The Character of a Representation

Equation (3) shows that the character system is invariant under a similarity transformation and therefore is the same for all equivalent representations. If for some S c G, S R S 1 7 , 

Equation (5) provides a simple test as to whether or not a representation is reducible. Example 4.4-2 Is the 2-D representation T3 of C3v reducible  

Generally, we would take advantage of the fact that all members of the same class have the same character and so perform the sums in eqs. (4), (5), and (6) over classes rather than over group elements. 

It is customary to include ck in the column headings along with the symbol for the elements in 4k (e.g. 3 rv in Table 4.3). Since E is always in a class by itself, E = Ey is placed first in the list of classes and ci = 1 is omitted. The first representation is always the totally symmetric representation T i- 

Example 4.4-3 Using the partial character table for C3v in Table 4.3, show that the character systems ixi and xf satisfy the orthonormality condition for the rows. 

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Since the characters of a representation must be uniquely defined, we see that those we would obtain by the above procedure when J is half-integral cannot belong to true representations. 

In particular, the character of a representation is the same as its block form (with the maximum number of blocks that correspond to the irreducible representations) ... 

Information on the irreducible representations of the various point groups is presented customarily in tables that are called character tables because they give the character of the irreducible representatimi of each symmetry operation in a point group. The character of a representation is the trace (the sum of the diagonal elements) of the matrix that represents that operatimi. Character tables for the D2h, C2v and 4 point groups are presented in Tables 4.2, 4.3 and 4.4. The symmetry elements of the point group are displayed in the top row of each table, and the conventional names, called Mulliken symbols, of the irreducible representations are given in the first column. Letters A and B are used as the Mulliken symbols for... 

The dimension of a representation is the same as the order of the matrix. To reduce a representation it is necessary to reduce its order. It is noted that the dimension of a matrix representation corresponds to the character of the identity (E) matrix. 

A standard tableau is defined as one in which the numbers increase when one reads from left to right in each row and from top to bottom in each column. It can be shown Mi) that the dimension of a representation is equal to the number of standard tableaux associated with the corresponding diagram. The reader is also referred to the literature Mi) for methods of calculating characters of the representations from the diagrams. 

To get the characters for the representation of subduced by a given representation of S, we just copy down the characters of that representation for the elements of which are also in . This is done for the irreducible representations of 4 subduced onto D2< in Table 3. Comparing Tables 2 and 3, and using the standard formula for finding the irreducible parts of a representation by means of the characters, we see that the representations subduced by. T<8> and /1<5) contain / ( )... 

If we consider the sets of functions (xi, ti, zi) and (xa, yi, zf), both of which belong to the same representation T of the <9 group, the product functions constitute a nine-dimensional space. Consequently, these product functions belong to a nine-dimensional representation, denoted by Ti x Ti. It can be shown that the character of this representation is given by =... 

An extremely useful mode of calculation with characters arises from considering the characters of any representation as a vector. Thus the four character vectors of C2v are ... 

In the text, when the character of a set of orbitals is deduced to give a reducible representation, the reduction to a sum of irreducible representations has been carried out by inspection of the appropriate character table. In some instances this procedure can be lengthy and unreliable. The formal method can also be lengthy, but it is highly reliable, although not to be recommended for simple cases where inspection of the character table is usually sufficient. The formal method will be explained by doing an example. 

The character of a dual representation is the complex conjugate of the character of the original. 

Proof. We calculate the scalar product by constructing a linear operator P whose trace is equal to the scalar product. Consider the representation (G, HomCVi, V2), cr) dehned in Proposition 5.12. Let / denote the character of this representation. By Proposition 5.14 we know that / = XiX2- Consider the linear operator... 

One-dimensional irreducible representations are labeled either A or B according to whether the character of a 2irjn (proper or improper) rotation about the symmetry axis of highest order n is +1 or —1, respectively. For the point groups Wl9 and which have no symmetry axis, all one-dimensional representations are labeled A. For... 

Thus we have an explicit expression for the number of times the ixh irreducible representation occurs in a reducible representation where we know only the characters of each representation. 

Now, to find the representation of a wave function that is a product of two other functions we must obtain the representation of the direct product of the two functions. The characters of this representation are the products of... 

Explain why the point group D2 = E C2z C2x C2y is an Abelian group. How many IRs are there in D2 Find the matrix representation based on (e e2 e31 for each of the four symmetry operators R e D2. The Jones symbols for R 1 were determined in Problem 3.8. Use this information to write down the characters of the IRs and their bases from the set of functions z xy. Because there are three equivalent C2 axes, the IRs are designated A, B1 B2, B3. Assign the bases Rx, Ry, Rz to these IRs. Using the result given in Problem 4.1 for the characters of a DP representation, find the IRs based on the quadratic functions x2, y2, z2, xy, yz, zx. 

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