The Concept of Symmetry

The various experimental tools that are utilized today to solve structural problems in chemistry, such as Raman, infrared, NMR, magnetic measurements and the diffraction methods (electron, X-ray, and neutron), are based on symmetry considerations. Consequently, the symmetry concept as applied to molecules is thus very important. 

Symmetry may be defined in a nonmathematical sense, where it is associated with beauty—with pleasing proportions or regularity in form, harmonious arrangement, or a regular repetition of certain characteristics (e.g., 

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In recent times, the study of molecular non-rigidity has been characterized by new and important developments in both experimental and theoretical directions. From the theoretical point of view, the pioneer work of Longuet-Higgins initiated numerous further studies The aim of this work was to extend the concept of symmetry to non-rigid molecules undergoing certain internal movements, the so-called feasible transformations. Some recent discussion of these problems may be found in Refs. for example. 

A popular but serious treatment of the concept of symmetry and its implications, applied to many topics, some of them from everyday hfe, such as the possible geometries of quilting patterns, and the analysis of the gaits of animals. 

Together, this famous couple, Pierre Curie, 1859-1906, and Mme. Marie Sklodowska Curie, 1867-1934, discovered radium and polonium, and founded the beneficent science of radioactivity. Pierre served as professor of physics at the Sorbonne, and collaborated with his brother, Jacques Curie, in the discovery and investigation of piezo-electricity. He introduced the concept of symmetry in physical phenomena and studied magnetic properties as a function of temperature. Marie served as professor of radioactivity at the University of Paris. 

Symmetry has for many years played a vital role in the elucidation of molecular structure although apart from some special situations it was not thought to have a dominant influence on the structure or chemical properties of molecules. In recent years however it has played a large part in the interpretation of many organic reactions through the work of Woodward and Hoffman, and the concept of symmetry allowed or forbidden reaction is now an important part of mechanistic organic chemistry. 

However, to start with we must get a clear idea what it is we mean by the symmetry of a molecule. In the first place it means consideration of the arrangement of the atoms (or, more precisely, the nuclei) in their equilibrium positions. Now, when we look at different nuclear arrangements, it is obvious that we require a much more precise and scientific definition of symmetry than any of those given previously in Chapter 1, for clearly there are many different kinds of symmetry, for example the symmetry of benzene is patently different from that of methane, yet both are in some sense symmetric. Only when we have put the concept of symmetry on a sound basis, will we be able to classify molecules into various symmetry types (see the next chapter). 

Tlie conservation laws and the concept of symmetry acquired importance in the area of elementary panicle physics. The conservation laws act as "selection rules to determine which reactions may take place between the many existing particles out of the very large number of otherwise conceivable reactions. 

Bonino s lecture met with success and gained a nod of approval from various colleagues, including Walter Hiickel, the brother of Erich. [72] Furthermore, Bonino renewed his friendship with Peter Debye and formed an acquaintance with Richard Kuhn, the president of the German Chemical Society. Nevertheless, in Paris he only touched upon the relationships between structure, valency, stereochemistry and the concept of symmetry. He took these arguments into deeper consideration the following year on a different occasion in Germany. 

The concept of symmetry and chirality in chemistry has a well-defined meaning only in relation to experiment.18 Consider a system of one or more molecules subject to experimental observation. The properties of any such system are invariant with respect to its symmetry operations.42 In Pierre Curie s famous dictum, c est la dissymetrie qui cree le phenomfcne. 43 That is, a phenomenon is expected to exist—and can in principle be observed—only because certain elements of symmetry are absent from the system. It follows that all manifestations of chirality flow from a single source the absence of symmetry elements of the second kind in the group describing the system under observation. Accordingly, if... 

In the study of the structure and properties of molecules and crystals, the concept of symmetry is of fundamental importance. Symmetry is an abstract concept associated with harmony and balance in nature or in social relationship. Yet in chemistry this ever-evolving concept does have a very practical role to play. 

In this chapter, we first discuss the concept of symmetry and the identification of the point group of any given molecule. Then we present the rudiments of group theory, focusing mainly on the character tables of point groups and their use. 

The impact of symmetry control on the homogeneity of film properties for large-area ZnO Al coating is shown in Fig. 5.30. Homogeneous coatings with acceptable variation of transmittance and sheet resistance have been achieved. The concept of symmetry control described here is well known in the glass industry. Other implementations have been realized using multiple PEM sensors [43],... 

The concept of symmetry is of ancient vintage and in many ways almost identical with the equally elusive concepts of beauty and harmony, i.e. beauty of form arising from balanced proportions. Although symmetry can be described in mathematically precise terms, symmetry in the physical world, like beauty1, never absolutely obeys the mathematical requirements of group theory even the most perfect crystal has a surface that spoils the symmetry. 

Let us now illustrate the symmetry operations for various familiar molecules as examples. As this is done it will be convenient to employ also the concept of symmetry elements. A symmetry element is an axis (line), plane, or point about which symmetry operations are performed. The existence of a certain symmetry operation implies the existence of a corresponding symmetry element, and conversely, the presence of a symmetry element means that a certain symmetry operation or set of operations is possible. 

The booklet tells how the concept of symmetry can be applied in quantum chemistry. Due to this concept much information on the electronic structure and other properties of a molecule can be obtained without resorting to the solution of complex equations. The author presents the main principles and touches upon some interesting results obtained in recent years. 

So far we used both geometrical and verbal tools to describe symmetry elements (e.g. plane, axis, center and translation) and operations (e.g. reflection, rotation, inversion and shift). This is quite convenient when the sole purpose of this description is to understand the concepts of symmetry. However, it becomes difficult and time consuming when these tools are used to work with symmetry, for example to generate all possible symmetry operations, e.g. to complete a group. Therefore, two other methods are usually employed ... 

Chemical symmetry has been noted and investigated for centuries in crystallography, which is at the border between chemistry and physics. It was probably more physics when crystal morphology and other properties of the crystal were described, and more chemistry when the inner structure of the crystal and the interactions between the building units were considered. Later, discussion of molecular vibrations, the selection rules, and other basic principles in all kinds of spectroscopy also led to a uniquely important place for the concept of symmetry in chemistry with equally important practical implications. 

This book surveys chemistry from the point of view of symmetry. We present many examples from chemistry as well as from other fields, in order to emphasize the unifying nature of the concepts of symmetry. 

Geometries of di- and poly-nuclear metal complexes having bridging halide, OR, SR, NRj, PRj, H, alkyl, and aryl groups have been discussed, and rationales of their stereochemistries based on the concepts of symmetry of MO s and their electron occupancy are proposed. The authors are pessimistic... 

The concepts of symmetry breaking discussed here with the example of molecular chirality find its analog in the investigation of time reversal symmetry and irreversibility in chemical processes [15-20, 38]. 

One important aspect of MO theory— the concept of symmetry-correct molecular orbitals— not only makes it possible to explain this experimental result but, in fact, requires it. A fimdamental property of molecular orbitals is that they have the full symmetry of the basis set of atomic orbitals used to generate the molecular orbitals. This means that the molecular orbitals must be either symmetric or antisymmetric with respect to the symmetry operations provided for by the symmetry group of the atomic orbitals.If we consider each C-H bond formed by overlap of an sp orbital on a carbon atom with a Is orbital on a hydrogen atom to be an MO, then clearly each of these MOs lacks the full symmetry of the basis set of s and p atomic orbitals. 

The concept of symmetry is equally important for understanding properties of individual molecules, crystals and liquid crystals [1]. The symmetry is of special importance in physics of liquid crystal because it allows us to distinguish numerous liquid crystalline phases from each other. In fact, all properties of mesophases are determined by their symmetry [2], In the first section we consider the so-called point group symmetry very often used for discussion of the most important hquid crystalline phases. A brief discussion of the space group symmetry will be presented in Section 2.2. 

Knowledge of basic crystallography starts from the conception of symmetry, symmetry planes and symmetry operations necessary to identify the parameters, like lattice, lattice planes, crystal lattices (i.e. Bravis lattice) describing different crystal symmetries. Miller indices h, k, 1) to identify crystal planes etc. are explained here. 

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