Sulfur dioxide,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

Sulfur dioxide, SO2\mathrm{SO_2}, has point-group symmetry C2vC_{2v}, with the molecular plane taken as yzyz (so xx is out of plane) and the two-fold axis zz bisecting the O-S-O angle. The character table (classes EE, C2C_2, σv(xz)\sigma_v(xz), σv′(yz)\sigma_v'(yz)) is A1=(1,1,1,1)A_1=(1,1,1,1) with basis function zz, A2=(1,1,−1,−1)A_2=(1,1,-1,-1), B1=(1,−1,1,−1)B_1=(1,-1,1,-1) with basis function xx, B2=(1,−1,−1,1)B_2=(1,-1,-1,1) with basis function yy. ((a)) Write down the 3×33\times3 matrix representation D(g)D(g) of (x,y,z)(x,y,z) displacement of the sulfur atom under each class, and give its character. ((b)) Decompose DD into irreducible representations.

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