A spectroscopy study reports a way to reconstruct molecular energy levels from transition frequencies without first deciding which quantum states produced the lines. Applied to HDO, a water isotopologue, the graph-based method reconstructed 158 energy levels from 686 unassigned transitions. The authors report that these levels were determined with kilohertz-level precision, more precise than the MARVEL HDO energies used for comparison. The demonstration is confined to the measured spectra, so it does not establish how the approach will perform on other molecules.
A network hidden in the frequency gaps
Instead of a pre-existing physics model, the procedure starts with measured line positions. It groups recurring pairwise frequency differences as candidate connections, then searches for matching four-line loops. Those loops instantiate local graph structures and allow the data to be assembled into an energy-level network. The lines do not need labels before the analysis begins.
The instrument behind the result
SCALS supplied the measurements. It is an automated cavity-enhanced spectrometer that scans continuously across tens of terahertz while combining broadband coverage, high sensitivity and kilohertz-level accuracy. The laboratory spectra covered acetylene, methane and HDO at 296 K.
Absolute frequency calibration used an optical frequency comb referenced to a GPS-disciplined rubidium clock. The reported fractional accuracy exceeded 1 × 10−12, corresponding to 0.19 kHz at a wavelength of 1.6 µm.
How much of the spectrum was recovered
The HDO network was substantial but incomplete. Of the 686 measured transitions, 425 were assigned to the reconstructed network and 261 remained unassigned; the latter predominantly involved other water isotopologues. The 158 levels were recovered in two subnetworks, so the result is a map of part of the measured spectrum rather than a complete accounting of every line.
Against a reference set called MARVEL, the reconstructed levels agreed within its stated uncertainties except for eight levels that exceeded 0.0005 cm⁻¹. The largest reported deviation was 0.00256 cm⁻¹. The no-assignment claim applies to the input: the calculation began without labels, while MARVEL supplied the comparison used afterward. A complete uncertainty distribution for all reconstructed levels was not supplied, so the precision result should be read as a reported comparison, not a guarantee that every level carries the same accuracy.
Checks in other molecules
Acetylene provided a separate check on repeatability. Repeated line-position measurements showed a power-dependent shift of approximately -12 kHz per mW, and the standard deviation of a single measurement ranged from 11 to 30 kHz. Reported variability therefore changed with the injection power used in the experiment.
A broadband acetylene survey found 129 spectral features, including weak transitions. Eighteen could be compared with literature positions, and their deviations ranged from -180 to 80 kHz. Because the literature comparison covered 18 of the 129 detections, it was a partial check on line positions rather than a full external validation of the survey.
Methane offered a different test of spectral interpretation. A feature at 6114.6707 cm⁻¹ was identified as a crossover resonance rather than a genuine transition, while the actual transition at 6114.664840 cm⁻¹ had not been detected. The broadband scan also found one two-photon absorption transition in each of 12C2H2 and HD16O, both identified for the first time. The HD16O line was located at 6288.755116 cm⁻¹.
What the evidence leaves open
These findings do not show that the algorithm works for arbitrary molecules. The focal reconstruction used HDO, while acetylene and methane supplied validation and feature searches. Performance beyond these measured spectra remains unresolved, as does the method's broader generalizability.
Taken together, the preprint supports a narrow conclusion: frequency patterns alone can be used to reconstruct a sizeable HDO energy-level network without prior assignments, but the current network captured only part of the measured data. It is a methods result, not evidence of complete recovery across molecular spectroscopy.
Status and supporting material
The document is an arXiv version 1 preprint dated 28 Aug 2026. It reports joint support from the National Natural Science Foundation of China, the Independent Deployment Project of HFNL, the Quantum Science and Technology - National Science and Technology Major Project, and the Chinese Academy of Sciences. The authors declare no competing interests.
Supplementary Information includes an acetylene accuracy assessment, broadband C2H2 and HDO survey spectra, and details of the inverse graph construction for HDO energy levels.
Paper data and sources
Original title: Model-free Reconstruction of Molecular Energy Levels by Broadband Kilohertz-accurate Cavity-enhanced Spectroscopy
Authors: S. Vasilchenko, A. -W. Liu, C. -X. Zuo et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text