Some connections between Dirac-Fock and Electron-Positron Hartree-Fock

TL;DR

This paper explores the relationship between Dirac-Fock and electron-positron Hartree-Fock models, revealing conditions under which their solutions coincide or diverge, using bifurcation theory and variational methods.

Contribution

It establishes a connection between Dirac-Fock solutions and a max-min problem from electron-positron field theory, highlighting cases of agreement and discrepancy.

Findings

Dirac-Fock ground states solve the max-min problem for certain electron numbers.

Bifurcation theory is used to analyze solutions in the weak repulsion regime.

Situations are identified where max-min levels do not correspond to Dirac-Fock solutions.

Abstract

We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a min-max problem. Then we investigate a max-min problem coming from the electron-positron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this max-min problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated self-consistent projector.