Convergence purification procedures

MZ purification procedure (Tables I and II). Although the results are very similar from a global point of view, it has been found that in certain cases (e.g., BeH2 molecule) the latter procedure yields 2-RDMs and 2-HRDMs that oscillate markedly before converging toward positive matrices. Another important difference betweeen the results concerns the spin G-conditions. Thus, although these conditions are not imposed in the I-MZ procedure, the negativity/positivity of the spin components of the 2-G matrix are corrected as effectively as the... 

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. 

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