Abstract We have recently (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) reported the form of the G‐particle‐hole hypervirial equation, which can be identified with the anti‐Hermitian part of the correlation contracted Schrödinger equation (Alcoba, Phys Rev A, 2002, 65, 032519), as a tool to obtain the second‐order reduced density matrix of an N‐electron system without previous knowledge of the wave‐function. The results which have been obtained when solving the G‐particle‐hole hypervirial equation with an iterative method also described in (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) have been highly accurate. The convergence of these test calculations has been very smooth, though rather slow. One of the factors which determines the performance of the method is the accuracy with which the 3‐order correlation matrices (3‐CM) involved in the calculations are approximated. It is, therefore, necessary to optimize to the utmost the construction algorithms of these 3‐order matrices in terms of the 2‐CM. In this article, the main theoretical features of the p‐CM are described. Also, some aspects of the correlation contracted Schrödinger equation and of the G‐particle‐hole hypervirial equation are revisited. A new theorem, concerning the sufficiency of the hypervirial of the 3‐order correlation operator to guarantee a correspondence between its solution and that of the Schrödinger equation, and some preliminary results concerning the constructing algorithms of the 3‐CM in terms of the 2‐CM, are reported in the second part of this article. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
A purification procedure that simultaneously corrects the
N- and
S-representability main defects of a second-order reduced density matrix (2RDM), second-order hole reduced density matrix (2HRDM), and second-order
G matrix is presented here. In this purifying procedure, the generalized unitarily invariant second-order matrix decomposition [D. R. Alcoba, Int. J. Quantum Chem. 97, 776 (2004)] for the 2RDM and the 2HRDM as well as for the
G matrix is combined with the
S-representability conditions. In particular, here we will focus our attention on the RDMs corresponding to doublet states. We will thus explicitly give the
S-representability conditions that a two-body correlation matrix has to satisfy when an
N-electron system is in a doublet spin-state in the spin-component of maximum projection. Furthermore, as a consequence of the
G-matrix spin properties (which directly affect the
S-representability of the 2RDM), we show that a different contracting form for the 2RDM is possible. The numerical results presented in this work confirm the efficiency of our purifying procedure.