Symmetry Breaking and the Generation of Spin Ordered Magnetic States in Density Functional Theory Due to Dirac Exchange for a Hydrogen Molecule

We study symmetry breaking in the mean field solutions to the electronic structure problem for the 2 electron hydrogen molecule within the Kohn Sham (KS) local spin density functional 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 matter and molecular systems. The KS solutions to the constrained SDFT variation problem undergo spontaneous symmetry breaking leading to the formation of spin ordered states as the relative strength of the non-convex exchange term increases. Numerically, we observe that with increases in the internuclear bond length, the molecular ground state changes from a paramagnetic state (spin delocalized) to an antiferromagnetic (spin localized) ground state and a symmetric delocalized (spin delocalized) excited state. We further characterize the limiting behavior of the minimizer when the strength of the exchange term goes to infinity both analytically and numerically. This leads to further bifurcations and highly localized states with varying character. Finite element numerical results provide support for the formal conjectures. Various solution classes are found to be numerically stable. However, for changes in the R parameter, numerical Hessian calculations demonstrate that these are stationary but not stable solutions.

Duke Scholars

Author Jianfeng Lu Mathematics