CO Fundamental Band from HITRAN

What this research found

Line positions for the fundamental vibrational band of carbon monoxide, the transition from v = 0 to v = 1, were fitted as a function of the running rotational index and two models were compared: a quadratic form corresponding to the rigid-rotor approximation, and a quartic expansion. The quadratic fit leaves a systematic S-shaped residual that grows toward both ends of the band, while the quartic fit flattens the residuals almost completely, cutting the root-mean-square error from 0.293 to 2.33 × 10⁻⁴ cm⁻¹.

• The rigid-rotor quadratic model leaves structured rather than random residuals. They trace an S-shaped curve that swings to roughly plus and minus 0.7 cm⁻¹ at the highest rotational lines, giving a root-mean-square error of 0.293 cm⁻¹.
• Extending the fit to quartic order removes that structure entirely. The residual root-mean-square error drops to 2.33 × 10⁻⁴ cm⁻¹, better by more than a thousandfold, and the residuals sit flat across the whole band.
• The misfit is concentrated where the rigid-rotor picture is weakest, at high absolute values of the running index, rather than being spread evenly across the band.
• The fitted band covers the P-branch out to a running index near -52 and the R-branch out to about +34, spanning roughly 1900 to 2250 cm⁻¹, with the expected gap at the centre where the index is zero and the transition is forbidden.

How it was done

Transition wavenumbers for the ¹²C¹⁶O isotopologue of carbon monoxide were taken from the HITRAN spectroscopic line list and indexed by the standard running variable, which equals minus the lower-state rotational quantum number for P-branch lines and that quantum number plus one for R-branch lines. That choice places both branches on a single smooth curve with a forbidden gap at the origin. A quadratic and a quartic polynomial were then fitted to the curve and compared through their residuals across the full extent of both branches, producing a two-panel diagnostic that pairs the fitted band with the residuals of each model.

Data sources

• HITRAN spectroscopic line list - transition wavenumbers for the ¹²C¹⁶O fundamental band (v = 0 to v = 1)