Representations reducible

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... 

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.)... 

S. Sun, Reduced representation model of protein structure prediction statistical potential and genetic algorithms. Protein Sci. 2 (1993), 762-785. 

S. Sun, Reduced representation approach to protein tertiary structure prediction statistical potential and simulated annealing, J. Theor. Biol. 172 (1995), 13-32. 

United Atom Force Fields and Reduced Representations... 

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. 

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

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... 

Reduced representation of normal modes the symmetry species which describe the symmetry of the (3N — 6) normal modes (where N is the number of atoms in the molecule). 

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 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. 

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, ... 

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... 

Whereas the number of i.r. s is fixed, reducible representations are unlimited in number and generally made up of matrices. As an example, the symmetry operations of C2v may be shown to correspond to the transformations described by the following 3rd order matrices ... 

The CMOs transform as irreducible representations of the molecular point group, whereas the LMOs form a basis for a reducible representation. 

In this case, it can be proved that the canonical SCF orbitals, being solutions of Eq. (26), are symmetry orbitals, i.e. that they belong to irreducible representations of the symmetry group. 12) If the number of molecular orbitals is larger than the dimension of the largest irreducible representation of the symmetry group, it must then be concluded that the set of all N molecular orbitals form a reducible representation of the group which is the direct sum of all the irreducible representations spanned by the CMO s. 

---