Dnh point groups

A Dnh point group contains a Cn axis, n C2 axes perpendicular to C and at equal angles to each other, a ah plane and n other a planes. For n even the point group contains a centre of inversion i. It also contains other elements which may be generated from these. 

A Dnh point group is related to the corresponding Cnv point group by the inclusion of a ah plane. 

The D h point group is derived from C, by the inclusion of oh therefore, all linear molecules with a plane of symmetry perpendicular to the axis belong to Di0ch. Acetylene 

The Td point group contains four C3 axes, three C2 axes and six ad planes. It also contains elements generated from these. 

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Bipyramidal molecules belong to Dnh point groups, where Z and (X, Y) no longer belong to the same irreducible representation. This means that the moment ellipsoid is no longer spherical and so Iz lx = Iy. The atoms lie on a surface defined by an ellipsoid with semi major and semi minor axes of dimension a = b c. The corresponding relationships between the dimensions... 

The second expression is simply a list of the six /s, in numerical order, each multiplied by the character for one of the six operations, in the conventional order , CA, Cl,..., C. This must be true for each and every representation. Hence, the sets of characters of the group are the coefficients of the LCAO-MOs. The argument is obviously a general one and applies to all cyclic (CH) systems belonging to the point groups Dnh, each of which has a uniaxial pure rotation subgroup, C . 

Other carbocyclic systems of the type (CH) , belonging to the point groups ) l/l, which are of interest are those with n equal to 3, 4, 5, 7, and 8. In some cases the actual systems (which may be cations, neutral molecules, or anions) are not necessarily of Dnh symmetry, but there is usually some purpose in... 

Type 3. One n-fold axis and n twofold axes point groups Dn, Dnh, Dnd. 

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. 

In any case the search is for the highest order C axis. Then it is ascertained whether there are n C2 axes present perpendicular to the C axis. If such C2 axes are present, then there is D symmetry. If in addition to D symmetry there is a a plane, the point group is Dnh, while if there are n symmetry planes (arf) bisecting the twofold axes,... 

When to the above is added a reflection perpendicular to the principal axis, 6h, one obtains the group Dnh generated by Cn, C2 and 6h. Again, as for Cnh, for n even we can instead generate the point group from the operators C, Q and I. 

Prisms and antiprisms have a minimum of Dnh or Dod point group symmetry respectively, and in some cases, such as the octahedron (trigonal antiprism) and cube (square prism), the symmetry is higher. For these polyhedra, the corresponding moments of inertia are ... 

Table 3.2 a,jc,S representations on the distinct orbits of the molecnlar point gronps Dn, Dnh Dnd Sn, up to n = 6 Oj is the orbit of order i and m is the number of times it occurs in a particular molecule as explained in the text. For each group the regular representation direct sum is given explicitly as the a representation for the largest orbit, but otherwise is abbreviated as r ulari in column 1 identifies a group in which a central Oj orbit can be present. For this orbit the a, n and 8 analysis does not apply. 

In molecules and polyatomic complex ions, the equilibrium positions of the nuclei determine point-groups8) which are finite (one of the seven cubic groups, or belonging to one of the seven series Dnh, Dn(j, D , Cnh, Cnv, Cn and S2n including the isolated plane of symmetry Cs, the isolated centre of inversion C and, finally,... 

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