Spectroscopic-network-assisted precision spectroscopy and its application to water

Frequency combs and cavity-enhanced optical techniques have revolutionized molecular spectroscopy: their combination allows recording saturated Doppler-free lines with ultrahigh precision. Network theory, based on the generalized Ritz principle, offers a powerful tool for the intelligent design and validation of such precision-spectroscopy experiments and the subsequent derivation of accurate energy differences. As a proof of concept, 156 carefully-selected near-infrared transitions are detected for H216O, a benchmark system of molecular spectroscopy, at kHz accuracy. These measurements, augmented with 28 extremely-accurate literature lines to ensure overall connectivity, allow the precise determination of the lowest ortho-H216O energy, now set at 23.794 361 22(25) cm−1, and 160 energy levels with similarly high accuracy. Based on the limited number of observed transitions, 1219 calibration-quality lines are obtained in a wide wavenumber interval, which can be used to improve spectroscopic databases and applied to frequency metrology, astrophysics, atmospheric sensing, and combustion chemistry.

on the manuscript by R.Tobias et al « Spectroscopic-network-assisted precision spectroscopy: application to water " The paper reports very accurate measurements of water vapor near infrared line positions by cavity-enhanced optical heterodyne molecular spectroscopy combined with frequency modulation methods . This permitted measurements of 156 saturated Doppler-free absorption lines in the range 7000-7300 cm-1 with the intensities between 10-20 and 10-26 cm molecule-1.
The reported work represents a very significant effort in the extension of precise measurements of water vapor transitions towards the exceptional 10-7 cm-1 accuracy in line positions.
This was achieved by approaching nearly zero pressure limit in the infrared range experiments together with a careful choice of transitions for a consistent energy level determination. This in turn permitted generating new transitions via well known Ritz-Bohr-Einstein relation between the photon frequency and the upper and lower energy levels.
Information for accurate line positions is important for calibration purposes because water molecule is ubiquitous in various environments in terrestrial and astrophysical applications. They could also serve as reference values in the pressure line shifts studies.
The authors thoroughly discuss the sources of data as well as their validation and supply Supplementary files for observed and calculated data. The manuscript is in general clearly written.
I would recommend the authors to consider the following comments prior publication: 1. When describing the theoretical part of the paper ( Ritz-SNAPS procedure) it would be fair to say that the Ritz principle had been already widely employed in the spectroscopic literature to determine rovibrational levels from observed transitions for various types of molecules accounting for the error propagations ( for example *1-*5 below and refs therein as well as MARWELL series of the authors). The corresponding comments and citation are to be added. The original part at this part of the work is a consistent choice of target lines for laborious experimental measurements rather than a development of a new Ritz-based procedure for the data extension.
2. The new terminology "magic number" employed at page 14 instead of the conventional term "ortho-para energy splittings for J=1" would possibly be justified in case of a new physical content or a new method of its determination. It is not the case. I would recommend using this old physically meaningful term for clarity. The corresponding value had been computed in many previous works, and here it comes out from a standard polynomial fit using the same very old effective model.
The question is whether this model ( Watson effective rotational Hamiltonian) originally based on a perturbation theory in frame of Born-Oppenheimer approximation could really provide an accuracy of 10-7 cm-1 ? In many previous works it was shown that this is not a best model for rotational levels of the water molecule ( for example *5 below and refs therein) suffering from severe convergence and extrapolation issues.
3. This comment is linked to the previous one. The authors introduced `artificial' transitions in the effective Hamiltonian fit that looks similar to a combination differences approach. A well-known risk of a polynomial fit with a large number of parameters concerns possible "fluctuations" for missing data points. The authors have to specify, how many energy differences of newly determined ground vibrational state (000) levels were used for the fit , and how many parameters of 14-th order Watson Hamiltonian were used ?
4. What about line intensities ? Was it possible improving these improve these important line parameters using reported high-precision measurements? *1. J-M. Flaud, at al. Higher ro-vibrational levels of H2O deduced from high resolution oxygenhydrogen flame spectra between 2800-6200 cm-*2. JKG Watson . The use of term-value fits in testing spectroscopic assignments. J. Mol. Spectrosc. This manuscript addresses the disconnect between the accuracy of the (mostly) Doppler-limited measurements used for spectroscopic databases (uncertainties often > 30 MHz even for water) with the extremely high accuracy now available with sub-Doppler and Doppler-Free experiments with frequency comb calibration (~3 kHz). Investigations of molecules at this high level of accuracy usually only result in a handful of transition frequencies and are unable to contribute much to databases composed of thousands of transitions over a broad frequency range. The authors argue that network-theory applied to spectroscopic networks (based on the Ritz principle) can determine a set of transitions which maximize the scientific utility of highly accurate measurements given the experimental constraints. They use a new network-theory based routine (SNAPS) to select 156 transitions of water within the sensitivity and frequency coverage of a NICE-OHMS instrument (7000-7350 cm-1) and measure these transitions with 1.5-10 kHz uncertainties. From this limited data set, SNAPS predicted the frequency of 1219 transitions with uncertainties of ~10 kHz, including ~600 rotational transitions. To extend this to absolute energy levels, they connect the ortho and para manifolds using both nuclear motion calculations of ortho/para doublets which are linked with paths to the lowest ortho/para levels and with a fit of ground state rotational levels to an effective Hamiltonian.
This result is impressive, although measuring 156 Doppler-Free transitions with <10 kHz uncertainties by itself is not a trivial feat, it is especially significant that it could provide such an improvement to a large number of transitions at THz frequencies and energy levels relative to the ground rotational state. Spectroscopic databases are heavily utilized by a number of fields, including atmospheric science and astronomy communities, and it is valuable to have methods which can reveal systematic errors in past experiments and to improve the accuracy of available data. Absolute energy levels of molecules such as water are also useful as benchmarks for quantum chemistry and for models of hot exoplanetary atmospheres. The SNAPS routine will benefit other high-precision spectroscopy groups who are trying to perform spectroscopic surveys with the best possible efficiency. The manuscript is well written, with clear figures showing the working principles of SNAPS and spectroscopic networks. The statistical analysis for the error determination of the predicted transitions and energy levels is done appropriately. I recommend the manuscript for publication after addressing a few minor issues and considering some suggested edits.
(Minor issues) 1. Introduction, paragraph 2, sentence 4: A NICE-OHMS setup is described as being disciplined to a Cs clock and to a GPS reference. It does not make sense to me for an instrument to use both frequency standards simultaneously and should either be described as "or" or one of the two.
2. Results and discussion, paragraph 2, sentence 2: It would be helpful to provide what uncertainty constitutes "former precision measurements".
3. Transfer of measurement accuracy, paragraph 1, sentence 5: I do not directly see the connection between points a,b,c and being superior to an effective Hamiltonian. Points a and b (and in some cases c) are often true for predictions from molecular constants. The reason SNAPS would be superior to an effective Hamiltonian is that it only relies on the experimental values, and therefore will not suffer from issues like overfitting/choice of parameters. I suggest this sentence be revised to reflect this. 4. Results and discussion, paragraph 9, sentence 3: A typical path is shown, but as seen in Fig. 3 there are multiple paths to the (000) 1_11 level [through (101) 1_01 or (200) 1_11]. It would be helpful to include a statement regarding how the energy levels in redundant cases are determined. If a weighted average is taken or if the lowest uncertainty path is used. 5. Results and discussion, paragraph 10, sentence 3: It is unclear to me what the states were redefined with respect to. If possible, this sentence should be reworded for clarity.
6. Results and discussion, paragraph 10, sentence 3: Should the phrase " too weak target lines" instead be "two weak target lines" or "too weak of target lines"? 7. Experiment: Although a thorough description of the instrument is available in another publication, the manuscript would be more self-contained if even a short description were included either in the Experiment section or in the SM.
(Comments and suggestions) 1. Introduction, paragraph 2, sentence 4: This sentence is unclear. It could be misinterpreted as "the NICE-OHMS" setup being a general NICE-OHMS setup. Either rephrase it to read "In the NICE-OHMS setup used in this work…" or phrase it such that it reads as an example of the capability of NICE-OHMS.
2. NICE-OHMS precision spectroscopy of H2O: In my opinion, it seems that the flow would be more logical to have paragraphs 2 and 3 switched. Paragraph 3 describes the experimental results related to paragraph 1. Paragraph 2 describes the confirmation of the uncertainty in paragraph 3 with SNAPS, and would logically follow descriptions of the experimental results.
3. Transfer of measurement accuracy, paragraph 1, sentence 4: It states that it "may have an accuracy characteristics of the NICE-OHMS setup". Should there be reason to doubt this, or is it accounting for the accumulation of uncertainties from long paths? 4. Results and Discussion: It seems the closest thing to a concluding paragraph is paragraph 5 of "Magic number". In my opinion, the manuscript would greatly benefit from a strong summary after "Frequency standards". 5. Theory, paragraph 3, sentence 3: Was there a reason for weighting transitions by uncertainty instead of strength or a combination of factors? I could envision cases where it would be experimentally easier to measure one relatively uncertain value vs multiple more certain transitions, or measuring a stronger transition vs a weaker transition.
6. Theory paragraph 7: Typo, weigths" General notes: The manuscript has been carefully revised in response to the reviewers' comments and suggestions, responses to the comments are detailed below. Our reply is highligthed in blue, whereas the reviewers' specific comments are in plain text. All changes introduced in the manuscript are also highlighted in blue. We are greatly indebted to the reviewers for their suggestions, which have significantly improved our manuscript.

REVIEWERS' COMMENTS:
Reviewer #1 ( The application of predictive networks to inform the targeted sub-Doppler spectroscopy of small molecules like water will have immediate impact, as current spectroscopic databases and radiative transfer codes (e.g., HITRAN) as well as Doppler-broadened comb-reference measurements [e.g., Sironneau and Hodges, JQSRT 152, 1-15 (2015)] are all less accurate by at least 3 orders of magnitude. Clearly, this comprehensive result immediately improves the line-by-line data for arguable the most important small molecule (water) and, in my opinion, will undoubtedly inspire further advances in interdisciplinary approaches at the cross-roads of computer science, first-principles quantum chemistry, and optical physics.
The statistical analysis appears consistent with best practices in reporting experimental uncertainty, and the details provide should allow a knowledgeable researcher with expertise in NICE-OHMS to reproduce the experimental results.
The accuracy and sensitivity of the NICE-OHMS experiments, as detailed in the methods section, merit acknowledgement in their own right as a noteworthy achievement.
The application of sub-Doppler spectroscopy to >150 transitions is a painstaking task, and therefore having such experiments informed by a predictive network theory is powerful. I have several minor comments below, but overall rate the manuscript as one of the highest examples of interdisciplinary work in accurate spectroscopy in recent memory. Therefore, I believe that this paper will immediately achieve high impact -and I strongly recommend it's acceptance.
The empirical assignment of connections in the network appears to be a power approach to efficiently identifying transitions with local perturbations. This is currently beyond the grasp of standard spectroscopic models derived from an effective Hamiltonian.  4. What about line intensities? Was it possible to improve these important line parameters using reported high-precision measurements? It should be noted that the current study is based on Doppler-free saturation measurements of spectral lines.
Hence it is a study in non-linear optics. Although the line intensity in saturation follows somehow the transition dipole moment, it is not an obvious step to extract information of the Einstein-A coefficients, that is usually included in databases like HITRAN. Our study focuses fully on the quantum level structure of the water molecule, and this is done at high precision. Other important parameters, usually included in molecular databases, concern line broadening effects. We find in our study a different collisional effect than reported in HITRAN. This is illustrated in Fig 4c and  1. Introduction, paragraph 2, sentence 4: A NICE-OHMS setup is described as being disciplined to a Cs clock and to a GPS reference. It does not make sense to me for an instrument to use both frequency standards simultaneously and should either be described as 'or' or one of the two.
We have moved this statement from the introduction to the Methods section, where we have elaborated on the experiment, also explaining this issue. In fact the spectroscopy laser is locked to the optical cavity for short term stability, then locked also to the frequency comb (OFC), which itself is locked to an atomic clock (Cs-clock) for long-term stability and frequency determination, and further corrected by signals from the GPS system. This is mentioned in the text of Methods now.
2. Results and discussion, paragraph 2, sentence 2: It would be helpful to provide what uncertainty constitutes 'former precision measurements'.
Response: The accuracy of former precision measurements is on the order of (some) kHz, which is now specified in the manuscript.