Many data, collected from the literature, support the postulation of two general rules which describe the static and dynamic results of having in a molecule or generating in an intermediate adjacent electron pairs and/or polar bonds.In disagreement with current chemical intuition, structures which contain the maximum number of gauche-interactions between lone electron pairs or polar bonds represent energy minima and, frequently, the lowest minima. The stereochemical implications of the phenomenon are discussed, along with possible physical explanations. It is suggested that partitioning of the total energy of the system into attractive-dominant and repulsive-dominant interactions provides the most helpful framework for the construction of a physical picture of the phenomenon. The total energy and its components can be obtained by ab initio molecular quantum mechanical calculations.The only apparent exception to the phenomenon that has been found corresponds to a polar bond adjacent to two lone pairs (as in the case of two hetero-atoms attached to the same carbon atom). This exception, here termed the Edward-Lemieux effect, has been examined theoretically by an ab initio (Hartree-Fock) calculation using fluoromethanol as a model compound. The calculation has reproduced the Edward-Lemieux effect; the stable conformation has the C-F bond trans to one ' electron pair ' and gauche to another, and the conformation in which the C-F bond bisects the ' electron pairs ' is the energy maximum. The partitioning of the total energy into its components of attraction and repulsion and comparison of the results with other systems for which barriers to internal rotation have been obtained by ab initio methods reveals a similarity between fluoromethanol, hydrazine, hydroxylamine, and hydrogen peroxide. An interpretation of this result is provided in which an early suggestion by Lemieux and Chu is supported and the concept of ' rabbit-ears ' is not.

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.

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.

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.

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