In (octahedral, ) place the six fluorines at , , , , , (bond length set to 1)

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

In SF6\mathrm{SF_6} (octahedral, OhO_h) place the six fluorines at 1=(1,0,0)1=(1,0,0), 2=(−1,0,0)2=(-1,0,0), 3=(0,1,0)3=(0,1,0), 4=(0,−1,0)4=(0,-1,0), 5=(0,0,1)5=(0,0,1), 6=(0,0,−1)6=(0,0,-1) (bond length set to 1). Consider the improper rotation σhR\sigma_h\mathsf{R}: rotate by 90∘90^\circ about the zz axis, then reflect in the xyxy plane, with (Chapter 1 matrices) R(x,y,z)=(−y,x,z),σh(x,y,z)=(x,y,−z).\mathsf{R}(x,y,z)=(-y,x,z),\qquad \sigma_h(x,y,z)=(x,y,-z). ((a)) Compute det⁡R\det\mathsf{R}, det⁡σh\det\sigma_h and det⁡(σhR)\det(\sigma_h\mathsf{R}) using the multiplicativity of the determinant (Chapter 1). ((b)) Apply R\mathsf{R} then σh\sigma_h to each fluorine in turn, and write the composite permutation of {1,…,6}\{1,\ldots,6\} in cycle notation. ((c)) Give the parity of this permutation and compare it with det⁡(σhR)\det(\sigma_h\mathsf{R}).

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