PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra
Deriving the Spectrum, Ch Polyatomic molecules: normal modes and symmetry
Problem
The strong carbon dioxide fundamentals are ν2=667cm−1 (bend, Πu, infrared active) and ν3=2349cm−1 (antisymmetric stretch, Σu+, infrared active); the symmetric stretch ν1 (Σg+) is Raman-only and, because of the Fermi resonance, appears as the Raman doublet at 1285 and 1388cm−1, whose mean is 1336.5cm−1. A laboratory infrared spectrum of carbon dioxide gas also shows two weaker absorption lines, at 3609cm−1 and 3714cm−1. (a) Anharmonicity allows combination bands whose wavenumber is approximately the sum of the wavenumbers of the modes involved. Using the Raman-doublet mean for ν1, evaluate ν1+ν3 and 2ν2+ν3, and give the symmetry species of the level ν1+ν3 and of the Σg+ component of 2ν2 combined with ν3. Explain why these two sums cannot by themselves assign one weak line to each combination. (b) Use the two-level coupling that explained the Raman doublet to explain why two lines about 105cm−1 apart appear instead of two nearly coincident ones, and estimate the coupling W, taking the unperturbed gap Δ as the difference of your two sums. Compare it with the coupling found from the Raman doublet. (c) Carbon dioxide has only 3N−5=4 normal modes. Explain why neither weak line can be a fifth fundamental. (d) A 288K surface emits, per unit wavenumber, with its maximum near 565cm−1 and about three quarters of its power between 400 and 1400cm−1. Say whether the 3609 and 3714cm−1 lines fall inside or outside the range that matters for trapping Earth's outgoing thermal radiation, and contrast this with ν2 at 667cm−1.