the s segment as follows:
 
image file: c8cp05169k-t102.tif (28)
S.6.20.4 if ϕdecMAR[s]rec < RSU[s], where ϕdec (= 2) is the declination factor, then[X]
S.6.20.4.1 substitute RSU[s] with max(ϕdecMAR[s]rec, ϕsepESU[s]);
   S.6.20.4.2 for all 1 ≤ kNT with image file: c8cp05169k-t103.tif: set δ0,k = ϕusRSU[s] and Bk = ⌊log10(RSU[s])⌋;
S.7 end procedure.
Notes:

IThe recalibration classes are recalibrated in decreasing order of their minimum ESU values to reduce the distortion effects caused by highly uncertain spectral lines on the recalibration factors.

IIAt stage S.7, image file: c8cp05169k-t104.tif denotes whether image file: c8cp05169k-t249.tif is fully recalibrated, i.e., none of the recalibration classes are recalibrated during the ABR procedure.

IIIThese empirical conditions, namely steps S.6.3, S.6.6, and S.6.10, have to be violated so that the image file: c8cp05169k-t105.tif class can be recalibrated with the ABR algorithm.

IVThe bridges of the SN are not included in image file: c8cp05169k-t250.tif because they are reproduced with zero residuals.

VIf this condition is true, then there are too few lines in image file: c8cp05169k-t251.tif for the safe recalibration of image file: c8cp05169k-t106.tif.

VIThe fbest value is obtained from an iterative refinement scheme similar to that presented in Appendix B.

VIIIf the image file: c8cp05169k-t107.tif recalibration class is well calibrated, the condition of step S.6.10 must be satisfied.

VIIIIf image file: c8cp05169k-t108.tif is recalibrated, then the wavenumbers of its lines should be multiplied with image file: c8cp05169k-t109.tif.

IXThe constrained energy levels are recalculated in an analogous way as described in Appendix B, the only difference is that the wavenumber uncertainties are not refined here. In what follows, stages S.6.15–S.6.18 are designated with image file: c8cp05169k-t290.tif and are referred to as the fixed blockMARVEL procedure.

XIf necessary, the RSU values, the initial line uncertainties, and the block indices associated to the recalibrated image file: c8cp05169k-t110.tif class should be modified to properly describe the accuracy of the improved wavenumbers.

Appendix D: stages of the local synchronization method (see Section 2.4.4)

S.1 set flag image file: c8cp05169k-t111.tif to 1;[I]
S.2 store the indices of the energy levels in t1, t2,…,tNL in the order that either comp(ti) < comp(tj) or comp(ti) = comp(tj) and ĒtiĒtj is valid for all 1 ≤ ijNL;
S.3 for all 1 ≤ iNL:
S.3.1 set T = ti;
S.3.2 if bT = 100 or core(comp(T)) = T, then continue S.3;[II]
S.3.3 for all 1 ≤ jNT: if Tχj and image file: c8cp05169k-t252.tif, then set image file: c8cp05169k-t112.tif, otherwise use image file: c8cp05169k-t113.tif,[III] where χj = {up(j), low(j)};
S.3.4 place the indices of the lines in image file: c8cp05169k-t253.tif into u1, u2,…,uNloc;
S.3.5 if Nloc < μsync, where μsync (= 4) is the synchronization margin, then continue S.3;[II,IV]
S.3.6 for all 1 ≤ jNloc: calculate ej by means of eqn (16);
S.3.7 search for the minimum ESU value in image file: c8cp05169k-t254.tif:
 
image file: c8cp05169k-t114.tif (29)
S.3.8 for all 1 ≤ jNloc: initialize δloc,j as ESUloc,min;
S.3.9 ‘infinite’ loop:[V]
  S.3.9.1 calculate ebest with the help of eqn (17);
  S.3.9.2 for all 1 ≤ jNloc: determine the Δloc,j residual and the dloc,j = |Δloc,j| − δloc,j defect;
  S.3.9.3 seek for image file: c8cp05169k-t115.tif;
  S.3.9.4 if dloc,maxϕnoi, then break S.3.9;
  S.3.9.5 for every 1 ≤ jNloc: if dloc,j > ϕdiscdloc,max, then replace δloc,j with ϕincδloc,j;
S.3.10 define the adjusted and non-adjusted MAR values of the Tth energy level in the following form:
 
image file: c8cp05169k-t116.tif (30)
 
image file: c8cp05169k-t117.tif (31)
S.3.11 if MAR(T)naϕsyncMAR(T)adj, where ϕsync (= 4) is the synchronization factor, then continue S.3;[II,VI]
S.3.12 reset image file: c8cp05169k-t118.tif;
S.3.13 exchange ĒT with ebest;
S.4 if image file: c8cp05169k-t119.tif, then:
S.4.1 initialize image file: c8cp05169k-t255.tif as image file: c8cp05169k-t120.tif for all 1 ≤ jNT;
S.4.2 for all 1 ≤ jNT:
  S.4.2.1 update Δj and dj;
  S.4.2.2 if Pj = 0, then:
   S.4.2.2.1 reset δj = max(δ0,j,ϕinc|Δj|);
   S.4.2.2.2 if image file: c8cp05169k-t121.tif, then[VII]
    S.4.2.2.2.1 set Pj = 1;
    S.4.2.2.2.2 assign image file: c8cp05169k-t122.tif;
  S.4.2.3 if Pj = 1 and dj > 0, then reassign image file: c8cp05169k-t123.tif;[VIII]
  S.4.2.4 reset δj = max(δj,ϕinc|Δj|);
S.4.3image file: c8cp05169k-t256.tif;
S.4.4image file: c8cp05169k-t257.tif where ϕld (= 0.9) is the local discrimination factor;
S.4.5 for all 1 ≤ jNT: replace δj with max(δj,ϕinc|Δj|);
S.5 end procedure.
Notes:

IUpon completion of the local synchronization procedure, image file: c8cp05169k-t124.tif will designate whether all the constrained empirical energy levels are supported by this algorithm (image file: c8cp05169k-t125.tif = 1) or there are energy values adjusted during the synchronization process (image file: c8cp05169k-t126.tif = 0).

IIIf any of the conditions S.3.2, S.3.5, and S.3.11 are not met, the synchronization of the Tth energy level with its image file: c8cp05169k-t258.tif subnetwork is ignored.

IIIThe image file: c8cp05169k-t259.tif subnetwork contains all the non-bridge transitions of the whole SN which are included in the CD relations of the Mth energy level. Bridges are not inserted into image file: c8cp05169k-t260.tif for the same reason detailed in Note IV of Appendix C.

IV μ sync provides a reasonable lower limit for Nloc.

VThe ebest value is determined by the aid of an iterative refinement technique (see also Appendices B and C).

VIIf the ĒT value conforms with their trial energies, then the inequality of stage S.3.11 has to be fulfilled. In the opposite case, ebest is used instead of ĒT in the rest of the local synchronization process.

VIIIn the case that certain constrained empirical energy levels were altered during the local synchronization, there may be transitions which can be reincluded in image file: c8cp05169k-t261.tif.

VIIIIf there are energy levels adjusted by the local synchronization procedure, then there must be lines of positive (updated) defects, for which the image file: c8cp05169k-t262.tif algorithm needs to be called.

Appendix E: stages of the FNC method (see Section 2.4.5)

S.1 for all 1 ≤ iNT: if Pi = 0 and image file: c8cp05169k-t127.tif, then reset Pi = 1,[I] where ϕrei(= 100) is the reinclusion factor;
S.2 detect the bridges of image file: c8cp05169k-t263.tifvia the BFS method;
S.3 for all 1 ≤ iNT: if Pi = 1 and the ith transition is a bridge of image file: c8cp05169k-t264.tif, then set image file: c8cp05169k-t265.tif, otherwise use image file: c8cp05169k-t266.tif;
S.4 build the maximum bridgeless subnetwork (MBS)[II] of image file: c8cp05169k-t267.tif, denoted with image file: c8cp05169k-t268.tif, as follows: image file: c8cp05169k-t128.tif for all 1 ≤ iNT;
S.5 identify the components of image file: c8cp05169k-t269.tif (i.e., the bridge components of image file: c8cp05169k-t270.tif);
S.6 archive the compB(1), compB(2),…, compB(NL) indices of image file: c8cp05169k-t271.tif;
S.7 save the image file: c8cp05169k-t272.tif component sizes of NB, where NB,c is the number of components in image file: c8cp05169k-t273.tif;
S.8 for all 1 ≤ iNL:
S.8.1 specify the image file: c8cp05169k-t274.tif resistance[III] of the ith energy level as
 
image file: c8cp05169k-t129.tif (32)
S.8.2 define the following ‘connectivity’ parameters,[IV]
 
image file: c8cp05169k-t130.tif (33)
where χj is defined in stage S.3.3 of Appendix D, and the ith rovibrational energy level is associated with the LTDi leading transition degree, the BTDi block transition degree, the LSDi leading source degree, and the BSDi block source degree;
S.8.3 assign the dependability grade[V] to the ith energy level as
 
image file: c8cp05169k-t131.tif (34)
where LTDc(= 10) and LSDc(= 6) are the critical LTD and LSD, respectively;
S.8.4 if core(comp(i)) = i, then assign εi = 0[VI] and continue S.8;
S.8.5 construct a image file: c8cp05169k-t275.tif fractional subnetwork[VII] of image file: c8cp05169k-t276.tif in the following form: if Pj = 1, image file: c8cp05169k-t277.tif, and iχj, then set image file: c8cp05169k-t132.tif, otherwise use image file: c8cp05169k-t133.tif for all 1 ≤ jNT;
S.8.6 save the indices of the transitions in image file: c8cp05169k-t278.tif into v1, v2,…, vNfrac such that δvjδvk holds for all 1 ≤ jkNfrac, where Nfrac is the number of lines in image file: c8cp05169k-t288.tif;
S.8.7 if Nfrac<ncct or |S(Nfrac)| < nccs, then set εi = −1[VIII] and continue S.8, where
   (a) image file: c8cp05169k-t134.tif is the set of sources including transitions with the v1, v2,…, vNfrac indices,
   (b) image file: c8cp05169k-t135.tif is the lth source of the SN,
   (c) image file: c8cp05169k-t136.tif is the mth source index for which image file: c8cp05169k-t137.tif contains the mth transition,
   (d) ncct(= 5) is the critical number of confirmative lines, and
   (e) nccs(= 3) is the critical number of confirmative sources;
S.8.8 specify the minimally required number of confirmative transitions, nmrct, in this way:[IX]
 
n mrct = min{k: ncctkNfrac and |S(k)| ≥ nccs}; (35)
S.8.9 introduce the number of confirmative transitions,[X]
 
image file: c8cp05169k-t138.tif (36)
  where ϕfe(= 3) is the fractional extension factor;
S.8.10 estimate the εi uncertainty as[XI]
 
image file: c8cp05169k-t139.tif (37)
S.9 for all image file: c8cp05169k-t140.tif:
S.9.1 set image file: c8cp05169k-t141.tif;
S.9.2 provide the (A[s], V[s], E[s]) triplet[XII] for the s segment as
 
image file: c8cp05169k-t142.tif (38)
where
  (a) bj* = max(bup(j), blow(j)), and
  (b) A[s], V[s], and E[s] symbolize the number of assigned, validated, and exploited transitions corresponding to the s segment, respectively;
S.9.3 define the following statistical quantities:[XIII]
 
image file: c8cp05169k-t143.tif (39)
where ASU[s] and MSU[s] are the average and maximum segment uncertainties of the s segment, respectively;
S.10 for all 1 ≤ iNT: represent the dependability grade of the ith transition, Γi, with the lower[V] grade of γup(i) and γlow(i);
S.11 end procedure.
Notes:

IAlthough the reincluded lines may have relatively large uncertainties, they could be useful for corroborating the dependability of the energy levels.

image file: c8cp05169k-t279.tif is built by placing the non-bridge lines of image file: c8cp05169k-t280.tif into this subnetwork.

IIIThe resistance of the ith rovibrational energy level is (a) protected (P), if it lies in the same bridge component as the core of its component in image file: c8cp05169k-t281.tif, (b) unprotected (U), if it is alone in its bridge component, or (c) semiprotected (S), otherwise.

IVThese quantities can be interpreted in the following way: (a) LTDi is the total number of lines in image file: c8cp05169k-t282.tif incident to the ith energy level, (b) BTDi is the number of exploited lines from image file: c8cp05169k-t283.tif including this rovibrational state, and (c) LSDi and BSDi correspond to the numbers of sources containing transitions used in the definitions of LTDi and BTDi, respectively. The jth transition of the SN is exploited if Bj ≤ max(bup(j), blow(j)), that is, if this line is utilized during the determination of its upper or lower energy level.

VThe particular grades are listed starting from the best grade down to the worst one. In fact, energy levels with grade D are undefined, as there is no line in image file: c8cp05169k-t284.tif incident to these energy levels.

VIThe uncertainties of the cores are set to zero.

image file: c8cp05169k-t285.tif contains all the non-bridge lines from image file: c8cp05169k-t286.tif which are incident to the ith energy level.

VIII ε i = −1 indicates that a reliable uncertainty cannot be assigned to the ith energy level.

IX n mrctncct is the smallest positive integer for which lines indexed with v1, v2,…, vnmrct arise from at least nccs sources.

XAs a result of this stage, a collection of nctnmrct transitions, called confirmative linelist (CL), is obtained, whose lines are suitable for the estimation of the uncertainty of the ith energy level.

XIIf nmrct > 0 and the ith energy level is not a core, then εi is calculated via eqn (37).

XIIThis triplet can be denoted with ‘A/V/E’, as well.

XIIIThe [σ[s]min, σ[s]max] interval represents the measurement range of the wavenumbers in the s segment, while the parameters ASU[s] and MSU[s] describe the estimated accuracy of s at the end of the intMARVEL procedure.

Acknowledgements

The authors thank Dr Csaba Fábri for fruitful discussions. The Budapest group gratefully acknowledges the financial support they received from NKFIH (grant number K119658). The Budapest group also received support from the grant VEKOP-2.3.2-16-2017-00014, supported by the European Union and the State of Hungary and co-financed by the European Regional Development Fund. The collaboration between ELTE and UCL was supported by the CM1405 COST action, MOLIM: Molecules in Motion.

References

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