Molecular symmetry, classification

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

In molecules with little or no symmetry, it may still be possible to recognize the main localized-orbital component of certain molecular orbitals. It is then convenient to adopt the label of this localized type as the label of the molecular orbital, even though the molecular symmetry does not coincide with the local symmetry. For instance, in methylenimine again, the 5A orbital is clearly built out of the in-plane 7rc 2 group orbital, with a small NH component. We therefore label the orbital t CU2, although the molecule does not have a vertical symmetry plane. Similarly, the orbitals 7A and 8A of propylene are labeled 7TqH3, tt CU2 (111.49).a Other examples where the local symmetry is sufficiently preserved and only weakly perturbed by the molecular environment are hydrazine (111.34) and methylamine (III.31). In some cases we have omitted the label as no unambiguous classification is possible. 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

The classification of molecular symmetry operations that we shall follow here is the conventional one (see, e.g., Tinkham [2]), involving rotations about a specified axis, denoted Cn for a counterclockwise rotation through reflections in a plane, de-... 

It will be realized that the values of n and m of A will depend on the metal site symmetry and n will only have even values for states of the same parity. In a frequently overlooked paper Eisenstein [554] tabulated the symmetry classifications of the metal ion and ligand orbitals for most of the point group site symmetries of interest. These classifications are often very useful in constructing a molecular orbital energy diagram. Predictions regarding the number and classification of the excited electronic states can then easily be made with the help of such diagrams. We will, however, resist the temptation to reproduce those tables here, in order to conserve space, as they are easily available. 

Figure 11.8 Classification of the reacting molecular orbitals of butadiene and cyclobutene for the conrotatory process. Symmetry classifications are with respect to the C2 axis, S indicating symmetric and A antisymmetric orbitals. The correlation lines are obtained by connecting orbitals of the same symmetry.

The PI group operations are defined by their effect on the space-fixed coordinates of the atomic nuclei and electrons. Since our molecular wavefunctions are written in terms of the vibrational coordinates, the Euler angles and the angle p, we must first determine the effect of the PI group operations on these variables. In the case of inversion this can lead to certain problems both in the understanding of the concepts of molecular symmetry and in the proper use of group theoretical methods in the classification of the states of ammonia. 

Besides the classification of the elements and compounds, there are related chemical systems that illustrate the intricacies and not quite hierarchical structure of scientific classification schemes. Molecular systems that crystallize are further classified by crystal structure. Organic molecules have an elaborate system of classification called "nomenclature" based on both their chemical composition and their symmetries. The classifications are correlated with properties that do not define the classification. Thus, the classification of the elements is associated with a numerical property—atomic weight—and molecular symmetries are associated with optical activity. What are the formal structures that make multiple classifications correct, and how does the existence of one classification scheme constrain others What is the purpose, methodological or otherwise, of corresponding properties Chemistry provides our most familiar examples of settled scientific classifications, and when we want to know what makes for such certainty, chemistry is where we should look. 

In this chapter we have introduced the matrix as a means of handling sets of objects and discussed the key aspects of matrix algebra. A great deal of this chapter has involved a cataloguing of the properties and types of matrices, but we have also tried to emphasize the chemical importance of matrices, in particular in the vital role they play in the classification of molecular symmetry and the development of group theory. The key points discussed include ... 

D. D. Nelson, Jr. and W. Klemperer,/. Chan. Phys., 87, 139 (1987). Classification of the Tunneling-Rotational Energy Levels of Ammonia Dimer in the Molecular Symmetry Group. 

The cycloaddition of alkenes and dienes is a very useful method for forming substituted cyclohexenes. This reaction is known as the Diels-Alder reaction The concerted nature of the mechanism was generally accepted and the stereospecificity of the reaction was firmly established before the importance of orbtial symmetry was recognized. In the terminology of orbital symmetry classification, the Diels-Alder reaction is a [AUg + lUg] cycloaddition, an allowed process. The transition state for a concerted reaction requires that the diene adopt the s-cis conformation. The diene and substituted alkene (which is called the dienophile) approach each other in approximately parallel planes. The symmetry properties of the n orbitals permit stabilizing interations between C-1 and C-4 of the diene and the dienophile. Usually, the strongest interaction is between the highest occupied molecular orbital (HOMO) of the diene and the lowest unoccupied molecular orbital (LUMO) of the dienophile. The interaction between the frontier orbitals is depicted in Fig. 6.1. 

This was the first example of classification based on molecular shape and gave some indication of the physical properties of molecules that were classified as symmetric compared with those that were labelled asymmetric. However, chirality is not the only manifestation of molecular symmetry, and so a more complete classification of molecular shape has been developed the system of point groups. To classify the symmetry of a molecule we derive its point group, which carries much more geometric information than Pasteur s symmetric or asymmetric designation. 

In earlier chapters we classified the symmetry of atomic orbitals (AOs) in a number of example molecules. It is now time to develop the ideas of molecular orbital (MO) theory and use it to describe chemical bonding. Symmetry classifications help in the MO description of chemical bonding because symmetry controls how the AOs on neighbouring atoms mix together. MOs are the wavefunctions for electrons in the complex field of the many nuclei and other eleetrons that make up a molecule. The complexity of MOs can be dealt with by constructing them from the AOs of the isolated atoms. The MOs are formed by mixing the AOs based on the idea of interference described by the superposition of waves when waves come together in the same phase they reinforce one another, whereas waves of opposite phase will tend to cancel each other out. 

This classification is surely related to the molecular symmetry of the molecule, but it is mostly based on the relative values of principal moments of inertia ly, and 4 [1] (where x, y, and z are the principal axes of a molecule-fixed coordinate system). As a result, the classification given above can be explained as follows ... 

What is symmetry In this book we are interested in two uses of this word. First, we are interested in the symmetiy of a molecule. When looking at objects, we invariably have some feel as to whether they are highly symmetric or alternatively not very symmetric. This needs to be quantified in some way. Second, we need to classify, in terms of some symmetry description, the molecular orbitals of molecules and fragments. It is just this aspect that tremendously simplifies the construction of molecular orbitals rather than blindly, mechanically solving the secular determinant and equations of Chapter I. Once this symmetry classification has been done, with the use of a few mathematical tools, we will be in a good position to understand how symmetry controls the orbital structure in molecules. 

Since the related Hamiltonian needs to remain invariant under all the symmetry operations of the molecular symmetry (point) group, the potential energy expansion, see equation (5), may contain only those terms which are totally symmetric under all symmetry operations. Consequently, a simple group theoretical approach, based principally on properties of the permutation groups can be devised, " which yields the number and symmetry classification of anharmonic force constants. The burgeoning number of force constants at higher orders can be appreciated from the entries given in Table 4. 

For polymers, the additional measurements of polarization in Raman spectroscopy and dichroic behavior in IR spectroscopy can be used to aid in the classification of the vibrational properties. In combination, the two techniques can be used to measure the vibrational modes and to classify these vibrational modes into types that are unique to the molecular symmetry of the polymer chain. 

According to one classification (15,16), symmetrical dinuclear PMDs can be divided into two classes, A and B, with respect to the symmetry of the frontier molecular orbital (MO). Thus, the lowest unoccupied MO (LUMO) of class-A dyes is antisymmetrical and the highest occupied MO (HOMO) is symmetrical, and the TT-system contains an odd number of TT-electron pairs. On the other hand, the frontier MO symmetry of class-B dyes is the opposite, and the molecule has an even number of TT-electron pairs. 

From this information, general principles for the design of spherical molecular hosts have been developed. [11] These principles rely on the use of convex uniform polyhedra as models for spheroid design. To demonstrate the usefulness of this approach, structural classification of organic, inorganic, and biological hosts - frameworks which can be rationally compared on the basis of symmetry - has revealed an interplay between symmetry, structure, and function. [53]... 

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