Sulfur dioxide,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra

Invariance, Ch Character tables at work: vibrations, selection rules and splitting

Problem

Sulfur dioxide, SO2\mathrm{SO_2}, is bent with point group C2vC_{2v}: the sulfur atom sits on the C2C_2 axis and both oxygen atoms lie in the molecular plane, taken as yzyz by the usual convention. Use the C2vC_{2v} character table (classes EE, C2C_2, σv(xz)\sigma_v(xz), σv′(yz)\sigma_v'(yz); basis functions z→A1z\to A_1, x→B1x\to B_1, y→B2y\to B_2, Rz→A2R_z\to A_2, Ry→B1R_y\to B_1, Rx→B2R_x\to B_2). ((a)) Find the character of the displacement representation Γ3N\Gamma_{3N} and decompose it into irreducible species. ((b)) Remove translations and rotations to find Γvib\Gamma_{\rm vib}, and state the number of distinct vibrational frequencies. ((c)) Decompose the direct product B1⊗B2B_1\otimes B_2. ((d)) An excited electronic state of SO2\mathrm{SO_2} has symmetry B1B_1 and the ground state is A1A_1. Using the vanishing integral theorem, decide whether the transition is allowed by one-photon electric-dipole absorption and, if so, state which Cartesian polarisation of light drives it.

Browse the problem library