Second-order equation purification

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

V. Purification Within the Framework of the Second-Order Contracted Schrodinger Equation... 

V. PURIFICATION WITHIN THE FRAMEWORK OF THE SECOND-ORDER CONTRACTED SCHRODINGER EQUATION... 

From the begining of the development of the RDM theory the need to render N-and 5-representable a 2-RDM obtained by an approximative method was patent. The development first of the spin-adapted reduced Hamiltonian methodology and, more recently, that of the second-order contracted Schrodinger equation rendered the solution of this problem urgent. The purification strategies... 

Another extension of this theoretical smdy is the consideration of both an economical and an effective purification strategy for the 4-RDM. The need for such a purification scheme is motivated by the need to have an N- and 5-representable 4-RDM if one wishes to solve the fourth-order modified contracted Schrodinger equation [62, 64, 87]. There have already been several attemps to purify both the 3-RDM and 4-RDM [18, 34, 52]. In particular, a set of inequalities that bound the diagonal and off-diagonal elements of these high-order matrices have been reported [18]. However, the results obtained with this approach within the framework of the fourth-order modified contracted Schrodinger equation (and the second-order contracted Schrodinger equation) were not fully satisfactory because the different spin-blocks of the matrices did not appear to be properly balanced [87, 114]. 

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