Dnh Groups

For even values of n the Dnh groups are direct products of and Cf. When n is odd is isomorphic with Da . 

When the symmetry of a given molecule has been determined, electronic transitions fall into the following three categories i a. electric allowed — magnetic forbidden b. magnetic allowed — electric forbidden [Groups C, Cnh, Dnh (n 2), S2n (n, odd), Oh, Td]... 

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. 

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. 

Group D. Tf the horizontal plane containing the twofold axes of Dn is also a plane of symmetry, then n vertical planes containing the main axis of symmetry and a twofold axis are necessarily also planes of symmetry. This system of axes and planes of symmetry gives rise to the group Dnh which, in addition to the elements of... 

Dnh. ADnh, group will also, necessarily, contain n vertical planes these planes contain the C2 axes. If there is no oh but there is a set of n vertical planes which pass between the C2 axes, the group is Dnd. 

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,... 

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. 

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