Chloroform, , has symmetry with the bond along . Place the three chlorines at , , for…

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

Invariance, Ch Groups: the algebra of symmetry operations

Problem

Chloroform, CHCl3\mathrm{CHCl_3}, has C3vC_{3v} symmetry with the C−H\mathrm{C{-}H} bond along zz. Place the three chlorines at Cl1=(0,1,h)\mathrm{Cl}_1=(0,1,h), Cl2=(−32,−12,h)\mathrm{Cl}_2=(-\tfrac{\sqrt3}{2},-\tfrac12,h), Cl3=(32,−12,h)\mathrm{Cl}_3=(\tfrac{\sqrt3}{2},-\tfrac12,h) for some h<0h<0. The mirror plane containing Cl1\mathrm{Cl}_1 and the zz axis is the yzyz plane, represented (Chapter 1) by M=(−100010001).\mathsf{M} = \begin{pmatrix}-1&0&0\\0&1&0\\0&0&1\end{pmatrix}. ((a)) State det⁡M\det\mathsf{M} and say what kind of operation this identifies. ((b)) Apply M\mathsf{M} to Cl1\mathrm{Cl}_1, Cl2\mathrm{Cl}_2, Cl3\mathrm{Cl}_3 and write the induced permutation of the three chlorines. ((c)) Give its parity and relate the result to the isomorphism C3v≅S3C_{3v}\cong S_3.

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