Solutions of the multiconfiguration Dirac-Fock equations

TL;DR

This paper proves the existence of solutions to the multiconfiguration Dirac-Fock equations in the weakly relativistic regime using variational methods and critical point theory.

Contribution

It introduces a new variational approach to establish the existence of solutions for the MCDF equations in the weakly relativistic setting.

Findings

Existence of critical points for the energy functional under certain constraints.

Removal of constraints in the weakly relativistic regime allows for solutions without restrictions.

Application of advanced variational principles to relativistic quantum models.

Abstract

The multiconfiguration Dirac-Fock (MCDF) model uses a linear combination of Slater determinants to approximate the electronic $N$-body wave function of a relativistic molecular system, resulting in a coupled system of nonlinear eigenvalue equations, the MCDF equations. In this paper, we prove the existence of solutions of these equations in the weakly relativistic regime. First, using a new variational principle as well as results of Lewin on the multiconfiguration nonrelativistic model, and Esteban and S\'er\'e on the single-configuration relativistic model, we prove the existence of critical points for the associated energy functional, under the constraint that the occupation numbers are not too small. Then, this constraint can be removed in the weakly relativistic regime, and we obtain non-constrained critical points, i.e. solutions of the multiconfiguration Dirac-Fock equations.