Reduction of a representation

The internal coordinates for the water molecule are chosen as changes in the structural parameters defined in Fig. 3. The effect of each symmetry operation of the symmetry group ( 2 on these internal coordinates is specified in Table 2. Clearly, the internal coordinate Ace is totally symmetric, as the characters xy(Aa) correspond to those given for the irreducible representation (IR) Ai. On die other hand, the characters x/(Ar), as shown, can not be identified with a specific IR. By inspection of Table 2, however, it is apparent that the direct sum Ai B2 corresponds to the correct symmetry of these coordinates. In more complicated cases the magic formula can always be employed to achieve the correct reduction of the representation in question. 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

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... 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

The distribution of the molecular orbitals can be derived from the patterns of symmetry of the atomic orbitals from which the molecular orbitals are constructed. The orbitals occupied by valence electrons form a basis for a representation of the symmetry group of the molecule. Linear combination of these basis orbitals into molecular orbitals of definite symmetry species is equivalent to reduction of this representation. Therefore analysis of the character vector of the valence-orbital representation reveals the numbers of molecular orbitals... 

Fig. 2.23 Schematics representation of the electroactive layer in the case of the electrochemical reduction of a solid when ion diffusion through the solid is allowed—electron diffusion is highly hindered...

Fio. 6. Three-dimensional representation of in situ XRD patterns recorded during reduction of a Cu/ZnO methanol synthesis catalyst. The collection time per diagram was 60 s [adapted from Clausen et at. (32)]. 

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. 

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... 

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... 

F.6), there appears the possibility to consider the latter to be the reduction of a many-body fermionic pure state to an N-representable two-matrix. Since the density matrix above, if adapted appropriately, consequently is essentially N-representable through its relation to Coleman s extreme case [107], one might, via appropriate projections, completely recover the proper information, cf. corresponding partitioning procedures depicted in Appendix A. The structure described here is also of fundamental importance in connection with the phenomena of superconductivity and superfluidity through its intimate connections with Yang s concept of ODLRO [106], see more under Section 3.2. 

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... 

Figure 3.4 Intracellular activation of a self-silenced peptide inhibitor, (a) Schematic representation of intracellular reduction of a disulfide bond to release the activated...

To this point in our discussion of molecular vibrations, we have shown how to generate the symmetries of the vibrational modes by reduction of the representation and subtraction of translational and rotational modes. When pictures of the normal modes of vibration are available, we have also been able to demonstrate how to assign the different symmetries to the correct normal modes of vibration. However, we have not yet attempted to show the origins of the normal modes themselves. The reason for this is because a full-blown norma/ mode analysis (NAM) is a... 

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