N-representability of two-electron densities and density matrices and the application to the few-body problem

TL;DR

This paper develops a comprehensive basis for N-representable two-electron densities, solves the inverse problem for wavefunctions, and proposes an algorithm for accurately computing ground-state energies in few-body quantum systems.

Contribution

It introduces a dense basis for N-representable two-electron densities and solves the inverse problem, advancing the understanding of density matrices and providing a new computational method.

Findings

All N-representable two-electron densities can be expanded with positive coefficients.

Two-electron densities form a convex set in a vector space.

Density matrices are more complex and do not form a convex set.

Abstract

We have found a (dense) basis for the N-representable, two-electron densities, in which all N-representable two-electron densities can be expanded, using positive coefficients. The inverse problem of finding a representative wavefunction, giving a prescribed two-electron density, has also been solved. The two-electron densities are found to lie in a convex set in a vector space. We show that density matrices are more complicated objects than densities, and density matrices do not seem to lie in a convex set. An algorithm to compute the ground-state energy of a few-particle system is proposed, based on the obtained results, where the correlation is treated exactly.