Iteration/iterative purification procedures

The regulated CSE iterative process has been linked with several iterative purification procedures [50, 70, 88, 111]. In the analysis carried out here the performance of the regulated CSE is compared with the one obtained by coupling the AV purification procedure to the regulated CSE—the regulated CSE-NS iterative process. 

This error was originally approximated by an iterative purification renormalizing procedure, focusing on rendering the 2-RDM and the 2-HRDM positive-semide-finite and correctly normalized [19]. 

Thus we propose to use the new decomposition given by Eqs. (95)-(97) instead of that given by Eqs. (57)-(59) for correcting the ajS-block of an approximated 2-RDM. This leads to a new iterative procedure, hereafter called the I-MZ purification procedure, which can be summarized as follows ... 

Figure 1, Lowest eigenvalue of the 2-RDM and the 2-HRDM matrices at each iteration of the I-MZ purification procedure for the ground state of the berylUum atom.

When the 2-CM is exact, all the 1-RDMs obtained from Eqs. (135)-(138) coincide however, in practice one can only hope that the differences among these matrices are small. These latter properties constitute important 5-representability conditions in the singlet case and are at the center of the N-and S -representability purification procedure, which will now be described. In what follows we will identify ly , D, D, and with the solutions of Eqs. (135), (136), (137), and (138), respectively while keeping the symbol D for the initial 1-RDM, which remains fixed throughout the iterations of the AV purification procedure. 

In view of all the results presented here it can be concluded that the coupling of any RDM-oriented method with the purification procedure augments its applicability in a significant way. In particular, the coupling of the purification procedure and regulating device with the iterative solution of the 2-CSE renders this approach not only reliable but also highly effective. 

Therefore the 4-MCSE is not only determinate but, when solved, its solution is exact. As already mentioned, the price one has to pay is the fact of working in a four-electron space and the difficulty, as in the 2-CSE case, is that the matrices involved must be A-representable. Indeed, in order to ensure the convergence of the iterative process, the 4-RDM should be purified at each iteration, since the need for its A-representability is crucial. In practice, the optimizing procedure used is to antisymmetrize the at each iteration. This operation would not be needed if aU the matrices were A-representable but, if they are not, this condition is not satisfied. In order to impose that the 4-RDM, from which all the lower-order matrices are obtained, be positive semidefinite, the procedure followed by Alcoba has been to diagonalize this matrix and to apply to the eigenvalues the same purification as that applied to the diagonal elements in the 2-CSE case, by forcing the trace to also have a correct value. 

Surprisingly, high enantioselectivities were observed when using BSA as the host harboring 32 (85-98% ee in favor of the endo-products 35) [122], In the case of 32/HSA (Fig. 15), the reaction of 33a with 34 led to an ee-value of only 85%. BSA or HSA alone did not catalyze the reaction. Since a good expression system for HSA has been reported [126,127], this system could be employed in a future study regarding the directed evolution of enantioselective hybrid catalysts. Iterative CASTing (Fig. 2) based on the model in Fig. 15 could then be used to increase the enantioselectivity of the present Diels-AIder cycloaddition or of other Cu"-catalyzed processes. However, a procedure for en masse purification of HSA mutants needs to be developed first. 

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