# 1. Human beings have a natural perception and appreciation for symmetry.

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1 I. SYMMETRY ELEMENTS AND OPERATIONS A. Introduction 1. Human beings have a natural perception and appreciation for symmetry. a. Most people tend to value symmetry in their visual perception of the world. b. Example: Cars and buildings tend to be very symmetrical. 2. Usually we just naturally make symmetry evaluations but never really stop to ask why something is symmetrical or what symmetry properties it has. 3. In this chapter we will develop tools with which to evaluate symmetry and then link this symmetry to a mathematical tool called group theory which will be very valuable in illustrating bonding and spectroscopic properties of molecules. B. Symmetry Elements 1. Definition: a line, point, or plane with respect to which one or more symmetry operations may be performed. 2. The symmetry elements we will be concerned with are: Name Type Symbol Rotation Axis line C n Mirror Plane plane σ Inversion Center point i Improper Rotation Axis line S n C. Symmetry Operations 1. Definition: The movement of a molecule relative to some symmetry element such that every atom in the molecule after the operation is performed coincides with an equivalent atom (or itself) of the molecule before the operation. 2. If a particular symmetry operation can be performed on a molecule, then the molecule is said to posess that symmetry element. Symmetry Page 1

2 3. The symmetry operations: Symmetry Element Symmetry Operation Actual Operation Performed C n C n m Clockwise rotation about the n-fold axis by 2pm/n radians Reflection through the σ v, σ d, σ h σ v, σ d, σ h symmetry plane i i Inversion through the center of symmetry Clockwise rotation about the n-fold axis by 2pm/n radians S n S n m followed by reflection in a plane perpendicular to the axis --- E ne tes: a. The C n and S n axes generate n-1 operations while s and i only generate one. b. σ d and σ v are mirror planes coincident with the principle rotation axis (highest n value) of the molecule and σ h is perpendicular to that axis. 4. Example: Identify the symmetry elements in the BF 3 molecule. F a F c B F b a. Rotation relative to an axis perpendicular to the plane of the molecule through the B atom by 120 will move F a to F b, F b to F c, F c to F a and B into itself. Also rotation by 240 will move F a to F c, F c to F b, F b to F a and again B into itself so BF 3 has a C 3 axis. b. Rotation relative to an axis through F a and B by 180 will exchange F c and F b while F a and B go into themselves. Likewise such axes exist through F b and B as well as F c and B. Thus, BF 3 has three C 2 axes perpendicular to the C 3 axis. Symmetry Page 2

3 c. A mirror plane through F a and B will also exchange F b and F c while F a and B are reflected into themselves. Such mirror planes also exist through F b and B as well as F c and B. Thus, BF 3 has three mirror planes designated as σ v since they are coincident with the C 3 axis.te that the C 3 axis is the principle axis since it has the highest n value. d. BF 3 also contains a mirror plane designated as σ h which is the plane of the molecule such that all atoms are reflected into themselves. e. Finally, if the molecule is rotated by 120 and reflected in the plane of the molecule all atoms go into themselves or equivalent atoms (in this case, since the molecule is planer, same result as C 3 ). Thus, the molecule also posesses an S 3 axis. II. POINT GROUPS A. The Stereographic Projection 1. At this time we will introduce a handy tool for working with symmetry operations called the stereographic projection. te that this is not in the textbook. 2. The stereographic projection is constructed around a circle with the principle axis (generally the z axis) perpendicular to the surface of the paper. Axes of different n value are indicated by symbols as shown. Principle Rotation Axis (z axis) Symbols: n=2 n=3 n=4 Other symbols used on the diagram: = mirror plane = point above the plane of the paper X = point below the plane of the paper Symmetry Page 3

4 3. Example: Use a stereographic representation to represent a structure that has a C 3 and 3 σ v 's (such as in the case of the ammonia molecule. The basic diagram begins with: Placing a point on the diagram and performing all of the operations associated with the symmetry elements produces: σv σv σv' σv" σv" σv' tice in this case, the C 3 and C 3 2 operations would lead to redundancy which is often the case and not a problem (i.e., s v followed by C 3 and C 3 2 would give same result. B. Matrices 1. In truth, the entire concept of group theory is a mathematical one and the performing of a "symmetry operation" is really a matrix multiplication on a coordinate. 2. Example: Consider the C 2 operation. C2 = Operating (multiplying) on a point (x,y,z) then gives: y x X y = z -x -y z or x Symmetry Page 4

5 3. Thus, we see that although this is a mathematical concept, we can get the same information from the more mechanical stereographic projection as from the matrix multiplication process. Hence, we will rely on the former since it is (hopefully) a conceptually easier method of representing the operations and studying their interrelationships. B. The Concept of a Point Group 1. The point group can be defined as a closed set of symmetry operations. That is, a set of operations such that if any two operations of a group are multiplied together their product must also be a member of the group. 2. Example: Consider the example above of a structure which posesses a C 3 axis and three s v planes. To look at this we will construct a multiplication table consisting of all possible products of operations. It should be pointed out that multiplying operations is the same as performing them in succession on a point. C3 σv σv' or C3 X σv = σv' The entire multiplication table is shown below. te that the order of the product can (but will not always) make a difference in the result. E C 3 C 3 2 v v' v" E E C 3 C3 2 σ v σ v ' σ v " C 3 C 3 C3 2 E σ v " σ v σ v ' C 3 2 C 3 2 E C 3 σ v ' σ v " σ v v σ v σ v ' σ v " E C 3 C3 2 v' σ v ' σ v " σ v C 3 2 E C 3 v" σ v " σ v σ v ' C 3 C3 2 E From this table, it can be seen that if the E operation is included in the set of operations that a closed group is obtained. This is, in fact, called the C 3v point group. Symmetry Page 5

6 C. Selecting a Point Group 1. There are a limited number of point groups that are important in chemistry and the one to which a molecule belongs can be selected by noting the key symmetry elements that the molecule possesses and following the chart below. 2. To determine the point group simply answer the questions in the flow chart and take the appropriate path depending on the answer. More than one axis with n>2? One of the special point groups such as Td, Oh, or Ih. C1 Rotation axis, Cn, present? σ? n C2 Cn? σh? i? Cs nσv? Cnh Ci Cn Cnv nσv? Dn σh? Dnd Dnh 3. Examples: BF 3 D 3h NH 3 C 3v H 2 C=C=CH 2 D 2d Symmetry Page 6

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