 Rotational spectral line assignment

A rotational spectral line assignment associates a measured spectral line with a molecule, an internal quantum state, and an initial and final set of quantum numbers. For the spin-free linear rigid rotor, absorption is labelled $J\to J+1$. A regularly spaced group suggests a candidate rotational constant, but the lowest line inside a scanned band need not be $0\to 1$. Several molecules can contribute to the same spectrum, and an apparent missing member can lie outside the band or below the detection threshold.

First determine line centers and their uncertainties from the instrument's readout, accounting for resolved Doppler shifts or hyperfine structure rather than treating every peak as a separate rotational transition. Then propose integer assignments, fit a rotational Hamiltonian, and predict additional spectral lines. For a weakly distorted linear molecule within one vibrational quantum state, the centrifugal distortion model gives the useful diagnosticHere $B,D,{\nu }_m$ are all frequencies in $\mathrm{Hz}$. Correct assignments make the second expression a straight line against $m^2$, with intercept $B$ and slope $-2D$. This follows by division of the transition formula; no additional model of molecular geometry is required. A direct weighted least squares fit of ${\nu }_m$ to $2Bm-4D{m}^3$ uses the measured uncertainties. If instead the transformed values are fitted, their uncertainties must also be divided by $2m$.^[ref]

For an original example, stipulate $B=5.000000\mathrm{GHz}$ and $D=2.000\mathrm{kHz}$, and suppose a scan covers $38$–$72\mathrm{GHz}$. The model places four consecutive transitions at $39.999488$, $49.999000$, $59.998272$, and $69.997256\mathrm{GHz}$. Their assignments are $m=4,5,6,7$, hence $J=3,4,5,6$. An extra stipulated line at $53.2\mathrm{GHz}$ is unrelated to this series. All these frequencies are calculated example inputs, not measured OCS data.

At $m=4$ and $m=7$, the transformed values are $4999.936$ and $4999.804\mathrm{MHz}$. Their straight line has slope $(4999.804-4999.936)/(49-16)=-0.004\mathrm{MHz}$, recovering $D=0.002\mathrm{MHz}$ and $B=5000\mathrm{MHz}$. Both intermediate transitions satisfy the same line exactly. Merely relabelling the first visible transition $0\to 1$ fails this diagnostic. The model predicts a withheld $7\to 8$ transition at $79.995904\mathrm{GHz}$: an additional measurement there would test the assignment rather than merely refit the original four lines.

An assignment earns confidence when several independently observed spectral lines, including lines outside the fitted interval, agree within uncertainty and expected strong accessible lines are present. Inspect the residual of an inverse problem for each line against $J$: systematic curvature can signal inadequate centrifugal distortion, whereas isolated discrepancies suggest a blend or wrong assignment. Comparing a known sample, a blank, and an isotope shift of a linear rotor supplies different checks. These are general inference controls; the actual reported controls of the NIST trace-gas FTMW validation experiment are described separately. One isolated match or one fitted rotational constant alone cannot certify molecular identity.