Molecular symmetries

Molecular symmetry originates in the fact that there exist symmetry operations (transformations of the nuclear coordinates) which transform the molecule into a nuclear configuration identical with an initial one. The symmetry elements (axis, plane, inversion centre) remain unchanged. Molecules belong to the point groups of symmetry as all the symmetry operations have at least one point in common (this point is not necessarily identified with any atom of the molecule) [5, 12, 19-38]. 

A brief summary of the properties of the symmetry point groups is presented in Table 1.17. Some more definitions follow  

The symmetry point groups, along with their important characteristics, are classified in Table 1.18. 

A set of matrices A(Rk), transforming coordinates in the same way as the symmetry operator Rk, forms a representation r of the group G. The 

Elementary terms in symmetry point group theory 

The theory of molecular symmetry provides a satisfying and unifying thread which extends throughout spectroscopy and valence theory. Although it is possible to understand atoms and diatomic molecules without this theory, when it comes to understanding, say, spectroscopic selection rules in polyatomic molecules, molecular symmetry presents a small barrier which must be surmounted. However, for those not needing to progress so far this chapter may be bypassed without too much hindrance. 

Rotation of the molecule about this axis by 180°, that is, the operation C2 applied to the molecule, takes one hydrogen atom into the other, an interchange of equivalent centers. On the other hand, a C4 operation changes things such that the water molecule is in a different plane. We conclude that C2 is an operation in the collection of symmetry operations appropriate for the water molecule, whereas C4 is not. Later, we shall refer to such a collection as a mathematical group. 

Rotation of fhe molecule about the axis of the carbon atoms by 90° changes the planes of the two CH2 groups. However, if a reflection operation is then performed, with the reflection plane being in the middle of the molecule, the original appearance of the molecule in space is achieved. The collection of symmetry operators appropriate for allene includes the S4 operation. 

The set of all distinct symmetry operations that take a molecule into an equivalent arrangement constitutes a mathematical structure called a group. An area of mafhemafics called group theory helps organize information we collect about molecular symmetry operations, although group theory has much broader application. 

A group exists only if certain conditions are met. The first requirement is that there is an operation associated with the elements of a group. We might contemplate a mathematical group composed of all positive integers with the operation being addition. In the case of molecular symmetry, the operation is successive application of the elements, something we may term a multiplication. We have already seen that the improper rotation operator 

A special symmetry operation that we have not yet considered has the role of an identity operation in a group. Its designation is E. If the allene molecule were rotated about the carbon atom axis by 180°, and then if this C2 operation were performed again, there would be no interchange of equivalent atomic centers. The result would be the same as if the molecule had not been rotated at all. By closure, this identity operation must be in the group because it corresponds to the successive application of two symmetry operations (both C2) in the group. 

The contents of this chapter are fundamental in the applications of molecular orbital theory to bond lengths, bond angles and molecular shapes, which are discussed in Chapters 3-6. This chapter introduces the principles of group theory and its application to problems of molecular symmetry. The application of molecular orbital theory to a molecule is simplified enormously by the knowledge of the symmetry of the molecule and the group theoretical rules that apply. 

A mathematical group consists of a set of elements which are related to each other according to certain rules, outlined later in the chapter. The particular kind of elements which are relevant to the symmetries of molecules are symmetry elements. With each symmetry element there is an associated symmetry operation. The necessary rules are referred to where appropriate. 

There are seven elements of symmetry which are commonly possessed by molecular systems. These elements of symmetry, their notations and their related symmetry operations are given in Table 2.1. 

Proper axis cr Rotate the molecule by 360/n degrees around the axis 

Horizontal plane Reflect the molecule through the plane which is perpendicular to the major axis 

Stacking the maps one above the other. The amount of detail that can be seen depends upon the resolving power of the instrument if the resolving power is sufficiently good the atoms appear as individual peaks in the image map. At lower resolutions there are groups of unresolved atoms, which can be frequently recognized by their characteristic shapes. 

An important contribution to the analysis of X-ray patterns for complex molecules was made by the British molecular physicist Max Ferdinand Perutz in 1953. His method, known as isomorphous replacement involves the preparation and study of crystals into which heavy atoms, such as atoms of uranium, have been introduced without altering the crystal structure. This technique led rapidly to the detailed analysis of the structures of a number of protein molecules. 

A completely different X-ray approach to the problem of molecular structure was proposed in 1971 by the American physicists Dale E. Sayers, Edward A. Stern, and Parrel W. Lytle. This method is based not on the diffraction of X rays by an array of atoms or ions, but on the absorption of X rays by individual atoms or ions. This new technique measures how the absorption is affected by the atoms in the immediate neighborhood of the atom which is absorbing the X rays, and yields a knowledge of the atomic environment of each type of atom in a molecule. This technique promises to provide valuable information about biological structures and the reactions they undergo. 

2 X 10 m, or 1.2 x 10 nm. This is somewhat less than the usual distances between neighboring atoms in a molecule, and therefore it is possible to diffract an electron beam in the same way as X rays, using a crystal as a diffraction grating. 

Important information about the symmetry properties of molecules is provided by measuring their optical rotation and optical rotatory dispersion. 

1 A student sets up the coordinates for NH3 rather unconventionally, as shown on the right. In this coordinate system, the c-axis is the three-fold axis. One of the a-planes, (Th, makes an angle of 45 with they-axis. By considering the transformation of a general point in space, deduce the six 3 X 3 matrices which form a set of reducible representations for the C3V point group. 

2 There are eight equivalent lobes for the f orbital. They point toward the comers of a cube with the origin as its center and have alternating signs, as shown in Fig. A below. The cube is orientated in such a way that x-, y-, and z-axis pass through the centers of the six faces, as shown in Fig. B below. Rotation of ixyz orbital about z-axis by —45 results in 4( 2-f) orbital. The top views of fxyz and 4( 2 2) orbitals are shown below in Figs. C and D, respectively. 

Assign these two orbitals to their proper irreducible representation in point groups Czv 

3 The square pyramidal [InCls] anion has C y symmetry. Of the four vertical mirror planes, one ffy is in the z-plane, the other ay in theya plane, and the two ct s lie in between them. 

4 Point groups S , have only two symmetry elements E and S . Explain why n must be an even integer and n 4. 

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Several groups have studied the structure of chiral phases illustrated in Fig. IV-15 [167,168]. These shapes can be understood in terms of an anisotropic line tension arising from the molecular symmetry. The addition of small amounts of cholesterol reduces X and produces thinner domains. Several studies have sought an understanding of the influence of cholesterol on lipid domain shapes [168,196]. 

Remarkable chiral patterns, such as those in Figs. IV-15 and XV-8, are found in mixtures of cholesterol and 5-dipalmitoyl PC (DPPC) on compression to the plateau region (as in Fig. XV-6). As discussed in Section IV-4F, this behavior has been modeled in terms of an anisotropic line tension arising from molecular symmetry [46-49]. 

An interesting point is that infrared absorptions that are symmetry-forbidden and hence that do not appear in the spectrum of the gaseous molecule may appear when that molecule is adsorbed. Thus Sheppard and Yates [74] found that normally forbidden bands could be detected in the case of methane and hydrogen adsorbed on glass this meant that there was a decrease in molecular symmetry. In the case of the methane, it appeared from the band shapes that some reduction in rotational degrees of freedom had occurred. Figure XVII-16 shows the IR spectrum for a physisorbed H2 system, and Refs. 69 and 75 give the IR spectra for adsorbed N2 (on Ni) and O2 (in a zeolite), respectively. 

Infrared Spectroscopy. The infrared spectroscopy of adsorbates has been studied for many years, especially for chemisorbed species (see Section XVIII-2C). In the case of physisorption, where the molecule remains intact, one is interested in how the molecular symmetry is altered on adsorption. Perhaps the conceptually simplest case is that of H2 on NaCl(lOO). Being homo-polar, Ha by itself has no allowed vibrational absorption (except for some weak collision-induced transitions) but when adsorbed, the reduced symmetry allows a vibrational spectrum to be observed. Fig. XVII-16 shows the infrared spectrum at 30 K for various degrees of monolayer coverage [96] (the adsorption is Langmuirian with half-coverage at about 10 atm). The bands labeled sf are for transitions of H2 on a smooth face and are from the 7 = 0 and J = 1 rotational states Q /fR) is assigned as a combination band. The bands labeled... 

Bunker P R 1979 Molecular Symmetry and Spectroscopy (New York Academic)... 

We collect syimnetry operations into various syimnetry groups , and this chapter is about the definition and use of such syimnetry operations and symmetry groups. Symmetry groups are used to label molecular states and this labelling makes the states, and their possible interactions, much easier to understand. One important syimnetry group that we describe is called the molecular symmetry group and the syimnetry operations it contains are pemuitations of identical nuclei with and without the inversion of the molecule at its centre of mass. One fascinating outcome is that indeed for... 

This introductory section continues with a subsection that presents the general motivation for using symmetry and ends with a short subsection that lists the various types of molecular symmetry. 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

Bunker P R and Jensen P 1998 Molecular Symmetry aiid Spectroscopy 2nd edn (Ottawa NRC)... 

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. 

Within physical chemistry, the long-lasting interest in IR spectroscopy lies in structural and dynamical characterization. Fligh resolution vibration-rotation spectroscopy in the gas phase reveals bond lengths, bond angles, molecular symmetry and force constants. Time-resolved IR spectroscopy characterizes reaction kinetics, vibrational lifetimes and relaxation processes. 

J. S. Griffith, The Irreducible Tensor Method for Molecular Symmetry Groups, Prentice-Hall, Englewood Cliffs, NJ, 1962, p. 20. 

Figure 4-10 [he Poteiitiiil Energy Form lor Ethylene. The midpoint of the range of (>) is (T and the end points -F180 . that is. [ a. Tt], The mid point and end points are identical by molecular symmetry. 

Figure 4-11 The Potential Energy Form for Ethane. The midpoint of the range of oj is m =0° and the end points are 180°. The end points and the minima are identical by molecular symmetry and correspond to the stable staggered form.

Enforcing the molecular symmetry will also help orbital-based calculations run more quickly. This is because some of the integrals are equivalent by symmetry and thus need be computed only once and used several times. 

In discussing molecular symmetry it is essential that the molecular shape is accurately known, commonly by spectroscopic methods or by X-ray, electron or neutron diffraction. 

The point groups discussed here are all those that one is likely to use, but there are a few very uncommon ones that have not been included descriptions of these are to be found in the books on molecular symmetry mentioned in the bibliography at the end of this chapter. 

Looking at all the examples in Figure 4.18 suggests that molecular symmetry can be used to formulate a mle which will tell us whether any molecule has a non-zero dipole moment. 

Ogden, J. S. (2001) Introduction to Molecular Symmetry, Oxford University Press, Oxford. Schonland, D. (1965) Molecular Symmetry, Van Nostrand, London. 

Vincent, A. (2000) Molecular Symmetry and Group Theory, 2nd edn., John Wiley, Chichester. 

In Chapter 4, on molecular symmetry, 1 have added two new sections. One of these concerns the relationship between symmetry and chirality, which is of great importance in synthetic organic chemistry. The other relates to the connection between the symmetry of a molecule and whether it has a permanent dipole moment. 

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