The reduction of a reducible representation

When we come to apply the results we have so far discovered to quantum mechanical situations, we will find that the application usually revolves around the reduction of some reducible representation for the point group concerned. We have already seen how to find out which irreducible representations appear in the reduction of a reducible representation, namely if we write 

Now we ask the parallel question—what is the new choice of basis functions for the function space (the one which produced rred) which will produce matrices in their fully reduced form Once again we are looking at the opposite side of the coin whose two faces are a similarity transformation and a change of basis functions. To answer the question we have posed, we will invoke the Great Orthogonality Theorem and carry out a certain amount of straightforward algebra. 

Suppose that we have found Jc different function spaces for a given point group, where k is the number of classes or irreducible representations for the point group, and suppose that each function space provides the basis functions for one of the Jc irreducible representations. If the dimension of the rth irreducible representation is nv, there will be nv orthonormal basis functions describing the rth function space. We will write these sets of basis functions as 

K is an operator which is a definite linear combination of the transformation operators O, with coefficients which are related to the matrices of r it is (for reasons which will be clear later) called a projection operator. If p = and q = j, eqn (7-6.4) becomes 

Furthermore, we can create another projection operator P by the equation  

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When such a complete reduction has been achieved, the component representations rF),r(2 are called the irreducible representations of the group G and the representation T is said to be fully reduced. An irreducible representation may occur more than once in the reduction of a reducible representation T. Symbolically... 

Our next task is to discover the relationship between the matrix elements of non-equivalent irreducible representations, the restrictions on the number of such representations, simple criteria for testing for irreducibility and a method for readily carrying out the reduction of a reducible 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 ... 

Ru 2,2 -bipyridine complexes can form a large number of colored compounds upon successive reduction, with the formal Ru oxidation state from +2 to -4. In the case of highly reduced complexes, proper representation of the electrochromic reaction is actually the reduction of the hgand, not that of the metal center. 

Schematic representation of the various reaction modes for the dissolution of Fe(III)(hydr)oxides a) by protons b) by bidentate complex formers that form surface chelates. The resulting solute Fe(III) complexes may subsequently become reduced, e.g., by HS c) by reductants (ligands with oxygen donor atoms) such as ascorbate that can form surface complexes and transfer electrons inner-spheri-cally d) catalytic dissolution of Fe(III)(hydr)oxides by Fe(II) in the presence of a complex former e) light-induced dissolution of Fe(III)(hydr)oxides in the presence of an electron donor such as oxalate. In all of the above examples, surface coordination controls the dissolution process. (Adapted from Sulzberger et al., 1989, and from Hering and Stumm, 1990.)...

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. 

Let us now consider the n-dimensional reducible representation r d which is produced from the function space whose basis functions are Qi> 9i> - Qn> d let us assume that in the reduction of I 1 no irreducible representation of the point group occurs more than once. One way of looking at the reduction is to see it as a change of basis functions from gl9 9i> gn to... 

In applying the methods of group theory to problems related to molecular structure or dynamics, the procedure that is followed usually involves deriving a reducible representation for the phenomenon of interest, such as molecular vibration, and then decomposing it into its irreducible components. (A reducible representation will always be a sum of irreducible ones.) Although the decomposition can sometimes be accomplished by inspection, for the more general case, the following reduction... 

It is interesting to note that the method of molecular orbitals leads to identical results, but by a rather different route. In this method we consider first the set of orbitals on the atoms surrounding the central atom. If this set consists of orbitals symmetrical about the line joining each external atom to the central atom, then these external orbitals form a basis for a representation of the symmetry group which is identical with the o- representation. The reduction of this representation then corresponds to the resonance of these external orbitals among themselves. The formation of molecular orbitals then takes place by the interaction between these reduced external orbitals and the orbitals of the central atom. This interaction can only take place, however, between orbitals belonging to the same representation. Hence, to obtain a set of molecular orbitals equal in number to the... 

As a check, we calculate the total vibrational degeneracy from eq. (4) as 6, which is equal, as it should be, to 3 N — 6. The arithmetic involved in the reduction of the direct sum for the total motion of the atoms can be reduced by subtracting the representations for translational and rotational motion from T before reduction into a direct sum of IRs, but the method used above is to be preferred because it provides a useful arithmetical check on the accuracy of T and its reduction. 

Representations. From the formal point of view, the full-curve and reduced representation of the spectra as well as all handling with them are identical. The only difference is the length (dimensionality) of the representations. The most important aspect of the reduced spectral representation (for the reduction of the spectral curves see Chapter 5) is the possibility to work with a significantly smaller number of variables compared to the number of intensity values of the full-curve representation. It is assumed, of course, that the reduced representation carries only slightly less information than the original full-curve spectrum. 

Due to the fact that the first phase of manipulation of such data is usually a fast scanning of the entire collection, a highly compressed representation of uniformly coded data is essential in order to accelerate the handling. After the search reduces the collection to a smaller group in which the target object is supposed to be, the full (extended) representation of objects can be invoked if necessary for further manipulation. In the next sections we shall discuss the use of two methods, Fast Fourier Transformation (FFT) and Fast Hadamard Transformation (FHT), for the reduction of object representations and show by some examples in 1- and 2-dimensional patterns (spectra, images) how the explained procedures can be used... 

The original representation of infrared spectrum in this example is a set of 512 equidistant intensity values (ref. 6). In order to show the reduction of the information content, the reduction of Fourier and Hadamard coefficients (FCs and HCs) in the transform is carried out to the extreme. In Figure 5.3 the spectrum is reproduced from reduced number of coefficients obtained with the FFT and FHT of the 512-intensity-point curve. 

The main purpose of FFT or FHT of discrete spectra is not the reduction of storage space because a tabular representation of peaks would be usually short enough (see Chapter 4). Nevertheless, an efficient (w ith a respect to the computation time) manipulation and comparison of spectra calls for short and uniform representation and the use of reduced transforms is justified and helpful, especially w hen searching spectral collections resident personal computers. 

The set of matrices Us Pr) in equation (41) form a reducible representation of the group which is reduced into its irreducible components A by the coefficients a above. If we denote the irreducible representations of iAn and 2 by 17 Is- iVl and Z)(/1>, respectively, then this reduction can be written symbolically as... 

This is, of course, a reducible representation. Reduce it now with the reduction formula (see Chapter 4) ... 

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