Trans-1,2-dichloroethene,

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

trans-1,2-dichloroethene,

\mathrm{ClHC=CHCl}

, is planar and has the symmetry group

C_{2h} = \{E, C_2, \sigma_h, i\}

. Here

C_2

is the half-turn about the axis through the midpoint of the

\mathrm{C{=}C}

bond, perpendicular to the molecular plane. The reflection

\sigma_h

in the molecular plane leaves every atom where it is, because all atoms lie in that plane, but it is still a distinct, non-identity operation. The inversion

i

is through the bond midpoint. ((a)) State how many elements lie in each conjugacy class of

C_{2h}

, and give the reason. ((b)) Label the two carbons

\mathrm{C_1, C_2}

, the two chlorines

\mathrm{Cl_1, Cl_2}

and the two hydrogens

\mathrm{H_1, H_2}

, with

C_2

sending

\mathrm{C_1 \to C_2}

\mathrm{Cl_1 \to Cl_2}

and

\mathrm{H_1 \to H_2}

. Write the permutation of these six atoms produced by

C_2

in cycle notation and give its sign (

+1

if even,

-1

if odd). ((c)) Given that the two chlorine atoms form a single orbit under

C_{2h}

, use the orbit-stabiliser theorem to find the order of the subgroup that fixes

\mathrm{Cl_1}

in place.

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