From reducible representations

Group Orbitals from Reducible Representations Each of the representations from Step 3 can be rednced by the procednre described in Section 4.4.2. For example, the representation F(25 ) reduces to Ag + 5i ... 

Figure C2.5.10. The figure gives tire foldability index ct of 27-mer lattice chains witli sets containing different number of amino acids. The sets are generated according to scheme described in [27], The set of 20 amino acids is taken as a standard sample. Each sequence witli 20 amino acids is optimized to fulfil tire stability gap [5]. The residues in tire standard samples are substituted witli four different sets containing a smaller number of amino acids [27]. The foldability of tliese substitutions is indicated by tire full circles. The open diamonds correspond to tire sequences witli same composition. However, tire amino acids are chosen from tire reduced representation and tire resultant sequence is optimized using tire stability gap [5].

Levitt Warshel [17, 18] were the first to show that reduced representations may work they used Ca atoms and virtual atoms at side chain centroids. OOBATAKE Crippen [24] simplified further by only considering the Ca atoms. This is snfficient since there are reasonably reliable methods (Holm Sander [11, 12]) that compute a full atom geometry from the geometry of the Ca atoms. (All atom representations are used as well, but limited to the prediction of tiny systems such as enkephalin.)... 

The empirical representation of the PVT surface for pure materials is treated later in this section. We first present general equations for evaluation of reduced properties from such representations. 

In judging hindrance, it is useful to view the molecule in its three-dimensional, folded configuration. For instance, 17 can be reduced without undue difficulty, whereas 18 requires extreme conditions (Raney Ni, 2(WC, 200 atm) (7), a difference not expected from planar representations of the molecule. Saturation of A -octalin (17) may largely go through a prior isomerization to A -octa in, despite an unfavorable equilibrium 121). 

This statement is often taken as a basic theorem of representation theory. It is found that for any symmetry group there is only one set of k integers (zero or positive), the sum of whose squares is equal to g, the order of the group. Hence, from Eq. (29), the number of times that each irreducible representation appears in the reduced representation, as well as its dimension, can be determined for any group. 

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. 

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... 

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 regular representation is a reducible representation composed of matrices constructed as follows first write down the group multiplication table in such a way that the order of the rows corresponds to the inverses of the operations heading the columns in this way will appear only along the diagonal of the table. For example, from Table 3 4.2 we would have... 

It is always possible to form a new, and in general reducible, representation r of a given point group from any two given representations T and r of the group. This is done by forming a new function space for which the basis functions are all possible products of the basis functions of T and T Let the basis functions of T and t9 be... 

First then, for methane, we must obtain I 71. To do this let us associate with each carbon hybrid orbital a vector pointing in the appropriate direction and let us label these vectors vv vs, v , v4 (see Fig. 11-3.1). All of the symmetry properties of the four hybrid orbitals will be identical to those of the four vectors. The reducible representation using these vectors (or hybrids) as a basis can be obtained from 4... 

The six necessary hybrid orbitals on the boron atom can also be assigned vectors. If w-bonds are to be formed, these vectors must have the same orientation as the six vectors on the chlorine atoms. If we followed in the footsteps of 11-3, we would now construct the reducible representation Th7b from a consideration of how the six vectors on the boron atom change under the symmetry operations of the B point group. However, it is clear that since the six vectors on the chlorine atoms match the six on the boron atom, exactly the same representation rhyb can be found by using these vectors instead. Since it is less confusing to have three pairs of vectors separated in space than six originating from one point, we will take this latter approach. 

The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

Clearly, the only way that any such matrix can have 1 rather than 0 at any diagonal position is when it carries one of the basis functions into itself. The C2(z) operation moves every basis function from its original position to somewhere else, and thus has all of its l s in off-diagonal positions. Its character could thus have been recognized to be 0 without the bother of writing out the entire matrix. The criterion is Any basis function that moves contributes nothing to the character. Thus the only operations in Dlh that leave any basis function in place are the two that leave them all in place, E and o(xy), and these have a character of 4. Hence we obtain the following reducible representation ... 

In carrying out the procedure for a tetrahedral species, it is convenient to let four vectors on the central atom represent the hybrid orbitals we wish to construct (Fig. 3.26). Derivation of the reducible representation for these vectors involves performing on them, in turn, one symmetry operation from each class in the Td point group. As in the analysis of vibrational modes presented earlier, only those vectors that do not move will contribute to the representation. Thus we can determine the character for each symmetry operation we apply by simply counting the number of vectors that remain stationary. The result for AB4 is the reducible representation, I",. 

In 1989, however, a practical resolution of this problem was derived independently by Haser and by Almlof [16]. They obtain the necessary matrix representation information by utilizing the reducible representation matrices obtained from symmetry transformations on the AO basis as in Eq. 5.1. Full details of their procedure is beyond the scope of this course, but, as would be expected, it has many similarities to the non-totally symmetric operators discussed in the previous section. 

Example 6.1-1 This example describes a bonding in tetrahedral AB4 molecules. The numbering of the B atoms is shown in Figure 6.1. Denote by ar a unit vector oriented from A along the bond between A and Br. With ((TX matrix representatives (MRs) T(T) from T(rr (a V(T) since we only need the character system -/T of the representation To-. Every ar that transforms into itself under a symmetry operator T contributes +1 to the character of that MR T(T), while every oy that transforms into point group. The values of, t( T) for the point group Td are given in Table 6.1. This is a reducible representation, and to reduce it we use the prescription... 

Figure 15 Linking models at various scales using ROMs and deriving lower scale specifications through an inverse optimization formulation. The ROM included at each scale is a reduced representation of the model at the scale below that could range from a set of parameters such as, for example, elementary rate constants to complex models derived from proper orthogonal decomposition and perhaps even to the full lower scale model. This is symbolized by coloring the ROM box with the same color as that of the box representing the adjacent lower scale model.

Figure 23. Schematic representation of flavomyoglobin. (a) Reconstituted myoglobin. Sequential electron transfer occurs from reduced nicotinamide adenosine dinucleotide (NADH) to the hemin via flavin moiety, (b) Structure of an artificial flavin-linked hemin, a flavohemin.

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. 

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. 

With the regard to the interpretability of the infrared spectra, it was experimentally found (ref. 6 and 7) that the representation of somewhere between 64 and 128 coefficients (closer to the 128 side) is the minimum required by an experienced spectroscopist for the determination of specific structural features. This does not mean that any infrared spectrum could be completely interpreted from a curve consisting of only 128 intensity points however, it has been shown recently that identical clustering in a small group of spectra can be obtained even by a far more reduced representation containing only about 3% of the original set (ref. 

It was discussed before that the irreducible representations can be produced from the reducible representations by suitable similarity transformations. Another important point is that the character of a matrix is not changed by any similarity transformation. From this it follows that the sum of the characters of the irreducible representations is equal to the character of the original reducible representation from which they are obtained. We have seen that for each symmetry operation the matrices of the irreducible representations stand along the diagonal of the matrix of the reducible representation, and the character is just the sum of the diagonal elements. When reducing a representation, the simplest way is to look for the combination of the irreducible representations of that group—that is, the sum of their characters in each class of the character table—that will produce the characters of the reducible representation. 

This is a reducible representation of the D point group which reduces to ug + uu. Two molecular orbitals must be generated, one with ag and the other with antibonding orbitals which can be formed from the two 1. s atomic orbitals. 

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