Dn point groups

A T) point group contains a C axis and n C2 axes. The C2 axes are perpendicular to C and at equal angles to each other. It also contains other elements which may be generated from these. 

The puckered nature of this ligand has been neglected in the figure. 

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Molecules that belong to Cn or Dn point groups are also chiral. For instance, tra i-2,5-dimethylpyrrolidine (Fig. 1-9), containing a twofold rotation axis, belongs to the point group C2 and is chiral.5 7... 

Atom n d"s2 Point group Occupation dn+1s1 Point group Occupation LSD BLYP B3LYP Exp.a... 

Type 3. One -fold axis and n twofold axes point groups Dn, ) /, Dnrj. 

Enantiomers The same in all scalar properties and distinguishable only under chiral conditions. Only molecules of which the point groups are Cn (n> 1), Dn (n> 1), T, O, or / are chiral and can exist in enantiomeric forms. 

Particular care should be taken to check for two-fold axes perpendicular to the principal axis. Overlooking these is probably the commonest error made in identifying point groups. Note also that the group that results from the product of a centre of inversion with Dn is Dnh when n is even but Dnd when n is odd. 

With respect to immobile chiroids, the appropriate symmetries are given by the familiar finite point groups C, (nonaxial), C (monoaxial), Dn (dihedral), T (tetrahedral), 0 (octahedral), and I (icosahedral). Molecules that belong to the first three groups are commonplace molecules with ground-state symmetries T [example tetrakis(trimethylsilyl)silane],39 O (example appoferritin 24-mer),40 and 1 (example human rhinovirus 60-mer),40 are relatively uncommon. 

In 2D there are two classes of point symmetry groups the class Cn having rotational symmetry of order n, and the class Dn having rotational symmetry of order n and n reflection axes. The problem of finding the minimizing orientation is irrelevant for the Cn symmetry groups and R is usually taken as 7 (the identity matrix). We derive here a solution for the orientation in the case where G is a Dn symmetry group. 

Molecules possessing one or more C can be dissymmetric but not asymmetric. They build point groups Cn and Dn (Table 3). 

In the discussion of [Cd(OAr)2(thf)2l. it was implied that it does not have true Dn symmetry. Look closely at Fig. 12.4 and assign a point group symmetry to it. 

As written, the CIDs (2.3) and (2.5) apply to Rayleigh scattering. The same expression can be used for Raman optical activity if the property tensors are replaced by corresponding vibrational Raman transition tensors. This enables us to deduce the basic symmetry requirements for natural vibrational ROA 15,5) the same components of aap and G p must span the irreducible representation of the particular normal coordinate of vibration. This can only happen in the chiral point groups C , Dn, O, T, I (which lack improper rotation elements) in which polar and axial tensors of the same rank, such as aaP and G (or e, /SAv6, ) have identical transformation properties. Thus, all the Raman-active vibrations in a chiral molecule should show Raman optical activity. 

If instead of a reflection one adds a two-fold rotation Q perpendicular to the principal rotation axis, one obtains the point group Dn, generated by Cn and C2. 

It is apparent that Dn Ca) is in block form and since the same block form appears for all the other symmetry operations of the point group, the r° representation has been reduced by the change to the normal coordinate basis. That such a reduction will always occur, is a point taken up in the next section. Needless to say V9 and T" have identical characters i.e. 2%R) = /"(J ), for all R. 

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