A linear symmetric triatomic molecule A-B-A lies along the axis at equilibrium
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra
Action and Symmetry, Ch Small oscillations: stability, degeneracy, symmetry and spectra
Problem
A linear symmetric triatomic molecule A-B-A lies along the x axis at equilibrium, with B at the centre; by symmetry its equilibrium dipole moment is exactly zero. Three of its normal modes are described by the following arrow diagrams (displacements at one instant of the oscillation, with the opposite displacement occurring half a period later): (a) Mode P: atom A1 moves in −x, atom B stays fixed, atom A2 moves in +x (both outer atoms move away from the centre, by equal amounts). (b) Mode Q: atom A1 moves in +x, atom B moves in −x, atom A2 moves in +x (both outer atoms move the same way, atom B the opposite way, so that the total momentum stays zero). (c) Mode R: atom A1 moves in +y, atom B moves in −y, atom A2 moves in +y (the same pattern as Q, but perpendicular to the molecular axis). For each mode, decide whether the oscillating charge distribution gives the molecule a dipole moment that oscillates at the vibration's frequency, and hence whether that mode can absorb infrared radiation.