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Symmetry Breaking in Density Functional Theory due to Dirac Exchange for a Hydrogen Molecule

We study symmetry breaking in the mean field solutions to the 2 electron hydrogen molecule within Kohn Sham (KS) local spin density function theory with Dirac exchange (the XLDA model). This simplified model shows behavior related to that of the (KS) spin density functional theory (SDFT) predictions in condensed and molecular systems. The Kohn Sham solutions to the constrained SDFT variation problem undergo spontaneous symmetry breaking as the relative strength of the non-convex exchange term increases. This results in the change of the molecular ground state from a paramagnetic state to an antiferromagnetic ground states and a stationary symmetric delocalized 1st excited state. We further characterize the limiting behavior of the minimizer when the strength of the exchange term goes to infinity. This leads to further bifurcations and highly localized states with varying character. The stability of the various solution classes is demonstrated by Hessian analysis. Finite element numerical results provide support for the formal conjectures.