Classes, of symmetry operations

In Section 2.4 the concept of classes of elements within a group was introduced. This concept is utilized in dealing with symmetry groups. As we shall see in Chapter 4, it is convenient and customary in writing what is called the character table of a group to consider all the elements of a given class together, 

using this to carry out all of the possible similarity transformations, we find that there are the following classes  

In coordinate system (6), however, the effects of C4 (clockwise) and C A (counterclockwise) are 

In short, the roles of C4 and C4 are interchanged in coordinate systems (b) from what they are in coordinate system (a). Now (and this is the important point) there is a symmetry operation in the group C4. which will convert coordinate system (a) into coordinate system (b), namely, cr jK Thus, in the 

A practical consequence of collecting all operations in the same class when writing down the complete set, for example, at the head of a character table, is that the notation used is a little different from what we have beep using thus far. This new and final form of notation will now be explained and illustrated for the four kinds of symmetry operations. 

---

The tables of characters have the general form shown in Table 5. Each colipua represents a class of symmetry operation, while the rows designate the different irreducible representations. The entries in the table are simply the characters (traces) of the corresponding matrices. Two specific properties of the character tables will now be considered. 

The column headings are the classes of symmetry operations for the group, and each row depicts one irreducible representation. The +1 and —1 numbers, which... 

Now the trace of the new representation should correspond to a character of the irreducible representation of the double group. By inspecting all classes of symmetry operations, the given z-th energy level is unambiguously classified according to the character table of the double group. 

A complete character table is given in Table 4-5 for the C3v point group. The classes of symmetry operations are listed in the upper row, together with the number of operations in each class. Thus, it is clear from looking at this character table that there are two operations in the class of threefold rotations and three in the class of vertical reflections. The identity operation, E, always forms a class by itself, and the same is true for the inversion operation, i (which is, however, not present in the C3v point group). The number of classes in C3v is 3 this is also the number of irreducible representations, satisfying rule 5 as well. 

The character tables usually consist of four main areas (sometimes three if the last two are merged), as is seen in Table 4-5 for the C3v and in Table 4-7 for the C2h point group. The first area contains the symbol of the group (in the upper left corner) and the Mulliken symbols referring to the dimensionality of the representations and their relationship to various symmetry operations. The second area contains the classes of symmetry operations (in the upper row) and the characters of the irreducible representations of the group. 

The group order g equals the number of symmetry operators of this group. The summation is extended over all classes of symmetry operators R. k R) is the number of elements in each class (number of conjugate symmetry operators / in a class). 

A symmetry element is defined as an operation that when performed on an object, results in a new orientation of that object which is indistinguishable from and superimposable on the original. There are five main classes of symmetry operations (a) the identity operation (an operation that places the object back into its original orientation), (b) proper rotation (rotation of an object about an axis by some angle), (c) reflection plane (reflection of each part of an object through a plane bisecting the object), (d) center of inversion (reflection of every part of an object through a point at the center of the object), and (e) improper rotation (a proper rotation combined with either an inversion center or a reflection plane) [18]. Every object possesses some element or elements of symmetry, even if this is only the identity operation. 

The A 2 representation of the C2v group can now be explained. The character table has four columns it has four classes of symmetry operations (Property 2 in Table 4-7). It must therefore have four irreducible representations (Property 3). The sum of the products of the characters of any two representations must equal zero (orthogonality. Property 6). Therefore, a product of A and the unknown representation must have 1 for two of the characters and — 1 for the other two. The character for the identity operation of this new representation must be 1 [x(i ) = 1 ] in order to have the sum of the squares... 

To determine the n and 5 characters, it is not necessary to evaluate the component traces separately for the individual classes of symmetry operations of the group G, which would often involve laborious trigonometry using standard methods. The application of equation 3.4 depends only on knowledge of the permutation character p for each orbit and the readily determined transformation properties of the central harmonics under G. 

The Dn groups (n = 2, 3, 4, 6), with additional rotations of an angle n through n axes perpendicular to the main axis (one kind, C2, for n = 3, two kinds, C2 and C2 for n even). For the groups including these additional rotations, the Cn and C l rotations about the main axis are equivalent (bilateral) and they belong to the same class of symmetry operations. 

Many common objects are said to be symmetrical. The most symmetrical object is a sphere, which looks just the same no matter which way it is turned. A cube, although less symmetrical than a sphere, has 24 different orientations in which it looks the same. Many biological organisms have approximate bilateral symmetry, meaning that the left side looks like a mirror image of the right side. Symmetry properties are related to symmetry operators, which can operate on functions like other mathematical operators. We first define symmetry operators in terms of how they act on points in space and will later define how they operate on functions. We will consider only point symmetry operators, a class of symmetry operators that do not move a point if it is located at the origin of coordinates. 

This preference originates from the group theory requirement of maximizing the number of unique classes of symmetry operations associated with a molecule, to be discussed in Section 4.3.3. 

The transformation Eq. 2.38 is called a similarity transformation and should be compared with Eq. 1.32 which generates classes of symmetry operations. Suppose that the transformation Eq. 2.38 yields a matrix c, then... 

The number of IRRs fora point group is always equal to the number of classes of symmetry operations. 

There are four classes of symmetry operations in the C2V point group, so there must be four IRRs, as shown in Table 8.6. At least one of these IRRs must be totally symmetric and have all of its characters equal to one. This is the A IRR. The sum of the squares of the characters for the identity operation across all the IRRs must equal the order of the group (h = 4) and each character for the identity operator must be a positive number. Thus, all four IRRs must have I as the character for their identity operation. Furthermore, the sum of the squares of the characters for each IRR must also equal the order of the group (h = 4). Thus, the characters for the other three operations in the group can only be I or —I. In order for the other... 

Solution. The C3, point group has three classes of symmetry operations ( , 2C3, and 3(7,) thus, there will be three IRRs. The order of the group is fi = 6 because there are a total of six symmetry operations. Because the character for the identity operation must always be positive and the sum of the squares of the characters for the identity element must equal the order of the group and there are only three IRRs, these characters must be 1, 1, and 2. There must be a totally symmetric IRR with all the characters equal to 1, as shown by T below. Note that although there are only three classes of symmetry operations, the sum of the squares of this IRR is still equal to six because we must take the sum over all the symmetry operations in the group. The other nondegenerate IRR must... 

EXAMPLE 13-7 How many classes of symmetry operation can you count for (planar) BH3 Can you use this to select one of the possibilities from Example 13-12 ... 

Consult the character table of the point group of the molecule. (Note that in step 2 we have constructed our own table to mimic the structure of the character table.) Consider each irreducible representation in the character table. For each individual symmetry operation (you may have to separate classes of symmetry operations), multiply the character by the result of the corresponding symmetry operation in each row of your table. 

Figure 3.12. Illustration of two classes of symmetry operations of the cube, (a) The C3 class consisting of three-fold rotation axes which correspond to the main diagonals of the cube, indicated by double-headed arrows and labeled 1-4. (b) and (c) The C2 class consisting of two-fold rotation axes that are perpendicular to, and bisect the angles between, the cartesian axes the rotation axes are shown as single-headed arrows and the points at which they intersect the sides of the cube are indicated by small circles.

---