Roland
Tóbiás
^{a},
Tibor
Furtenbacher
^{a},
Jonathan
Tennyson
^{b} and
Attila G.
Császár
*^{c}
^{a}Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, ELTE Eötvös Loránd University and MTAELTE Complex Chemical Systems Research Group, P.O. Box 32, H1518 Budapest 112, Hungary
^{b}Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
^{c}MTAELTE Complex Chemical Systems Research Group, P.O. Box 32, H1518 Budapest 112, Hungary. Email: csaszarag@caesar.elte.hu
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 leastsquares 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 H_{2}^{16}O transition measurements. As a result, almost 300 highly accurate energy levels of H_{2}^{16}O 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: (v_{1}v_{2}v_{3}) = (0 0 0), (0 1 0), and (0 2 0), where v_{1}, v_{2}, and v_{3} 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.
Complementing the experimental advances in highresolution 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 welldefined 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 ^{12}C_{2},^{15} nine isotopologues of water,^{16–20} three isotopologues of SO_{2},^{21} three isotopologues of H_{3}^{+},^{22,23}^{14}NH_{3},^{24} and parent ketene.^{25} A webbased version of the MARVEL code has allowed the involvement of highschool students^{26} in spectroscopic research projects, leading to published studies on ^{48}Ti^{16}O,^{27}^{90}Zr^{16}O,^{28} H_{2}^{32}S,^{29} and ^{12}C_{2}H_{2},^{30} molecules of considerable astronomical interest.
Energy levels derived using the MARVEL procedure are increasingly being used to improve not only transition wavenumbers^{31} but also partition functions^{32} present in standard information systems. Just considering water, MARVEL energy levels have been used extensively by the ExoMol project^{33} to improve predicted line positions in line lists for H_{2}^{n}O (n = 16, 17, 18).^{34,35} Similarly, the most recent edition of the HITRAN database, HITRAN2016,^{31} makes use of MARVEL energy levels for H_{2}^{n}O (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 H_{2}^{16}O,^{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 secondgeneration MARVEL code^{12} runs extremely fast, treats transition data and the related matrices (vide infra) on the order of 100000 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 MARVELbased 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 stateoftheart of highresolution spectroscopy for the H_{2}^{16}O 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 highestquality 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 H_{2}^{16}O dataset of JPL energy levels is based on a study of Lanquetin et al.^{45} In particular, the experimental JPL energy levels of H_{2}^{16}O 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.^{16–20}Ref. 20 gives a summary of this work. Part III of this series^{18} contains a validated and recommended set of measured H_{2}^{16}O transitions (about 200000) and empirical energy levels (about 20000), 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 opticalfrequencycombbased 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 ortho–para rovibrational transitions have been observed in water vapor.^{67} This has the consequence that water spectra define two principal components^{10} within the measured spectroscopic network of H_{2}^{16}O, corresponding to ortho and paraH_{2}^{16}O. Note that the number q = v_{3} + K_{a} + K_{c} is even for para and odd for ortho rovibrational states, where v_{3} is the vibrational quantum number corresponding to the antisymmetric stretch motion, while K_{a} and K_{c} are the usual rigidrotor quantum numbers of an asymmetrictop 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 experimentally^{18}), the best frequency standards correspond to paraH_{2}^{16}O, as there is no hyperfine splitting of the paraH_{2}^{16}O lines and the minimumenergy level of the para PC can properly be set to zero with zero uncertainty.
Highresolution radialvelocityshift transition measurements used to detect molecular species in the atmospheres of exoplanets^{68,69} has greatly increased the need for accurate laboratory transition frequencies. While this technique has been used successfully to identify water in exoplanets^{70,71} 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 list^{73} of the astrophysically most important spectral lines, the International Astronomical Union listed 13 H_{2}^{16}O 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 conservation^{13} to the accuracy of the best measurements, which calls for opticalfrequencycombbased remeasurement of certain rovibrational transitions on the ground vibrational state of water.
AF/GHz  Component  Assignment  extMARVEL/kHz  01LaCoCa^{45}/kHz  Expt./kHz 

22.235  para  6_{16} ← 5_{23}  22235079.85(6)  22235007(4240)  22235079.85(6)^{48} 
22235080(29)^{49}  
22235200(600)^{50}  
22235220(154)^{51}  
22235000(190)^{52}  
183.310  ortho  3_{13} ← 2_{20}  183310090.4(1)  183309897(2544)  183310090.6(1)^{53} 
183310087(4)^{54}  
183310117(29)^{55}  
183310150(65)^{49}  
183310200(190)^{52}  
183311300(1330)^{56}  
325.153  para  5_{15} ← 4_{22}  325152899(2)  325153101(3392)  325152899(2)^{54} 
325152888(13)^{57}  
325152919(22)^{55}  
325153700(882)^{52}  
380.197  para  4_{14} ← 3_{21}  380197359.8(6)  380197395(3610)  380197359.8(6)^{58} 
380197356(4)^{54}  
380197365(13)^{57}  
380197372(16)^{55}  
380196800(621)^{52}  
439.151  para  6_{43} ← 5_{50}  439150794.8(6)  439151582(5528)  439150794.8(6)^{58} 
439150795(2)^{54}  
439150812(19)^{55}  
448.001  para  4_{23} ← 3_{30}  448001077.5(6)  448001155(3610)  448001077.5(6)^{58} 
448001075(16)^{55}  
448000300(904)^{52}  
474.689  ortho  5_{33} ← 4_{40}  474689108(2)  474689879(3816)  474689108(2)^{54} 
474689127(21)^{55}  
556.936  para  1_{10} ← 1_{01}  556935987.6(6)  556935841(1824)  556935987.7(6)^{58} 
556935985(3)^{59}  
556935995(8)^{54}  
556936002(16)^{55}  
556935819(185)^{60}  
556935800(190)^{52}  
556935800(206)^{61}  
620.700  para  5_{32} ← 4_{41}  620700954.9(6)  620701697(4664)  620700954.9(6)^{58} 
620700950(35)^{62}  
620700844.1(150)^{63}  
620700807(163)^{55}  
752.033  ortho  2_{11} ← 2_{02}  752033113(15)  752032978(2544)  752033113(15)^{64} 
752033104(19)^{60}  
752033227(125)^{55}  
752033300(190)^{52}  
916.172  ortho  4_{22} ← 3_{31}  916171581(21)  916171448(3392)  916171580(21)^{65} 
916171582(21)^{66}  
916171405(194)^{60}  
970.315  ortho  5_{24} ← 4_{31}  970315021(21)  970315165(3392)  970315022(21)^{66} 
970315020(21)^{65}  
970314968(58)^{60}  
987.927  ortho  2_{02} ← 1_{11}  987926762(21)  987926473(2341)  987926764(21)^{66} 
987926760(21)^{65}  
987926743(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 H_{2}^{16}O, the admittedly most important polyatomic molecule for highresolution 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 H_{2}^{16}O 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.
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 linebyline uncertainties, only an average “expected” accuracy is provided, usually corresponding to an intense unblended line, i.e., to a bestcase 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 leastsquares technique (e.g., robust reweighting^{104}) 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 lowestlying energy level (core) of each component to zero. If the core of a component corresponds to the lowestenergy level of a particular nuclearspin 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 nonbridge 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(P_{1}, P_{2},…, P_{NT}), where N_{T} is the number of lines in the SN, and P_{i} = 1 if the ith transition of the SN is inserted in the subnetwork, otherwise P_{i} = 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 , which contains all the nonexcluded lines of the database. In the present description, P will always denote the participation matrix assigned to the subnetwork.
During a MARVEL analysis of measured rovibronic transitions the following objective function is minimized:
Ω(E) = (σ − RE)^{T}PW(σ − RE),  (1) 
Since the minimum of Ω(E), denoted by Ē = {Ē_{1}, Ē_{2},…, Ē_{NL}}^{T}, is not unique,
Ē_{core(1)} = Ē_{core(2)} = ⋯ = Ē_{core(Nc)} = 0  (2) 
GĒ = F,  (3) 
To eliminate the linear dependencies of the blockdiagonal 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 are not necessarily known to the user. Then, from any solution, , of eqn (3), the core(i) index can be obtained for all 1 ≤ i ≤ N_{c} as follows:
(4) 
(5) 
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 linebyline uncertainties.
As soon as Ē is determined, it is necessary to examine the Δ_{i} = σ_{i} − _{i} (fitting) residuals and the d_{i} = Δ_{i} − δ_{i} (fitting) defects, where _{i} = Ē_{up(i)} + Ē_{low(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. is called consistent if the largest defect of , d_{max}, is not positive. In the case of d_{max} > 0, we have to either exclude certain lines or increase their uncertainties in order to ensure the consistency of .
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 format^{11,12} 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.
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.
After this revision, reassignments can be made for the uncorrected, but still excluded lines, using (a) the wavenumbersorted 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”.
Designate the core indices in and with core_{p}(1), core_{p}(2),…, core_{p}(N_{p,c}) and core_{a}(1), core_{a}(2),…, core_{a}(N_{a,c}), respectively, where N_{p,c} and N_{a,c} are the number of components in and , respectively. Let be the participation matrix of . With the notation introduced, eqn (2) and (3) can be reformulated for as follows:
Ē_{corea(1)} = Ē_{corea(2)} = ⋯ = Ē_{corea(Na,c)} = 0  (6) 
(7) 
Ē_{i} − Ē_{corep(compp(i))} = β_{i},  (8) 
Ē = C + β,  (9) 
(10) 
_{1} = _{2} = ⋯ = _{Na,c} = 0,  (11) 
(12) 
Note that 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 260000 H_{2}^{16}O 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, 71990, 174965, 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 speedup of about four. Nevertheless, in contrast to the G matrix, is not diagonally dominant; thus, a pivoting strategy must be employed to yield the solution of eqn (11) and (12). Once is determined, Ē can easily be calculated by means of eqn (9).
Utilizing eqn (11) and (12), a divideandconquerstyle 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) 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 . 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).
To recalibrate the segments, they have to be placed into the R(1), R(2),…, R(N_{R}) recalibration classes, where N_{R} 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, , which contains the nonbridge lines of the segments in R(i), is introduced. To assign a recalibration factor to R(i), the following objective function should be optimized:
(13) 
(14) 
Starting from proper initial values, the δ_{reg,j} uncertainties are iteratively adjusted to provide an f_{best} value satisfying δ_{reg,j} ≤ Δ_{reg,j}, where Δ_{reg,j} = f_{best}σ_{pj} − _{pj} for all 1 ≤ j ≤ N_{reg}. If (a) the recalibration of R(i) is permitted, (b) N_{reg} is sufficiently large, and (c) the f_{best} parameter significantly reduces the distance of the residuals from zero in , then f_{best} 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 accuracybased 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 reexecuted (only one more time) at the termination of the ABR method.
(15) 
(16) 
(17) 
As in Section 2.4.3, an iterative refinement can be performed for the δ_{loc,j} uncertainties for which reasonable guesses are employed. Obviously, the final e_{best} value should fulfill the relation δ_{loc,j} ≤ Δ_{loc,j}, where Δ_{loc,j} = e_{j} − e_{best}, for all 1 ≤ j ≤ N_{loc}.
If (a) N_{loc} is sufficiently large, and (b) the Δ_{uj} absolute residuals are systematically larger than the corresponding Δ_{loc,j} values, then (a) the ith energy level is synchronized with (i.e., Ē_{i} is substituted with e_{best}), (b) all the transitions of are excluded from whose recalculated fitting defects are positive, and (c) those lines of which were incorrectly excluded from are reincluded in . Based on eqn (17), a socalled 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 , 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.
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.
Segment tag  Range/cm^{−1}  A/V/E  ESU/cm^{−1}  RSU/cm^{−1}  ASU/cm^{−1}  MSU/cm^{−1} 

a Tags denote experimental datasource 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.  
69Kukolich^{48}  0.74168–0.74168  1/1/1  2.000 × 10^{−9}  2.000 × 10^{−}9  2.000 × 10^{−9}  2.000 × 10^{−9} 
09CaPuHaGa^{58}  10.715–20.704  7/7/7  2.000 × 10^{−8}  2.000 × 10^{−8}  2.000 × 10^{−8}  2.000 × 10^{−8} 
71Huiszoon^{53}  6.1146–6.1146  1/1/1  4.000 × 10^{−8}  4.000 × 10^{−8}  4.000 × 10^{−8}  4.000 × 10^{−8} 
95MaKr^{59}  18.577–18.577  1/1/1  7.000 × 10^{−8}  7.000 × 10^{−8}  9.475 × 10^{−8}  9.475 × 10^{−8} 
06GoMaGuKn^{54}  6.1146–18.577  12/12/11  2.000 × 10^{−7}  1.145 × 10^{−7}  8.827 × 10^{−8}  2.724 × 10^{−7} 
18KaStCaCa^{102}  7164.9–7185.6  8/8/8  4.000 × 10^{−7}  4.000 × 10^{−7}  4.000 × 10^{−7}  4.000 × 10^{−7} 
09CaPuBuTa^{96}  36.604–53.444  9/9/9  5.000 × 10^{−7}  5.000 × 10^{−7}  5.000 × 10^{−7}  5.000 × 10^{−7} 
11Koshelev^{64}  25.085–25.085  1/1/1  5.000 × 10^{−7}  5.000 × 10^{−7}  5.000 × 10^{−7}  5.000 × 10^{−7} 
07KoTrGoPa^{57}  10.715–12.682  3/3/3  6.000 × 10^{−7}  8.507 × 10^{−7}  8.584 × 10^{−7}  1.725 × 10^{−6} 
83HeMeLu^{65}  13.013–32.954  7/7/7  7.000 × 10^{−7}  7.000 × 10^{−7}  7.000 × 10^{−7}  7.000 × 10^{−7} 
83MeLuHe^{66}  16.797–32.954  5/5/5  7.000 × 10^{−7}  7.000 × 10^{−7}  7.000 × 10^{−7}  7.000 × 10^{−7} 
71StBe^{49}  0.74168–6.1146  3/3/3  9.000 × 10^{−7}  1.941 × 10^{−6}  5.186 × 10^{−6}  1.241 × 10^{−5} 
87BaAlAlPo^{77}  14.199–19.077  6/6/5  1.000 × 10^{−6}  1.000 × 10^{−6}  4.244 × 10^{−6}  2.046 × 10^{−5} 
95MaOdIwTs^{60}  18.577–162.44  139/138/135  1.000 × 10^{−6}  1.281 × 10^{−6}  2.520 × 10^{−6}  3.638 × 10^{−5} 
18ChHuTaSu^{47}  12.622–12.665  6/6/6  1.000 × 10^{−6}  1.000 × 10^{−6}  1.000 × 10^{−6}  1.000 × 10^{−6} 
54PoSt^{51}  0.74169–0.74169  1/1/1  2.000 × 10^{−6}  4.673 × 10^{−6}  5.142 × 10^{−6}  5.142 × 10^{−6} 
80Kuze^{75}  0.40057–4.0026  5/5/4  2.000 × 10^{−6}  2.000 × 10^{−6}  9.021 × 10^{−6}  3.440 × 10^{−5} 
91AmSc^{79}  8.2537–11.835  5/5/5  2.000 × 10^{−6}  2.000 × 10^{−6}  2.000 × 10^{−6}  2.000 × 10^{−6} 
97NaLoInNo^{88}  118.32–119.07  5/5/4  2.000 × 10^{−6}  2.000 × 10^{−6}  6.032 × 10^{−6}  2.216 × 10^{−5} 
91PeAnHeLu^{80}  4.6570–19.804  30/30/30  3.000 × 10^{−6}  3.000 × 10^{−6}  3.907 × 10^{−6}  1.652 × 10^{−5} 
83BuFeKaPo^{62}  16.797–21.545  5/5/5  4.000 × 10^{−6}  2.316 × 10^{−6}  2.120 × 10^{−6}  5.070 × 10^{−6} 
95Pearson^{85}  4.3300–17.220  9/8/8  4.000 × 10^{−6}  4.000 × 10^{−6}  4.000 × 10^{−6}  4.000 × 10^{−6} 
06MaToNaMo^{95}  28.685–165.31  104/104/102  4.000 × 10^{−6}  1.745 × 10^{−6}  2.637 × 10^{−6}  3.066 × 10^{−5} 
12YuPeDrMa^{98}  9.8572–90.767  103/102/101  4.000 × 10^{−6}  4.000 × 10^{−6}  6.643 × 10^{−6}  2.337 × 10^{−4} 
72LuHeCoGo^{55}  6.1146–25.085  14/14/13  5.000 × 10^{−6}  1.062 × 10^{−6}  1.451 × 10^{−6}  5.427 × 10^{−6} 
13YuPeDr^{63}  17.690–67.209  182/182/181  5.000 × 10^{−6}  5.000 × 10^{−6}  5.992 × 10^{−6}  6.152 × 10^{−5} 
11DrYuPeGu^{97}  82.862–90.843  26/25/25  6.000 × 10^{−6}  6.000 × 10^{−6}  6.734 × 10^{−6}  1.156 × 10^{−5} 
00ChPePiMa^{90}  28.054–52.511  17/17/17  8.000 × 10^{−6}  1.600 × 10^{−6}  3.184 × 10^{−6}  1.422 × 10^{−5} 
79HeJoMc^{74}  0.072049–0.072049  1/1/1  1.000 × 10^{−5}  1.000 × 10^{−5}  1.000 × 10^{−5}  1.000 × 10^{−5} 
51Jen^{50}  0.74169–0.74169  1/1/0  2.000 × 10^{−5}  2.000 × 10^{−5}  2.000 × 10^{−5}  2.000 × 10^{−5} 
72FlCaVa^{52}  0.74168–25.085  7/7/0  2.000 × 10^{−5}  1.266 × 10^{−5}  1.509 × 10^{−5}  3.015 × 10^{−5} 
06JoPaZeCo^{94}  1485.1–1486.2  2/2/2  2.000 × 10^{−5}  2.000 × 10^{−5}  3.116 × 10^{−5}  4.232 × 10^{−5} 
70StSt^{61}  18.577–18.577  1/1/0  3.000 × 10^{−5}  6.241 × 10^{−6}  6.883 × 10^{−6}  6.883 × 10^{−6} 
87BeKoPoTr^{78}  7.7616–19.850  5/5/5  3.000 × 10^{−5}  4.103 × 10^{−5}  2.858 × 10^{−5}  4.663 × 10^{−5} 
96Belov^{86}  28.054–30.792  5/5/5  3.000 × 10^{−5}  7.376 × 10^{−6}  7.849 × 10^{−6}  1.765 × 10^{−5} 
05HoAnAlPi^{92}  212.56–594.95  166/164/60  3.000 × 10^{−5}  1.640 × 10^{−5}  2.668 × 10^{−5}  3.643 × 10^{−4} 
13LuLiWaLi^{99}  12.573–12.752  73/73/65  3.000 × 10^{−5}  9.660 × 10^{−5}  6.039 × 10^{−5}  5.243 × 10^{−4} 
54KiGo^{56}  6.1146–6.1146  1/1/0  5.000 × 10^{−5}  3.710 × 10^{−5}  4.437 × 10^{−5}  4.437 × 10^{−5} 
96BrMa^{87}  5206.3–5396.5  28/28/28  5.000 × 10^{−5}  1.327 × 10^{−5}  2.534 × 10^{−5}  9.346 × 10^{−5} 
91Toth^{81}  1072.6–2265.3  740/738/722  6.000 × 10^{−5}  2.117 × 10^{−5}  5.183 × 10^{−5}  1.460 × 10^{−3} 
97MiTyKeWi^{89}  2507.2–4402.8  935/935/190  6.000 × 10^{−5}  1.639 × 10^{−4}  4.774 × 10^{−4}  1.050 × 10^{−2} 
95PaHo^{84}  177.86–519.59  246/246/5  7.000 × 10^{−5}  1.072 × 10^{−4}  2.200 × 10^{−4}  9.827 × 10^{−3} 
93Totha^{82}  1316.1–4260.4  587/587/575  8.000 × 10^{−5}  3.317 × 10^{−5}  8.755 × 10^{−5}  9.449 × 10^{−4} 
93Tothb^{83}  1881.1–4306.7  1076/1076/1061  8.000 × 10^{−5}  3.428 × 10^{−5}  8.151 × 10^{−5}  1.485 × 10^{−3} 
05Toth^{93}  2926.5–7641.9  1895/1895/1832  8.000 × 10^{−5}  6.031 × 10^{−5}  1.396 × 10^{−4}  4.447 × 10^{−3} 
85BrTo^{76}  1323.3–1992.7  71/71/69  1.000 × 10^{−4}  2.042 × 10^{−5}  8.836 × 10^{−5}  1.877 × 10^{−3} 
03ZoVa^{91}  3010.2–4044.9  469/469/456  1.000 × 10^{−4}  9.533 × 10^{−5}  2.385 × 10^{−4}  2.465 × 10^{−3} 
15SiHo^{101}  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 orthoH_{2}^{16}O, 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 linebyline 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.
One of the important practical results of this study, a set of highly accurate empirical energy levels of H_{2}^{16}O, 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 (v_{1}v_{2}v_{3}) = (0 0 0), (0 1 0), and (0 2 0), where v_{1}, v_{2}, and v_{3} stand for the symmetric stretch, bend, and antisymmetric stretch vibrational quantum numbers. All energy levels reported have at least an 8digit accuracy, often considerably better (up to 10 digits of accuracy). J_{max} is 13/14 for the most dependable ortho/para rovibrational energy levels presented in Table 3.
Energy/cm^{−1}  (v_{1}v_{2}v_{3})J_{Ka,Kc}  Energy/cm^{−1}  (v_{1}v_{2}v_{3})J_{Ka,Kc}  Energy/cm^{−1}  (v_{1}v_{2}v_{3})J_{Ka,Kc}  Energy/cm^{−1}  (v_{1}v_{2}v_{3})J_{Ka,Kc} 

0  (0 0 0)0_{0,0}  1806.671529(30)  (0 0 0)13_{1,13}  0  (0 0 0)1_{0,1}  1765.248467(64)  (0 0 0)8_{8,1} 
37.13712384(68)  (0 0 0)1_{1,1}  1810.583280(79)  (0 0 0)9_{7,3}  18.57738488(39)  (0 0 0)1_{1,0}  1782.875649(18)  (0 0 0)13_{0,13} 
70.09081349(65)  (0 0 0)2_{0,2}  1813.787601(17)  (0 1 0)3_{2,2}  55.7020277(99)  (0 0 0)2_{1,2}  1786.7935598(52)  (0 0 0)9_{7,2} 
95.1759380(31)  (0 0 0)2_{1,1}  1817.451194(21)  (0 1 0)4_{0,4}  111.1072838(60)  (0 0 0)2_{2,1}  1789.429033(13)  (0 0 0)11_{3,8} 
136.1639195(12)  (0 0 0)2_{2,0}  1843.029604(33)  (0 0 0)11_{4,8}  112.967305(10)  (0 0 0)3_{0,3}  1795.5407549(85)  (0 1 0)3_{2,1} 
142.2784859(10)  (0 0 0)3_{1,3}  1875.461821(19)  (0 0 0)10_{6,4}  149.5714542(92)  (0 0 0)3_{1,2}  1797.802452(11)  (0 1 0)4_{1,4} 
206.301428(21)  (0 0 0)3_{2,2}  1875.469719(18)  (0 1 0)4_{1,3}  188.36200986(47)  (0 0 0)3_{2,1}  1851.178614(11)  (0 0 0)10_{6,5} 
222.052757(13)  (0 0 0)4_{0,4}  1907.451421(19)  (0 1 0)3_{3,1}  201.04402869(42)  (0 0 0)4_{1,4}  1875.213801(11)  (0 0 0)11_{4,7} 
275.497042(18)  (0 0 0)4_{1,3}  1922.829071(12)  (0 1 0)5_{1,5}  261.6242179(88)  (0 0 0)3_{3,0}  1883.821404(23)  (0 1 0)3_{3,0} 
285.219339(10)  (0 0 0)3_{3,1}  1922.901125(11)  (0 1 0)4_{2,2}  276.56792527(55)  (0 0 0)4_{2,3}  1884.221976(10)  (0 1 0)4_{2,3} 
315.77953341(82)  (0 0 0)4_{2,2}  1960.207413(33)  (0 0 0)12_{2,10}  301.5535473(94)  (0 0 0)5_{0,5}  1896.972180(16)  (0 1 0)5_{0,5} 
326.62546666(58)  (0 0 0)5_{1,5}  1985.784894(12)  (0 0 0)11_{5,7}  358.722528(11)  (0 0 0)4_{3,2}  1938.712538(10)  (0 0 0)12_{3,10} 
383.842515(13)  (0 0 0)4_{3,1}  2005.917050(16)  (0 1 0)4_{3,1}  375.663155(11)  (0 0 0)5_{1,4}  1975.200945(19)  (0 0 0)11_{5,6} 
416.2087402(12)  (0 0 0)5_{2,4}  2024.152654(39)  (0 1 0)5_{2,4}  422.71630882(88)  (0 0 0)5_{2,3}  1977.068674(18)  (0 1 0)5_{1,4} 
446.696589(13)  (0 0 0)6_{0,6}  2041.780551(28)  (0 1 0)6_{0,6}  423.4579912(56)  (0 0 0)6_{1,6}  1981.0213341(95)  (0 1 0)4_{3,2} 
488.134178(15)  (0 0 0)4_{4,0}  2042.374098(77)  (0 0 0)13_{2,12}  464.313340(12)  (0 0 0)4_{4,1}  1986.010725(72)  (0 0 0)9_{8,1} 
503.9681027(59)  (0 0 0)5_{3,3}  2054.368667(66)  (0 0 0)10_{7,3}  485.01769568(72)  (0 0 0)5_{3,2}  2018.516215(38)  (0 0 0)13_{1,12} 
542.905778(13)  (0 0 0)6_{1,5}  2105.867908(44)  (0 0 0)12_{3,9}  529.117030(21)  (0 0 0)6_{2,5}  2018.958981(16)  (0 1 0)6_{1,6} 
586.4791835(77)  (0 0 0)7_{1,7}  2126.407724(37)  (0 1 0)5_{3,3}  562.449182(22)  (0 0 0)7_{0,7}  2030.174344(14)  (0 1 0)5_{2,3} 
602.7734936(28)  (0 0 0)6_{2,4}  2129.618682(26)  (0 1 0)4_{4,0}  586.546806(40)  (0 0 0)5_{4,1}  2030.550848(17)  (0 0 0)10_{7,4} 
610.114430(14)  (0 0 0)5_{4,2}  2142.597661(55)  (0 0 0)11_{6,6}  625.184327(15)  (0 0 0)6_{3,4}  2101.157030(17)  (0 0 0)12_{4,9} 
661.5489133(67)  (0 0 0)6_{3,3}  2146.263726(36)  (0 1 0)6_{1,5}  680.4196881(12)  (0 0 0)7_{1,6}  2105.804845(43)  (0 1 0)4_{4,1} 
709.608213(12)  (0 0 0)7_{2,6}  2205.652716(69)  (0 0 0)12_{4,8}  718.281923(24)  (0 0 0)5_{5,0}  2106.699939(14)  (0 1 0)5_{3,2} 
742.073025(29)  (0 0 0)5_{5,1}  2211.1906371(89)  (0 1 0)6_{2,4}  720.368303(20)  (0 0 0)8_{1,8}  2120.251898(17)  (0 0 0)11_{6,5} 
744.063661(23)  (0 0 0)8_{0,8}  2248.064567(84)  (0 0 0)13_{3,11}  732.93041648(96)  (0 0 0)6_{4,3}  2137.4916811(42)  (0 1 0)6_{2,5} 
757.78018789(88)  (0 0 0)6_{4,2}  2300.685002(61)  (0 0 0)12_{5,7}  758.6154628(87)  (0 0 0)7_{2,5}  2223.090456(41)  (0 0 0)13_{2,11} 
816.694236(52)  (0 0 0)7_{3,5}  2321.813015(61)  (0 0 0)11_{7,5}  818.562227(10)  (0 0 0)7_{3,4}  2228.068156(19)  (0 1 0)5_{4,1} 
882.890327(21)  (0 0 0)8_{1,7}  2327.883775(49)  (0 0 0)14_{1,13}  861.805844(21)  (0 0 0)8_{2,7}  2230.489516(24)  (0 0 0)10_{8,3} 
888.632650(14)  (0 0 0)6_{5,1}  2399.165477(17)  (0 1 0)6_{4,2}  864.804377(26)  (0 0 0)6_{5,2}  2247.917879(18)  (0 1 0)6_{3,4} 
920.210001(25)  (0 0 0)9_{1,9}  2426.19618(21)  (0 0 0)13_{4,10}  896.373983(24)  (0 0 0)9_{0,9}  2251.578491(52)  (0 0 0)12_{5,8} 
927.743902(11)  (0 0 0)7_{4,4}  2522.261331(64)  (0 0 0)11_{8,4}*  907.442739(12)  (0 0 0)7_{4,3}  2285.935851(16)  (0 1 0)7_{1,6} 
982.911714(13)  (0 0 0)8_{2,6}  2613.104573(35)  (0 0 0)12_{7,5}*  982.321572(21)  (0 0 0)8_{3,6}  2298.111383(77)  (0 0 0)11_{7,4} 
1045.0583403(41)  (0 0 0)6_{6,0}  2670.789689(15)  (0 1 0)8_{3,5}  1021.2635806(10)  (0 0 0)6_{6,1}  2304.119654(21)  (0 0 0)14_{2,13} 
1050.157663(19)  (0 0 0)8_{3,5}  2748.099560(76)  (0 0 0)13_{6,8}*  1036.0410822(48)  (0 0 0)7_{5,2}  2313.872430(23)  (0 1 0)8_{1,8} 
1059.646655(15)  (0 0 0)7_{5,3}  2920.132087(14)  (0 1 0)8_{5,3}  1055.285214(19)  (0 0 0)9_{1,8}  2368.7982056(85)  (0 1 0)7_{2,5} 
1080.385444(19)  (0 0 0)9_{2,8}  1090.755547(36)  (0 0 0)10_{1,10}  2374.587138(32)  (0 1 0)6_{4,3}  
1114.532248(50)  (0 0 0)10_{0,10}  1098.914168(13)  (0 0 0)8_{4,5}  2390.929063(14)  (0 0 0)13_{3,10}  
1131.775573(22)  (0 0 0)8_{4,4}  1178.127134(11)  (0 0 0)9_{2,7}  2410.006047(55)  (0 0 0)12_{6,7}  
1216.189769(53)  (0 0 0)7_{6,2}  1192.4001374(68)  (0 0 0)7_{6,1}  2439.0809341(94)  (0 1 0)7_{3,4}  
1216.231260(21)  (0 0 0)9_{3,7}  1231.3723839(61)  (0 0 0)8_{5,4}  2471.371510(47)  (0 1 0)8_{2,7}  
1255.911545(14)  (0 0 0)8_{5,3}  1259.12474186(98)  (0 0 0)9_{3,6}  2498.470791(42)  (0 0 0)11_{8,3}  
1293.018138(13)  (0 0 0)10_{1,9}  1269.8396771(11)  (0 0 0)10_{2,9}  2509.998800(16)  (0 0 0)13_{4,9}  
1327.117604(24)  (0 0 0)11_{1,11}  1303.315623(28)  (0 0 0)11_{0,11}  2527.689134(62)  (0 0 0)14_{3,12}  
1340.884880(14)  (0 0 0)9_{4,6}  1336.440968(13)  (0 0 0)9_{4,5}  2548.3448242(97)  (0 1 0)7_{4,3}  
1394.814159(54)  (0 0 0)7_{7,1}  1371.0198349(43)  (0 0 0)7_{7,0}  2589.005435(18)  (0 0 0)12_{7,6}  
1411.641890(12)  (0 0 0)8_{6,2}  1387.8170727(57)  (0 0 0)8_{6,3}  2605.540121(60)  (0 0 0)13_{5,8}  
1437.968586(14)  (0 0 0)10_{2,8}  1422.333875(12)  (0 0 0)10_{3,8}  2606.398257(16)  (0 1 0)8_{3,6}  
1474.980787(12)  (0 0 0)9_{5,5}  1453.502995(12)  (0 0 0)9_{5,4}  2607.474510(66)  (0 0 0)15_{1,14}  
1525.135991(43)  (0 0 0)11_{2,10}  1501.053523(22)  (0 0 0)11_{1,10}  2664.285572(38)  (0 1 0)9_{1,8}*  
1538.149477(27)  (0 0 0)10_{3,7}  1534.053335(22)  (0 0 0)12_{1,12}  2700.372781(12)  (0 1 0)7_{5,2}  
1557.844418(26)  (0 0 0)12_{0,12}  1557.541614(18)  (0 0 0)10_{4,7}  2794.603744(30)  (0 1 0)9_{2,7}  
1590.69071(14)  (0 0 0)8_{7,1}  1566.89570(11)  (0 0 0)8_{7,2}  2894.450561(85)  (0 0 0)14_{5,10}*  
1616.453054(22)  (0 0 0)10_{4,6}  1594.762769(66)  (0 1 0)1_{0,1}  2895.8386511(81)  (0 1 0)8_{5,4}  
1631.245487(18)  (0 0 0)9_{6,4}  1607.588639(18)  (0 0 0)9_{6,3}  3368.954987(18)  (0 2 0)3_{2,1}  
1634.967095(11)  (0 1 0)1_{1,1}  1616.711503(44)  (0 1 0)1_{1,0}  4383.251954(52)  (0 2 0)6_{6,1}  
1664.964587(11)  (0 1 0)2_{0,2}  1653.267093(20)  (0 1 0)2_{1,2} 
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 4cycles, both formed by highly accurate transitions, connected by a bridge of lower measured accuracy. If one of the 4cycles 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 4cycle, 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 energylevelbased 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 paraH_{2}^{16}O one of the most important lines is the 5_{1,5} ← 4_{2,2} 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 8digitaccuracy determination gives 325152899(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} 95MaOdIwTs^{60} and 06MaToNaMo^{95} 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 extMARVELpredicted wavenumbers, are plotted in Fig. 3. The estimated ESU of the transitions in 95MaOdIwTs^{60} 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.
Fig. 3 Unsigned residuals for 95MaOdIwTs^{60} 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 orthoH_{2}^{16}O, two often quoted pure rotational transitions are 321.23 GHz for 10_{2,9} ← 9_{3,6} on (0 0 0) and 336.23 GHz for 5_{2,3} ← 6_{1,6} on (0 1 0). The extMARVEL counterpart of the first transition determined in this study is 321225677(44) kHz, while the frequency of the second transition is determined to be 336228131(140) kHz, both of which can be considered dependable due to the fact that they are based on energy levels of A^{+} grade.
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.
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
(18) 
(19) 
Fig. 5 Weighted unsigned deviations [see eqn (19)] of the extMARVEL (this work) and the IUPAC Part III^{18} 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
Ē_{01LaCoCa,i} − Ē_{extMARVEL,i} ≤ ε_{extMARVEL,i}  (20) 
(21) 
Fig. 6 Relative unsigned deviations [see eqn (21)] of the energy levels found in 01LaCoCa^{45} from the extMARVEL energy levels. 
Fig. 7 shows the 01LaCoCa^{45} 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}
Fig. 7 Uncertainties of the energy levels in 01LaCoCa^{45} 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}
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 MARVELtype 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 linebyline analysis yielding individual initial line uncertainties would be ideal but this appears to be unrealistic). Second, the inversion and weighted leastsquares 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 conservation^{13} 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 H_{2}^{16}O 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 opticalfrequencycomb spectroscopic techniques. For ortho and paraH_{2}^{16}O, 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 paraH_{2}^{16}O and 0–1800 cm^{−1} for orthoH_{2}^{16}O.
The present dataset of highly accurate energy levels is larger than the experimental energylevel 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.
S.1 for all : set , where  
(a) is the estimated segment uncertainty (ESU) of the segment,  
(b) is the (actual) refined segment uncertainty (RSU) of , and  
(c) is the number of segments in the SN;  
S.2 for all 1 ≤ i ≤ n_{mh}, where n_{mh} (= 10) is the maximum number of HSP iterations:  
S.2.1 save the indices of the segments into so that is enforced for all ;  
S.2.2 for all :  
S.2.2.1 set ;  
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 and s;  
S.2.2.4 create a perturbation subnetwork^{[I]} with its participation matrix from as follows: if P_{m} = 1 and either or , else for all 1 ≤ m ≤ N_{T}, where  
(a) is the mth segment index for which includes the mth transition, and  
(b) ϕ_{per} ∈ (0,1] is the perturbation factor (ϕ_{per} = 0.1 is the default value);  
S.2.2.5 construct a cycle basis (CB) for , designated with , using the breadthfirst search (BFS) method;^{112}  
S.2.2.6 calculate the reduced discrepancy^{[II]} of every as  


where  
(a) n^{[s]}_{C} is the number of transitions in C from s, and  
(b) is the discrepancy of C;  
S.2.2.7 collect the representative cycles, for which RD^{[s]}_{C} > ϕ_{hin}RSU^{[s]}, into the set ρ, where ϕ_{hin} (= 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:  




where ρ is the cardinality of the set ρ;  
S.2.2.9 if MRD^{[s]} > ϕ_{ter}RSU^{[s]}, where ϕ_{ter} (= 200) is the termination factor, then exit from intMARVEL;^{[IV]}  
S.2.2.10 if n^{[s]}_{nb} < ⌊ϕ_{rep}n^{[s]}⌋,^{[V]} then set RSU^{[s]} = RSU^{[s]}_{prev}, otherwise use RSU^{[s]} = max(ARD^{[s]},ϕ_{sep}ESU^{[s]}), where  
(a) ⌊⌋ is the floor operation,  
(b) n^{[s]}_{nb} is the number of nonbridge transitions from s in , and  
(c) ϕ_{rep} (= 0.1) and ϕ_{sep} (= 0.2) are the representativity and the separation factors, respectively;  
S.2.3 if r_{max} < ϕ_{conv}, then break S.2, where  
(a) ϕ_{conv} (= 3) is the convergence factor,  
(b) , and  
(c) ;  
S.3 for all 1 ≤ j ≤ N_{T}: in the case that , apply , else set , where  
(a) δ_{0,j} is the initial uncertainty of the jth transition, and  
(b) ϕ_{us} (= 0.5) is the uncertainty scaling factor;  
S.4 for all 1 ≤ j ≤ N_{T}: if P_{j} = 1, then set , otherwise use B_{j} = 100,^{[VI]} where B_{j} is the block index of the jth line;  
S.5 set and ;  
S.6 identify the components of ;  
S.7 save the core(j)^{[VII]} and comp(k) indices for all 1 ≤ j ≤ N_{c} and 1 ≤ k ≤ N_{L}, respectively, by means of eqn (4);  
S.8 end procedure. 
^{I}The 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]}.
^{II}The 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.
^{III}In what follows, the default values of the socalled control parameters, needed for the extMARVEL procedure, are mostly given in parentheses (see, e.g., ‘ϕ_{hin} (= 0.01)’ in stage S.2.2.7).
^{IV}In this case, the prior cleansing should be continued to decrease the reduced discrepancies.
^{V}If there are at least ⌊ϕ_{rep}n^{[s]}⌋ lines from s in the cycles of , then its RSU is modified by the corresponding ARD value, else this RSU will remain unchanged in the given iteration.
^{VI}If the block index of a transition is 100, this line will not participate in the blockMARVEL refinement (see Appendix B).
^{VII}The core indices are determined via a traditional MARVEL analysis, using eqn (4). To accelerate this identification process, a logical variable, u_{ST}, can be introduced: if u_{ST} = 1, then only lines related to a BFSbased ST of the LS are employed in eqn (4); otherwise, all the transitions of the LS are utilized in eqn (4). Obviously, u_{ST} = 0 is a more stable choice than u_{ST} = 1.
S.1 for all 1 ≤ i ≤ N_{L}: set b_{i} = 100, where b_{i} is the block index^{[I]} of the ith energy level; 
S.2 set to an empty subnetwork, where for all 1 ≤ j ≤ N_{T}; 
S.3 assign N_{p,c} = N_{L}; 
S.4 for all 1 ≤ i ≤ N_{L}: set core_{p}(i) = comp_{p}(i) = i and β_{i} = 0; 
S.5 for all 1 ≤ j ≤ N_{T}: initialize δ_{j} as δ_{0,j}; 
S.6 for all : 
S.6.1 construct a subnetwork of in the following fashion: if , then , otherwise for all 1 ≤ j ≤ N_{T}; 
S.6.2 if , then continue S.6; 
S.6.3 find the components of ; 
S.6.4 archive the comp_{a}(1), comp_{a}(2),…,comp_{a}(N_{L}) component indices of the energy levels in ; 
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 ≤ j ≤ N_{T} with : calculate Δ_{j} and d_{j}; 
S.6.5.3 search for ;^{[III]} 
S.6.5.4 if d_{max} ≤ ϕ_{noi}, then break S.6.5, where ϕ_{noi} (= 10^{−10}) designates the numerical noise factor; 
S.6.5.5 for all 1 ≤ j ≤ N_{T}: if and d_{j} > ϕ_{disc}d_{max}, then substitute δ_{j} with ϕ_{inc}Δ_{j}, where 
(a) ϕ_{disc} (= 0.1) is the discrimination factor, and 
(b) ϕ_{inc} (= 1.1) denotes the increase factor; 
S.6.5.6 create a filtered subnetwork with its participation matrix as follows: if and , then set , otherwise use for all 1 ≤ j ≤ N_{T}, where 
(a) ϕ_{fil} (= 10) is the filtration factor, and 
(b) ϕ_{co} (= 0.05 cm^{−1}) is the cutoff factor; 
S.6.5.7 save the indices of the transitions in into l_{1}, l_{2},…, l_{NF} in the order^{[IV,V]} that either log_{10}(d_{li}) > log_{10}(d_{lj}) or log_{10}(d_{li}) = log_{10}(d_{lj}) and B_{li} ≥ B_{lj} holds for all 1 ≤ i ≤ j ≤ N_{F}, where N_{F} is the number of lines in (exclusion sort); 
S.6.5.8 for all 1 ≤ j ≤ N_{T}: set to ; 
S.6.5.9 for all 1 ≤ i ≤ : 
S.6.5.9.1 identify the components of ; 
S.6.5.9.2 if the l_{i}th line connects components in , then reset , otherwise assign P_{li} = 0 and B_{li} = 100; 
S.6.5.10 for all 1 ≤ j ≤ N_{T}: set to ; 
S.6.5.11 if , then set δ_{j} = δ_{0,j} for every 1 ≤ j ≤ N_{T} with , whereby is the trace of ; 
S.6.6 for all 1 ≤ i ≤ N_{L}: whenever b_{i} = 100 and core_{a}(comp_{a}(i)) = core(comp(i)) together with , then set the b_{i} index to , where 
(a) core_{a}(j) is the jth core index of , and 
(b) is the jth component size;^{[VI]} 
S.6.7 replace N_{p,c} with N_{a,c}, where N_{a,c} is the number of components in ; 
S.6.8 for all 1 ≤ i ≤ N_{p,c}: overwrite core_{p}(i) with core_{a}(i); 
S.6.9 for all 1 ≤ i ≤ N_{L}: set comp_{p}(i) = comp_{a}(i) and β_{i} = Ē_{i}; 
S.7 for all 1 ≤ j ≤ N_{T} with P_{j} = 0: 
S.7.1 recalculate Δ_{j} and d_{j}; 
S.7.2 set δ_{j} = max(δ_{0,j},ϕ_{inc}Δ_{j}); 
S.8 end procedure. 
^{I}At the end of the blockMARVEL process, the b_{i} block index denotes the value when (a) the ith energy level becomes a nonisolated node in , and (b) the component of this energy level contains at least two vertices, among which at least one core of can also be found. In the case that b_{i} remains 100, the ith energy level became an isolated node of during the course of prior cleansing.
^{II}The core indices of are set in analogy to eqn (4).
^{III}Since the residuals of the lines in cannot be changed, it is sufficient to restrict d_{max} to those lines satisfying .
^{IV}At this point, the values of the defects are the same as in stage S.6.5.2.
^{V}For convenience, steps S.6.5.7–S.6.5.10 will be referred to as the exclusion procedure and denoted with in the remaining of this Appendix. During the exclusion procedure, lines are excluded one by one from so as to avoid increasing the number of components in .
^{VI}It is the number of energy levels in the jth component of .
S.1 find the bridges of the (whole) SN via the BFS method;  
S.2 for all 1 ≤ i ≤ N_{T}: if the ith transition is a bridge of the SN, then set , otherwise use ;  
S.3 for all 1 ≤ i ≤ N_{R}: determine the values as  


where ℜ_{j} is the recalibration class index of the segment, for which R(ℜ_{j}) contains ;  
S.4 rearrange the recalibration classes such that should be satisfied for all 1 ≤ i ≤ j ≤ N_{R};^{[I]}  
S.5 set the flag^{[II]} to 1;  
S.6 for 1 ≤ i ≤ N_{R}:  
S.6.1 assign to R(m_{i});  
S.6.2 set , where is the recalibration factor for ;  
S.6.3 in the case that simple recalibration is not permitted for , continue S.6;^{[III]}  
S.6.4 for all 1 ≤ j ≤ N_{T}: if and , then set , otherwise use ;^{[IV]}  
S.6.5 save the indices of the lines in into p_{1}, p_{2},…, p_{Nreg};  
S.6.6 if , then continue S.6;^{[III,V]}, where  
(a) is the number of transitions in the segments of ,  
(b) μ_{rec} ≥ 2 is the recalibration margin with the default value of μ_{rec} = 5, and  
(c) ϕ_{cdd}(= 0.2) is the critical datadensity factor;  
S.6.7 for all 1 ≤ j ≤ N_{reg}: initialize δ_{reg,j} as ;  
S.6.8 ‘infinite’ loop:^{[VI]}  
S.6.8.1 determine f_{best} by means of eqn (14);  
S.6.8.2 for all 1 ≤ j ≤ N_{reg}: calculate the Δ_{reg,j} residual and the d_{reg,j} = Δ_{reg,j} − δ_{reg,j} defect;  
S.6.8.3 search for ;  
S.6.8.4 if d_{reg,max} ≤ ϕ_{noi}, then break S.6.8;  
S.6.8.5 for all 1 ≤ j ≤ N_{reg}: if d_{reg,j} > ϕ_{rd}d_{reg,max}, where ϕ_{rd}(= 0.1) is the regional discrimination factor, then substitute δ_{reg,j} with ϕ_{inc}δ_{reg,j};  
S.6.9 specify the recalibrated and nonrecalibrated absolute median residuals (AMR) as follows:  




S.6.10 whenever holds for , then continue S.6,^{[III,VII]} where  
(a) ϕ_{acc} (= 0.8) is the acceptability factor, and  
(b) ϕ_{ass} (= 3) is the assimilation factor;  
S.6.11 reassign ;  
S.6.12 reset ;  
S.6.13 for 1 ≤ j ≤ N_{T} with : replace σ_{j} with ;^{[VIII]}  
S.6.14 for 1 ≤ j ≤ N_{reg}: exchange δ_{pj} with δ_{reg,j};  
S.6.15 build an empty subnetwork in the following form: for all 1 ≤ j ≤ N_{T};^{[IX]}  
S.6.16 set N_{p,c} = N_{L};  
S.6.17 for every 1 ≤ j ≤ N_{L}: assign b_{j} = 100, core_{p}(j) = comp_{p}(j) = j, and β_{j} = 0;  
S.6.18 for all :  
S.6.18.1 construct a subnetwork of according to the following scheme: if , then , else for all 1 ≤ j ≤ N_{T};  
S.6.18.2 if , then continue S.6.18;  
S.6.18.3 find the components of the subnetwork;  
S.6.18.4 save the comp_{a}(1), comp_{a}(2),…,comp_{a}(N_{L}) component indices in ;  
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 ≤ j ≤ N_{L}: if b_{j} = 100, , and core_{a}(comp_{a}(j)) = core(comp(j)), then set ;  
S.6.18.8 overwrite N_{p,c} with N_{a,c};  
S.6.18.9 for all 1 ≤ j ≤ N_{p,c}: exchange core_{p}(j) with core_{a}(j);  
S.6.18.10 for all 1 ≤ j ≤ N_{L}: set comp_{p}(j) = comp_{a}(j) and β_{j} = Ē_{j};  
S.6.18.11 for all 1 ≤ j ≤ N_{T}: update Δ_{j};  
S.6.19 for all 1 ≤ j ≤ N_{T}: replace δ_{j} with max(δ_{j},ϕ_{inc}Δ_{j});  
S.6.20 for all with R(ℜ_{j}) = :  
S.6.20.1 set ;  
S.6.20.2 place the indices of the nonbridge transitions in s into q_{1}, q_{2},…,q_{N[s]}, where N^{[s]} is the number of nonbridge lines in s;  
S.6.20.3 define the recalibated median absolute residual (MAR) for the s segment as follows:  


S.6.20.4 if ϕ_{dec}MAR^{[s]}_{rec} < RSU^{[s]}, where ϕ_{dec} (= 2) is the declination factor, then^{[X]}  
S.6.20.4.1 substitute RSU^{[s]} with max(ϕ_{dec}MAR^{[s]}_{rec}, ϕ_{sep}ESU^{[s]});  
S.6.20.4.2 for all 1 ≤ k ≤ N_{T} with : set δ_{0,k} = ϕ_{us}RSU^{[s]} and B_{k} = ⌊log_{10}(RSU^{[s]})⌋;  
S.7 end procedure. 
^{I}The 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.
^{II}At stage S.7, denotes whether is fully recalibrated, i.e., none of the recalibration classes are recalibrated during the ABR procedure.
^{III}These empirical conditions, namely steps S.6.3, S.6.6, and S.6.10, have to be violated so that the class can be recalibrated with the ABR algorithm.
^{IV}The bridges of the SN are not included in because they are reproduced with zero residuals.
^{V}If this condition is true, then there are too few lines in for the safe recalibration of .
^{VI}The f_{best} value is obtained from an iterative refinement scheme similar to that presented in Appendix B.
^{VII}If the recalibration class is well calibrated, the condition of step S.6.10 must be satisfied.
^{VIII}If is recalibrated, then the wavenumbers of its lines should be multiplied with .
^{IX}The 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 and are referred to as the fixed blockMARVEL procedure.
^{X}If necessary, the RSU values, the initial line uncertainties, and the block indices associated to the recalibrated class should be modified to properly describe the accuracy of the improved wavenumbers.
S.1 set flag to 1;^{[I]}  
S.2 store the indices of the energy levels in t_{1}, t_{2},…,t_{NL} in the order that either comp(t_{i}) < comp(t_{j}) or comp(t_{i}) = comp(t_{j}) and Ē_{ti} ≤ Ē_{tj} is valid for all 1 ≤ i ≤ j ≤ N_{L};  
S.3 for all 1 ≤ i ≤ N_{L}:  
S.3.1 set T = t_{i};  
S.3.2 if b_{T} = 100 or core(comp(T)) = T, then continue S.3;^{[II]}  
S.3.3 for all 1 ≤ j ≤ N_{T}: if T ∈ χ_{j} and , then set , otherwise use ,^{[III]} where χ_{j} = {up(j), low(j)};  
S.3.4 place the indices of the lines in into u_{1}, u_{2},…,u_{Nloc};  
S.3.5 if N_{loc} < μ_{sync}, where μ_{sync} (= 4) is the synchronization margin, then continue S.3;^{[II,IV]}  
S.3.6 for all 1 ≤ j ≤ N_{loc}: calculate e_{j} by means of eqn (16);  
S.3.7 search for the minimum ESU value in :  


S.3.8 for all 1 ≤ j ≤ N_{loc}: initialize δ_{loc,j} as ESU_{loc,min};  
S.3.9 ‘infinite’ loop:^{[V]}  
S.3.9.1 calculate e_{best} with the help of eqn (17);  
S.3.9.2 for all 1 ≤ j ≤ N_{loc}: determine the Δ_{loc,j} residual and the d_{loc,j} = Δ_{loc,j} − δ_{loc,j} defect;  
S.3.9.3 seek for ;  
S.3.9.4 if d_{loc,max} ≤ ϕ_{noi}, then break S.3.9;  
S.3.9.5 for every 1 ≤ j ≤ N_{loc}: if d_{loc,j} > ϕ_{disc}d_{loc,max}, then replace δ_{loc,j} with ϕ_{inc}δ_{loc,j};  
S.3.10 define the adjusted and nonadjusted MAR values of the Tth energy level in the following form:  




S.3.11 if MAR^{(T)}_{na} ≤ ϕ_{sync}MAR^{(T)}_{adj}, where ϕ_{sync} (= 4) is the synchronization factor, then continue S.3;^{[II,VI]}  
S.3.12 reset ;  
S.3.13 exchange Ē_{T} with e_{best};  
S.4 if , then:  
S.4.1 initialize as for all 1 ≤ j ≤ N_{T};  
S.4.2 for all 1 ≤ j ≤ N_{T}:  
S.4.2.1 update Δ_{j} and d_{j};  
S.4.2.2 if P_{j} = 0, then:  
S.4.2.2.1 reset δ_{j} = max(δ_{0,j},ϕ_{inc}Δ_{j});  
S.4.2.2.2 if , then^{[VII]}  
S.4.2.2.2.1 set P_{j} = 1;  
S.4.2.2.2.2 assign ;  
S.4.2.3 if P_{j} = 1 and d_{j} > 0, then reassign ;^{[VIII]}  
S.4.2.4 reset δ_{j} = max(δ_{j},ϕ_{inc}Δ_{j});  
S.4.3;  
S.4.4 where ϕ_{ld} (= 0.9) is the local discrimination factor;  
S.4.5 for all 1 ≤ j ≤ N_{T}: replace δ_{j} with max(δ_{j},ϕ_{inc}Δ_{j});  
S.5 end procedure. 
^{I}Upon completion of the local synchronization procedure, will designate whether all the constrained empirical energy levels are supported by this algorithm ( = 1) or there are energy values adjusted during the synchronization process ( = 0).
^{II}If 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 subnetwork is ignored.
^{III}The subnetwork contains all the nonbridge transitions of the whole SN which are included in the CD relations of the Mth energy level. Bridges are not inserted into for the same reason detailed in Note IV of Appendix C.
^{IV} μ _{sync} provides a reasonable lower limit for N_{loc}.
^{V}The e_{best} value is determined by the aid of an iterative refinement technique (see also Appendices B and C).
^{VI}If the Ē_{T} value conforms with their trial energies, then the inequality of stage S.3.11 has to be fulfilled. In the opposite case, e_{best} is used instead of Ē_{T} in the rest of the local synchronization process.
^{VII}In the case that certain constrained empirical energy levels were altered during the local synchronization, there may be transitions which can be reincluded in .
^{VIII}If there are energy levels adjusted by the local synchronization procedure, then there must be lines of positive (updated) defects, for which the algorithm needs to be called.
^{I}Although the reincluded lines may have relatively large uncertainties, they could be useful for corroborating the dependability of the energy levels.
is built by placing the nonbridge lines of into this subnetwork.
^{III}The 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 , (b) unprotected (U), if it is alone in its bridge component, or (c) semiprotected (S), otherwise.
^{IV}These quantities can be interpreted in the following way: (a) LTD_{i} is the total number of lines in incident to the ith energy level, (b) BTD_{i} is the number of exploited lines from including this rovibrational state, and (c) LSD_{i} and BSD_{i} correspond to the numbers of sources containing transitions used in the definitions of LTD_{i} and BTD_{i}, respectively. The jth transition of the SN is exploited if B_{j} ≤ max(b_{up(j)}, b_{low(j)}), that is, if this line is utilized during the determination of its upper or lower energy level.
^{V}The 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 incident to these energy levels.
^{VI}The uncertainties of the cores are set to zero.
contains all the nonbridge lines from 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 _{mrct} ≥ n_{cct} is the smallest positive integer for which lines indexed with v_{1}, v_{2},…, v_{nmrct} arise from at least n_{ccs} sources.
^{X}As a result of this stage, a collection of n_{ct} ≥ n_{mrct} transitions, called confirmative linelist (CL), is obtained, whose lines are suitable for the estimation of the uncertainty of the ith energy level.
^{XI}If n_{mrct} > 0 and the ith energy level is not a core, then ε_{i} is calculated via eqn (37).
^{XII}This triplet can be denoted with ‘A/V/E’, as well.
^{XIII}The [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.
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