In this paper we analyse the relevant role played by the third-order correlation terms in the contracted Schrödinger equation (CSE) methodology. The quality of the approximations used when evaluating these terms influence significantly both the convergence of the iterative procedure and the accuracy of the final energy value obtained. But where the performance of these approximating algorithms for the third-order terms becomes crucial is in the study of those states whose description, at first-order, needs more than one Slater determinant. This is still an unsolved problem, which is analysed here. Two possible ways for approximately solving this problem are outlined here.

A a procedure is proposed by which, in the course of an iterative solution of the second-order contracted Schrödinger equation, The N representability of the second-order reduced density matrix can be tested with increasing stringency. This procedure was suggested by an extended study of the G conditions and from the contraction into the two-electron space of an N-fermion relation, expressing in matrix form the antisymmetry and normalization properties of the N-electron wave function. Several relations are reported.

We have recently (Valdemoro et al., Sixth International Congress of theInternational 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.

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