S4 point group

Direct products for S4 point group (Bethe symbols)... 

Bethe symbol and Mulliken symbol for the irreducible representation of S4 point group... 

Fig. 3.21 A tetrafluoro derivative of spiropentane which belongs to the S4 point group. This is an example of a molecule that contains no inversion centre and no mirror plane but is, nonetheless, achiral.

Figure 3.7 The top lines of the character tables for (a) C4 and (b) S4 point groups.

Figure 3.11 Chemical compounds with S4 symmetry (a) chemical structure and (b) three-dimensional model of 1,3,5,7-tetramethylcyclooctatetraene in both cases H atoms are omitted for clarity, (c) A cyclic pbospbazene with the formula N4P4CIS also belongs to the S4 point group.

Collectively, the symmetry elements present in a regular tetrahedral molecule consist of three S4 axes, four C3 axes, three C2 axes (coincident with the S4 axes), and six mirror planes. These symmetry elements define a point group known by the special symbol Td. 

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

In accordance with the overwhelming tendency of type 3 molecules to take on an S4 ground state symmetry, tetracyclohexylsilane (3, M = Si, R s = cyclo-CgHii) was confirmed by X-ray analysis as belonging to this point group. MM calculations of other conformers are available (71). 

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

The examples 6a and 6b are added as a test of spatial vision one is achiral (point group. S ), one is chiral (point group D2). This problem is best solved (6a ), 6b S4) by projecting the models on a plane through the spiro-carbon and perpendicularly arranged to the obvious two-fold axis. 

Figure 3.5 Drawings showing that 1,3,5,7-tetramethylcydooctatetraene, which belongs to point group S4, is converted by the SA operation into its mirror image. The direct result of the SA operation needs to be turned 90° to become coincident with the mirror image at lower left.

The complete set of point symmetry operators that is generated from the operators Ri R2... that are associated with the symmetry elements (as shown, for example, in Table 2.2) by forming all possible products like R, Ry and including E, satisfies the necessary group properties the set is complete (satisfies closure), it contains E, associativity is satisfied, and each element (symmetry operator) has an inverse. That this is so may be verified in any particular case we shall see an example presently. Such groups of point symmetry operators are called point groups. For example, if a system has an S4 axis and no... 

Table 16.22. Quaternion and Cayley-Klein parameters for the symmetry operators of the point group S4.

On the basis of the idealised structure depicted, copper(I) benzoate belongs to the symmetry point group D2a. The symmetry elements present are three C2 axes, an S4 axis and two dihedral planes of symmetry o (these are referred to as dihedral because they are vertical planes of symmetry which... 

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