Geometrical picture of the electron-electron correlation at the large-D limit

TL;DR

This paper explores a geometric approach to understanding electron-electron correlation energy in high-dimensional spaces, linking it to entanglement and providing potential bounds on correlation effects.

Contribution

It introduces a novel geometric framework at the large-D limit to describe and quantify electronic correlations, connecting it with quantum entanglement.

Findings

Distinct geometries for mean field and exact solutions at large-D

Correlation effects can be characterized by geometric differences

Feasibility of bounding correlation energy using geometric insights

Abstract

In electronic structure calculations, the correlation energy is defined as the difference between the mean field and the exact solution of the non relativistic Schr\"odinger equation. Such an error in the different calculations is not directly observable as there is no simple quantum mechanical operator, apart from correlation functions, that correspond to such quantity. Here, we use the dimensional scaling approach, in which the electrons are localized at the large-dimensional scaled space, to describe a geometric picture of the electronic correlation. Both, the mean field, and the exact solutions at the large-D limit have distinct geometries. Thus, the difference might be used to describe the correlation effect. Moreover, correlations can be also described and quantified by the entanglement between the electrons, which is a strong correlation without a classical analog. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics and bounded by the area law for local gapped Hamiltonians of interacting many-body systems. This study opens up the possibility of presenting a geometrical picture of the electron-electron correlations and might give a bound on the correlation energy. The results at the large-D limit and at D=3 indicate the feasibility of using the geometrical picture to get a bound on the electron-electron correlations.