Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

TL;DR

This paper proves the existence of ground states for a mean-field model of no-photon QED, under certain conditions, and connects the relativistic model to the non-relativistic Hartree-Fock theory.

Contribution

It establishes the existence of minimizers in the Bogoliubov-Dirac-Fock model under binding conditions and analyzes their properties and limits.

Findings

Existence of minimizers under binding conditions.

Derivation and interpretation of the minimizer equation.

Convergence to Hartree-Fock ground state in the non-relativistic limit.

Abstract

The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of no-photon Quantum Electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of $N$ electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant $\alpha$ is small whereas $\alpha Z$ and the particle number $N$ are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently $\alpha$ tends to zero) and $Z$, $N$ are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree-Fock ground state.