Yves Alain, Bernard

### Description

The context of this Thesis is a new approach to the electron correlation problem based on two ideas. The first idea states that the correlation energy can be approximated simply by a linear operator which contains information about both the position, r, and the momentum, p, of an electron, i.e. a phase-space operator. The second idea proposes the use of two-electron operators, as the electron correlation occurs between pairs of electrons. The combination of these two ideas gave birth to...[Show more] Intracule Function Theory, where intracules are two-electron distributions. To include position and momentum information, Intracule Functional Theory uses the Wigner distribution, a quasi-phase-space distribution, as a true phase-space distribution does not exist because of the Heisenberg Uncertainty Principle. In this Thesis, we study two new phase-space variables, the one-particle Posmom variable, s=r.p, and the two-particle Posmom intracule variable, x=(r1- r2).(p1- p2), and their respective distributions, the Posmom density, S(s), and the Posmom intracule, X(x). The one-particle Posmom variable, s, and its associated operator are known in physics and have been used, for example, in the development of scattering theory. However, they have never been used in quantum chemistry and we present, for the first time, the quantum distribution S(s) for several relevant systems. The two-particle equivalent variable, x, has been introduced previously in Intracule Functional Theory and an intracule, based on the Wigner distribution, has been proposed; the Dot intracule D(x). Within Intracule Functional Theory, we use the Dot intracule to calculate the correlation energy of small atoms and molecules. Furthermore, we derive the exact two-particle Posmom operator and its exact distribution, and show that the Dot intracule appears as an approximation of the exact Posmom intracule. The Posmom variables, s and x, are dot products between position and momentum vectors and contain information about both the position and the momentum of one or two electrons. Furthermore, the associated Posmom operators are quantum mechanical observables, and thus respect the Heisenberg Uncertainty Principle. We show that they are connected to types of electron trajectories and give us new information about the behaviour of electrons. They could form the basis of a new type of spectroscopy. This Thesis introduces the first applications of Posmom in quantum chemistry. We first review the theory of quantum mechanics, quantum chemistry and Intracule Functional Theory. We then present an entire development of the Posmom intracule X(x) by starting from the simple and well known spherically averaged position and momentum densities, followed by the new Posmom density S(s) and the Wigner-based Dot intracule D(x).

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