Reducible representations characters

Reduce the representation. The sums formed in the application of the reduction formula are given in Table 6.6. In this table the symmetry classes that have reducible representation characters of 0 are omitted as they will not contribute to the totals in any irreducible representation. The order of the Da point group h = 8, so that the reduction process yields... 

In applications of group theory we often obtain a reducible representation, and we then need to reduce it to its irreducible components. The way that a given representation of a group is reduced to its irreducible components depends only on the characters of the matrices in the representation and on the characters of the matrices in the irreducible representations of the group. Suppose that the reducible representation is F and that the group involved... 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

The characters 4,1,0 form a reducible representation in the C3 point group and we require to reduce it to a set of irreducible representations, the sum of whose characters under each operation is equal to that of the reducible representation. We can express this algebraically as... 

Here, I as given by the direct sum, is a (reducible) representation of a given operation, R, Its trace is the character, a quantity that is independent of tire choice of basis coordinates. As xr is merely the sum of the diagonal elements of T, it is also equal to the sum of the traces of the individual submatrices... 

To illustrate the application of Eq. (37), consider the ammonia molecule with the system of 12 Cartesian displacement coordinates given by Eq. (19) as the basis. The reducible representation for the identity operation then corresponds to the unit matrix of order 12, whose character is obviously equal to 12. The symmetry operation A = Cj of Eq. (18) is represented by the matrix of Eq. (20) whore character is equal to zero. Hie same result is of course obtained for die operation , as it belongs to the same class. For the class 3av the character is equal to two, as exemplified by the matrices given by Eqs. (21) and (22) for the operations C and Z), respectively. The representation of the operation F is analogous to D (problem 12). 

This result (problem 14) allows the coefficients to be calculated. Thus, with a knowledge of the symmetry group and the corresponding table of characters, the structure of the reduced representation can be determined. Equation (37) is of such widespread applicability that it is referred to by many students of group theory as the magic formula. ... 

A planar molecule of point group 03b is shown in Fig. 5. The sigma orbitals i, <72 and (73 represented there will be taken as the basis set Application of the method developed in Section 8.9 yields the characters of the reducible representation given in Table 14. With the use of the magic formula (Eq. (37)] the structure of the reduced representation is of the form Ta — A, ... 

Thus, any representation Tcan be expressed as a function of its irreducible representations Pi. This operation is written as P = S a, Pi, where a, indicates the number of times that Pi appears in the reduction. In group theory, it is said that the reducible representation P is reduced into its Pi irreducible representations. The reduction operation is the key point for applying group theory in spectroscopy. To perform a reduction, we need to use the so-called character tables. 

Table 7.3 The character table of group C4 . The basis functions are not included for the sake of brevity. A reducible representation, F, is shown below...

Table 7.4 The character tables of group O and its subgroup >4. The irreducible representation T] of group O appears written below as a reducible representation in >4...

Equation (7.7) can be used to determine the characters of the symmetry operations C , where n = In ja, and then, to construct the reducible representations in the group G of the ion in the crystal. Some of the most common character elements are listed below ... 

At this point, we are able to construct the reducible representations D- of a group composed only of rotational elements. For instance, let us consider that the ion in the crystal has a symmetry group G = 0, whose character table (Table 7.4) consists of only rotational symmetry elements classes C . 

Products between the irreducible representation characters within a group will produce representations which are often reducible. A simple calculation can decompose this product to a sum of the irreducible representation characters, as is demonstrated in Table V for two representations from the S3-DP-S2 group. 

Schematic representation of manganese nodule end-member morphologies. The size of the arrows Indicates the proportion and direction of metal supply, (a) Typical situation In the open ocean with the nodules lying on an oxidized sediment substrate dominant mode of formation Is hydrogenous, (b) Typical situation In nearshore and freshwater environments with nodules lying on a sediment substrate that Is partly reducing In character. Dominant supply of metals Is via Interstitial waters from below the substrate surface. Source From Chester, R. (2003). Marine Geochemistry, 2nd ed. Blackwell, p. 425.

These vectors form the basis for a reducible representation. Evaluate the characters for this reducible representation under the symmetry operations of the D h group. 

Using the orthogonality of characters taken as vectors we can reduce the above set of characters to Ai + E. Hence, we say that our orbital set of three Ish orbitals forms a reducible representation consisting of the sum of Ai and E IR s. This means that the three Ish orbitals can be combined to yield one orbital of Ai symmetry and a pair that transform according to the E representation. 

The sequence of numbers arrived at constitutes the representation of the two Is orbitals with respect to symmetry. Such a combination of numbers is not to be found in the character table it is an example of a reducible representation. Its reduction to a sum of irreducible representations is, in this instance, a matter of realizing that the sum of the a,+ and gu+ characters is the representation of the two Is orbitals ... 

The character of the reducible representation of the 2p orbitals of E2 molecules may be obtained by writing down, under each symmetry element of the group, the number of such orbitals which are unchanged by each symmetry operation. This produces the representation ... 

The reducible representation of the six 2p orbitals may be seen, by inspection of the character table and carrying out the following exercise, to be equivalent to the sum of the irreducible representations ... 

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 coefficients of the symmetry elements along the top of the above classification (the same as those across the top of the C3v character table), Le. 1,2 and 3, give a total of six which is the order of the point group, denoted by h. The relationship used to test the hypothesis that the reducible representation contains a particular irreducible representation is ... 

They form a reducible representation with the characters ... 

The characters x r f°rm a reducible representation of the molecular orbitals of this molecule that is, the orbitals still have finite interaction elements between them, and the secular determinant must be reduced further to diagonal form. The number of times af that a reducible representation occurs in an irreducible representation is given by (140)... 

We are now in a position to show that two representations with a one-to-one correspondence in characters for each operation, are necessarily equivalent (see 7-3). If we consider two different nonequivalent irreducible representations then, since the characters are orthogonal (eqn (7-3.4)), there cannot be a one-to-one correspondence. If we consider two different reducible representations T° and Tb then, by eqn (7-4.2), if the characters are the same, the reduction will also be the same, that is the number of times occurs in P (a ) will, by the formula, be the same as the number of times T occurs in Fb. The reduced matrices can therefore be brought to the same form by reordering the basis functions of either Ta or Tb. The reduced matrices are therefore equivalent and necessarily Ta and Tb from whence the reduced matrices came (via a similarity transformation) must also be equivalent. Hence, we have proved our proposition. 

Hence we have a method of finding basis functions which belong to a given irreducible representation, if we are given some function space which produces a reducible representation. Notice that in addition to the 0M, the construction of P (eqn (7-6.6)) requires only the knowledge of the characters of the T representation. 

From Table 7-9,2 and using eqn (5-7,2) we can find the diagonal elements of the matrices which represent the 4h point group in the p-orbital basis and in the d-orbital basis. From these elements we get the characters of two reducible representations they are shown in Table 7-9,3, By applying eqn (7-4.2)... 

The characters of the reducible representations using p-orbitals and d-orbitals... 

Given the characters x of a reducible representation T of the indicated point group 9 for the various classes of 9 in the order in which these classes appear in the character table, find the number of times each irreducible representation occurs in T. 

As a second example, let us consider a molecule with the formula AB6 having the symmetry of a trigonal bipyramid Ih. The vector system is shown in Fig. 11-3.2. The set of five hybrid orbitals (or vectors) on A form a basis for a reducible representation of the point group, with the following character ... 

Except for this change, we find hyb(l ) in the same way as before. We note, however, that this time the direction of a vector may be reversed as the result of a symmetry operation and in such a case there will be a contribution of — 1 to the character of that operation. Furthermore, we immediately see that in carrying out the different symmetry operations, no vector perpendicular to the molecular plane is ever interchanged with one in the molecular plane and vice versa. This implies two things the representation rhyb is at once in a partially reduced form (the matrices are already in block form, each consisting of two blocks) and the vectors perpendicular to the molecular plane on their own form a basis for a reducible representation of (which we will call rby ) and the vectors in the molecular plane on their own also form a basis for a reducible representation of 9 (which we will call iy.T) necessarily... 

The ten rr-type p-orbitals form the basis for a reducible representation Tr of the 6d point group with the following character ... 

The following are the characters of certain reducible representations. Use (9.39) to write each representation as the direct sum of irreducible representations. 

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