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
Formaldehyde, H2CO, is planar and belongs to the point group C2v; by the convention that the molecular plane is yz, the C2 axis is z (it passes through the carbon and the oxygen) and the direction perpendicular to the molecular plane is x. Use the C2v character table of this chapter (operations E, C2, σv(xz), σv′(yz)). (a) Count the unshifted atoms under each operation, construct χ3N, reduce it with the reduction formula, and remove the translations and rotations to find Γvib. Check that its dimension is 3N−6. (b) Using the rule that the infrared matrix element ⟨1k∣d∣0⟩ is non-zero only if the mode's species Γk matches the species of x, y or z, state which of formaldehyde's species are infrared active. (c) Using the same argument for the Raman matrix element ∂(αpol)ij/∂Qk, which requires Γk to match the species of a quadratic, state which are Raman active. (d) Hence state how many of formaldehyde's fundamentals are infrared active, how many are Raman active, and whether any is silent in both spectra. (e) Carbon dioxide shows mutual exclusion: no fundamental is active in both spectra. Explain, from the point groups of the two molecules, why formaldehyde does not.