Continuous Point Groups

The continuous point groups are developed by systematic desymmetrization of the full group of the sphere (Kh, Figure 6).41 A symmetric stretch along any one of... 

Figure 6. Complete subgroup lattice of continuous point groups. Solid circles represent point goups that can be represented by geometrical figures Ki, (sphere), (cylinder), Cw (cone). Open circles represent point goups that cannot be represented by geometrical figures. Schonflies notations are accompanied by Hermann-Mauguin (international) notations in brackets.

The reader will realize that the nature of the operator Tr has hardly been specified. The foregoing theory applies equally well to point groups and to permutation groups, and without much alteration to continuous point groups. It is clear that we require the irreducible representations of these groups before tackling actual applications of the theory to chemistry. These are the subject of Chapter 6. 

The isotropic phase formed by achiral molecules has continuous point group symmetry Kh (spherical). According to the group representations [5], upon cooling, the symmetry Kh lowers, at first, retaining its overall translation symmetry T(3) but reduces the orientational symmetry down to either conical or cylindrical. The cone has a polar symmetry Coov and the cylinder has a quadrupolar one Dooh- The absence of polarity of the nematic phase has been established experimentally. At least, polar nematic phases have not been found yet. In other words, there is a head-to-tail symmetry taken into account by introduction of the director n(r), a unit axial vector coinciding with the preferred direction of molecular axes dependent on coordinate (r is radius-vector). 

In addition to the above crystallographic symmetry groups, in fluids one also needs to consider continuous point groups introduced by P. Curie, which are C , C , D , SO(3), 0(3). Note that the last two terms represent... 

This process could be continued so that all the combinations of symmetry operations would be worked out. Table 5.3 shows the multiplication table for the C3 point group, which is the point group to which a pyramidal molecule such as NH3 belongs. 

As an example, the group of rotations about an axis is a connected group. The property of connectedness is not the same as the continuous nature of a group. A continuous group, for instance the rotation-inversion group in three dimensions may be disconnected. The parameter space of a continuous disconnected group consists of two or more disjoint subsets such that each subset is a connected space, but where it is impossible to go continuously from a point in one subset to a point in another without going outside the parameter space. 

A continuous connected group may be simply connected or multiply connected, depending on the topology of the parameter space. A subset of the euclidean space Sn is said to be k-fold connected if there are precisely k distinct paths connecting any two points of the subset which cannot be brought into each other by continuous deformation without going outside the subset. A schematic of four-fold connected space is shown in the lower diagram. 

Notice that the symmetry operations of each point group by continued repetition always bring us back to the point from which we started. Considering, however, a space crystalline pattern, additional symmetry operations can be observed. These involve translation and therefore do not occur in point groups (or crystal classes). These additional operations are glide planes which correspond to a simultaneous reflection and translation and screw axis involving simultaneous rotation and translation. With subsequent application of these operations we do not obtain the point from which we started but another, equivalent, point of the lattice. The symbols used for such operations are exemplified as follows ... 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

Since we will continually be requiring the characters of the irreducible representations of the point groups, it is convenient to put them together in tables known as character tables- In the character table of a point group each row refers to a particular irreducible representation and, since the characters of operations of the same class are identical, only a single entry (C,) is made for all the operations of a given class. The columns are headed by a representative element from each class preceded by the number of elements or operations in that class gf. 

This completes the derivation of the point groups that are important in molecular symmetry, with the exception of the two continuous rotation groups and L) x h, which apply to linear molecules. 

Discrete Rotational Symmetry This is a subset of continuous rotations and reflections in three-dimensional space. Since rotation has no translational components their symmetry groups are known as point groups. Point groups are used to specify the symmetry of isolated objects such as molecules. 

The discontinuous symmetry changes and the binary nature of the presence or absence of symmetry elements hinders the application of point group symmetry methods for general molecular structures. In the syntopy approach, based on fuzzy set theory, the discrete concept of point symmetry is replaced by a continuous concept and is applicable to cases of almost symmetric or quasisymmetric molecular arrangements. When replacing symmetry with syntopy, some of the advantages of the group... 

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