Point groups subgroups

Cs subgroup which was used above in the allyl ease) has no degenerate representations. Moleeules with higher symmetry sueh as NH3, CH4, and benzene have energetieally degenerate orbitals beeause their moleeular point groups have degenerate representations. 

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

Fig. 12-1. Subgroup Decomposition of the Thirty-Two Point Groups. A heavy line indicates that the subgroup is not invariant.

Strong operator convergence, 616 Sturm sequence, 77 Subgraph, 256 maximal connected, 256 Subgroup decomposition of point groups, 738... 

The vibrations of the free molecule can be correlated with the vibrations of the crystal by group theoretical methods. Starting with the point group of the molecule Did)> the irreducible representations (the symmetry classes) have to be correlated with those of the site symmetry (C2) in the crystal and, as a second step, the representations of the site have to be correlated with those of the crystal factor group (D2h) [89, 90]. Since the C2 point group is not a direct subgroup of of the molecule and of D211 of the crystal, the correlation has to be carried out in successive steps, for example ... 

A suitable way to represent group-subgroup relations is by means of family trees which show the relations from space groups to their maximal subgroups by arrows pointing downwards. In the middle of each arrow the kind of the relation and the index of the symmetry reduction are labeled, for example ... 

It has turned out that the most concise description of the symmetry species compatible with a molecular point group, that still includes enough iirformation for useful predictions, is the group character table. The character table of a group is a list of the traces of sets of matrices that form groups isomorphic to the group or to one of its subgroups. 

Of course, the point group of a molecule is always a subgroup of the point group of its ellipsoid of inertia. For example, asymmetric tops can belong to the point groups... 

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 . 

The list of point groups is split into two classes seven infinite families and seven sporadic cases. Every point group contains a normal subgroup formed by its rotations. 

The point groups Td,Oh, and / are the respective symmetry group of Tetrahedron, Cube, and Icosahedron the point groups T, O, and I are their respective normal subgroup of rotations. The point group 7 is generated by T and the central symmetry inversion of the centre of the Isobarycenter of the Tetrahedron. 

As was discussed in Chapter 2, the need to have full matrix representations available to obtain basis functions adapted to symmetry species is something of a handicap. Although character projection itself is not adequate for this task, Hurley has shown how the use of a sequence of character projectors for a chain of subgroups of the full point group can generate fully symmetry-adapted functions. Further discussion of this approach is beyond the scope of the present course, but interested readers may care to refer to the originad literature [6]. 

The second method is applicable to proper point groups P that have an invariant subgroup Q of index 2, so that... 

Table 2.8. The relation of the point groups 0 and T dto their invariant subgroup T.

---