Symmetry elements and

List the symmetry elements and point groups of these molecules in both electronic states. 

The retrosynthetic analysis presented in Scheme 6 (for 1, 2, and 16-19) focuses on these symmetry elements, and leads to the design of a strategy that utilizes the readily available enantiomers of xylose and tartaric acid as starting materials and/or chiral auxiliaries to secure optically active materials.14 Thus by following the indicated disconnections in Scheme 6, the initially generated key intermediates 16-19 can be traced to epoxide 23 (16,19 =>23),... 

Space lattices and crystal systems provide only a partial description of the crystal structure of a crystalline material. If the structure is to be fully specified, it is also necessary to take into account the symmetry elements and ultimately determine the pertinent space group. There are in all two hundred and thirty space groups. When the space group as well as the interatomic distances are known, the crystal structure is completely determined. 

Inversion. Reflection through a point (Fig. 3.2). This point is the symmetry element and is called inversion center or center of symmetry. 

Adenine as an isolated molecule has no symmetry elements and therefore might mathematically be considered chiral however, as in the case of glycine (Section 1.2.1), this description is not useful in chemistry since the enantiomers differ only by inversion through the weakly pyramidal nitrogen atom of the amine functionality, the main body of the molecule being planar. The inversion corresponds to a low-frequency vibration and a low-energy barrier such that single enantiomers... 

Fig. 3. Symmetry elements and torsion angles of cyclohexane (chair conformation) and cyclodecane (stable BCB-conformation)...

During the study of inorganic chemistry, the structures for a large number of molecules and ions will be encountered. Try to visualize the structures and think of them in terms of their symmetry. In that way, when you see that Pt2+ is found in the complex PtCl42 in an environment described as D4h, you will know immediately what the structure of the complex is. This "shorthand" nomenclature is used to convey precise structural information in an efficient manner. Table 5.1 shows many common structural types for molecules along with the symmetry elements and point groups of those structures. 

Obviously, a molecule having a different structure (symmetry elements and operations) would require a different table. 

To provide further illustrations of the use of symmetry elements and operations, the ammonia molecule, NH3, will be considered (Figure 5.6). Figure 5.6 shows that the NH3 molecule has a C3 axis through the nitrogen atom and three mirror planes containing that C3 axis. The identity operation, E, and the C32 operation complete the list of symmetry operations for the NH3 molecule. It should be apparent that... 

D raw structures for the following. List all symmetry elements and determine the point group for each species,... 

The minimal basis calculation on the hydrogen molecule is a well-worn but eminently suitable example for our purposes. It has a convenient symmetry element and orbital basis calculations can be carried through which are quantitatively acceptable and yet not prohibitively unwiedly to report. We give below variational calculations on the H2 molecule using the familiar simplest AO basis in the one-electron-group (MO) model and the electron-pair (VB) model. These calculations have been performed explicitly to investigate the effect of symmetry constraints . 

Notice, indeed, that a space group is a group of symmetry elements. If an atom is placed in a quite general position in the unit cell it is multiplied by the symmetry elements and thus other atoms, exactly equivalent to the first, are found at other positions precisely related to those of the first. Each space group has its own characteristic number of equivalent general positions. 

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. 

The rac-isomers have a twofold axis and therefore C2-symmetry. The meso-isomer has a mirror plane as the symmetry element and therefore Cs-symmetry. For polymerisation reactions the racemic mixture can be used since the two chains produced by the two enantiomers are identical when begin- and end-groups are not considered. Note When catalysts of this type are to be used for asymmetric synthesis, e.g. as Lewis acids in Diels-Alder reactions, separation of the enantiomers is a prerequisite [25],... 

For each initiator, indicate the (1) type of symmetry, (2) symmetry elements, and (3) stereochemistry of polymer produced. 

Both isomers lack symmetry elements and, consequently, the two three-coordinate phosphorus atoms are inequivalent. The chemical environments of the two P(III) centres are similar in both isomers, but the large value of /ab=263 Hz indicates a P-P bond as found in 3.11b. ... 

It should be remembered that when a reference atom is moved, all the atoms related to it by symmetry elements move also in a manner determined by the symmetry elements and the problem is to know, for any particular reflection, the direction in which to move the reference atom so that the contribution of the whole group of related atoms either increases or decreases. This problem is best solved by the use of charts which show at a glance the magnitude of the structure amplitude for such a group of atoms for all coordinates of the reference atom. 

This is the operation of inverting all points in a body abont some centre, i.e. if the centre is 0, then any point A is moved to A on the line AO such that OA = OA or put another way, if a set of Cartesian axes have their origin at 0, a point with coordinates (x, y, z) is moved to (— z, —y, —z), If this operation brings the nnclear framework into coincidence with itself, the molecule is said to have a centre of symmetry as a symmetry element and this is symbolized by i (no relation to y —l). In Fig. 2-3.4 we show the inversion operation for an octahedral framework. 

Fig, 7 9.1. Symmetry elements and axes for i h Except for Oh, the symmetry planes oontain the z axis and the axis alongside which the plane s label is written. A and B represent two different atoms. 

The two things, symmetry elements and symmetry operations, are inextricably related and therefore are easily confused by the beginner. They are, however, different kinds of things, and it is important to grasp and retain, from the outset, a clear understanding of the difference between them. 

Symmetry elements and symmetry operations are so closely interrelated because the operation can be defined only with respect to the element, and at the same time the existence of a symmetry element can be demonstrated only by showing that the appropriate symmetry operations exist. Thus, since the existence of the element is contingent on the existence of the operation(s) and vice versa, we shall discuss related types of elements and operations together. 

In treating molecular symmetry, only four types of symmetry elements and operations need be considered. These, in the order in which they will be discussed, are listed in Table 3.1. 

TABLE 3.1 The Four Kinds of Symmetry Elements and Operations Required in Specifying Molecular Symmetry... 

GENERAL RELATIONS AMONG SYMMETRY ELEMENTS AND OPERATIONS... 

We present here some very general and useful rules about how different kinds of symmetry elements and operations are related. These deal with the way in which the existence of some two symmetry elements necessitates the existence of others, and with commutation relationships. Some of the statements are presented without proof the reader should profit by making the effort to verify them. 

Although we have followed conventional practice—and for general purposes will continue to do so—in setting out four kinds of symmetry elements and operations, a, /, C , and Sny we should note that the list can in principle be reduced to only two C and S . A reflection operation can be regarded as an... 

Inspection of the tetrahedron (see Fig. 3.7) reveals the following symmetry elements and operations. 

Finally, we turn to the pentagonal dodecahedron and the icosahedron. These two polyhedra have the same symmetry. They are related to each other as the cube and octahedron are related. The symmetry elements and operations are as follows. 

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