Symmetry Elements and Point Groups

As noted before, polyatomic molecules have 3 AT-6 or, if linear, 3JV—5 normal vibrations. For any given molecule, however, only vibrations that are permitted by the selection rule for that molecule appear in the infrared and Raman spectra. Since the selection rule is determined by the symmetry of the molecule, this must first be studied. 

This symmetry element is possessed by every molecule no matter how unsymmetrical it is, the corresponding operation being to leave the molecule unchanged. The inclusion of this element is necessitated by mathematical reasons which will be discussed in Sec. 1-6. 

If reflection of a molecule with respect to some plane produces a configuration indistinguishable from the original one, the plane is called a plane of symmetry. 

A molecule may have more than one of these symmetry elements. Combination of more and more of these elements produces systems of higher and higher symmetry. Not all combinations of symmetry elements, however, are possible. For example, it is highly improbable that a molecule will have a C3 and C4 axis in the same direction because this requires the existence of a 12-fold axis in the molecule. It should also be noted that the presence of some symmetry elements often implies the presence of other elements. For example, if a molecule has two cr-planes at right angles to each other, the line of intersection of these two planes must be a C2 axis. A possible combination of symmetry operations whose axes intersect at a point i.s called a point group  

So far, we have discussed only the vibrations whose displacements occur along the molecular axis. There are, however, two other normal vibrations in which the displacements occur in the direction perpendicular to the molecular axis. They are not treated here, since the calculation is not simple. It is clear that the method described above will become more complicated as a molecule becomes larger. In this respect, the GF matrix method described in Sec. 1.12 is important in the vibrational analysis of complex molecules. 

By using the normal coordinates, the Schrodinger wave equation for the system can be written as 

Since the normal coordinates are independent of each other, it is possible to write 

As noted before, polyatomic molecules have 3N-6 or, if linear, 3N-5 normal vibrations. For any given molecule, however, only vibrations that are permitted by 

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

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. 

Draw the formulae of all the possible isomeric butenes and determine their symmetry elements and point groups. Use the flow chart in the appendix to assist you. 

How many isomers are there of diamminedichloridoplatinum(II), [PtCl2NH3)2] Determine the symmetry elements and point groups for all isomers and assign appropriate stereodescriptors. 

Draw formulae for all the isomers of dichloro cyclopropane and predict the relative signal intensities in their H NMR spectra. Compare your answer with results obtained from a determination of the symmetry elements and point groups. 

Pig. Symmetry elements and point groups of three type of molecules, <7, av ad denote verticalhorizontal and diagonal plane of symmetry. 

The symmetry elements and point groups of molecules and ions in the free state have been discussed in Sec. 1.5. For molecules and ions in crystals, however, it is necessary to consider some additional symmetry operations that characterize translational symmetries in the lattice. Addition of these translational operations results in the formation of the space groups that can be used to classify the symmetry of molecules and ions in crystals. 

SPECTRA OF INORGANIC AND COORDINATION COMPOUNDS 1-4. SYMMETRY ELEMENTS AND POINT GROUPS " ... 

The standard notation for symmetry elements and point groups used in chemistry is the following ... 

Symmetry Elements, Symmetry Operations and Point Groups... 

If none of these groups is appropriate, and if there is no rotation axis, there are three possibilities the molecule (t) does not possess and symmetry element (the point group C, in which the identity operation is the only symmetry operation) (it) possesses a plane of symmetry (the point group Cj) (tit) possesses an inversion centre (the point group C,). 

The set of symmetry elements and operations that characterize the symmetry of an individual molecule defines its point group. If only one rotational symmetry operation (besides E) is possible then the point group bears the same name (C2, S3, etc.) Otherwise, if there is just one symmetry element, the point group is called Cs if there is a mirror plane, a, and Q if there is an inversion center, i. Finally, if no other symmetry elements are present then the point group is Ci. 

We have established two things about symmetry operations. First, they are operators and expressed mathematically in terms of a 3 X 3 matrix for operations on a point in 3-D space. Second, we have stated that only certain collections of symmetry elements, called point groups, are possible for real objects. 

Point group. This group has the same symmetry elements as point group O plus a center of symmetry and nine planes of symmetry. As a consequence of this center of symmetry there are also three 4 axes coincident with the three C4 axes. Some typical chemical examples of selected point groups are tabulated in Table 3.2. 

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

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 snoutene skeleton is shown below. Locate all of the elements of symmetry (state the point group if you know it), and identify the stereochemical relationship between the specified pairs of groups or faces. 

It is dear that the elements of point groups do not necessarily commute, that is the order in which one combines two symmetry operations can be important (see, for example, Fig. 2-4.1 and Table 3-4.1 where for the symmetric tripod Ctar j o C ). A group for which all the elements do commute is called an Abelian group. 

The point given at the top of this list (and all such lists) is called the general position for that space group. It does not lie on any symmetry element and... 

Just as we have previously done with point groups and 2D space groups, we must test all the possibilities, beginning first with those where we add only one symmetry element, and then those with combinations. For the latter there can (and will) be redundancies. Two combinations may seem different, but because a pair of symmetry elements in combination generates a third one, they may be no more than two ways of defining the same final result. 

Figure 11.19. Diagrams showing symmetry elements and general point positions for space groups PI, P2, /42, and /42, (which is not different from A2 except for placement of the origin).

A flowchart for assigning point symmetry. The symmetry elements, and the rules and procedures for their use in determining the symmetry of molecules, can be formalized in a flow chart such as that shown in Fig. 3.16. It contains all of the point groups discussed above (enclosed in square boxes) as well as a few others not commonly encountered. In addition, the symmetries assigned above by inspection may be derived in a more systematic way by the use of this diagram. 

Any set with the four properties (a)-(d) forms a group therefore the set G is a group for which the group elements are point symmetry operators. This point group is called C3v or 3m, because the pyramid has these symmetry elements a three-fold principal axis and a vertical mirror plane. (If there is one vertical plane then there must be three, because of the three-fold symmetry axis.)... 

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