A new definition of the Dirac-Fock ground state

TL;DR

This paper introduces a simpler, more natural definition of the Dirac-Fock ground state inspired by Lieb's variational principle, along with an iterative method to construct admissible density matrices.

Contribution

It proposes a new, more straightforward definition of the Dirac-Fock ground state and an iterative procedure to construct admissible density matrices, improving upon previous approaches.

Findings

Defined a new notion of Dirac-Fock ground state.

Proved existence of the ground state under the new definition.

Developed an iterative method to construct admissible density matrices.

Abstract

The Dirac-Fock (DF) model replaces the Hartree-Fock (HF) approximation in quantum chemistry when relativistic effects cannot be neglected. Since the Dirac operator is not bounded from below, the notion of ground state is problematic in this model, and several definitions have been proposed in the literature. We give a new definition for the ground state of the DF energy, inspired of Lieb's relaxed variational principle for HF. Our definition and existence proof are simpler and more natural than in previous works on DF, but remains more technical than in the nonrelativistic case. One first needs to construct a set of physically admissible density matrices that satisfy a certain nonlinear fixed-point equation: we do this by introducing an iterative procedure, described in an abstract context. Then the ground state is found as a minimizer of the DF energy on this set.