Microwave / Rotational Spectroscopy

Pure rotational spectrum of a diatomic molecule: B from the masses and bond length, B_v with α_e, line frequencies 2B(J+1) − 4D(J+1)³, Boltzmann populations and absolute line intensities S in HITRAN units, with CO and HCl presets.

Model: Textbook approximation

Load molecule (¹Σ diatomic)Atomic Mass m₁ (u)Atomic Mass m₂ (u)Equilibrium Bond Length r_e (Å)Vibration–Rotation Constant α_e (MHz)Centrifugal Distortion D (MHz)Dipole Moment μ (D)Temperature (K)Vibrational Level v (0–5)

Rotational Constant B_e

1.93128 cm⁻¹

57.8984 GHz; I = μr_e² = 1.449e-46 kg·m²

B_v (v = 0)

57.63598 GHz

B_e − α_e(v + ½) = 1.92253 cm⁻¹

J = 1 ← 0 Line

115.2712 GHz

2B_v − 4D; λ = 2.601 mm

Partition Function Q_rot

108.87

kT/hcB + ⅓ = 108.79; most populated J = 7

Strongest Line

J = 13 ← 12

1496.92 GHz

Its Intensity S

1.522e-21 cm/molecule

HITRAN units, per molecule in v = 0, not abundance-weighted; JPL log₁₀ I = -2.3408

J = 1 ← 0 Intensity

3.257e-24 cm/molecule

JPL log₁₀ I = -5.0104 (nm²·MHz)

Lines Listed

down to 10⁻⁶ of the strongest

B = h/(8π²cμr²), ν(J+1 ← J) = 2B_v(J+1) − 4D(J+1)³, S = 2π²σμ²(J+1) e^(−E_J/kT)(1 − e^(−hν/kT)) / (3hcε₀Q)

The line intensity is the HITRAN integrated cross-section per molecule in level v (multiply by the vibrational population for v ≥ 1, and by the isotopologue abundance for a gas sample). Its (J+1) factor is the dipole matrix element summed over M; with the frequency factor and 1 − e^(−hν/kT) ≈ hν/kT it grows as (J+1)³ e^(−E_J/kT), so the strongest line lies above the most populated level. For ¹²C¹⁶O at 300 K the J = 1 ← 0 line is 115.2712 GHz and log₁₀ I = −5.0105 in the JPL catalogue; this page gives 115.2712 GHz and −5.0104.

Linear ¹Σ molecules only: no hyperfine structure (HCl shows three Cl-quadrupole components ≈ 30 MHz apart), no Λ-doubling, no symmetric-top K structure, no nuclear-spin statistics and no vibrational partition function. Homonuclear molecules (μ = 0) have no pure rotational spectrum.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Linear ¹Σ rotor with first-order centrifugal distortion; intensities from the permanent dipole with exact Boltzmann and stimulated-emission factors. Intensities are per molecule in the chosen level v, not weighted by vibrational population or isotopic abundance. No hyperfine structure or nuclear-spin statistics. CO matches the JPL catalogue to 0.2 ppm in frequency and 0.001 in log intensity.

Gordy W., Cook R. L. (1984). Microwave Molecular Spectra, 3rd ed. Wiley.
Pickett H. M., Poynter R. L., Cohen E. A., Delitsky M. L., Pearson J. C., Müller H. S. P. (1998). Submillimeter, millimeter, and microwave spectral line catalog. JQSRT 60, 883–890. doi:10.1016/S0022-4073(98)00091-0
NIST Chemistry WebBook, SRD 69: constants of diatomic molecules (after Huber & Herzberg 1979). webbook.nist.gov
JPL Molecular Spectroscopy catalogue, entries 28001 (CO) and 36001 (HCl). spec.jpl.nasa.gov

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