Allene, , has a linear backbone (taken as the axis) with the two groups lying in…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra

Invariance, Ch Point groups and crystallographic symmetry

Problem

Allene, H2C=C=CH2\mathrm{H_2C=C=CH_2}, has a linear C=C=C\mathrm{C=C=C} backbone (taken as the zz axis) with the two CH2\mathrm{CH_2} groups lying in mutually perpendicular planes. Besides EE, its symmetry elements include an S4S_4 axis along zz, two C2C_2 axes perpendicular to zz, and two σd\sigma_d planes; it has no σh\sigma_h. ((a)) Write the 3×33\times3 matrix of the S4S_4 operation about zz (a 90∘90^\circ rotation followed by reflection in the plane perpendicular to zz). State its trace and determinant and use tr=det⁡+2cos⁡θ\mathrm{tr}=\det+2\cos\theta to confirm it is an improper rotation through 90∘90^\circ. ((b)) Using the point-group determination sequence, together with the elements listed above, find allene's point group and order. ((c)) Using the polarity and chirality rules, state whether allene is polar and whether it is chiral.

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