Point Group Symmetry of Rigid ObjectsDefinitionsSymmetry ope ration: Rigid rotation about an axis, or rotation about an axis fol-lowed by reflection through a plane normal to the axis, such that, after the operationhas been performed, the “new” object is indistinguishable from the original. In thiscontext, indistinguishable objects can be superimposed in all their parts by rigidtranslation with no rotation required.Symmetry e lement: A rotation axis or rotation/reflection axis which is a symmetryoperation of an object.Symmetry Element Symbol Symmetry OperationsIdentityn-Fold a xisMirror planeCenter of symmetry(inversion c enter )Higherrotation/reflection axisECnσ = S1i = S2S4360° rotation = no operation360°/n rotationreflection through a plane =360° rotation/reflectionInversion through a point =360°/2 rotation/reflection360°/4 rotation/reflectionRotation axes are often termed proper axes of symmetry; rotation/reflection axesare t erme d improper axes of symmetry. Objects possessing an improper symme-try axis are said to possess reflection symmetry, and are achiral. Object lackingreflection sym metry are chiral. The point group s ymmetry of an abject is a groupwhich includes all symmetry elements of the object. For example, if an object has Eas i ts only symmetry element, it’s Schoenflies point group is C1. A selection ofpoint groups, c lassified according to chirality and polarity, follows.Cs= S1C2v... C∞v(a cone; E, C∞, ∞σv)Ci =S2C2h... Cnh(all C2halso have i)S4DndDnh... D∞h(a cylinder)TdOhIhKh(a sphere - achiral isotropic)C1CnDnTK (chiral isotropic)Chiral AchiralPolarNonpolarBiphenyl with a 90°dihedral between therings (not a minimum)BrHFClC1BrBrC2BrBrGas phase minimumof biphenyl (dihdralangle between therings ~ 57°)D2BrClCsBrOHHC2vH BrH BrC∞vCiC2hS4CHHHHD2dBrBrH
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