Iteration/iterative purification

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]. 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

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. 

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. 

In addition to reconstruction within the CSE, it is important to constrain the 2-RDM to remain approximately Al-representable. The process of correcting a 2-RDM to satisfy A-representability constraints is known as purification. In the context of an iterative solution of the CSE, early algorithms by Valdemoro checked that the 2-RDM satisfies a number of fundamental inequahties such as... 

Here we synthesize the concepts of the last four sections, (i) CSE, (ii) reconstruction, (iii) purification, and (iv) a contracted power method, to obtain an iterative algorithm for the direct calculation of the 2-RDM. 

B. Analysis of the Effects of the N- and S -Purification on the Regulated CSE Iterative Process... 

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. 

As an illustrative example, the energy curves showing the convergence of the overall process for both the regulated CSE process without purification and the regulated CSE process with 15 purification iterations of the calculations of the Li2 molecule are shown in Fig. 9. The two horizontal lines correspond to the Hartree-Fock and the full Cl energy values. As can be noticed, both the accuracy and the convergence rate are remarkably improved when purification is carried out. 

Figure 9. The ground-state electronic energy for O2 is shown as a function of the regulated CSE-NS iterations both with and without purification. The regulated CSE-NS with purification captures 99.12% of the correlation energy, while the regulated CSE-NS without purification achieves only 51.61% of the correlation energy before diverging.

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. 

Earlier iterative solutions of the CSE for the 2-RDM often required that the 2-RDM be adjusted to satisfy important A-representabUity conditions in a process called purification [18, 24]. The solution of the ACSE automatically maintains the A-representability of the 2-RDM within the accuracy of the 3-RDM reconstruction. Necessary A-representability conditions require keeping the eigenvalues of three different forms of the 2-RDM, known as the D, Q, and... 

---