Representations, completely reduced reducible

If it is impossible to find a transformation which can further reduce all submatrices R. .. R , then all matrices D i) of a given representation are said to be completely reduced. 

The submatrices are also representations, i.e., so-called irreducible representations. The completely reduced representation can be regarded as the direct sum (represented by the symbol Q)) of n, multiples of irreducible representations /, where u, is a positive number or zero ... 

From eqn [74] it can be seen that the trace of the Ci matrix xiCz) is — 1. Using the same procedure for the other symmetry operations, the complete reducible representation formed from the traces of the matrices corresponding to all the symmetry operations of the Civ group is given by... 

It will be shown (Sec. 6-2) that when the transformations representing the symmetry operations of a molecule are written in terms of the normal coordinates [see (6) and (7)] the representation will be completely reduced. 

When the representation of SDsa formed by the transformations of the displacement coordinates of C0 is completely reduced in one of the many possible ways, the transformation representing [Pg.54]

Pig. 5-10. Form of the completely reduced representation for the plane equilateral triangular model of the carbonate ion, CO3 , an example of the group Bsj. The symbols indicate the irreducible representation corresponding to each nonmixing block. 

This normal coordinate representation is at least partially, and usuall completely, reduced, in that the transformations do not mix normal coordinates corresponding to different frequencies. Suppose that this is not true so that, for example, the transformation representing the operation R mixes the coordinates Q and Qk" corresponding to two different frequencies k and Xa" that is,... 

No change of coordinates can reduce such a representation any further it is therefore called an irreducible representationd A completely reduced representation is evidently made up of a number of irreducible representations, each of its noncombining sets of coordinates forming the basis for one irreducible representation. 

One of these R = methods given in Chap. 6. By inserting these values of xr id (3), together with values of xk taken from Table 5-2, the result is obtained (as may readily be verified) that the reduced representation contains the irreducible representation A[ once, A 2 once, twice, E three times, and E" once (see Fig. 5-10, Sec. 5-7). But since it has already been stated that the normal coordinates form the basis of a completely reduced representation, there must be one normal coordinate which transforms like A[, one like A, three pairs like E, etc. Referring to Fig. 5-9 and Table 5-1, one sees that Qi is of symmetry A[, Q2 of symmetry A, and the pairs Q3, Qi, and Qr>, Qe are of symmetry E. The other six coordinates represent translation and rotation. 

Therefore, barring cases of accidental degeneracy, 1 the representation formed by the normal coordinates is a completely reduced one. It is consequently legitimate to apply the equation [Eq. (1), Sec. 5-9]... 

This exclusion is not really necessary since even in this exceptional case the normal coordinates can be chosen so as to form a completely reduced representation. 

The coordinates Si, etc., used above are illustrations of symmetry coordinates, i.e., coordinates in terms of which the secular equation is factored to the maximum extent made possible by the symmetry. In Appendix XII it is proved formally that coordinates arc symmetry coordinates if (a) they form the basis of a completely reduced unitary representation of the point group of the molecule (5) sets of coordinates of the same degenerate symmetry species have identical transformation coefficients. 

Suppose at first that is the wave function for the ground state, Avhich is completely symmetrical. Furthermcre, consider that linear combinations of the dipole moment components, and of the AA ave functions of the upper state, have been chosen in such a way as to form completely reduced representations. Since these representations may be degenerate, the operation R Avill haA e the effect... 

Therefore, when the regular representation is completely reduced, each matrix will be diagonal. But in Eq. (6), Sec. 6-2, it was proved that = dy for the regular representation, which means that each species appears in the completely reduced form dy times. From these facts it then folloAvs that dy = 1 for all species of such a group. 

They constitute a basis for a completely reduced unitary representation of the point group, Q, of the molecule. 

Figure 9.3. Simplified representation of the state of the system during discharge of an organic-electrolyte Li-S battery (a) charged state (b) during discharge, (c) discharged state). The dissolution of the active material occurs at the beginning of discharge, whereas the completely reduced product - Li - is insoluble. During the reduction, the metal lithium electrode is, for its part, partially consumed...

If a transformation can be found which will put all the matrices of a given representation into this general form, the representation is said to be reducible. If no transformation can further diagonalize all submatrices such as c,/, and i in Eq. (3.16) then the set of matrices of a given representation is said to be completely reduced and the sets of submatrices are called the irreducible representations. The fact that the submatrices are also representations can be clarified by referring to Eq. (3.16). If the group multiplication table requires that AB = C then, following the rules for matrix multiplication, it is clear that ad = g be = h, and cf = i so that the set of submatrices a, d, and g, for example, form part of a representation which as described above cannot be further reduced. 

If this matrix characterizing C2 is compared with that obtained for the C2 operation in cartesian displacement coordinates [Eq. (3.10)], it can be seen that the character is — 1 as before but now there are no off-diagonal elements. We have only one-by-one matrices on the diagonal which obviously cannot be further reduced. Since the matrices for the other operations will be in this same form, the representation for the water molecule using normal coordinates as a basis is a completely reduced one. Each of the nine one-by-one matrices on the diagonal of the large matrix in Eq. (3.17) represents the C2 operation in one of the four possible irreducible representations illustrated in Fig. 3.7 and tabulated in Table 3.5. 

Notice here that the six>dimensionaI reducible representation is completely reduced to the direct sum of twice the one-dimensional irreducible representation and twice the two-dimensional irreducible representation. Or the number of times the one-dimensional irreducible representation appears in the reducible one is two and the two-dimensional irreducible representation appears twice. Concerning the character of the reducible representation, one can write that Xd(R) = 2xdi(R)+ 2xd2(R)i for every R of the point group C3.. Another result from group theory is that a necessary and sufficient condition for a representation to be irreducible is... 

Sessions were not balanced the first session presented small datasets, while the second presented large datasets. Sessions were split into three blocks, one for each representation. The order of the representations was complete and counterbalanced across subjects. Each representation block was split into three blocks of three datasets (small for the first session, large for the second) counterbalanced across subjects using a Latin square. We alternated the order of representations to reduce memorization effects subjects remembering the answer from the previous representation and dataset. However, we kept the order of datasets constant for each session and counterbalanced across subjects. 

Fig. 7. Schematic representation of the redox and spin states attained by center 1 and center 2 of D. desulfuricans Fuscoredoxin as indicated by EPR and Mossbauer spectroscopies. The fully reduced state indicated in the figure remains to be completely understood. In particular, the numbers of electrons accepted are still under debate. Filled circles represent Fe(II).

Figure 4.14 Diagrammatic representation of (a) oxy-radical>mediated S-thioiation and (b) thiol/disulphide-initiated S-thiolation of protein suiphydryl groups. Under both circumstances mixed disuiphides are formed between glutathione and protein thiois iocated on the ion-translocator protein resulting in an alteration of protein structure and function. Both of these mechanisms are completely reversible by the addition of a suitabie reducing agent, such as reduced glutathione, returning the protein to its native form.

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

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... 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

The Slater hull constraints represent the entire family of A -representabUity constraints that can be expressed using only the diagonal elements of the reduced density matrix [25, 43]. That is, the complete set of (g, K) conditions is necessary and sufficient to ensure the A -representability of the g-density. 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

Derive the complete matrices for all representations of the group C3t>. Hint Write out and reduce matrices for expressing the transformations of the general point (x, y, z), and use matrix multiplication. 

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