Dichloromethane,

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

Dichloromethane, CH2Cl2\mathrm{CH_2Cl_2}, is a tetrahedral-centred molecule of point group C2vC_{2v}: the twofold axis bisects both the H-C-H and Cl-C-Cl angles, one mirror plane σ(Cl)\sigma(\mathrm{Cl}) contains the carbon and both chlorine atoms, and the other mirror plane σ(H)\sigma(\mathrm{H}), perpendicular to it, contains the carbon and both hydrogen atoms. The classes, each of size 11, are EE, C2C_2, σ(Cl)\sigma(\mathrm{Cl}), σ(H)\sigma(\mathrm{H}), with irreducible characters on (E,C2,σ(Cl),σ(H))(E,C_2,\sigma(\mathrm{Cl}),\sigma(\mathrm{H})): A1A_1: (1,1,1,1)(1,1,1,1) A2A_2: (1,1,−1,−1)(1,1,-1,-1) B1B_1: (1,−1,1,−1)(1,-1,1,-1) B2B_2: (1,−1,−1,1)(1,-1,-1,1) Let Γ\Gamma be the representation generated by the permutation of the two chlorine labels. (C2C_2 and σ(H)\sigma(\mathrm{H}) both swap the two chlorines, since neither chlorine lies on the axis or in the plane σ(H)\sigma(\mathrm{H}); σ(Cl)\sigma(\mathrm{Cl}) contains both chlorines and so fixes them.) ((a)) Give the character of Γ\Gamma on each class. ((b)) Decompose Γ\Gamma into irreducible representations.

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