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Inorganic Chemistry, Gary L. Miessler, Donald A. Tarr Textbook Chapter 4

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CHEM 212
Richard Oakley

Symmetry is a phenomenon of the natural world as well as the world of human inven tion Figure 41 In nature many types of flowers and plants snowflakes insects cer tain fruits and vegetables and a wide variety of microscopic plants and animals exhibit characteristic symmetry Many engineering achievements have a degree of symmetry that contributes to their esthetic appeal Examples include cloverleaf intersections the pyramids of ancient Egypt and the Eiffel Tower Symmetry concepts can be extremely useful in chemistry By analyzing the sym metry of molecules we can predict infrared spectra describe the types of orbitals used in bonding predict optical activity interpret electronic spectra and study a number of additional molecular properties In this chapter we first define symmetry very specifi cally in terms of five fundamental symmetry operations We then describe how mole cules can be classified on the basis of the types of symmetry they possess We conclude with examples of how symmetry can be used to predict optical activity of molecules and to determine the number and types of infraredactive stretching vibrations In later chapters symmetry will be a valuable tool in the construction of molecu lar orbitals Chapters 5 and 10 and in the interpretation of electronic spectra of coordi 11 and vibrational spectra of organometallic compounds nation compounds Chapter Chapter 13 A molecular model kit is a very useful study aid for this chapter even for those who can visualize threedimensional objects easily We strongly encourage the use of such a kit 41 All molecules can be described in terms of their symmetry even if it is only to say they have none Molecules or any other objects may contain symmetry elements such as SYMMETRY mirror planes axes of rotation and inversion centers The actual reflection rotation or ELEMENTS AND inversion is called the symmetry operation To contain a given symmetry element a OPERATlONS molecule must have exactly the same appearance after the operation as before In other words photographs of the molecule if such photographs were possible taken from the same location before and after the symmetry operation would be indistinguishable If a symmetry operation yields a molecule that can be distinguished from the original in 41 Symmetry Elements and Operations 77 a FIGURE 41 Symmetry in Nature Art and Architecture any way then that operation is not a symmetry operation of the molecule The examples in Figures 42 through 46 illustrate the possible types of molecular symmetry operations and elements The identity operation E causes no change in the molecule It is included for mathematical completeness An identity operation is characteristic of every molecule even if it has no other symmetry The rotation operation C also called proper rotation is rotation through 360n about a rotation axis We use counterclockwise rotation as a positive rotation An example of a molecule having a threefold C3 axis is CHC13 The rotation axis is coinci dent with the CH bond axis and the rotation angle is 36013120 Two Cj opera tions may be performed consecutively to give a new rotation of 240 The resulting Chapter 4 Symmetry and Group Theory 78 H I C1C Cl c3gc1 C1 C1 Top view C3 rotations of CHCI C2 C and C6 rotations Cross section of protein disk of a snowflake design of tobacco mosaic virus FIGURE 42 Rotations The cross section of ilie tobacco mosaic virus is a cover diagram from Nature 1976259 O 1976 Macmillan Journals Ltd Reproduced with permission of Aaron Klug operation is designated c and is also a symmetry operation of the molecule Three suc C3 operations are the same as the identity operation cE The identity oper cessive ation is included in all molecules Many molecules and other objects have multiple rotation axes Snowflakes are a case in point with complex shapes that are nearly always hexagonal and nearly planar The line through the center of the flake perpendicular to the plane of the flake contains a twofold C2 axis a threefold C3 axis and a sixfold C6 axis Rotations 3 and 300 6 are also symmetry operations of the snowflake by 240 Rotation Angle Symmetry Operation There are also two sets of three C2 axes in the plane of the snowflake one set through opposite points and one through the cutin regions between the points One of each of these axes is shown in Figure 42 In molecules with more than one rotation axis the C axis having the largest value of n is the highest order rotation axis or principal The highest order rotation axis for a snowflake is the C6 axis In assigning Carte axis C axis is usually chosen as the z axis When neces sian coordinates the highest order sary the C2 axes perpendicular to the principal axis are designated with primes a single C2 indicates that the axis passes through several atoms of the molecule where prime as a double prime C2 indicates that it passes between the outer atoms Finding rotation axes for some threedimensional figures is more difficult but the same in principle Remember that nature is not always simple when it comes to symmetrythe protein disk of the tobacco mosaic virus has a 17fold rotation axis In the reflection operation o the molecule contains a mirror plane If details such as hair style and location of internal organs are ignored the human body has a left right minor plane as in Figure 43 Many molecules have mirror planes although they may not be immediately obvious The reflection operation exchanges left and right as if each point had moved perpendicularly through the plane to a position cxactly as far from the plane as when it started Linear objects such as a round wood pencil or molecules
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