Infrared (IR) Spectroscopy
Vibration–rotation band of HCl or CO: anharmonic band origin, P and R branch lines, Boltzmann intensities at the instrument resolution, force constant and the isotope shift of H³⁷Cl or ¹³CO.
Band Origin ν₀
2885.98 cm⁻¹
G(1) − G(0); λ = 3.4650 µm
R(0) / P(1)
2906.24 / 2865.10
cm⁻¹; gap ≈ 2(B₀ + B_v) = 41.15 cm⁻¹, no Q branch
Strongest R / P Lines
R(2) / P(3)
2944.87 / 2821.55 cm⁻¹; most populated J = 3
Isotope Shift of ν₀
-2.106 cm⁻¹
H³⁷Cl − H³⁵Cl; ρ = √(μ/μᵢ) = 0.999243
Harmonic Frequency ω_e
2990.95 cm⁻¹
fundamental 2885.98 cm⁻¹ (anharmonic)
Force Constant k
516.3 N/m
k = μ(2πcω_e)²
Anharmonicity ω_e x_e
52.819 cm⁻¹
Morse well depth ω_e²/(4ω_e x_e) = 5.25 eV (rough)
B₀ / B_v
10.4399 / 10.1327
cm⁻¹; B_v = B_e − α_e(v + ½)
ν = G(v) − G(0) + F_v(J′) − F₀(J″), G(v) = ω_e(v+½) − ω_e x_e(v+½)² + ω_e y_e(v+½)³ + ω_e z_e(v+½)⁴, F_v(J) = B_vJ(J+1) − DJ²(J+1)²
R(J″): J′ = J″ + 1, P(J″): J′ = J″ − 1; a ¹Σ molecule has no ΔJ = 0 line, hence the gap at ν₀. Line strengths are relative: σ × Hönl–London factor (J″+1 for R, J″ for P) × Boltzmann fraction × (1 − e^(−hcσ/kT)), convolved with a Gaussian of the chosen resolution. The isotopologue uses the same potential with ω_e ∝ μ^(−½), ω_e x_e ∝ μ⁻¹, B_e ∝ μ⁻¹, and its band is scaled by ρ, the harmonic ratio of squared transition dipoles (|⟨1|x|0⟩|² ∝ 1/μω); for overtones that ratio is only approximate. Constants: NIST WebBook (Huber & Herzberg).
No Herman–Wallis factor (vibration–rotation interaction makes real P and R lines of equal J unequal, by several per cent at moderate J in the HCl fundamental and more in overtones), no pressure or Doppler broadening, no absolute band strength: the y-axis shows shape, not absorbance. Polyatomic molecules have 3N − 6 modes and group frequencies (C=O ≈ 1700, O–H ≈ 3300, C–H ≈ 2900 cm⁻¹) that this page does not model.
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
Anharmonic oscillator to (v + ½)⁴ plus vibrating rotor (B_v = B_e − α_e(v + ½), one distortion constant) with NIST constants (including ω_e y_e and ω_e z_e); relative line strengths from Hönl–London, Boltzmann and stimulated-emission factors, without Herman–Wallis factors, pressure broadening or absolute band strengths. With one D term, HCl is limited to 1200 K.
- Herzberg G. (1950). Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules, 2nd ed. Van Nostrand.
- NIST Chemistry WebBook, SRD 69: constants of diatomic molecules (after Huber & Herzberg 1979). webbook.nist.gov
- Wang M., Huang W. J., Kondev F. G., Audi G., Naimi S. (2021). The AME 2020 atomic mass evaluation (II). Chin. Phys. C 45, 030003. doi:10.1088/1674-1137/abddaf
- Meija J. et al. (2016). Isotopic compositions of the elements 2013 (IUPAC Technical Report). Pure Appl. Chem. 88, 293–306. doi:10.1515/pac-2015-0503